Copositive Duality for Discrete Markets and Games
CCopositive Duality for Discrete Markets and Games
Cheng Guo, Merve Bodur
Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON M5S 3G8, [email protected],[email protected]
Joshua A. Taylor
The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, ON M5S 3G4,[email protected]
Optimization problems with discrete decisions are nonconvex and thus lack strong duality, which limits theusefulness of tools such as shadow prices and the KKT conditions. It was shown in Burer (2009) that mixed-binary quadratic programs can be written as completely positive programs, which are convex. Completelypositive reformulations of discrete optimization problems therefore have strong duality if a constraint qualifi-cation is satisfied. We apply this perspective in two ways. First, we write unit commitment in power systemsas a completely positive program, and use the dual copositive program to design a new pricing mechanism.Second, we reformulate integer programming games in terms of completely positive programming, and usethe KKT conditions to solve for pure strategy Nash equilibria. To facilitate implementation, we also designa cutting plane algorithm for solving copositive programs exactly.
Key words : Copositive programming, unit commitment, integer programming game
1. Introduction
Discreteness is a challenge in many contexts. Examples include optimization, markets with discretedecisions, which can lack efficient equilibria, and games. A basic difficulty is the lack convexity,which precludes the use of tools like strong duality and the Karush-Kuhn-Tucker (KKT) condi-tions. Many discrete problems can be written as mixed-integer programs (MIPs). Burer (2009) hasshown that mixed-binary quadratic programs (MBQPs), a generalization of MIPs, can be writtenas completely positive programs (CPPs). CPPs and their dual copositive programs (COPs) areconvex but NP-hard. While this does not point to a better way of solving MBQPs, it does providea new notion of duality. In this paper, we explore the use of copositive duality for discrete prob-lems, focusing on two applications: pricing in nonconvex electricity markets, and characterizing theequilibria of discrete games.An immediate challenge in using copositive duality is the relatively small number of options forsolving COPs exactly. Our first contribution is the design of a novel cutting plane algorithm thatexactly solves COPs when it terminates. The algorithm consists of a sequence of MIPs, which canthus be implemented using standard industrial solvers. It also accommodates discrete variables, a r X i v : . [ m a t h . O C ] J a n and can thus solve COPs with integer constraints. This feature is the first in the literature anduseful for obtaining the equilibria of integer programming games (IPGs) in Section 6.Our first application of copositive duality is unit commitment (UC) in power systems, in whichthe decision to turn a generator on or off is binary. It is commonly formulated as an MIP, for whichefficient prices are difficult to construct; see, e.g., Liberopoulos and Andrianesis (2016). We rewritethe MIP as a CPP, and use the dual COP to design a pricing mechanism based on shadow prices.The mechanism is budget balanced and, under certain conditions, individually rational. It is alsostraightforward to incorporate additional features, such as revenue adequacy of the generators, byadding constraints directly to the dual COP.In our second application, we use copositive duality to characterize the Nash equilibria (NE) ofIPGs with both binary and continuous decisions. There are well-known conditions guaranteeingthe existence and uniqueness of NE in convex (concave) games with convex strategy sets (Debreu1952, Fan 1952, Glicksberg 1952, Rosen 1965). Also, due to the convexity of strategy sets, the KKTconditions can be used to compute the NE. We use copositive duality to extend some these resultsto IPGs, and use the KKT conditions to compute their equilibria.The reminder of the paper is organized as follows. In Section 2 we review the literature oncopositive programming, pricing in nonconvex markets, and discrete games. In Section 3 we providethe necessary background on CPPs and COPs. In Section 4 we present the new cutting planealgorithm for solving COPs. In Section 5 we design a COP-based pricing scheme for UC, and inSection 6 we derive new results on the equilibria of IPGs. In Section 7 we present the results ofour computational experiments. We conclude the paper in Section 8.All proofs are in Section EC.3 of the e-companion.
2. Literature Review
In this section, we review the relevant literature. Section 2.1 reviews the relationship between CPPs,COPs, and integer programs, and solution methods for COPs. Section 2.2 surveys the pricingschemes for nonconvex markets. Section 2.3 reviews on the literature on IPGs.
Copositive programming has been shown to generalize a number of NP-hard problems, such asquadratic optimization (Bomze and De Klerk 2002), two-stage adjustable robust optimization (Xuand Burer 2018, Hanasusanto and Kuhn 2018), and MBQPs (Burer 2009). In particular, Burer(2009) shows that an MBQP can be written as a CPP. This reformulation is the basis of our work.At present, no industrial software can directly solve COPs. Parrilo (2000) constructs a hierarchyof semidefinite programs (SDPs), which is widely used to approximate COPs. For example, it is used by De Klerk and Pasechnik (2002) to find the stability number of a graph, and by Hanasusantoand Kuhn (2018) for two-stage distributionally robust linear programs. An exact algorithm forCOPs based on simplicial partitions is proposed by Bundfuss and D¨ur (2009). Bomze et al. (2008)and Bomze et al. (2010) use cutting planes to strengthen the SDP relaxation for COPs, but bothare problem specific and can only be used for quadratic programming and clique number problems,respectively. In this paper, we develop a purely MIP-based cutting plane algorithm that exactlysolves COPs when it terminates. Also, to the best of our knowledge, this is the first algorithm thatcan solve COPs with discrete variables.
Many markets have nonconvexities, e.g., due to binary decisions and indivisible goods. It is difficultto design efficient pricing mechanisms in such markets because of the lack of strong duality. Thishas now been a subject of research for decades. We refer the reader to Liberopoulos and Andrianesis(2016) for a more thorough review of pricing in nonconvex markets.A basic source of nonconvexity in electricity markets is the binary startup and shutdown decisionsof generators. These decisions are optimized via UC, which is commonly formulated as an MIP(Carri´on and Arroyo 2006). The basic idea of most current approaches is to construct approximateshadow prices for the MIP. O’Neill et al. (2005) eliminate the nonconvexity by fixing the binarydecisions at their optimal values and pricing the resulting linear program (LP). We call this schemerestricted pricing (RP). RP and its variants are used by some independent system operators (ISOs)in the US, such as the Pennsylvania-New Jersey-Maryland Interconnect. However, RP is often toolow to cover the costs of generators, in which uplift or make-whole payments are needed to ensuregenerator profitability. The convex hull pricing (CHP) of Hogan and Ring (2003) and Gribik et al.(2007) uses the Lagrangian multipliers of the demand constraints as prices, and has been shownshown to minimize uplift. A modified version of CHP called extended locational marginal pricingis used by the Midcontinent ISO. Ruiz et al. (2012) propose a primal-dual approach for pricing,which combines the UC problem and the dual of its linear relaxation, as well as revenue adequacyconstraints that ensure nonnegative profit for each generator. We note that such revenue adequacyconstraints eliminate uplift, but also modify the original UC problem and may therefore result insuboptimal decisions.We use the CPP reformulation of MBQP to design a new pricing scheme, which we refer toas copositive duality pricing (CDP). CDP is budget balanced, supports the optimal UC, and isflexible in that it provides direct access to the dual COP. We make use of this flexibility by addinga revenue adequacy constraint. We refer to the augmented pricing scheme as revenue-adequateCDP (RCDP).
It is desirable for a pricing scheme to ensure individual rationality, i.e., that individual generatorshave no incentive to deviate from the optimal UC solution. O’Neill et al. (2005) prove that RP isindividually rational. CHP satisfies individual rationality in some special cases, but in general itdoes not support each individual generator’s profit-maximizing solution (Gribik et al. 2007). Theprimal-dual approach does not guarantee individual rationality either. CDP does not in generallead to individual rationality, but we provide a simple sufficient condition under which it does.Table 1 compares the properties of several pricing schemes, where “ ◦ ” indicates the existence ofa sufficient condition in the literature for ensuring the property. Table 1
Features of Pricing Schemes
Uplift Optimal IndividualScheme free UC rationalityRP × (cid:88) (cid:88) CHP × × ◦
Primal-dual (cid:88) × ×
CDP × (cid:88) ◦ RCDP (cid:88) × ×
We also mention that there is a related literature stream focusing on markets with indivisiblegoods. Recent papers include Danilov et al. (2001) and Baldwin and Klemperer (2019), which usediscrete convexity to prove the existence of equilibria.
IPGs are a class of games where each player’s action set contains integer decisions. Some examplesof IPGs include bimatrix games, Nash-Cournot energy production games (Gabriel et al. 2013),and Cournot games with indivisible goods (Kostreva 1993). Most papers focus on algorithms forcomputing NE, and are limited to IPGs with only integer decisions (K¨oppe et al. 2011, Sagratella2016, Carvalho et al. 2017). Theoretical results on IPGs include Mallick (2011) and Sagratella(2016), which respectively provide NE existence conditions for two-person discrete games and2-groups partitionable discrete games. In this work we provide conditions for the existence anduniqueness of pure NE (PNE) in IPGs with both continuous and discrete variables, as well as KKTconditions that can be used to compute the PNE.Similar to our approach, in the literature there are several applications of convex optimization toIPGs. Parrilo (2006) reformulates zero-sum polynomial games as a single SDP. Ahmadi and Zhang(2020) use SDP to find the additive (cid:15) -approximate NE of bimatrix games. Closely related to ourwork, Sayin and Basar (2019) study a Stackelberg game with binary decisions in the context ofoptimal hierarchical signaling. They reformulate each player’s optimization problem as a CPP, and solve the CPP and its dual COP approximately with SDP. Our work differs in that we analyze theequilibria of general IPGs, and for certain classes we can obtain the PNE by solving a single COPexactly.
3. Background
In this section, we define CPP and COP and state some basic results.Throughout the paper, we use bold letters for vectors. Tr( · ) denotes the trace of a matrix and( · ) (cid:62) denotes the transpose of a vector or matrix. e k ∈ R n is the k th unit vector. Let S n be the set of n -dimensional real symmetric matrices. The copositive cone C n is defined as: C n = (cid:8) X ∈ S n | y (cid:62) X y ≥ y ∈ R n + (cid:9) . (1)The dual cone of C n is the completely positive cone C ∗ n : C ∗ n = (cid:8) XX (cid:62) | X ∈ R n × r + (cid:9) . (2)In COP (CPP), we optimize a linear function of the matrix X subject to linear constraints and X ∈ C n ( X ∈ C ∗ n ).Because COP and CPP are convex, strong duality holds if a regularity condition is satisfied,e.g., Slater’s condition, which requires the feasible region to have an interior point. In this section, we state the two CPP reformulations of MBQP given in Burer (2009). We alsoderive their COP duals.Consider the MBQP: P MBQP : min x (cid:62) Q x + 2 c (cid:62) x (3a)s . t . a (cid:62) j x = b j ∀ j = 1 , ..., m (3b) x k ∈ { , } ∀ k ∈ B (3c) x ∈ R n + (3d)where B ⊆ { , ..., n } is the set of indices of the binary elements of x . Without loss of generality weassume the matrix Q is symmetric.Burer (2009) gives two CPP reformulations of P MBQP , (C) and (C’), which here we refer toas P CPP o and P CPP r . P CPP o is obtained by squaring the linear constraints and substituting liftedvariables for the bilinear terms. P CPP o : min Tr( QX ) + 2 c (cid:62) x (4a) s . t . a (cid:62) j x = b j ∀ j = 1 , ..., m (4b) a (cid:62) j X a j = b j ∀ j = 1 , ..., m. (4c) x k = X kk ∀ k ∈ B (4d) (cid:20) x (cid:62) x X (cid:21) ∈ C ∗ n +1 . (4e)We now derive the dual of (4). For convenience, define: Y = (cid:20) x (cid:62) x X (cid:21) ; ˜ Q = (cid:20) × n n × Q (cid:21) ; C = (cid:20) c (cid:62) c 0 n × n (cid:21) ; A j = (cid:20) / a (cid:62) j / a j n × n (cid:21) , ∀ j = 1 , ..., m. (5)We also define the matrices ˜ A j = [0 , a (cid:62) j ] (cid:62) [0 , a (cid:62) j ] , ∀ j = 1 , ..., m , and, for each k ∈ B , B k such that( B k ) l ,l = / l = k + 1 , l = 1 or l = 1 , l = k + 1 − l = k + 1 , l = k + 10 otherwise . Then, P CPP o can be written as min Tr( ˜ QY ) + Tr( CY ) (6a)s . t . Tr( A j Y ) = b j ∀ j = 1 , ..., m (6b)Tr( ˜ A j Y ) = b j ∀ j = 1 , ..., m (6c)Tr( B k Y ) = 0 ∀ k ∈ B (6d) Y ∈ C ∗ n +1 . (6e)Let γγγ o , βββ o , δδδ o ,and Ω o be the respective dual variables of constraints (6b) - (6e). The dual of P CPP o is the following COP: P COP o : max γγγ o ,βββ o ,δδδ o , Ω o m (cid:88) j =1 (cid:0) γ oj b j + β oj b j (cid:1) (7a)s . t . ˜ Q + C − m (cid:88) j =1 γ oj A j − m (cid:88) j =1 β oj ˜ A j − (cid:88) k ∈B δ ok B k − Ω o = 0 (7b)Ω o ∈ C n +1 . (7c)Burer (2009) shows that P MBQP and P CPP o are equivalent, in the sense that (i) opt( P MBQP ) =opt( P CPP o ), where opt( · ) is the optimal objective value; and (ii) if ( x ∗ , X ∗ ) is optimal for P CPP o ,then x ∗ is in the convex hull of optimal solutions of P MBQP . The second point indicates that x ∗ isnot necessarily feasible for P MBQP , i.e., it is possible for some x ∗ k with k ∈ B to be fractional. Remark 1. If x ∗ is an optimal solution of P MBQP , then ( x ∗ , x ∗ x ∗(cid:62) ) is optimal for P CPP o . Remark 2.
Let ( x ∗ , X ∗ ) be an optimal solution for P CPP o . If Q (cid:23) x ∗ is feasible for P MBQP ,then x ∗ is an optimal solution of P MBQP . Note that the condition Q (cid:23) P MBQP is optimal at x ∗ . In Section 6 we also make use of the second CPP reformulation given in Burer (2009), P CPP r .The advantages of P CPP r over P CPP o are that (i) it has a lower dimensional cone constraint, X ∈ C ∗ n ,which may improve computational efficiency, and (ii) it may have an interior, whereas P CPP o neverdoes. P CPP r is a valid reformulation of P MBQP if the following Key Property is satisfied: ∃ y ∈ R m s . t . m (cid:88) j =1 y j a j ≥ , m (cid:88) j =1 y j b j = 1 . (KP)Setting ααα := (cid:80) mj =1 y j a j , the CPP is given by: P CPP r : min Tr( QX ) + 2 c (cid:62) x (8a)s . t . a (cid:62) j x = b j ∀ j = 1 , ..., m (8b) a (cid:62) j X a j = b j ∀ j = 1 , ..., m (8c) x k = X kk ∀ k ∈ B (8d) x l = Tr (cid:18) ααα e (cid:62) l + e l ααα (cid:62) X (cid:19) ∀ l = 1 , ..., n (8e) X ∈ C ∗ n . (8f)Note that P CPP r , as written above, looks slightly different from (C’) in Burer (2009). This is be-cause we have included the variable x via the constraint x = Xα , and made several correspondingsubstitutions. This makes the proofs in Section 6 more straightforward. We have written the con-straint x = Xα in symmetric form in (8e) because taking the dual of a CPP with non-symmetriclinear constraints will bring further complications; we discuss the details of this in Section EC.2 ofthe e-companion. Also note that Remarks 1 and 2 straightforwardly extend to P CPP r .Condition (KP) is usually not restrictive. For example, if P MBQP contains or implies a constraintof the form (cid:80) i ∈I a i x i = b , where a i ≥ i ∈ I , and b >
0, and this is the (cid:96) th constraint of P MBQP ,then we can construct a valid y with 1 /b in the (cid:96) th entry and 0 elsewhere.Despite its lack of an interior, P CPP o may be useful because it does not require (KP) to hold.Also, strong duality may still hold for P CPP o , which we often observe in our numerical examples.Finally, we write the dual of P CPP r . Let γγγ , βββ , δδδ , ξξξ , and Ω be the respective dual variables ofconstraints (8b) - (8f). The dual of P CPP r is the COP: P COP r : max γγγ,βββ,δδδ,ξξξ, Ω m (cid:88) j =1 (cid:0) γ j b j + β j b j (cid:1) (9a)s . t . c − m (cid:88) j =1 γ j a j − (cid:88) k ∈B δ k e k − n (cid:88) l =1 ξ l e l = (9b) Q − m (cid:88) j =1 β j a j a (cid:62) j + (cid:88) k ∈B δ k e k e (cid:62) k + n (cid:88) l =1 ξ l ααα e (cid:62) l + e l ααα (cid:62) − Ω = 0 (9c)Ω ∈ C n . (9d)
4. Cutting plane algorithm
To make use of copositive duality, we must be able to solve COPs. At present, no industrial solvercan handle COPs. In the literature they are often approximately solved via SDPs (see Section EC.1of the e-companion). To solve COPs exactly, we design a novel cutting plane algorithm, whichreturns the optimal solution when it terminates.The cutting plane algorithm is applicable for the following general type of COPs with mixed-integer linear constraints over a copositive cone:max q (cid:62) λλλ + Tr( H (cid:62) Ω) (10a)s . t . d (cid:62) λλλ + Tr( D (cid:62) i Ω) = g i ∀ i = 1 , ..., m (10b) λλλ ≥ (10c)Ω ∈ C n c (10d) λ k ∈ Z ∀ k ∈ L (10e)where λλλ is an n l -dimensional vector, L is the index set for integer variables in λλλ , Ω ∈ R n c × n c . C n c is an n c -dimensional copositive cone. Note that the COP problems P COP o (7) and P COP r (9) arespecial cases of the COP (10). We make use of the integer constraint, (10e), in solving for PNE ofIPGs in Section 6.The algorithm starts by removing the conic constraint (10d) to obtain the initial master prob-lem, which is iteratively refined by the addition of cuts. At each iteration, we solve the masterproblem to obtain an optimal solution, denoted by (¯ λλλ, ¯Ω). To determine whether ¯Ω is copositive,we employ the MIP problem proposed by Anstreicher (2020), which checks copositivity using arecent characterization of copositive matrices given in Dickinson (2019). This MIP problem, whichserves as the separation problem in our cutting plane algorithm, is given by: SP (Ω) : max w (11a)s . t . Ω z ≤ − w + M (cid:62) (1 − u ) (11b) (cid:62) u ≥ q (11c) w ≥ ≤ z ≤ u (11e) u ∈ { , } n c (11f)where q = 1 (or a larger integer, depending on problem structure), is a vector of all ones, and M ∈ R n c × n c ++ is a matrix of large numbers. By Theorem 2 of Anstreicher (2020), ¯Ω is copositive ifand only if the optimal objective of SP ( ¯Ω) is zero. At any iteration, if the optimal value of the subproblem is zero, then the master problem solutionis feasible and optimal for the COP (10). Otherwise, as ¯Ω is not copositive, we add the followingcut to the master problem: ¯z (cid:62) Ω ¯z ≥ ¯z is an optimal solution of SP ( ¯Ω). Proposition 1.
If the optimal value of SP ( ¯Ω) is nonzero, then (12) cuts off ¯ΩNote that the cut (12) does not eliminate any feasible solutions from (10). This is because for any ¯z ∈ R n c + , any copositive matrix Ω satisfies ¯z (cid:62) Ω ¯z ≥ SP ( ¯Ω) is an MIP, we can strengthen its LP relaxation to improve its computationalefficiency. Anstreicher (2020) suggests doing so by setting M ik = 1 + (cid:80) n c j =1 , j (cid:54) = i { ¯Ω ij : ¯Ω ij > } , ∀ k =1 , ..., n c . Another way to strengthen the separation problem is to let q = 2, which is valid if diag( ˆΩ) ≥ ≥ q = 2because Ω ∈ C n c in the initial master problem.In some cases the master problem is unbounded at initialization. There are several ways to dealwith this. One is to impose a large upper bound on the elements of Ω. This bound can be graduallyrelaxed and then removed eventually.If the original MBQP can be solved with reasonable efficiency and (10) is the dual of its com-pletely positive reformulation, we can use strong duality in several ways. Suppose x ∗ is the optimalsolution of the MBQP. • If the optimal objective of the MBQP and the cutting plane algorithm are equal, then thealgorithm has terminated at the optimal solution. • Suppose x ∗ is optimal for the MBQP. Then we can tighten the master problem by adding thecomplementary slackness constraint Tr( x ∗ x ∗(cid:62) Ω) = 0.We can also tighten the master problem by incorporating the semidefinite relaxations of the copos-itive cone given in (Parrilo 2000).We were unfortunately unable to prove that the cutting plane algorithm terminates in finitesteps. However, note that the other exact algorithm for COPs, the simplicial partition method(Bundfuss and D¨ur 2009), is also shown to be exact only in the limit. We find that in our numericalexperiments, the cutting plane algorithm usually converges in a reasonable computational time,and when it does not the approximate solution obtained from the last iteration is often still useful.
5. Pricing unit commitment
In this section we use copositive duality to design a pricing mechanism for UC. UC optimallyschedules the startups and shutdowns of the generators in a power system, typically over a 48hour horizon. The problem is usually formulated as an MIP, in which the startup and shutdowndecisions are binary variables. We reformulate the MIP as a CPP in Section 5.1, and use the dualto define prices in Section 5.2, which we modify in Section 5.3 to guarantee revenue adequacy forindividual generators.Let G be the set of generators and T the set of time periods. c pg , c ug , and d t are respectively theproduction cost, startup cost, and the load. For generator g ∈ G at time t ∈ T , p gt is the productionlevel, u gt is the binary decision to startup, z gt is equal to one if online and zero if offline, and ψ jgt is the slack variable for the constraint that is not power balances.UC can be written (Carri´on and Arroyo 2006, Taylor 2015): U C : min (cid:88) g ∈G (cid:88) t ∈T (cid:0) c pg p gt + c ug u gt (cid:1) (13a)s . t . (cid:88) g ∈G p gt = d t ∀ t ∈ T (13b) a (cid:62) jgt x = b jgt ∀ j = 1 , ..., m, g ∈ G , t ∈ T (13c) z gt ∈ { , } ∀ g ∈ G , t ∈ T , (13d)where x (cid:62) = ( u (cid:62) , z (cid:62) , p (cid:62) , ψψψ (cid:62) ), with bold letters of variables denoting vectors. For example, u denotesthe vector of variables u gt for all g ∈ G , t ∈ T . The objective (13a) is the total production andstartup cost. Constraints (13b) ensure that the total production satisfies the load at each hour.Constraints (13c) are individual generator’s operational constraints, which can include productionlevel constraints, minimum up/down time, ramping constraints, and energy storage. The binaryconstraint on u gt is implied by constraints that link u gt and z gt in (13c). We reformulate (13) in the form of P CPP o . Let X be the matrix of lifted variables for x . Y is asdefined in (5). To make the correspondence between elements of X and variables in vector x moreexplicit, we denote by X vwk,q the element of X corresponding to the row of the v k variable and thecolumn of the w q variable. That is, X vw represents the block of X with rows corresponding to thevariables v and columns to the variables w by X vw . For example, for the UC above, we have X = X uu , X uu , ... X uψ ,rl X uu , X uu , ...... . . . ... X ψurl, .. ... X ψψrl,rl . Let h t be the coefficient vector for the left-hand side of constraints (13b). The CPP reformulationis as follows: U C
CPP : min (cid:88) g ∈G (cid:88) t ∈T (cid:0) c pg p gt + c ug u gt (cid:1) (14a)s . t . (cid:88) g ∈G p gt = d t ∀ t ∈ T ( λ t ) (14b) a (cid:62) jgt x = b jgt ∀ j = 1 , ..., m, g ∈ G , t ∈ T ( φ jgt ) (14c)Tr( h t h (cid:62) t X ) = d t ∀ t ∈ T (Λ t ) (14d)Tr( a jgt a (cid:62) jgt X ) = b jgt ∀ j = 1 , ..., m, g ∈ G , t ∈ T (Φ jgt ) (14e) z gt = Z gt ∀ g ∈ G , t ∈ T ( δ gt ) (14f) Y ∈ C ∗ n +1 (Ω) . (14g)Dual variables are shown to the right of the constraints.The dual of (14) is: U C
COP : max (cid:88) t ∈T (cid:32) d t λ t + d t Λ t + m (cid:88) j =1 (cid:88) g ∈G (cid:0) b jgt φ jgt + b jgt Φ jgt (cid:1)(cid:33) (15a)s . t . ( λλλ, φφφ, ΛΛΛ , ΦΦΦ , δδδ, Ω) ∈ F COP , (15b)where F COP denotes the feasible region of the dual problem, which can be written in the form ofconstraints (7b) - (7c).
We now describe CDP, a pricing mechanism for UC. CDP is budget balanced and, under cer-tain conditions, individually rational. Let x ∗ be the optimal solution of U C , and set X ∗ = x ∗ x ∗(cid:62) .According to Remark 1, ( x ∗ , X ∗ ) is an optimal solution to U C
CPP (14). The CDP mechanism isdefined as follows:
Definition 1 (CDP).
Let ( λλλ ∗ , φφφ ∗ , ΛΛΛ ∗ , ΦΦΦ ∗ ) be an optimal solution for U C
COP . Under the CDPmechanism, at hour t the system operator (SO): • collects π L t = λ ∗ t d t + Λ ∗ t d t + (cid:80) mj =1 (cid:80) g ∈G (cid:0) b jgt φ ∗ jgt + b jgt Φ ∗ jgt (cid:1) from the load, and • pays π G gt = λ ∗ t p ∗ gt + Λ ∗ t X pp ∗ gt,gt + (cid:80) mj =1 (cid:0) φ ∗ jgt a jgt x ∗ + Φ ∗ jgt Tr( a jgt a (cid:62) jgt X ∗ ) (cid:1) + (cid:80) g ∈− g f gg (cid:48) t to generator g at time t , where − g = G \ { g } . f gg (cid:48) t is the share of g ’s revenue from the cross-term payment2Λ ∗ t X ppgt,g (cid:48) t . It must satisfy f gg (cid:48) t + f g (cid:48) gt = 2Λ ∗ t X ppgt,g (cid:48) t , and if X ppgt,g (cid:48) t = 0, then f gg (cid:48) t = 0. π L t consists of volumetric price payments, d t λ ∗ t and d t Λ ∗ t , and payments that depend on theshadow prices of operational constraints with non-zero right-hand sides. Note that the quadratic term d t Λ ∗ t corresponds to the lifted power balance (14d). The shadow price payments for operationalconstraints are roughly comparable to transmission congestion rents. They could represent, forexample, a payment corresponding to a ramping constraint, which, if loosened, would improve theobjective. π G gt depends on generators’ optimal production levels and on/off statuses. It is obtained bysumming the products of the left-hand sides of constraints (14b) - (14e) with their correspondingdual prices. As in the load payment, λ ∗ t p ∗ gt and Λ ∗ t X pp ∗ gt,gt are volumetric payments for which theprices are uniform for all generators. Λ ∗ t X pp ∗ gt,gt corresponds to the lifted power balance (14d). Theterms in the first sum similarly correspond to shadow prices of various constraints. The shadowprices are not uniform as they depend on the generator index, g . The second sum corresponds tothe off-diagonal entries in the lifted matrix X ∗ . Since this payment involves two generators, it is anopen question as to how this payment should be divided between those generators. For example,we can assign it to the generator that loses money, or divide it evenly as in Section 5.3.The following example examines non-uniform payments resulting from shadow prices. We usethis example in our numerical experiments in Section 7.2. Example 1.
Suppose constraint (13c) is given by: u gt − θ gt = z gt − z g,t − ∀ g ∈ G , t ∈ T \ { } (16a) p gt − µ gt = z gt P g ∀ g ∈ G , t ∈ T (16b) p gt + γ gt = z gt P g ∀ g ∈ G , t ∈ T (16c) z gt + η gt = 1 ∀ g ∈ G , t ∈ T , (16d)where θ gt , µ gt , γ gt , and η gt are slack variables, and P g and P g are respectively the lower and upperproduction limits of generator g ∈ G . Constraint (16d) implies upper bounds on z gt , and is the onlyconstraint with a non-zero constant on the right-hand-side.The portion of the payment resulting from constraint (16d) and the corresponding lifted con-straint are included in π G t . We refer to the shadow prices of these constraints the availability prices ,as they signal the availability of the generators at each hour. These are the only non-uniform pricesin this example.The following theorem shows that CDP is revenue neutral for both the SO and the generators.Note that the individual generators are in general not revenue neutral under CDP. Theorem 1.
If strong duality holds for
U C
CPP , then CDP balances the revenue and the aggregatecost of the generators. It also balances the revenue and payment of the SO. Next, we provide a sufficient condition guaranteeing the individual rationality of CDP. A pricingmechanism is individually rational if each market participant has no incentive to deviate from thecentrally optimal solution.The profit-maximization problem of generator g is given by: π Gen g ( p − g ) := max x g (cid:88) t ∈T (cid:16) λ ∗ t p gt + Λ ∗ t p gt + m (cid:88) j =1 (cid:0) φ ∗ jgt b jgt + Φ ∗ jgt b jgt (cid:1) + (cid:88) g (cid:48) ∈− g f gg (cid:48) t − c pg p gt − c ug u gt (cid:17) (17a)s . t . a (cid:62) jgt x = b jgt ∀ j = 1 , ...m, t ∈ T (17b)where p − g denotes the production decision of all generators other than g . x g ∈ R n g denotes theportions of x that corresponds to generator g ∈ G . The first five terms in the objective representthe total revenue of generator g . π Gen g ( p − g ) should not contain entries of X , which are liftedvariables that do not have direct physical meanings. As discussed in Remark 1, at optimality wecan equivalently use ( p ∗ gt ) in place of X pp ∗ gt,gt and p ∗ gt p ∗ g (cid:48) t in place of X pp ∗ gt,g (cid:48) t .If the optimal solution of the profit-maximization problem (17) matches the corresponding por-tion of the optimal solution of U C for all generators g ∈ G , then the market mechanism is individ-ually rational.Theorem 2 provides a sufficient condition for when CDP is individually rational. Theorem 2.
Assume strong duality holds for
U C
CPP , and let ( x ∗ , X ∗ ) be an optimal solution. Ifall items in X ∗ in the forms of X vw ∗ g t,g t , ∀ v, w, g (cid:54) = g , g , g ∈ G , t ∈ T are equal to zero, then themarket mechanism with CDP is individually rational. The condition X vw ∗ g t,g t = 0 , ∀ v, w, g (cid:54) = g , g , g ∈ G , t ∈ T ensures that the conic constraint in U C
CPP is decomposable by g , which is a key property needed in the proof.More generally, there may be groups of generators that are coupled to each other, but not tothose of other groups. We refer to this as subset rationality. There could also, for example, begroups of time periods that are similarly decoupled. It may be possible to identify more complicateddecomposable structures using matrix completion (Drew and Johnson 1998). This is a topic offuture work. CDP does not guarantee each individual generator’s revenue adequacy, i.e., nonnegative profit.This is also the case with some other schemes, including RP and CHP. In CDP, we can enforcerevenue adequacy by adding constraints directly to the dual of
U C
CPP .Revenue adequacy can be enforced through the non-uniform prices φ ∗ jgt and Φ ∗ jgt , which can bedifferent for each generator, and/or through the uniform prices λ t and Λ t . We use uniform pricing because it is easier to implement in practice. In Section EC.4 of the e-companion, we present adifferent version that uses availability prices as well.We enforce revenue adequacy by adding a new constraint to the dual problem, which ensuresthat the payment to each generator is no less than its costs. The resulting augmented dual problemis given by: max (cid:88) t ∈T (cid:32) d t λ t + d t Λ t + m (cid:88) j =1 (cid:88) g ∈G ( b jgt φ jgt + b jgt Φ jgt ) (cid:33) (18a)s . t . (cid:88) t ∈T p ∗ gt λ t + p ∗ gt Λ t + (cid:88) g (cid:48) ∈− g p ∗ gt p ∗ g (cid:48) t Λ t ≥ (cid:88) t ∈T (cid:0) c pg p ∗ gt + c ug u ∗ gt (cid:1) ∀ g ∈ G (18b)( λλλ, φφφ, ΛΛΛ , ΦΦΦ , δδδ, Ω) ∈ F COP . (18c)The objective (18a) and the constraint (18c) are the same as in the original dual problem, U C
COP .The left-hand side of constraint (18b) is the total revenue of generator g , assuming that the cross-term payments are divided evenly between generators. The right-hand side of (18b) is the totalcost of generator g .The prices are computed by solving (18). The optimal values of primal variables, p ∗ gt and u ∗ gt ,could be obtained by solving U C . We refer to this pricing mechanism as Revenue-adequate CDP(RCDP). More formally, the RCDP mechanism is defined as follows.
Definition 2 (RCDP Mechanism).
Let ( λλλ ∗ , ΛΛΛ ∗ ) be an optimal solution for (18). Under theRCDP mechanism, at hour t the SO: • collects π L (cid:48) t = λ ∗ t d t + Λ ∗ t d t from the load, and • pays π G (cid:48) gt = (cid:0) λ ∗ t p ∗ gt + Λ ∗ t X pp ∗ gt,gt (cid:1) + (cid:80) g ∈− g p ∗ gt p ∗ g (cid:48) t Λ t to the generator g .A benefit of RCDP is that all generators are paid the same price in each hour, so that moreefficient generators with lower costs earn higher profit. The load also pays on a purely volumetricbasis. The SO remains revenue neutral, in that π L (cid:48) t = π G (cid:48) t . This can be proved by adapting theproof of Theorem 1.Because the addition of constraints (18b) restrict the original dual problem (15), it is worthchecking if the problem is still feasible. We known that if either the primal or dual is feasible,bounded, and has an interior point, then the other is also feasible (Luenberger et al. 1984). Wetherefore focus on the dual of (18). Let q g ≥ (cid:88) g ∈G (cid:88) t ∈T (cid:0) c pg p gt + c ug u gt − ( c pg p ∗ gt + c ug u ∗ gt ) q g (cid:1) (19a)s . t . (cid:88) g ∈G (cid:0) p gt − p ∗ gt q g (cid:1) = d t ∀ t ∈ T (19b) Tr( h t h (cid:62) t X ) − (cid:88) g ∈G p ∗ gt + (cid:88) g (cid:48) ∈− g p ∗ gt p ∗ g (cid:48) t q g = d t ∀ t ∈ T (19c)(14c) , (14e) , (14f) (19d) q g ≥ g ∈ G (19e) Y ∈ C ∗ n +1 . (19f)Observe that this is similar to U C
CPP , but with additional terms multiplying q g in the objectiveand constraints. Note that (19) includes both p gt , a variable, and p ∗ gt , part of the solution to U C
CPP .The CPP (19) is feasible because we recover
U C
CPP by setting all q g to zero. It is bounded dueto the equality (19b), and the fact that the production level p gt is usually bounded by generatorcapacity. If the problem has an interior, then its dual, (18), is feasible. Remark 3.
Since (19) is a relaxation of
U C
CPP , (19) should have an interior if
U C
CPP has aninterior. Also, if
U C
CPP has an interior, then
U C
COP is feasible. Thus, the RCDP problem (18) isfeasible whenever
U C
COP is feasible.
6. Mixed-Binary Quadratic Games
In this section, we use copositive duality to characterize the equilibria of mixed-binary quadraticgames (MBQGs). In an MBQG, each player solves an optimization of the form P MBQP , which arecoupled through the objectives, but not the constraints. The binary decisions preclude the directuse of existence and uniqueness results for continuous games, and the use of the KKT conditionsfor obtaining equilibria. We surmount these issues by converting each player’s MBQP to a CPP,as described in Section 3.2. We give conditions for existence and uniqueness in Section 6.2, andcompute PNE using the KKT conditions in Section 6.3.We study an n -person MBQG, G MBQP = (cid:104)I , ( X i ) i ∈I , ( x i ) i ∈I (cid:105) , where I is the set of indices for theplayers and |I| = n , X i is the strategy set of player i , and x i ∈ X i is the n i -dimensional decisionvector of player i , which contains both continuous and binary variables. We assume that X i isbounded throughout this section. If it is not, we can always add constraints to make it boundedwithout changing the optimal solution. X i is also closed because there are no strict inequalityconstraints in P MBQP i ( x − i ) below, and therefore compact. Let X − i and x − i denote the strategy setsand strategies of all players except i . Player i faces the following MBQP: P MBQP i ( x − i ) : min f i ( x i , x − i ) (20a)s . t . a ( i ) (cid:62) j x i = b ( i ) j ∀ j = 1 , ..., m i (20b) x ik ∈ { , } ∀ k ∈ B i (20c) x i ∈ R l + . (20d) f i ( x i , x − i ) = x (cid:62) Q ( i ) x + 2 c ( i ) (cid:62) x is the quadratic payoff function of player i , x the vector of allstrategies, and Q ( i ) ∈ S n i a symmetric matrix. Constraints (20b) are linear and independent of thedecisions of other players. B i is the subset of indices corresponding to the binary decisions of player i . A pure-strategy NE (PNE) of G MBQP is a vector of actions x ∗ ∈ ( X i ) i ∈I such that for each player i : f i ( x ∗ i , x ∗− i ) ≤ f i ( x i , x ∗− i ) , ∀ x i ∈ X i , which is equivalently stated f i ( x ∗ i , x ∗− i ) = opt( P MBQP i ( x ∗− i )) . (21) We reformulate G MBQP in terms of CPP, and show how the PNE of one can be obtained from theother, and vice versa. Throughout this section, we use the CPP reformulation P CPP r because it hasseveral advantages over P CPP o . For this reason we assume that (KP) holds throughout. However,we could in principle derive all of the results in this section using P CPP o , in which case (KP) neednot hold.We denote the reformulated game G CPP = (cid:104)I , ( X (cid:48) i ) i ∈I , ( x i , X i ) i ∈I (cid:105) , where X (cid:48) i is the strategy setof player i . Due to the relationship between the feasible regions of P MBQP and P CPP , as establishedby Corollary 2.5 of Burer (2009), the compactness of X i implies the compactness of X (cid:48) i . Player i solves: P CPP i ( x − i ) : min Tr( Q ( i ) ii X i ) + 2 x (cid:62)− i Q ( i ) − i,i x i + x (cid:62)− i Q ( i ) − i, − i x − i + 2 c ( i ) (cid:62) x (22a)s . t . a ( i ) (cid:62) j x i = b ( i ) j ∀ j = 1 , ..., m i (22b) a ( i ) (cid:62) j X i a ( i ) j = b ( i )2 j ∀ j = 1 , ..., m i (22c) x ik = X i,kk ∀ k ∈ B i (22d) x il = Tr (cid:18) ααα ( i ) e (cid:62) l + e l ααα ( i ) (cid:62) X i (cid:19) ∀ l = 1 , ..., n i (22e) X i ∈ C ∗ n i . (22f)Note that x i and X i are the only variables in P CPP i ( x − i ), and x − i is treated as given. ααα ( i ) is definedfor P MBQP i as in (KP). Let F i ( x i , X i , x − i ) denote the objective function of P CPP i ( x − i ). A PNE of G CPP is a set of strate-gies, ( x ∗ i , X ∗ i ) i ∈I ∈ ( X (cid:48) i ) i ∈I , such that for each player i : F i ( x ∗ i , X ∗ i , x ∗− i ) = opt( P CPP i ( x ∗− i )) . (23)Any game in the form of G MBQP can be written in the form of G CPP . Theorem 3 relates theequilibria of the games.
Theorem 3.
Let x ∗ be a PNE of G MBQP . Then ( x ∗ i , x ∗ i x ∗(cid:62) i ) i ∈I is a PNE of G CPP .Conversely, if ( x ∗ i , X ∗ i ) i ∈I is a PNE of G CPP , and if ( Q ( i ) ii (cid:23) i ∈I , ( x ∗ ik ∈ { , } , ∀ k ∈ B i ) i ∈I , then x ∗ is a PNE of G MBQP .Remark.
The requirement ( x ∗ ik ∈ { , } , ∀ k ∈ B i ) i ∈I ensures that a PNE of G CPP can be transformedinto G MBQP . We can replace this requirement with the (possibly more restrictive) condition thatthe matrix (cid:20) x ∗(cid:62) i x ∗ i X ∗ i (cid:21) is rank one for all i ∈ I . This is similar to the condition given in Ahmadi andZhang (2020) to ensure the validity of an SDP reformulation of a 2-person bimatrix game.The second part of Theorem 3 can also be replaced with the following (more restrictive) propo-sition: Proposition 2. If ( x ∗ i , X ∗ i ) i ∈I is a PNE of G CPP , and if ( Q ( i ) ii (cid:23) i ∈I , and each one of the opti-mization problems ( P MBQP i ( x ∗− i )) i ∈I has a unique optimal solution, then x ∗ is a PNE of G MBQP . This proposition is true because if P MBQP i ( x ∗− i ) has a unique optimal solution, then x ∗ i must befeasible for P MBQP i ( x ∗− i ). Thus we can follow a similar proof as in Theorem 3 to show that x ∗ is aPNE of G MBQP . G MBQP
We first prove that PNE for G MBQP exists under some conventional assumptions. To prove theexistence of PNE in G MBQP , first note that its corresponding CP game G CPP always has at least onePNE. This is because according to the classical PNE existence condition for games with convexstrategy sets (Debreu 1952, Glicksberg 1952, Fan 1952), G CPP should have a PNE as P MBQP i ( x − i )has a convex and compact feasible region and a linear objective function.Now we provide a sufficient condition for the existence of PNE in G MBQP . Proposition 3 (Existence of PNE) . G MBQP has at least one PNE if(i) Property (KP) is satisfied for P MBQP i ( x − i ) ,(ii) Q ( i ) ii (cid:23) , and(iii) there exists a PNE ( x ∗ i , X ∗ i ) i ∈I for the corresponding G CPP that satisfies ( x ∗ ik ∈ { , } , ∀ k ∈B i ) i ∈I . Conditions (ii) and (iii) are necessary to ensure that PNE of G CPP can be transformed to thePNE of G MBQP (see Theorem 3). If condition (iii) is not satisfied, then G MBQP does not have anyPNE. This is because if G MBQP did have a PNE, then, following Theorem 3, we could construct aPNE for G CPP that does satisfy (iii).Using Proposition 2, condition (iii) in Proposition 3 can also be replaced with the requirementthat the optimization problems in G MBQP all have unique optimal solutions, since this requirementalso ensures that a PNE of G CPP can be converted to a PNE of G MBQP . It is worth noting that asimilar requirement is proposed for the existence of PNE in two-person discrete games, which isa special type of MBQ game, by Mallick (2011). This paper proves that for a two-person discretegame, if both players have unique best responses and a condition called Minimal Acyclicity issatisfied, then the PNE must exist.We note that in principle the uniqueness theorem of Rosen (1965) can be applied to G CPP .In particular, it implies that if equilibria exist, certain regularity conditions hold for the playerproblems to have strong duality, and the payoff functions are diagonally strictly convex, then anyPNE of G MBQP is unique. It may be possible to use this perspective to identify new classes ofdiscrete games that have a unique equilibrium.
A PNE of G CPP satisfies the KKT conditions when a certain constraint qualification, such as Slater’scondition, is met. We write the KKT conditions for G CPP explicitly. For all i ∈ I , we have:(22b) − (22f) (24a)2 Q ( i ) (cid:62)− i,i x − i + 2 c i − m i (cid:88) j =1 γ ij a ( i ) j − (cid:88) k ∈B i δ ik e k − n i (cid:88) l =1 ξ il e l = 0 (24b) Q ( i ) ii − m i (cid:88) j =1 β ij a ( i ) j a ( i ) (cid:62) j + (cid:88) k ∈B i δ ik e k e (cid:62) k + n i (cid:88) l =1 ξ il ααα ( i ) e (cid:62) l + e l ααα ( i ) (cid:62) − Ω i = 0 (24c)Ω i ∈ C n i (24d)Tr(Ω i X i ) = 0 (24e)where (24a) are constraints of P CPP i ( x − i ). γγγ i , βββ i , δδδ i , ξξξ i , and Ω i are the respective dual variables for(22b) - (22f). Let L ( x i , X i , x − i ) be the Lagrangian function of P CPP i ( x − i ), i.e., L ( x i , X i , x − i ) = F i ( x i , X i , x − i ) + m i (cid:88) i =1 (cid:16) γ ij ( b ( i ) j − a ( i ) (cid:62) j x i ) + β ij (( b ( i ) j ) − a ( i ) (cid:62) j X i a ( i ) j ) (cid:17) + (cid:88) k ∈B i δ ik ( X i,kk − x ik ) + n i (cid:88) l =1 ξ il (cid:18) Tr (cid:18) ααα ( i ) e (cid:62) l + e (cid:62) l ααα ( i ) X i (cid:19) − x il (cid:19) + Tr(Ω i X i ) . Then ∂ L ( x i , X i , x − i ) /∂ x i = 0 and ∂ L ( x i , X i , x − i ) /∂X i = 0 respectively correspond to constraints(24b) and (24c). Constraint (24d) is the dual cone constraint. Finally, constraint (24e) enforcescomplementary slackness.Our goal is to obtain the PNE for G MBQP . By Proposition 3, if Q ( i ) ii (cid:23) , ∀ i ∈ I , then we can addthe following extra constraints to the KKT conditions of G CPP : x ik ∈ { , } , ∀ k ∈ B i , i ∈ I , (25)to ensure that the solution is a PNE for G MBQP .The KKT conditions (24) contain conic constraints X i ∈ C ∗ n i and Ω i ∈ C n i , as well as a bilinearconstraint Tr(Ω i X i ) = 0. As a result, it is difficult to solve (24) with current technology. However,in the special case where | B i | = l in P MBQP i ( x − i ) (20) ∀ i ∈ I , i.e., all variables in P MBQP i ( x − i ) arebinary (such as in bimatrix games), then the KKT conditions (24) could be replaced with a MIPproblem over copositive cone constraints. We now reformulate the KKT conditions for this specialcase.From the primal problem constraints (24a), we drop the completely positive conic constraint(22f). We keep constraints (22b), and add constraints (25). Observe that constraints (22c) - (22f) area relaxation of the constraint X i = x i x (cid:62) i , or, equivalently, X i,j ,j = x ij x i,j . Because all variables in P MBQP i ( x − i ) are binary, we can therefore replace (22c) - (22f) with the following linear constraints( ∀ i ∈ I ): X i,jj = x ij ∀ j = 1 , ..., l (26a) X i,j j ≤ x ij ∀ j (cid:54) = j ; j , j = 1 , ...l (26b) X i,j j ≤ x ij ∀ j (cid:54) = j ; j , j = 1 , ...l (26c) X i,j j ≥ x ij + x ij − ∀ j (cid:54) = j ; j , j = 1 , ...l (26d) X i,j j ≥ ∀ j (cid:54) = j ; j , j = 1 , ...l. (26e)Constraints (26b)-(26e) are the McCormick relaxation of the bilinear expression X i,j ,j = x ij x i,j ( j (cid:54) = j ).On the dual side, we keep all constraints (24b) - (24d). We linearize the bilinear complementaryslackness constraint, (24e), by defining a new matrix, Z i ∈ R l × l , and letting Z i,j j = Ω i,j j X i,j j .Constraint (24e) can then be replaced with the disjunctive:Tr( Z ) = 0 (27a) − mX i,j j ≤ Z i,j j ≤ mX i,j j ∀ j , j = 1 , ..., l (27b)Ω i,j j − m (1 − X i,j j ) ≤ Z i,j j ≤ Ω i,j j + m (1 − X i,j j ) ∀ j , j = 1 , ..., l (27c) where m is a sufficiently large number. Here we use the fact that elements in X i are binary, whichis due to constraints (25) and (26).We can now state the reformulation of the KKT conditions (24) for the case where all variablesare binary. We have, ∀ i ∈ I : a ( i ) (cid:62) j x i = b ( i ) j ∀ j = 1 , ..., m i (28a)(24b) , (24c) , (25) , (26) , (27) (28b)Ω i ∈ C n i . (28c)This is an MIP over the copositive cone, and can be solved exactly with the cutting plane algorithmin Section 4 using an MIP solver. If a solution to (28) exists, then it is a PNE for G MBQP . Notethat we cannot approximate (28) with SDP solvers such as MOSEK because they cannot readilysolve mixed-integer SDPs.
7. Numerical Results
In this section we present our numerical experiments. Section 7.1 implements several pricingschemes on the Scarf’s example. Section 7.2 compares different pricing schemes for a nonconvexelectricity market. In Section 7.3, we compute the PNE of bimatrix games using their KKT condi-tions. We also include a comparison of our COP cutting plane algorithm with other COP algorithmsin e-companion Section EC.5.All experiments are implemented in Julia v1.0.5 using the optimization package JuMP.jl v0.20.1.The COP cutting plane algorithm was implemented using CPLEX 12.8. We use Mosek 9.1 to solveSDPs approximations in Sections 7.1, 7.2, and EC.5.
Scarf’s example is often used to compare pricing schemes for nonconvex markets. We use themodified version from Hogan and Ring (2003) to compare CDP with RP and CHP, which arecurrently used by utilities in the U.S.. In the modified Scarf’s example, there are three types ofgenerators: smokestack, high technology, and medium technology. Let G i be the set of generatorsof type i = 1 , ,
3. We have |G | = 6 , |G | = 5 , |G | = 5. The binary variables u g i , g i ∈ G i , i = 1 , ,
3, rep-resent startup decisions, and the continuous variables p g i , g i ∈ G i , , i = 1 , ,
3, represent productiondecisions. Scarf’s example solves the following cost minimization problem:min (cid:88) g ∈G (53 u g + 3 p g ) + (cid:88) g ∈G (30 u g + 2 p g ) + (cid:88) g ∈G p g (29a)s . t . (cid:88) g ∈G p g + (cid:88) g ∈G p g + (cid:88) g ∈G p g = d (29b) p g ≥ u g ∀ g ∈ G (29c) p g ≤ u g ∀ g ∈ G (29d) p g ≤ u g ∀ g ∈ G (29e) p g ≤ u g ∀ g ∈ G (29f) p g ≥ , u g ∈ { , } ∀ g ∈ G , G , G (29g)where the objective is to minimize the total cost. Constraint (29b) ensures the total productionequals the demand. Constraints (29c) set the lower bound for the production of medium technologygenerators when they are on. Constraints (29d)-(29f) set the capacity of each generator.We experiment with various demand levels from 5 to 160, with a step length of 5. In Figure 1we compare the following aspects of RP, CHP, CDP pricing and RCDP:1. The uniform prices. Notice that for CDP and RCDP, both λ ∗ t and Λ ∗ t are uniform prices. Inall of our experiments Λ ∗ t equals zero. Therefore, we report only λ ∗ t for those schemes.2. Generator profits, as calculated by deducting costs from total revenue, which includes bothprice-based payments and uplift.3. The payments from generator-dependent (non-uniform) prices φ ∗ jgt and Φ ∗ jgt .4. The make-whole uplift payments. This payment is made when the revenue from electricityprices is not enough to cover the costs. It is equal to the difference between the revenue andcosts.Mosek solves all instances in under ten seconds. Because the cutting plane algorithm is signif-icantly slower, we only use it when the demand level is less than 100. When the demand level ishigher, we present results from the SDP approximation.Figure 1a shows that a small change in demand level can result in significant volatility in RP. Thisobservation is consistent with results in the literature. Interestingly, CHP and CDP are equivalentfor all demand levels. RCDP is higher than CDP for lower demand, and equals COP when thedemand is high.In Figure 1b, we find that RP and CDP have zero profit for all instances. CHP generates near-zero profits at lower demand level and higher profits at higher demand levels. RCDP generates thehighest profits among all pricing schemes, and match the profits of CHP at higher demand levels.In Figure 1c, both CHP and RCDP have no generator-dependent pricesdue to the fact that bothonly use uniform prices corresponding to the demand constraint. CDP produces near-zero negativegenerator-dependent prices at low demand levels, and more negative prices at higher demand levels.RP produces volatile and large generator-dependent prices in many instances. As explained byO’Neill et al. (2005), the negative generator-dependent prices are used to discourage the entry of Demand U n i f o r m p r i c e RPCHP CDPRCDP (a)
Demand P r o f i t RPCHP CDPRCDP (b)
Demand N o n - un i f o r m p r i c e p a y m e n t RPCHP CDPRCDP (c)
Demand M a k e - w h o l e u p li f t RPCHP CDPRCDP (d)
Figure 1
Comparison of different pricing schemes for (a) uniform prices, (b) profits, (c) payments fromgenerator-dependent prices, (d) make-whole uplift payments. marginal plants when it is uneconomic to do so. In practice, utilities usually disregard such negativeprices.Figure 1d shows that RP requires zero make-whole payment, which is also consistent with theresults in Azizan et al. (2020). RCDP ensures revenue adequacy and thus also needs no make-wholepayment. CHP requires make-whole payments, as expected, because Lagrangian duals of MIPs donot in general have strong duality. Interestingly, CDP also requires make-whole payments in manyinstances, as in those instances strong duality does not hold.The advantages of RCDP are that it does not rely on generator-dependent/make-whole upliftpayments to ensure revenue adequacy, and its uniform prices are less volatile than RP.
In this section we compare pricing schemes for unit commitment, as described in Section 5. Notethat in the experiments we use the formulation in Example 1 for the
U C problem. Our test instancesare based on the adapted California ISO dataset of Guo et al. (2020). The parameters for thegenerators are listed in Table 2. Table 2
Generator Parameters
Gen. 1 Gen. 2 Gen. 3 Gen. 4 c og c sg P g
297 238 198 198 P g
620 496 620 620We first consider 2 generators over 4 hours, which we refer to as Case 1. We set d t =[508 , , , x ∗ x ∗(cid:62) Ω) = 0 when solving the CDP COP.We compare the following aspects of RP, CHP, CDP and RCDP and present the results in Table3:1. Generator profits.2. Profits without the uplift payments.3. The payments from generator-dependent prices.4. The make-whole uplift payments.
Table 3
Comparison of Pricing Schemes for Case 1
RP CHP CDP RCDPGen. 1 profit 0 839.5 292.0 1393.6Gen. 1 profit (pre-uplift) 0 839.5 292.0 1393.6Gen. 2 profit 0 0 0 72.1Gen. 2 profit (pre-uplift) 0 -95.3 -292.0 72.1Total profit 0 839.5 292.0 1465.7Total profit (pre-uplift) 0 744.2 0 1465.7Gen. dep. payment 498.0 0 -201.5 0Make-whole uplift 0 95.3 292.0 0RP results in zero profit for both generators, but relies on the revenue from generator-dependentprices for Generator 2. CHP uses a make-whole uplift payment to avoid a loss for Generator 2.Interestingly, CDP results in positive profit for Generator 1, and uses a make-whole uplift equalto Generator 1’s profit to cover the loss of Generator 2. As expected, CDP is revenue neutralin aggregate, but not for individual generators. RCDP is the only pricing scheme that has nogenerator-dependent or make-whole uplift payments. It also ensures the generators with lower costsreceive higher profits, which is desirable and to be expected in a market with only uniform prices. In the second example, we include all 4 generators in Table 2, and 4 hours in the planninghorizon. We refer to this as Case 2. The demand is d t = [1469 , , , Table 4
Comparison of Pricing Schemes for Case 2
RP CHP CDP RCDPTotal profit 86859.5 86528.4 56560.1 129918.3Total profit (pre-uplift) 86859.5 86445.0 56560.1 129918.3Gen. dep. payment 86.3 0 -89354.5 0Make-whole uplift 0 83.4 0 0The results for Case 2 are similar to Case 1. RCDP is the only pricing schemes that does notresult in generator-dependent or make-whole uplift payments. It also results in larger profits thanthe other schemes. Although not solved to optimality, CDP still produces lower overall profit thanRP and CHP. This is to be expected because the strong duality of COP guarantees zero pre-upliftprofit. Given enough time to converge, CDP result in zero profit, excluding the make-whole uplift.
In this section, we provide a proof of concept for the copositive formulation of the KKT conditions,(28), with bimatrix games. A bimatrix game has two players, 1 and 2, and each player has n i strategies to choose from. We use the binary vector x i ∈ { , } n i to denote player i ’s decision. Theelements of x i are binary because we only look for PNE. Player i faces the following maximizationproblem: max x (cid:62) R i x (30a)s . t . n i (cid:88) j =1 x ij = 1 (30b) x ij ∈ { , } ∀ j = 1 , ..., n i , (30c) where R i ∈ R n × n is the payoff matrix. If Player 1 plays the k th1 strategy and Player 2 plays the k th2 strategy, then the payoff for Player i is R i,k k . Constraint (30b) ensures that each player canonly play one strategy.Problem (30) has all binary variables and only equality constraints. The upper bound of binaryvariables are implied by constraint (30b). We therefore don’t need to add upper bounds to the CPPreformulation. Because there are no quadratic terms between Player i ’s decisions in the objective, Q ( i ) ii in (22a) of P CPP i ( x − i ) is a zero matrix, which implies that Q ( i ) ii (cid:23)
0. Therefore, bimatrix gamessatisfy the requirements of the purely binary case in Section 6.3. We can use the KKT conditions(28) to solve for the PNE of its CP counterpart, and then recover the PNE of the bimatrix gamefrom the PNE of the CP game.We test the KKT conditions on bimatrix games with 2 to 5 strategies. For each number ofstrategies we test 5 instances. For all cases we use the cutting plane algorithm to solve KKTconditions (28). In each initial master problem, we set an upper bound of 100 for the absolutevalues of the elements of each copositive matrix. Our computational results show that this boundis large enough to obtain copositive matrices upon termination. We verify the correctness of ourresults by comparing with the results from the lrsnash bimatrix game solver (Avis et al. 2010).Note that our method is slower than specialized bimatrix game algorithms such as in Avis et al.(2010), but is applicable to more general classes of games.In Table 5 we show the average time (seconds) and number of iterations used by the cuttingplane algorithm in each case. All test cases are solved in 7 seconds and under 90 iterations.
Table 5
Performance of KKT Conditions for Bimatrix Games
8. Conclusion
MBQPs can be equivalently written as CPPs, which are NP-hard but convex. Given an MBQP, westraightforwardly derive its dual COP. Due to convexity, if a constraint qualification is satisfied, theCPP and COP have strong duality. This provides a new and general notion of duality for discreteoptimization problems.We apply this perspective in two ways. We first design a new pricing mechanism for nonconvexelectricity markets, which has several useful theoretical properties. One direction of future study is the design of economic mechanisms for other nonconvex markets, e.g., surge pricing in transporta-tion. Second, we reformulate MBQ games as CP games and provide conditions for the existenceand uniqueness of equilibria. We also use the KKT conditions to solve for the equilibria. To enableimplementation, we design a new cutting plane algorithm for COPs, which we use to solve ournumerical examples.There are several promising avenues of future work. It would be useful to identify classes ofMBQP that, when reformulated as CPPs, always have strong duality. By reformulating otherclasses of discrete games in terms of CPPs, existing results for convex games could potentiallyidentify new conditions for existence and uniqueness. We also intend to extend our results tomixed-strategy Nash equilibria. Finally, it may be possible to improve the cutting plane algorithmby strengthening the master problem, and by deriving conditions under which the algorithm isguaranteed to terminate. References
Ahmadi AA, Zhang J (2020) Semidefinite programming and Nash equilibria in bimatrix games. INFORMSJ. Comput. .Anstreicher KM (2020) Testing copositivity via mixed–integer linear programming .Avis D, Rosenberg GD, Savani R, Von Stengel B (2010) Enumeration of Nash equilibria for two-player games.Economic theory 42(1):9–37.Azizan N, Su Y, Dvijotham K, Wierman A (2020) Optimal pricing in markets with nonconvex costs.Oper. Res. 68(2):480–496.Baldwin E, Klemperer P (2019) Understanding preferences: demand types, and the existence of equilibriumwith indivisibilities. Econometrica 87(3):867–932.Bomze IM, De Klerk E (2002) Solving standard quadratic optimization problems via linear, semidefinite andcopositive programming. J. Global. Opt. 24(2):163–185.Bomze IM, Frommlet F, Locatelli M (2010) Copositivity cuts for improving sdp bounds on the clique number.Math. Program. 124(1-2):13–32.Bomze IM, Locatelli M, Tardella F (2008) New and old bounds for standard quadratic optimization: domi-nance, equivalence and incomparability. Math. Program. 115(1):31.Bundfuss S, D¨ur M (2009) An adaptive linear approximation algorithm for copositive programs. SIAMJ. Optim. 20(1):30–53.Burer S (2009) On the copositive representation of binary and continuous nonconvex quadratic programs.Math. Program. 120(2):479–495.Carri´on M, Arroyo JM (2006) A computationally efficient mixed-integer linear formulation for the thermalunit commitment problem. IEEE Trans. Power Syst. 21(3):1371–1378.Carvalho M, Lodi A, Pedroso JP (2017) Computing Nash equilibria for integer programming games .Danilov V, Koshevoy G, Murota K (2001) Discrete convexity and equilibria in economies with indivisiblegoods and money. Math. Soc. Sci. 41(3):251–273.7De Klerk E, Pasechnik DV (2002) Approximation of the stability number of a graph via copositive program-ming. SIAM J. 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Soc. 3(1):170–174.Gribik PR, Hogan WW, Pope SL (2007) Market-clearing electricity prices and energy uplift. ElectricityPolicy Group .Guo C, Bodura M, Papageorgioub DJ (2020) Generation expansion planning with revenue adequacy con-straints .Hanasusanto GA, Kuhn D (2018) Conic programming reformulations of two-stage distributionally robustlinear programs over Wasserstein balls. Oper. Res. 66(3):849–869.Hogan WW, Ring BJ (2003) On minimum-uplift pricing for electricity markets. Electr. Policy Gr. 1–30.K¨oppe M, Ryan CT, Queyranne M (2011) Rational generating functions and integer programming games.Oper. Res. 59(6):1445–1460.Kostreva MM (1993) Combinatorial optimization in Nash games. Comput. Math. Appl. 25(10-11):27–34.Liberopoulos G, Andrianesis P (2016) Critical review of pricing schemes in markets with non-convex costs.Oper. Res. 64(1):17–31.Luenberger DG, Ye Y, et al. 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Guo, Bodur, and Taylor:
Copositive Duality for Discrete Markets and Games ec1 e-Companion
EC.1. SDP Approximation for COP
Let S + n be the n -dimensional positive semidefinite (PSD) cone and N n be the cone of n -dimensionalentrywise nonnegative matrices, then we have C ∗ n ⊆ S + n ∩ N n and S + n + N n ⊆ C n (D¨ur 2010). Takingadvantage of these relationships, we can relax and approximately solve CPP as an optimizationproblem over the intersection of S + n and N n cones, and restrict the COP problem as an optimizationover the Minkowski sum of S + n and N n cones.In our work, we reformulate MBQP models to CPPs using the method by Burer (2009), toutilize the dual COP problems. We summarize the relationship between all those mathematicalprogramming problems in Figure EC.1.CPPCOPMBQP Optimization over S + n ∩ N n Optimization over S + n + N n Reformulation DualizationRelaxationRestriction
Figure EC.1
Relationships between MBQP, CPP, COP and their approximations
We need to solve COPs in this work. One method often used in the literature, as mentionedabove, is to use the relationship S + n + N n ⊆ C n for approximation. More specifically, we replace theconic constraint Ω ∈ C n with the following restriction:Ω − N ∈ S + n N ≥ , which can be solved with SDP solvers such as Mosek and SeDuMi.Another method to obtain a solution of a COP is to solve its dual CPP problem using a com-mercial solver, then query the duals of CPP constraints via the solver. However, there is not anysolver that directly solves CPPs, so we instead solve an SDP relaxation of the CPP problem, thenquery the duals of the SDP relaxation. More specifically, we relax the conic constraint X ∈ C ∗ n tothe following constraints: X ∈ S + n X ≥ , which can then be solved with SDP solvers. c2 e-companion to Guo, Bodur, and Taylor:
Copositive Duality for Discrete Markets and Games
EC.2. Symmetry of CPP
When taking of dual of conic programs, it is often assumed in the literature that the coefficientmatrices are symmetric. We find that this assumption may lead to a difference in the form of thedual problem, if it is also assumed that the conic matrix is in S n , the set of symmetric matrices.We include a discussion about this issue in this section, as we believe it is an important detail tonotice when deriving dual problems in copositive programming, and it is not something we finddiscussed in the literature.We study the following CPP: P CPP C : min Tr( C (cid:62) X ) (EC.1a)s . t . Tr( A (cid:62) i X ) = b i ∀ i = 1 , ..., m ( λ i ) (EC.1b) X ∈ C ∗ n (Ω) . (EC.1c)If C ∈ S n and A i ∈ S n , i = 1 , ..., m , then the dual of P CPP C is the following: P COP C : min λ (cid:62) b (EC.2a)s . t . C − m (cid:88) i =1 λ i A i − Ω = 0 (EC.2b)Ω ∈ C n . (EC.2c)However, if some of the parameter matrices are not symmetric, P COP C is not necessarily the correctdual formulation. To understand this, first notice that although C ∗ n contains only symmetric matri-ces, its dual cone in R n × n may contain non-symmetric matrices. For example, the non-symmetricmatrix Y = (cid:20)
10 1 − (cid:21) is in the dual cone of C ∗ n . This is because for any X ∈ C ∗ ,Tr( XY ) = Tr (cid:18) X (cid:20)
10 00 10 (cid:21)(cid:19) + Tr (cid:18) X (cid:20) − (cid:21)(cid:19) = 10Tr( X ) + 0 ≥ . where the last inequality follows from the fact that when n ≤ C ∗ n = S + n ∩ N n (D¨ur 2010) .To derive the dual problem of P CPP C in the general case, we define general completely positivecone and copositive cone that contain non-symmetric matrices, by using their symmetric parts: GC ∗ n := (cid:40) X ∈ R n × n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X + X (cid:62) (cid:88) k ∈K z k ( z k ) (cid:62) for some finite { z k } k ∈K ⊂ R n + \ { } (cid:41) , and GC n := (cid:40) X ∈ R n × n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y (cid:62) X + X (cid:62) y ≥ y ∈ R n + (cid:41) . A similar discussion for PSD cones can be found at https://github.com/cvxgrp/scs/issues/31 . -companion to Guo, Bodur, and Taylor:
Copositive Duality for Discrete Markets and Games ec3
It can be proved that GC ∗ n and GC n are dual cones of each other.Then P CPP C is equivalent to the following CPP: P CPP GC : min Tr( C (cid:62) X ) (EC.3a)s . t . Tr( A (cid:62) i X ) = b i ∀ i = 1 , ..., m ( λ i ) (EC.3b) X ∈ GC ∗ n (Ω) (EC.3c) X ij = X ji ∀ i = 1 , ..., n − , ∀ j = i + 1 , ..., n ( π ij ) . (EC.3d)Thus, the dual of both P CPP C and P CPP GC is: P COP GC : min λ (cid:62) b (EC.4a)s . t . C − m (cid:88) i =1 λ i A i + n − (cid:88) i =1 n (cid:88) j = i +1 π ij ( e i e (cid:62) j − e j e (cid:62) i ) − Ω = 0 (EC.4b)Ω ∈ GC n . (EC.4c)We have the following proposition: Proposition EC.1.
If matrices C and A i , i = 1 , ..., m are symmetric, then P COP GC and P COP C areequivalent.Proof. Let ˆ X be an optimal solution of P CPP C and thus also of P CPP GC . Since C and A i , i = 1 , ..., m are symmetric, ˜ X := ( ˆ X + ˆ X ) / P CPP GC without the symmetry constraints). Thus at ˜ X , constraints (EC.3d) are redundant, which meansthere exists an optimal dual solution with all π ij = 0.This means in P COP GC we can get rid of the term (cid:80) n − i =1 (cid:80) nj = i +1 π ij ( e i e (cid:62) j − e j e (cid:62) i ), while preservingat least one optimal solution. This makes P COP GC the same as P COP C except for the conic constraintbeing non-symmetric. However, the symmetry of Ω in P COP GC is ensured by the symmetry of thematrices in constraint (EC.4b). Thus, P COP GC is equivalent to P COP C . (cid:3) From the proof we can conclude that if some A i is not symmetric, then P COP C is generally notthe dual of P CPP C . EC.3. Proofs
Proposition 1.
If the optimal value of SP ( ¯Ω) is nonzero, then (12) cuts off ¯Ω .Proof of Proposition 1. Because of (11c) and the fact that q = 1, there is at least one elementin u that is nonzero, i.e. β = { i | u i = 1 , i = 1 , ..., n c } (cid:54) = ∅ . Denote Ω ββ as the submatrix of Ω thatconsists of rows and columns with indices in β . Similarly we define the subvector z β . From thefact that the optimal objective ¯ w >
0, we know from constraint (11b) that ¯Ω ββ ¯ z β < and thus¯ z β (cid:54) = . Therefore, ¯ z (cid:62) β ¯Ω ββ ¯ z β <
0. Also, let α = { , ..., n c } \ β , then ¯ z α = due to (11e), which means¯ z (cid:62) ¯Ω¯ z = ¯ z (cid:62) β ¯Ω ββ ¯ z β <
0. Thus, ¯Ω violates the cut (12). (cid:3) c4 e-companion to Guo, Bodur, and Taylor:
Copositive Duality for Discrete Markets and Games
Theorem 1.
If strong duality holds for
U C
CPP , then CDP balances the revenue and the aggregatecost of the generators. It also balances the revenue and payment of the SO.Proof of Theorem 1.
Fix all primal and dual variables at the optimal values. Multiplying con-straints (14b) - (14e) by their corresponding dual variables yields (cid:88) g ∈G λ ∗ t p ∗ gt = λ ∗ t d t ∀ t ∈ T (EC.5b) φ ∗ jgt a (cid:62) jgt x ∗ = φ ∗ jgt b jgt ∀ j = 1 , ..., m, g ∈ G , t ∈ T (EC.5c) (cid:88) g ∈G Λ ∗ t X pp ∗ gt,gt + 2 (cid:88) g CPP , and considering the optimalobjective of U C COP equals π L , we have: π G = (cid:88) g ∈G (cid:88) t ∈T (cid:0) c pg p ∗ gt + c ug u ∗ gt (cid:1) . Therefore, for the generators the total payment from the SO equals the total cost. (cid:3) Theorem 2. Assume CPP (14) satisfies some sufficient condition for convex programming strongduality. Additionally, let ( x ∗ , X ∗ ) be an optimal solution for the problem (14) , assume all items in X ∗ in the forms of X vw ∗ g t,g t , ∀ g (cid:54) = g , g , g ∈ G , t ∈ T are equal to zero. Then the market mechanismwith CDP is also individually rational.Proof of Theorem 2. In the CPP reformulation (14), we dualize demand constraints (14b) andlifted demand constraints (14d) with their respective optimal dual prices λ ∗ t and Λ ∗ t , we obtain thefollowing Lagrangian relaxation problem:min (cid:88) g ∈G (cid:88) t ∈T (cid:0) c og p gt + c sg u gt (cid:1) + (cid:88) t ∈T λ ∗ t (cid:32) d t − (cid:88) g ∈G p gt (cid:33) -companion to Guo, Bodur, and Taylor: Copositive Duality for Discrete Markets and Games ec5 + (cid:88) t ∈T Λ ∗ t d t − (cid:88) g ∈G X ppgt,gt + (cid:88) g (cid:48) ∈G ,g (cid:48) (cid:54) = g X ppgt,g (cid:48) t (EC.7a)s . t . (14c) , (14e) − (14g) . (EC.7b)Since we have strong duality, and because the Lagrangian multipliers in the Lagrangian relaxationproblem (EC.7) are fixed to their optimal values, an optimal solution ( x ∗ , X ∗ ) for CPP (14) is alsooptimal for its Lagrangian relaxation (EC.7).Because the terms in the form of X vwg t,g t are zeros, the term X ppgt,g (cid:48) t in the objective can be elim-inated. Also, because of this condition, by Lemma 1 of Drew and Johnson (1998), X is completelypositive if and only if X g , ∀ g ∈ G are completely positive. Then straightforwardly we obtain theequivalence between Y g ∈ C ∗ n g +1 , ∀ g ∈ G and Y ∈ C ∗ n +1 .Now, the Lagrangian relaxation (EC.7) can be separated into individual optimization problems,one per generator. Converting the minimization problem into maximization, and ignoring theconstant terms λ ∗ t d t and Λ ∗ t d t in the objective, solving problem (EC.7) is equivalent to solving thefollowing maximization problem for all g ∈ G :max (cid:88) t ∈T (cid:0) λ ∗ t p gt + Λ ∗ t X ppgt,gt − c og p gt − c sg u gt (cid:1) (EC.8a)s . t . a (cid:62) jgt x = b jgt (EC.8b)Tr( a jgt a (cid:62) jgt X ) = b jgt (EC.8c) z gt = Z gt (EC.8d) Y g ∈ C ∗ n g +1 . (EC.8e)Therefore, if a solution ( x ∗ , X ∗ ) is optimal for CPP (14), then its component corresponding to g , [ x ∗ g , X ∗ g ], also solves (EC.8) optimally.Notice that in the profit-maximization problem (17) we assumed that at optimality X pp ∗ gt,gt = p ∗ gt and X pp ∗ gt,g (cid:48) t = p ∗ gt p ∗ g (cid:48) t , in addition to the assumption of X vw ∗ g t,g t = 0, we can eliminate the cross-termpayment from the the objective of (17). Now if we reformulate (17) into a CPP, its objective is thesame as (EC.8a) (except for the constant terms φ ∗ jgt b jgt and Φ ∗ jgt b jgt ). Moreover, problem (EC.8) isexactly the CPP equivalence of the profit-maximization problem (17), which means x ∗ g is optimalfor (17). Therefore, if we charge electricity prices λ ∗ t and Λ ∗ t , individual generators will not have theincentive to deviate from the optimal dispatch levels set by the system operator, as those dispatchlevels also maximize their own profits. (cid:3) Theorem 3. Let x ∗ be a PNE of G MBQP . Then ( x ∗ i , x ∗ i x ∗(cid:62) i ) i ∈I is a PNE of G CPP .Conversely, if ( x ∗ i , X ∗ i ) i ∈I is a PNE of G CPP , and if ( Q ( i ) ii (cid:23) i ∈I , ( x ∗ ik ∈ { , } , ∀ k ∈ B i ) i ∈I , then x ∗ is a PNE of G MBQP . c6 e-companion to Guo, Bodur, and Taylor: Copositive Duality for Discrete Markets and Games Proof of Theorem 3. ( ⇒ ) We first prove the conversion from G MBQP to G CPP . Since x ∗ is a PNEof G MBQP , it satisfies f i ( x ∗ i , x ∗− i ) = opt( P MBQP i ( x ∗− i )) , ∀ i ∈ I . On the other hand, P MBQP i ( x ∗− i ) canbe reformulated to P CPP i ( x ∗− i ). From Remark 1, we know that ( x ∗ i , x ∗ i x ∗(cid:62) i ) is an optimal solutionof P CPP i ( x ∗− i ), which means ( x ∗ i , x ∗ i x ∗(cid:62) i ) i ∈I is a PNE for G CPP .( ⇐ ) For the opposite direction, assume ( x ∗ i , X ∗ i ) i ∈I is a PNE for G CPP , which means ( x ∗ i , X ∗ i ) isan optimal solution for P CPP i ( x ∗− i ), and x ∗ i is in the convex hull of optimal solutions for P MBQP i ( x ∗− i ).Now assume ( Q ( i ) ii (cid:23) i ∈I , ( x ∗ ik ∈ { , } , ∀ k ∈ B i ) i ∈I . We first show that x ∗ i is feasible to P MBQP i ( x ∗− i ). Note that x ∗ i satisfies (20b) because it satisfies (22b). x i ∈ R l + is guaranteed because X ∗ i ∈ C ∗ n i ⇒ X ∗ i ≥ ⇒ X ∗ i ααα ( i ) ≥ ⇒ x ∗ i ≥ 0, where the last step follows from the constraint (22e).Additionally, we assumed x ∗ ik ∈ { , } , ∀ k ∈ B i , which means (20c) is satisfied. Since x ∗ i is feasibleto P MBQP i ( x ∗− i ) and Q ( i ) ii (cid:23) 0, by Remark 2 we know that x ∗ i is an optimal solution for P MBQP i ( x ∗− i ),where x ∗− i are also ensured to be pure strategies. Thus, x ∗ is a PNE for G MBQP . (cid:3) Proposition 3. (Existence of PNE) G MBQP has at least one PNE if(i) Property (KP) is satisfied for P MBQP i ( x − i ) ,(ii) Q ( i ) ii (cid:23) , and(iii) there exists a PNE ( x ∗ i , X ∗ i ) i ∈I for the corresponding G CPP that satisfies ( x ∗ ik ∈ { , } , ∀ k ∈B i ) i ∈I .Proof of Proposition 3. Condition (i) ensures that the optimization problems of G MBQP canbe reformulated as CPPs: For each player i , because property (KP) is true, we can reformulate P MBQP i ( x − i ) as the corresponding CPP problem P CPP i ( x − i ). As we discussed in the text, the cor-responding CP game G CPP has at least one PNE.Let ( x ∗ i , X ∗ i ) i ∈I be an PNE that satisfies condition (iii), along with the condition (ii), all require-ments for the existence of PNE for G MBQP in Theorem 3 are met and thus x ∗ is a PNE for G MBQP .Therefore, the PNE of G MBQP exists with conditions (i)-(iii). (cid:3) EC.4. Individual Revenue Adequacy with Both Uniform and Availability Prices If we include both uniform and availability prices in the revenue adequacy constraints, we have thefollowing pricing problem to solve:max (cid:88) t ∈T (cid:32) d t λ t + d t Λ t + m (cid:88) j =1 (cid:88) g ∈G ( b jgt φ jgt + b jgt Φ jgt ) (cid:33) (EC.9a)s . t . (cid:88) t ∈T p ∗ gt λ t + p ∗ gt Λ t + (cid:88) g (cid:48) ∈G\{ g } p ∗ gt p ∗ g (cid:48) t Λ t + m (cid:88) j =1 (cid:0) a (cid:62) j x ∗ φ jgt + Tr( a j a (cid:62) j X ∗ )Φ jgt (cid:1) ≥ (cid:88) t ∈T (cid:0) c og p ∗ gt + c sg u ∗ jgt (cid:1) ∀ g ∈ G (EC.9b) -companion to Guo, Bodur, and Taylor: Copositive Duality for Discrete Markets and Games ec7 ( λλλ, φφφ, ΛΛΛ , ΦΦΦ , δδδ, Ω) ∈ F COP . (EC.9c)Again, we use ( p ∗ gt ) in place of X pp ∗ gt,gt and p ∗ gt p ∗ g (cid:48) t in place of X pp ∗ gt,g (cid:48) t .If (EC.9) is feasible, then prices from (EC.9) should satisfy revenue neutrality for generators. Thisis because if we sum up left-hand sides of (EC.9b) over g , then the value equals the objective (EC.9a)(proved by (EC.6)), which represents the total revenue of generators. One the other hand, (EC.9)can be viewed as imposing extra constraints on the original CDP problem (15), whose objectivevalue, according to weak duality, is no more than the total costs (cid:80) g ∈G (cid:80) t ∈T (cid:0) c og p ∗ gt + c sg u ∗ gt (cid:1) . But(EC.9) also restricts the total revenue of generators to be no less than the total costs. Therefore,a feasible solution of (EC.9) should ensure revenue neutrality for generators.In addition, we actually have revenue neutrality for every generator, so each generator is paidexactly its cost. To understand why this result is true, assume towards contradiction if any generatorhas a strictly positive profit, then because of revenue neutrality of the whole system, some othergenerator must have a strictly negative profit, which violates (EC.9b).In comparison, we do not have revenue neutrality for individual generators in RCDP, i.e., in thatcase generators could have strictly positive profits.Similar to the case of RCDP, with this pricing scheme, it can be proved that (EC.9) is guaranteedto be feasible if its dual problem has an interior. EC.5. COP Algorithms Comparison for Maximum Clique Problem To compare our COP cutting plane algorithm with commonly used approaches for COP in litera-ture, we present the results from solving the maximum clique problem with those algorithms. Themaximum clique problem tries to find the maximum clique number on a graph G = ( N , E ), whichis equivalent to finding the stability number of G ’s complementary graph ¯ G = ( N , ¯ E ). Let ω be themaximum clique number of graph G , then we can formulate the maximum clique problem as thefollowing MIP, which finds the stability number on graph ¯ G : ω = max n (cid:88) i =1 x i (EC.10a)s . t . x i + x j ≤ ∀ ( i, j ) ∈ ¯ E (EC.10b) x i ∈ { , } ∀ i = 1 , ..., n. (EC.10c)Let A be the adjacency matrix of G , then we have A = Q − ¯ A , where Q = ee (cid:62) − I , ¯ A is theadjacency matrix of ¯ G . Applying this relationship to the COP model in Corollary 2.4 of De Klerkand Pasechnik (2002), we obtain the following COP model to calculate the maximum clique numberof G : ω = min λ (EC.11a) c8 e-companion to Guo, Bodur, and Taylor: Copositive Duality for Discrete Markets and Games s . t . λ ( ee (cid:62) − A ) − ee (cid:62) = Y (EC.11b) Y ∈ C n . (EC.11c)In our experiment we use 10 max-clique problem instances from the second DIMACS challenge.We compare the following ways of solving the COP (EC.11):(1) Approximately solve the COP with SDP, as shown in Section EC.1. This is the method sug-gested by De Klerk and Pasechnik (2002). We use Mosek 9.1.2 to solve those SDP approximations.(2) Exactly solve the COP with the cutting plane algorithm of Section 4.Notice that we can strengthen the master problem in our cutting plane algorithm by providingbounds for Y in the initialization stage. Since the maximum clique number ω cannot exceed thenumber of total nodes |N | , and elements of ee (cid:62) are all 1’s while elements of A are either 0 or 1,from constraints (EC.11b) we have that the elements of Y should be in the range of [ − , |N |− |N | , number of edges |E| , and the maximum clique number of the graph ω for eachinstance. For the computational performance of solving the SDP approximation via Mosek, we listthe objective (“Obj”), optimality gap (“Gap”, compared with the true ω ), and the computationaltime (“Time”). For the performance of our cutting plane algorithm we list the computational timeand the number of iterations needed for convergence. There is no need to list the objectives becausethe cutting plane method always converges to an exact solution. Table EC.1 Algorithm Comparison for Maximum Clique COP Model Mosek Cutting planeInstance |N | |E| ω Obj Gap(%) Time(sec) Time(sec) a 64 1824 32 32 0 1.51 6.05 2hamming6-4 a 64 704 4 4 0 1.59 1.55 4johnson8-2-4 28 210 4 4 0 0.20 9.53 2johnson8-4-4 70 1855 14 14 0 2.47 11.82 2johnson16-2-4 a,b 120 5460 8 8 0 31.88 62.75 2keller4 171 9435 11 13.47 18.34 426.16 - -MANN a9 45 918 16 17.48 8.47 0.45 547.62 2 a Obtained by setting q = ¯ ω in the separation problem, see text for explanation. b Obtained by early termination of the separation problem. When solving instances “hamming6-2”, “hamming6-4” and “johnson16-2-4” with cutting planes,we encounter some very hard separation problems that take very long time to solve. To speedup theprocess, we use instead a strengthened version of the separation problem with q = ¯ ω in constraint(11c) (Anstreicher 2020), where ¯ ω is the current master problem solution for ω . In addition, even -companion to Guo, Bodur, and Taylor: Copositive Duality for Discrete Markets and Games ec9 after our enhancement of q = ¯ ω , the instance “johnson16-2-4” still has a hard separation problemwhich achieves a nonzero lower bound early on (thus proves that the matrix is not copositive), butcannot converge after an extended period of time. In this case we set a time limit of 1 minute tohelp the separation problem stop early. Finally, when solving the instance “keller4” we have a veryhard separation after a few iterations. CPLEX fails to find a feasible solution for this separationproblem after an extended period of time, and we had to stop the solution process because of highmemory usage. However, we can still get useful information from the master problem objective,as it always provides an upper bound for the COP objective. When we stopped the algorithm for“keller4” the master problem objective was already 11, which equals the correct value of ω is betterthan the bound provided by the SDP approximation (13.47).From the results we can observe that for some instances, the SDP approximation fails to providethe correct maximum clique number. Also, in certain instances such as “c-fat200-1”, “c-fat200-2”and “c-fat200-3”, the cutting plane algorithm is faster than the SDP approximation.We also compare our algorithm with the simplicial partition method of Bundfuss and D¨ur (2009),which we believe is the only exact algorithm for general linear COPs in the literature. Bundfussand D¨ur (2009) also solve the maximum clique instances from the second DIMICS challenge.They report that their computation time for “johnson8-2-4” and “hamming6-4” are respectively 1minutes 33 seconds and 57 minutes 52 seconds. For all the other instances, their algorithm producesonly poor bounds within two hours. Therefore, the performance of our algorithm is better thantheirs in all test instances.Notice that the cutting plane algorithm terminates in very few iterations for almost all the testinstances. It is not generally the case with the cutting plane algorithm when solving other COPproblems. One reason for the small number of iterations could be the use of a strong formulationfor the maximum clique problem. For example, if we use the weaker COP formulation (EC.12)below, then the cutting plane algorithm takes longer to terminate: the simplest instance (in termsof the number of nodes and edges) “johnson8-2-4” now takes 200.64 seconds and 690 iterations: ω = min λ (EC.12a)s . t . λI + (cid:88) ( i,j ) ∈ ¯ E x ij E ij − ee (cid:62) = Y (EC.12b) Y ∈ C n (EC.12c)where E ij ∈ R n × n is a matrix with ones at i th row and j th column and j th row and i th column,and with zeros for all other positions. c10 e-companion to Guo, Bodur, and Taylor: Copositive Duality for Discrete Markets and Games EC.6. Plots for Trends of Revenue and Profit In Figure EC.2 we show the revenue and profit trends in the cutting plane algorithm for the firstUC instance. We observe that for both CDP and RCDP, the revenue and profit are monotonicallydecreasing. Similar trends are observed for the second UC instance in Figure EC.3 (except for verysmall increases for a few data points). Those trends show a tendency of convergence, which meansthat the algorithm still produces meaningful results even if we terminate it early. L o g ( R e v e nu e f r o m p r i c e s ) (a) L o g ( P r o f i t ) (b) L o g ( R e v e nu e f r o m p r i c e s ) (c) L o g ( P r o f i t ) (d) Figure EC.2 Trends of the cutting plane algorithm for the first UC instance in (a) revenue of CDP, (b)profit of CDP, (c) revenue of RCDP, (d) profit of RCDP. -companion to Guo, Bodur, and Taylor: