Chance-Constrained Optimization: A Review of Mixed-Integer Conic Formulations and Applications
CChance-Constrained Optimization: A Review of Mixed-IntegerConic Formulations and Applications
Simge Küçükyavuz * Ruiwei Jiang † January 22, 2021
Abstract
Chance-constrained programming (CCP) is one of the most difficult classes of optimization problems thathas attracted the attention of researchers since the 1950s. In this survey, we first review recent developmentsin mixed-integer linear formulations of chance-constrained programs that arise from finite discrete distributions(or sample average approximation). We highlight successful reformulations and decomposition techniques thatenable the solution of large-scale instances. We then review active research in distributionally robust CCP, whichis a framework to address the ambiguity in the distribution of the random data. The focal point of our review isscalable formulations that can be readily implemented with state-of-the-art optimization software. However, wealso discuss alternative approaches and specialized algorithms. Furthermore, we highlight the prevalence of CCPswith a review of applications across multiple domains.
Most optimization models in practice involve problem parameters that are uncertain. Furthermore, in some casesthese uncertain parameters involve risky outcomes with low probability. Therefore, requiring feasibility of asolution for every possible outcome may lead to overly conservative solutions. To remedy this, chance-constrainedprogramming (CCP) has emerged as a powerful paradigm to model system failure/reliability considerations and toaddress the conservatism of a solution given a certain tolerance for risky outcomes.For example, in power systems, production levels need to be determined so as to meet peak load (demand) [93].This problem is complicated by uncertainties in both generator availabilities (especially with renewables) and loads.The utility company’s aim is to minimize the expected cost of power production while ensuring that the loss-of-loadprobability (i.e., the probability that the available generator capacity is insufficient to meet the peak load) is belowan acceptable reliability level [163]. In supply chain problems, service level constraints are introduced to limit theprobability of stock-outs [40]. In portfolio optimization problems, there is interest to restrict the downside risk at acertain threshold (value-at-risk) [53]. Finally, in communications network design problems, a certain quality of * Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL 60208, USA, [email protected] † Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, USA, [email protected] a r X i v : . [ m a t h . O C ] J a n ervice (QoS) with respect to packet losses needs to be ensured [148]. Such risk, service, or reliability constraintsare modeled using CCPs. We will discuss more applications of CCPs in Section 4. Formally, for a given probability space (Ω , F , P ) , a chance-constrained program (CCP) is given by min x c > x s.t. P ( x ∈ P ( ω )) ≥ − (cid:15), (1a) x ∈ X , (1b)where c ∈ R n is a cost vector, X ⊂ R n represents a compact set defined by deterministic constraints on the decisionvariables x , possibly including integrality restrictions on some variables, ω ∈ Ω ⊂ R d is a random vector with atrue distribution P , for a given ω , P ( ω ) represents the set of solutions that are safe or desirable, and (cid:15) ∈ (0 , isthe risk tolerance for the decision vector x being unsafe. For risk-averse decision makers typical choices for the risklevel are small values, e.g., (cid:15) ≤ . . In this survey, we mainly focus on linear chance constraints, i.e., polyhedral P ( ω ) . More precisely, let P ( ω ) := { x : T ( ω ) x ≥ r ( ω ) } , (2)where T ( ω ) is an m × n matrix of random constraint coefficients, and r ( ω ) ∈ R m is a vector of random right-handsides.Next, we introduce the taxonomy of CCPs. Constraint (1a) is said to be an individual chance constraint for m = 1 ,and a joint chance constraint for m > . If, for all ω ∈ Ω , we have T ( ω ) = T for some deterministic m × n matrix T , and only r ( ω ) is random, we say that the CCP has right-hand side (RHS) uncertainty. In contrast, if theso-called technology matrix T ( ω ) is random, we say that the CCP has left-hand side (LHS) uncertainty , regardlessof whether r ( ω ) is a fixed vector or is random. Most of the work in CCP can be seen as single-stage (i.e., static)decision-making problems where the decisions are made here and now, and there are no recourse actions once theuncertainty is revealed. In Section 2.4, we discuss extensions to two-stage CCPs . Finally, in many problems ofinterest, the decision vector x is pure binary and this structure can be exploited to obtain stronger formulations andspecialized algorithms. We refer to such CCPs with pure binary variables as chance-constrained combinatorialoptimization problems.CCP dates back to the early work of Charnes and Cooper [38], Charnes et al. [39], Miller and Wagner [152], Prékopa[182], and Prékopa [183], who first consider problems with individual or joint chance constraints. We refer the readerto [25, 59, 104, 185, 186, 202] for textbook treatment and detailed reviews that describe the earlier developments inthis area. This survey is aimed at reviewing the developments in the past two decades primarily from a mixed-integerconic reformulations perspective.Despite long-standing interest and ubiquity in practice, CCP remains one of the most challenging class of problemsin general. There are two main challenges with CCPs. 2. Difficulty of evaluating the probability of an undesirable solution.
In practice, the distribution P in thechance constraint is not fully specified. In rare cases when P is a known continuous distribution, calculatingthe joint probability of several events requires evaluation of a multi-dimensional integral, which is hardto compute accurately [4]. Ben-Tal and Nemirovski [19], Calafiore and Campi [29, 30], and Nemirovskiand Shapiro [161, 162] approximate the non-convex chance constraint with convex constraints such thatthe solution to this approximation is feasible with high probability. However, such methods could yieldhighly conservative solutions [4] (see Section 2.5). Finally, a black-box simulation model or an oracle maybe available to evaluate P for a given solution x , however it is not straightforward to integrate such anoracle within the optimization model and the number of feasible solutions to evaluate is typically huge [228].In this survey, we focus on two main approaches to address this difficulty, namely the Sample AverageApproximation (SAA) approach (Section 2) and the distributionally robust approach (Section 3).2. Non-convexity of the feasible set.
For certain special cases such as joint CCPs with RHS uncertaintyinvolving quasi-concave or log-concave distributions [182, 185, 226, 227], or individual chance constraintswith LHS uncertainty under a certain log-concave distribution and choice of (cid:15) [116], such as normal [105],there is an equivalent convex representation of the corresponding CCP. In general, however, chance constraintseven in the case with continuous x , polyhedral P , and only RHS uncertainty result non-convex feasibleregions in their original variable space. We illustrate this challenge with an example. Example 1. [Adapted from [198]] Let ω and ω are dependent random variables with joint probabilitydensity function given in Table 1. Consider the CCP with RHS uncertainty min x + x s.t. P (cid:26) x − x ≥ ω x + 2 x ≥ ω (cid:27) ≥ . x ≥ . The feasible region of this problem is non-convex as illustrated in Figure 1.Table 1: Joint probability density function of ω Scenario 1 2 3 4 5 6 7 8 9 ω ω (cid:3) Indeed, the resulting problems are NP-hard, in general [145, 162].There has been a renewed and growing interest in CCP since the early 2000s [61, 196] to tackle these challenges.Capitalizing on the enormous success of mixed-integer programming (MIP) and conic optimization solvers sincethe early 2000s, our focal point is on reformulations that aim to circumvent the aforementioned challenges andenable progress towards the solution of this difficult class of problems.3 = 2 ω = 1 . ω = . ω = . ω = . x x ω = 1 . Figure 1: The feasible region of the example CCP.
We next present two relevant definitions pertaining to the risk associated with a univariate random variable thatwill be used in our discussion. We refer the reader to [176, 177, 192] for a more detailed treatment of these riskmeasures.
Definition 1.
For a univariate random variable X , with cumulative distribution function F X , the value-at-risk ( VaR )at confidence level (1 − (cid:15) ) , also known as (1 − (cid:15) ) -quantile, is given by: VaR − (cid:15) ( X ) = min { η : F X ( η ) ≥ − (cid:15) } . (3) (cid:3) It follows from (3) that, for any x ∈ R , the inequalities VaR − (cid:15) ( X ) ≤ x and P ( X ≤ x ) ≥ − (cid:15) are equivalent.That is, a chance constraint on random variable X can be equivalently represented as a constraint on its VaR . Definition 2 ([193, 194]) . The conditional value-at-risk (CVaR) at confidence level (1 − (cid:15) ) ∈ (0 , is given by CVaR − (cid:15) ( X ) = min (cid:26) η + 1 (cid:15) E ([ X − η ] + ) : η ∈ R (cid:27) , (4)where ( a ) + := max { , a } . (cid:3) It is well known that the minimum in definition (4) is attained at the
VaR at confidence level (1 − (cid:15) ) . CVaR,introduced by Rockafellar and Uryasev [193], satisfies the axioms of coherent risk measures, such as law invarianceand sub-additivity, as defined in [9]. It has other desirable properties, such as tractability—for finite distributions,CVaR can be formulated as a linear program and embedded in an optimization model [192]. More precisely, suppose X is a random variable with realizations X , . . . , X N and corresponding probabilities p , . . . , p N . Throughout,for a ∈ Z + , let [ a ] := { , . . . , a } . The optimization problem in (4) can equivalently be formulated as the linearprogram (LP): min η + 1 (cid:15) X i ∈ [ N ] p i w i : w i ≥ X i − η, ∀ i ∈ [ N ] , w ∈ R N + . (5)4urthermore, let ρ denote an ordering of the realizations such that X ρ ≤ X ρ ≤ · · · ≤ X ρ N . Then, for a givenconfidence level (cid:15) ∈ (0 , we have VaR − (cid:15) ( X ) = X ρ q , where q = min j ∈ [ N ] : X i ∈ [ j ] p ρ i ≥ − (cid:15) . (6) Our survey is organized as follows. In the first part of this survey, in Section 2, we consider CCPs under a finitediscrete distribution. We consider a natural MIP formulation and valid inequalities for both RHS and LHS uncertaintyin Sections 2.1 and 2.2, respectively. In Section 2.3, we review alternative formulations and specialized methods forCCPs under a finite distribution. In Section 2.4, we describe a two-stage CCP and a Benders decomposition methodfor its solution. In Section 2.5 we describe approximations of CCPs. In the second part of this survey, in Section 3,we consider distributionally robust CCPs, primarily under two types of uncertainty sets: moment-based (Section3.1) and Wasserstein ambiguity sets (Section 3.2). We give an overview of a wide range of applications in Section 4,and conclude in Section 5.
In this section, we consider CCPs under a finite discrete probability space (Ω , Ω , P N ) , where Ω = { ω , . . . , ω N } ,where p i = P N ( ω = ω i ) . Of particular interest are such CCPs that result from the Sample Average Approximation(SAA) approach [144, 173], which approximates P via a finite empirical distribution, P N .For ease of exposition, we will assume that the samples are independent and identically distributed (i.i.d.) andconsider the SAA formulation of CCP (i.e., p i = N , i ∈ [ N ] ). The methods we discuss can be adapted to the caseof non-i.i.d. scenarios, for example those that are obtained via importance sampling [17].The SAA formulation of (1) is min x c > x (7a)s.t. N X i ∈ [ N ] ( x
6∈ P ( ω i )) ≤ (cid:15), (7b) x ∈ X , (7c)where ( · ) is the indicator function. From this formulation, it is apparent that the use of finite discrete distributioncircumvents the first difficulty of evaluating high-dimensional integrals. Under non-equal probability scenarios,constraint (7b) is simply X i ∈ [ N ] p i ( x
6∈ P ( ω i )) ≤ (cid:15). When P ( · ) is polyhedral as given by (2), formulation (7) for CCP under a discrete distribution lends itself to anequivalent mixed-integer linear program (MIP) via the introduction of binary variables and big-M constraints. Hence,the non-convex feasible region in the original space of variables can be represented as a MIP with additional binary5ariables. This addresses the second difficulty of non-convexity by enabling the immediate use of off-the-shelf MIPsolvers. Next we present such MIP formulations for the RHS and LHS uncertainty cases. First, let us consider the problem with RHS uncertainty. In this setting, the joint linear CCP (7) with RHS uncertaintyis reformulated as a mixed-integer linear program [196] min x,t,z c > x (8a)s.t. x ∈ X , T x = ¯ r + t, (8b) t j ≥ r i,j (1 − z i ) , ∀ i ∈ [ N ] , ∀ j ∈ [ m ] , (8c) N X i ∈ [ N ] z i ≤ (cid:15), (8d) t ∈ R m + , z ∈ { , } N , (8e)where ¯ r ∈ R m is chosen vector satisfying r ( ω i ) ≥ ¯ r for all i and r i = ( r i, , . . . , r i,m ) > denotes r ( ω i ) − ¯ r . Thechoice of ¯ r ensures that the data vector r i is nonnegative for all i ∈ [ N ] . For (cid:15) < , we have T x ≥ ¯ r from (8c)-(8d),hence t ≥ . The binary variable z i encodes the indicator function in (7b) to model the event T x ≥ r ( ω i ) . Inparticular, if z i = 0 , then constraints (8c) enforce that t ≥ r i holds and thus T x ≥ r ( ω i ) is satisfied. Otherwise, z i = 1 , and constraints (8c) reduce to the trivial relation t ≥ . Finally, (8d) enforces that the probability of x
6∈ P ( ω ) is within the risk threshold (cid:15) . Note that this constraint is equivalent to a cardinality constraint on thebinary variables P i ∈ [ N ] z i ≤ b (cid:15)N c =: k . In the non-equiprobable case, it is a knapsack constraint P i ∈ [ N ] p i z i ≤ (cid:15). In the case of individual chance constraints, when m = 1 , we can linearize the single inequality in the chanceconstraint as T x ≥ F − ω (1 − (cid:15) ) to lower bound the LHS with the (1 − (cid:15) ) -quantile. Therefore, under RHS uncertainty,problems with joint chance constraints ( m > are more challenging. In fact, Luedtke et al. [145] show that theproblem is NP-hard for m > . Constraints (8c) are referred to as big- M constraints. Often, formulations withbig-M constraints result in weak LP relaxation bounds, which hinder the convergence of the branch-and-boundmethods. Therefore, MIP approaches have focused on obtaining strong formulations for the SAA formulation toscale up the problem sizes that can be solved. To this end, an important substructure in the formulation (8) is givenby the constraints (8c) and (8e) for a fixed j . This particular substructure is a special case of the mixing set studiedin [83] that involve general integer variables. Its specific form involving only binary variables is first considered inAtamtürk et al. [14] in the context of vertex covering.We first consider strengthening based on an individual inequality in the chance constraint. More precisely, consider(8c) and (8e) for a fixed j . We will drop the dependence on j for notational convenience. The resulting system isnothing but a mixing set with binary variables given by M := (cid:8) ( t, z ) ∈ R + × { , } N : t + r i z i ≥ r i , ∀ i ∈ [ N ] (cid:9) . The (binary) mixing set M involves N inequalities that share a common continuous variable t , but independentbinary variables z i , i ∈ [ N ] . The so-called mixing inequalities of Günlük and Pochet [83] specialized to binary6ase, which is known to be equivalent to the so-called star inequalities introduced in [14], are an exponentialfamily of linear inequalities that provide the complete linear description of conv( M ) (see also, Pochet and Wolsey[179, Theorem 18]). Furthermore, this class of inequalities can be separated in polynomial time [10, 83], henceformulation (8) can be strengthened using the mixing inequalities within a branch-and-cut framework. Somewhatsurprisingly, Kılınç-Karzan et al. [106] uncover that mixing set M can be viewed as a polymatroid set correspondingto the epigraph of submodular functions. Indeed, the authors show that mixing inequalities are equivalent to extremalpolymatroid inequalities as defined in Lovász [139], Atamtürk and Narayanan [12, Proposition 1].Luedtke et al. [145] further strengthen formulation (8) by exploiting the cardinality constraint (8d) and by studyingthe resulting set given by (8c)–(8e) for a fixed j . In this case, an immediate strengthening is that of the big-M.Consider the set M C := ( t, z ) ∈ R + × { , } N : t + r i z i ≥ r i , ∀ i ∈ [ N ] , X i ∈ [ N ] z i ≤ k . Sort the values r i for i ∈ [ N ] , to obtain a permutation σ such that: r σ ≥ r σ ≥ · · · ≥ r σ N . Now observe that due to the cardinality constraint P i ∈ [ N ] z i ≤ k , we must have t ≥ r σ k +1 . Therefore, we deducethat M C = ( t, z ) ∈ R + × { , } N : t + ( r i − r σ k +1 ) z i ≥ r i , ∀ i ∈ [ N ] , X i ∈ [ N ] z i ≤ k . Note, here, that this is an immediate big-M coefficient strengthening that can be readily incorporated into the MIPformulation. This strengthening uses the quantile information that t ≥ r σ k +1 .Due to their common usage, we give a precise definition of the resulting mixing inequalities that make use of thecardinality-based strengthening next. Then, consider a subset S = { s , s , . . . , s ‘ } ⊆ { σ , σ , . . . , σ k } such that r s i ≥ r s i +1 for i = 1 , . . . , ‘ , where s = σ and s ‘ +1 = σ k +1 . Luedtke et al. [145] show that a strong mixinginequality valid for M C is given by t + ‘ X i =1 (cid:0) r s i − r s i +1 (cid:1) z s i ≥ r s . (9)This idea can be adapted to the non-equiprobable case by redefining k as k := arg min { j : P ji =1 p i ≤ (cid:15) } .Furthermore, inequality (9) can be strengthened by further use of the cardinality relation or for the case where thescenarios are not equiprobable when constraint (8d) is in the form of a knapsack inequality [1, 113, 145, 253].Next, we illustrate this concept on our numerical example (Example 1). Consider the first inequality inside thechance constraint and note that k = 3 with respect to ω . Note that the scenarios are already ordered in nonincreasingorder with respect to the possible values of r ( ω ) . Therefore, we have t ≥ .
25 = r ( ω ) . A possible strengthenedmixing inequality is for S = { , } given by t + (0 . − . z + (0 . − . z ≥ . .
7t is easy to see the validity of this inequality. If z = 0 , then we must have t ≥ . , which satisfies this inequality.If z = 1 and z = 0 , then we must have t ≥ . , which is also satisfied. Finally, when z = z = 0 , the inequalityreduces to t ≥ . , which holds due to the (1 − (cid:15) ) -quantile relation.So far, we reviewed inequalities based on an individual inequality inside the chance constraint. If we considermultiple inequalities inside the chance constraint jointly, the resulting set is an intersection of multiple mixing setsthat share a common set of binary variables z , but independent continuous variables t j , j ∈ [ m ] . For this case,Atamtürk et al. [14, Theorem 3] show that adding the mixing inequalities written for each set to the LP relaxation ofthe set defined by (8c) and (8e) is sufficient to obtain the convex hull of solutions. Furthermore, Kılınç-Karzan et al.[106] show how to extend their framework exploiting submodularity to recover this result, as well as extend it topropose the so-called aggregated mixing inequalities that incorporate lower bounds on the continuous variablesbased on the quantile relation. For the special case of two-sided chance constraints, the convex hull descriptionprovided in Liu et al. [133] are equivalent to the aggregated mixing inequalities. The aggregated mixing inequalitiesdo not directly use the cardinality information, but use it indirectly through the lower bound on the continuousvariables obtained from the quantile. In contrast, Küçükyavuz [113] and Zhao et al. [253] propose valid inequalitiesfor a joint chance constraint by directly considering the cardinality/knapsack constraint. Now consider the problem with uncertainty data in both LHS and RHS. In this setting, the joint linear CCP (7) withLHS uncertainty is reformulated as a mixed-integer linear program [196] min x,z c > x (10a)s.t. x ∈ X , (10b) T ( ω i ) x ≥ r ( ω i ) − M ( ω i )(1 − z i ) , ∀ i ∈ [ N ] , (10c) N X i ∈ [ N ] z i ≤ (cid:15), (10d) z ∈ { , } N , (10e)where M ( ω i ) , i ∈ [ N ] is a vector of big-M coefficients such that when z i = 1 , inequality (10c) is redundant.In Section 2.1 we exploited the mixing structure associated with (8c) and (8e) for a fixed j . In other words, weconsidered an individual inequality inside the (joint) chance constraint. Furthermore, we considered RHS uncertaintyonly. In contrast, in this section we will consider LHS as well as RHS uncertainty, and we will jointly consider theinequalities inside the chance constraints for any m ≥ .The mixing procedure described in Section 2.1 relies on the fact that all scenarios share the same LHS for a given j ∈ [ m ] , that is t = T j x , where T j is the j th row of T . Due to this, we arrive at a mixing structure with N constraints that share the same continuous variable t and different binary variables. In contrast, in LHS uncertaintycase, we no longer have a common continuous variable. Can we still apply the mixing procedure?As it turns out, we can indeed extend the mixing procedure to generate other classes of valid inequalities for joint8hance-constrained programs with LHS uncertainty. To do so, we solve the following single-scenario optimizationproblem for all scenarios ω ∈ Ω and for a given φ ∈ R n : q ω ( φ ) = min x φ > x (11a) x ∈ P ( ω ) , (11b) x ∈ X . (11c)We sort the values q ω ( φ ) for ω ∈ Ω , to obtain a permutation σ such that: q σ ( φ ) ≥ q σ ( φ ) ≥ · · · ≥ q σ N ( φ ) . Observe that φ > x ≥ q σ k +1 ( φ ) is a valid inequality. Furthermore, substituting t = φ > x and r = q ( φ ) in inequality(9), we obtain a valid inequality of the desired form. These inequalities are referred to as quantile cuts . Thisand related inequalities based on quantile information have been studied in [6, 131, 143, 189, 208, 235]. Theseinequalities consider the interaction between the decision variables across multiple inequalities in the chanceconstraint, which results in improved computational performance. In another line of work, Tanner and Ntaimo [212]propose a class of cuts based on the irreducibly infeasible subsystems (IIS) of an LP that requires that a subset ofscenarios are satisfied. The authors demonstrate the efficacy of this approach in a vaccine allocation application. While we focus on natural big-M formulations that can be easily adopted by practitioners, it is important to notethat there are alternative reformulations for this class of problems relying on the concept of (1 − (cid:15) ) -efficient points,which are an exponential number of points representing the multivariate value-at-risk associated with the chanceconstraint (12b) to be specified later. Definition 3. [184] Let ν ∈ R m be such that F ( ν ) ≥ − (cid:15) and F ( ν − ε ) < − (cid:15) for ε ≥ , ε = . The point ν iscalled (1 − (cid:15) ) -efficient . (cid:3) In Example 1, observe that ν ∈ { (0 . , , (0 . , . , (0 . , . } is (1 − (cid:15) ) -efficient. The (1 − (cid:15) ) -efficient pointsthen prescribe the extreme points of the non-convex feasible region as seen in Figure 1.There are several methods in the literature that rely on the enumeration of the exponentially many (1 − (cid:15) ) -efficientpoints [61, 111, 112, 119, 184, 198]. Such alternative formulations lead to specialized branch-and-bound algorithmsdescribed in [22, 23, 196, 197]. Sen [198] uses the (1 − (cid:15) ) -efficient points to give a disjunctive programmingreformulation of joint chance constraints with finite discrete distributions. Valid inequalities are proposed based onthe extreme points of the reverse polar of the disjunctive program, which can be separated by a cut generation linearprogram (CGLP) [15]. Küçükyavuz [113] gives a compact and tight extended formulation based on disjunctiveprogramming for m = 1 . Vielma et al. [217] extend this formulation for varying m > to obtain a hierarchyof stronger relaxations. Dentcheva et al. [61] use (1 − (cid:15) ) -efficient points to obtain various reformulations ofprobabilistic programs with discrete random variables, and to derive valid bounds on the optimal objective functionvalue. Ruszczy´nski [196] uses the concept of (1 − (cid:15) ) -efficient points to derive consistent orders on different scenarios9epresenting the discrete distribution. The consistent ordering is represented with precedence constraints, andvalid inequalities for the resulting precedence-constrained knapsack set are proposed. Beraldi and Ruszczy´nski[22] propose a branch-and-bound method for probabilistic integer programs using a partial enumeration of the (1 − (cid:15) ) -efficient points.Alternatively, Ahmed et al. [6] and Jiang and Xie [101] consider a Lagrangian relaxation of the MIP formulation bycreating copies of the variables, and relaxing the non-anticipativity constraint that these variables are equal. Theauthors derive extended formulations whose relaxations achieve the stronger bounds than the basic formulation(without mixing strengthening).Furthermore, for problems with pure binary variables and special structures, i.e., for combinatorial CCPs , strongerformulations have been developed (see, e.g., [21, 95, 130, 206, 208, 228]). For example, Song et al. [208] studychance-constrained bin packing problems, and propose a formulation that does not involve additional indicatorvariables to represent (7b) based on the so-called lifted probabilistic cover inequalities. Later, Wang et al. [225]consider a closely related formulation with multiple chance constraints and derive lifted cover, clique, and projectioninequalities based on a bilinear reformulation. In a related line of work, Wang et al. [224] consider a chance-constrained assignment problem and its distributionally robust variant, and propose lifted cover inequalities basedon a bilinear reformulation of the problem. For chance-constrained knapsack problems, Yoda and Prékopa [243]provide sufficient conditions for the convexity of the formulation, Klopfenstein and Nace [110], De [54], Han et al.[85], and Joung and Lee [103] derive approximate but more tractable formulations that can provide near-optimalsolutions, and Goyal and Ravi [82] derive a fully polynomial time approximation scheme when the random itemsizes are independent and Gaussian. In addition, Nikolova [164] studies approximation algorithms for generalchance-constrained combinatorial optimization problems with random parameters following either the Gaussiandistribution or a general distribution. Xie and Ahmed [236] provide a bicriteria approximation algorithm for aclass of chance-constrained covering problems and their distributionally robust variants that finds a solution withinconstant factor of the violation probability and a constant factor of the optimal objective.For chance-constrained set covering models with RHS uncertainty, Beraldi and Ruszczy´nski [23], Saxena et al.[197] propose a specialized branch-and-bound algorithm based on the enumeration of (1 − (cid:15) ) -efficient points.Subsequently, Saxena et al. [197] derive polarity cuts to improve the computational performance of this approach.For individual chance-constrained set-covering problems with LHS uncertainty, [73] developed cutting planeapproaches for the case that all components of the Bernoulli random vector ω i are independent. In addition, Wu andKüçükyavuz [228] propose an exact approach for a partial set covering problem for the case that there exists anoracle to retrieve the probability of any events under P . In another line of work, Goyal and Ravi [81] and Swamy[210] propose approximation algorithms for chance-constrained set-covering problems with optimality guarantees.In addition to the aforementioned combinatorial CCPs, Padberg and Rinaldi [172] and Campbell and Thomas [32]study chance-constrained traveling salesman problems, Song and Shen [207] incorporate a chance constraint into abi-level shortest path interdiction problem, and Ishii et al. [98] and Geetha and Nair [77] study chance-constraintvariants of the spanning tree problem.The focus of this survey is on mixed-integer conic reformulations of CCPs, which yield provably optimal solutions10t termination. However, it bears mentioning that there are recent nonlinear programming-based approaches toaddress the non-convexity of chance constraints. Cheon et al. [46] give a global optimization algorithm thatsuccessively partitions the non-convex feasible region until a global optimal solution is obtained. Tayur et al. [213]give an algebraic geometry algorithm for a scheduling problem with joint chance constraints that solves a seriesof chance-constrained integer programs with varying reliability levels. Peña-Ordieres et al. [175] derive smoothnon-convex reformulations of the chance constrained based on the sampled empirical distribution. Other nonlinearprogramming approaches, which may result in solutions that are stationary points, include difference-of-convexoptimization methods [94], sequential outer and inner approximations [78], and sequential cardinality-constrainedquadratic optimization methods [50].Finally, throughout, we have assumed that the risk level (cid:15) is fixed. However, in practice, the decision-maker maybe interested in the trade-offs between risk level and the optimal objective. One way to assess this would be tosolve the problem for multiple values of fixed (cid:15) . For example, Shen [204] proposes a novel v ariable risk thresholdmodel in which the risk tolerance is adjustable with an appropriate penalty function in the objective to preventhigh risk. The author proposes a MIP formulation for this problem for individual chance constraints. Xie et al.[237, Theorem 8] show that the corresponding optimization problem is strongly NP-hard. Elçi et al. [70] proposea stronger MIP formulation for this problem under RHS uncertainty. Finally, Lejeune and Shen [121] considerjoint chance constraints also with LHS uncertainty and propose a Boolean-based mathematical formulation for thismodel. Thus far, we have considered a decision-making problem that is static. In other words, the decisions are madehere-and-now before the revelation of the outcome of a random event. However, in most practical situations, thereare multiple decision stages—intervened by a probabilistic event—and the decision-maker takes recourse actions inthe later epochs based on the observed outcome of the event. In this section, we focus on problems that involvetwo stages. For example, in a power generation setting, the day-ahead problem determines the on/off status ofthe conventional generators a day before realizing the demand (load) or supply (in case of renewable generators).Then the second-stage problem ensures that the loss-of-load probability is no more than a pre-specified risk level (cid:15) ∈ (0 , . Therefore, a two-stage chance-constrained model is called for.As before, the random outcome ω is defined on a probability space (Ω , Ω , P N ) . Let E [ · ] denote the expectationoperator taken with respect to ω . Liu et al. [131] propose the two-stage chance-constrained mixed-integer program min x c > x + P N ( x ∈ P ( ω )) E [ h ( x, ω ) | x ∈ P ( ω )] , (12a) P N ( x ∈ P ( ω )) ≥ − (cid:15) (12b) x ∈ X , (12c)11here P ( ω ) = { x : ∃ y satisfying W ( ω ) y ≥ r ( ω ) − T ( ω ) x, y ∈ Y} and the second-stage problem is given by h ( x, ω ) = min y g ( ω ) > y (13a) W ( ω ) y ≥ r ( ω ) − T ( ω ) x (13b) y ∈ Y . (13c)Here, g ( ω ) is a vector of second-stage objective coefficients, Y is the domain of the second-stage decision vector y . For a related model that considers only the feasibility of the second-stage problem without an associatedsecond-stage cost function h ( x, ω ) , we refer the reader to [143].The two-stage chance-constrained problem can be formulated as a large-scale mixed-integer program by introducinga big- M term for each inequality in the chance constraint and a binary variable for each scenario. In particular,analogous to the static CCP, the deterministic equivalent formulation (DEF) of the two-stage CCP may be stated as min x,y,z c > x + 1 N X i ∈ [ N ] g ( ω i ) > y ( ω i ) z i (14a) T ( ω i ) x + W ( ω i ) y ( ω i ) ≥ r ( ω i ) − M ( ω i ) z i , i ∈ [ N ] (14b) N X i ∈ [ N ] z i ≤ (cid:15), (14c) x ∈ X , y ( ω i ) ∈ Y , i ∈ [ N ] (14d) z i ∈ { , } i ∈ [ N ] , (14e)where z i , i ∈ [ N ] is a binary variable that equals 0 only if the second-stage problem for scenario ω i has a feasiblesolution, and M ( ω i ) is a vector of large enough constants that makes constraint (14b) redundant if z i = 1 , i.e., ifthe second-stage problem for scenario ω i need not be feasible. The rest of the constraints are interpreted similarlyas before.This formulation poses multiple challenges in addition to the usual difficulties of a formulation with big-Mconstraints (14b). First, the objective function (14a) is nonlinear. Second, the problem is large scale due to thecopies of the variables y ( ω i ) and the large number of binary variables z i for i ∈ [ N ] . Nevertheless, the formulation(14) has a decomposable structure—for a fixed first-stage vector x , the problem decomposes into independentscenario problems. Furthermore, if y is a continuous decision vector and Y is polyhedral, then the second-stageproblems are linear programs. Next we describe a Benders-type decomposition algorithm that not only exploits thisdecomposable structure, but also replaces the weak big- M constraints (14b) with stronger optimality and feasibilitycuts, using the mixing structure. Benders method [20], or its specific use in the classical two-stage stochastic programming (without chanceconstraints) referred to as the L -shaped method [215], is the method of choice for problems that have a similarstructure and the second-stage problems are linear programs. However, these methods are not immediately applicableto (14), since both the feasibility and optimality cuts of the Benders method assume that all second stage problems12ust be feasible, which is not the case for two-stage CCPs. For general recourse problems, feasibility and optimalitycuts different from the traditional Benders cuts must be developed.Let η i represent a lower bounding approximation of the optimal objective function value of the second-stage problemunder scenario ω i , i ∈ [ N ] . Without loss of generality, we assume that η i ≥ , i ∈ [ N ] . At each iteration of aBenders decomposition method, a sequence of relaxed master problems (RMP) are solved: min x,z,η c > x + 1 N X i ∈ [ N ] η i (15a) N X i ∈ [ N ] z i ≤ (cid:15), (15b) ( x, z ) ∈ F , (15c) ( x, z, η ) ∈ O , (15d) x ∈ X (15e) z ∈ { , } N , (15f)where, F and O denote the set of feasibility and optimality cuts—to be specified later,—respectively.At iteration k , let ( x k , z k ) be the optimal solution to the RMP. Given this first-stage solution, suppose that wesolve the LP (13) for outcome ω to obtain h ( x k , ω ) . The feasibility cuts in set F are derived from the solutionto this LP. If z ki = 0 for some i ∈ [ N ] , then the second-stage problem must be feasible. If it is infeasible for ascenario j ∈ [ N ] , then there exists an extreme ray ψ ω j associated with the dual of (13) for scenario ω j that yieldsthe inconsistent solution. Then, letting φ = ψ > ω j T ( ω j ) in (11) and following the mixing procedure gives a violatedvalid inequality that cuts off this infeasible solution ( x k , z k ) . If, on the other hand, for all ω ∈ Ω , the second-stageproblem associated with scenario ω such that z k ( ω ) = 0 is indeed feasible, then the current solution ( x k , z k ) is afeasible solution and no feasibility cuts are necessary. However, optimality cuts may be needed. Next we describehow to obtain valid optimality cuts.Let ψ ω j be the dual vector associated with the optimal basis of the second-stage problem (13) for scenario ω j at thisiteration. One possible big-M optimality cut is given by [221, 222] η j + M j z j ≥ ψ > ω j ( r ( ω j ) − T ( ω j ) x ) , (16)where M j , j ∈ [ N ] is a big-M coefficient vector.Next we describe a stronger optimality cut proposed by [131] that leads to faster convergence to an optimal solution.Clearly, the traditional Benders optimality cut, η j ≥ ψ > ω j ( r ( ω j ) − T ( ω j ) x ) is a valid optimality cut for x ∈ X (infact for x ∈ P ( ω ) ) if z j = 0 . However, it may not be valid for all x ∈ X for solutions with z j = 1 . To obtain avalid optimality cut, we solve the following secondary problem with φ = ψ > ω j T ( ω j ) : ¯ v ω j ( φ ) = min x,y φxx ∈ X , y ∈ Y . η j + (cid:16) ψ > ω j r ( ω j ) − ¯ v ω j ( φ ) (cid:17) z j ≥ ψ > ω j ( r ( ω j ) − T ( ω j ) x ) . (17)To see the validity of this inequality at z j = 1 , note that in this case, the second-stage objective function contributionfor scenario ω j is zero. Furthermore, inequality (17) evaluated at z j = 1 reduces to η j ≥ ¯ v ω ( φ ) − φx . Because ¯ v ω ( φ ) − φx ≤ for all x ∈ X and η j ≥ , this inequality is trivially satisfied. The finite convergence of theresulting algorithm is proven in [131] under certain assumptions.In Table 2, we summarize a set of computational experiments that appear in [131] to show the effectiveness of theapproaches discussed so far. The instances are based on a resource planning problem adapted from [143]. In thefirst stage, the number of servers among s types of servers to employ is determined. The second-stage problem is toallocate the servers to clients of τ types, so that their demands are met with high probability ( − (cid:15) ). Instances withvarious choices of N, (cid:15), τ, s are tested and we report the average statistics for three random instances generated forthe combination reported in each row. We compare the proposed “Strong" decomposition algorithm which usesthe optimality cuts (17) with DEF (14) and the decomposition approach (referred to as “Basic") which uses themixing-based feasibility cuts and the big- M optimality cuts (16) with an appropriate choice of big- M as describedin [131]. We report the solution times (in seconds) only for Strong decomposition, because for DEF and Basic, allinstances tested reach the time limit of one hour. We also report the percentage optimality gap at termination underthe Gap column. In most cases, DEF is unable to find a feasible solution to the LP relaxation, as indicated by a ‘-’.In cases when it is able to find a feasible solution, it ends with a gap ranging from 4% to 8%. On the other hand,Basic is able to find a feasible solution for all instances, but is unable to prove optimality for any of the 36 instancestested. It ends after an hour with optimality gaps ranging from 2% to 7%. In contrast, the Strong decompositionalgorithm, based on the proposed strong optimality cuts, is able to solve most of the instances to optimality. Forthe two unsolved instances (indicated by a superscript under the Gap column), the average optimality gap is lessthan 0.1%. These results highlight the importance of using strong formulations and decomposition for large-scaleinstances.It is important to note that in this model, the undesirable outcomes ω such that x
6∈ P ( ω ) are simply ignored. Liuet al. [131] propose an extension of the two-stage model (12), where they allow so-called recovery decisions for theundesirable scenarios. They discuss how to resolve a potential time inconsistency in two-stage CCP. Furthermore,the Benders decomposition-based solution method is extended to operate in the case of recovery.Elçi and Noyan [69] extend this framework to a two-stage chance-constrained optimization model with a mean-riskobjective, using the conditional value-at-risk as a risk measure. They apply this framework to a humanitarian reliefnetwork design problem and demonstrate its effectiveness on a case study based on hurricane preparedness inSoutheastern United States. Lodi et al. [136] extend this two-stage framework to convex second-stage problems,motivated by hydro-power scheduling applications. They build an outer approximation of the nonlinear second-stageformulations to design a Benders-type algorithm that converges to an optimal solution under mild assumptions.They demonstrate the computational benefit of the decomposition algorithm on a case study based on hydroplantdata from Greece.We close this subsection by noting the assumption of continuous second-stage variables can be lifted by leveraging14able 2: Result for instances with random RHS.Instances DEF Basic Strong ( N, (cid:15) ) ( s, τ ) Gap (%) Gap (%) Time Gap (%) (2000, 0.05) (5,10) 4.60 2.34 166 0(10,20) - 2.93 483 0(15,30) - 2.69 1106 0(2500, 0.05) (5,10) 4.64 2.61 279 0(10,20) - 3.08 711 0(15,30) - 2.88 1819 0.09 (2000, 0.1) (5,10) 7.1 5.46 723 0(10,20) - 5.99 1069 0(15,30) - 6.27 1032 0(2500, 0.1) (5,10) 7.63 5.32 641 0(10,20) - 5.79 1198 0(15,30) - 6.03 2112 0.02 the developments for decomposition algorithms for classical two-stage stochastic mixed-integer programs, wherethe second-stage problems also involve integer decisions [35, 75, 115, 117, 167–169, 187, 199–201, 245]. Thesemethods rely on iteratively convexifying the second-stage problems and updating the feasibility and optimality cutsaccordingly. These methods can be combined with the Benders-type algorithm we described to enable the solutionof two-stage CCPs with integer variables at the second-stage. Given the difficulty of solving the exact formulations of CCPs or their SAA reformulations, one line of research hasfocused on inner and outer approximations of CCPs that are more tractable. This tractability often comes at theprice of conservatism in the resulting solutions. Here we briefly review these formulations and refer the reader to[5] for a review of relaxations and approximations for CCPs.•
Scenario approximation.
Scenario approximation (SA) [e.g., 29, 30, 33, 34, 55] entails sampling to ap-proximate the true distribution P with a finite distribution P N with a set of outcomes Ω = { ω , . . . , ω N } .However, unlike the SAA model (7), a usual stochastic program (not chance-constrained) is solved enforcingthat the relations inside the chance constraint hold for each scenario. Thus, the scenario approximationproblem is given by min x c > x s.t. x ∈ P ( ω ) , ω ∈ Ω , (18a) x ∈ X , (18b)As a result, for polyhedral P ( ω ) and continuous x , the resulting SA formulation is a large-scale LP. Theauthors give a finite sample guarantee that the solution to this problem is feasible to the original CCP with highprobability. Interestingly, this sample size does not depend on m , under certain assumptions. Unfortunately,15he required sample size is typically large and the resulting solution is overly conservative. The SAA approach[144, 173] is aimed at alleviating the conservatism of the SA approach by enforcing the chance constraint,with a smaller risk level, over the finite distribution P N , albeit as a MIP as opposed to an LP.• CVaR approximation.
From Definitions 2 and 3, it is readily apparent that for a univariate random variable X , CVaR − (cid:15) ( X ) ≥ VaR − (cid:15) ( X ) . Therefore, for individual chance constraints ( m = 1) , one can approximatethe constraint P ( r ( ω ) − T ( ω ) x ≤ ≥ − (cid:15) , or in other words, VaR − (cid:15) ( r ( ω ) − T ( ω ) x ) ≤ with CVaR − (cid:15) ( r ( ω ) − T ( ω ) x )) ≤ . For the case of finite discrete distributions, this approximation leads totractable reformulations due to the LP representation of CVaR given in (5). In particular, for individual chanceconstrained CCP (7), the CVaR approximation LP is min x c > x s.t. η + 1 (cid:15)N X i ∈ [ N ] w i ≤ ,w i ≥ r ( ω i ) − T ( ω i ) x − η, ∀ i ∈ [ N ] ,x ∈ X . In general, though, it is not possible to represent CVaR tractably [162]. Nevertheless, Nemirovski and Shapiro[162] give a family of safe (i.e., feasible with high probability) and, in some cases, tractable approximations—referred to as generator-based approximations—that include the Bernstein approximation [178]. They showthat the tightest such approximation is a CVaR approximation. However, CVaR approximation is alsoconservative in some cases [7]. We refer the reader to [160], and references therein, for a survey on relatedsafe tractable approximations for individual chance constraints.In the case of joint chance constraints ( m > ), it is worthwhile to note that even for the discrete case, while avector-valued multivariate VaR definition exists (Definition 3), there is no unified definition of multivariateCVaR [see, 150, and the discussions therein]. This poses challenges in formulating related CVaR-basedapproximations that are tractable. One approach is to scalarize the multivariate random vector r ( ω ) − T ( ω ) x and use the corresponding univariate CVaR. Considering the ambiguity of the scalarization weights leads to amultivariate CVaR definition that can be represented as a challenging MIP with big-M constraints [165]. MIPstrengthening techniques can be used to improve the computational performance of the resulting multivariateCVaR formulations [114, 132, 166].• Bonferroni approximation.
Given that joint chance constraints are significantly harder than individualchance constraints, one approximation scheme that is commonly considered replaces the joint chanceconstraint with m individual chance constraints. In this case, consider replacing the joint chance constraint P ( T j ( ω ) x ≥ r j ( ω ) , j ∈ [ m ]) ≥ − (cid:15) with P ( T j ( ω ) x ≥ r j ( ω )) ≥ − (cid:15) j , (19)where X j ∈ [ m ] (cid:15) j ≤ (cid:15). (20)16rom Bonferroni’s inequality, it follows that any solution satisfying constraints (19)–(20) also satisfies thejoint chance constraint [42, 162]. Because optimizing over (cid:15) j is, in general, difficult, a common choice is (cid:15) j = (cid:15)/m, j ∈ [ m ] . However, this is also known to be a conservative approach [41, 162].Note that while these approximations provide some statistical guarantees for feasibility, they are known to beconservative and do not come with optimality guarantees. Indeed, Xie and Ahmed [236] show an inapproximabilityresult for CCPs. Ahmed [2] uses a similar idea as [162], this time to obtain a convex (Bernstein) relaxation thatyield deterministic lower bounds. Integrated chance constraints proposed by Klein Haneveld [108] replaces the non-convex chance constraints with a quantitative measure of shortfalls that lead to polyhedral representations [109] inthe discrete case. In this case, they are equivalent to the LP relaxation of the MIP formulation of CCP. Alternatively,statistical lower bounds can be obtained by using order statistics based on SAA solutions [144, 173]. Suchdeterministic or statistical bounds are useful in assessing the quality of a solution obtained from an approximation.The finite sample guarantees of sampling based methods [29, 30, 34, 144, 173] are much too large and conservativein practice. On the other hand, for small N , the out-of-sample performance of the SAA solution may even beinfeasible to the true problem. For example, in [228], the authors consider a partial set covering problem when anoracle that can evaluate the true probability of the desired event is available. They observe that for sample sizesthat lend themselves to a tractable solution of the resulting MIP, the SAA solution is often infeasible to the trueproblem. This is related to the over-fitting phenomenon in machine learning when the solution of the problemis highly sensitive to the samples { ω i } i ∈ [ N ] used to obtain it. In the next section, we describe an approach thatalleviates this problem. Given the unavailability of the exact distribution P and the potential overfitting issues due to SAA-based approaches,there has been growing interest in modeling stochastic optimization problems that are distributionally robust [see,190, and references therein].Formally, a distributionally robust chance-constrained program (DRCCP) is modeled as min x c > x (21a)s.t. sup P ∈F ( β ) P ( x
6∈ P ( ω )) ≤ (cid:15) (21b) x ∈ X , (21c)where F ( β ) is an ambiguity set of distributions and β is a set of parameters that describe the ambiguity set.Accordingly, the distributionally robust chance constraint (21b) ensures that the chance constraint is satisfied withrespect to all distributions in F ( β ) , even the worst possible one.Several types of ambiguity sets have been studied in the literature based on various characteristics of the distribution,including moments, shape information (e.g., symmetry and unimodality), support, mixture models, and discrepancymeasures (e.g., Wasserstein and φ -divergence) [3, 31, 43, 68, 71, 87, 102, 118, 124, 162, 216, 232, 238, 240, 254].17hese ambiguity sets lead to different computational tractability and conservatism of the corresponding DRCCP. Inthis survey, we will focus on moment-based ambiguity sets (Section 3.1) and Wasserstein ambiguity sets (Section3.2). There are many successful developments on the tractability of single and joint chance constraints with momentambiguity sets, which characterize P based on moment information of P [31, 87, 88, 124, 233, 241, 254].For known mean value µ and covariance matrix Σ , El Ghaoui et al. [68] characterize a moment ambiguity set F ( µ, Σ) := { P : E [ ω ] = µ, E [( ω − µ )( ω − µ ) > ] = Σ } . All probability distributions in F ( µ, Σ) need to have the designated first two moments, and are otherwise allowedto have different distribution types (e.g., Gaussian, Gaussian mixture, etc.) or different support (e.g., discreteor continuous). Perhaps surprisingly, El Ghaoui et al. [68, Theorem 1] show that DRCCP is second-order conicrepresentable for individual chance constraints (i.e., m = 1 ). Specifically, if T ( ω ) := ω > A + T for some datamatrix A ∈ R d × n and vector T ∈ R × n and r ( ω ) := b > ω + r for some data vector b ∈ R d and constant r ∈ R ,then constraint (21b) is equivalent to µ > ( b − Ax ) + r − (cid:15)(cid:15) k Σ / ( b − Ax ) k ≤ T x − r . (22)This indicates that DRCCP may improve not only the out-of-sample performance of CCP when the sample size N is small but also the computational tractability. The same result is also discovered by Calafiore and El Ghaoui [31]and Wagner [219]. In addition, Zymler et al. [254] point out an interesting fact that, for m = 1 and ambiguity set F ( µ, Σ) , constraint (21b) is equivalent to its conservative approximation that replaces the chance constraint withCVaR, i.e., sup P ∈F ( µ, Σ) CVaR − (cid:15) ( r ( ω ) − T ( ω ) x ) ≤ .For individual chance constraints, the result of El Ghaoui et al. [68] can be extended in multiple directions whilemaintaining both exactness and computational tractability . For example, Cheng et al. [45] incorporate supportinformation into F ( µ, Σ) (e.g., specifying that P is supported on a convex set) and derive an exact reformulation of(21b) based on linear matrix inequalities. Zhang et al. [248] consider potential errors of estimating the mean value µ and covariance matrix Σ , e.g., when this is done based on inadequate historical data. To address this, they adopt analternative ambiguity set proposed by Delage and Ye [56] to allow the true mean value of ω to be within an ellipsoidcentered at µ and the true covariance matrix to be bounded from above by Σ . For this extended ambiguity set, Zhanget al. [248] show that constraint (21b) is still second-order conic representable. For ambiguity set F ( µ, Σ) , Xuet al. [238] study a distributionally robust variant of the stochastic dominance constraint (see, e.g., Dentchevaand Ruszczy´nski [60]), which requires different risk tolerances for violating a chance constraint with differentmagnitudes. More precisely, they study constraints sup P ∈F ( µ, Σ) P [ T ( ω ) x ≥ r ( ω ) − s ] ≤ (cid:15) − β ( s ) for all s ≥ , where β ( s ) is a pre-specified non-decreasing function of s , and show that these constraints are conic representable forvarious β ( s ) functions. Furthermore, Yang and Xu [241] and Xie and Ahmed [233] consider an extension thatallows the event x ∈ P ( ω ) to depend non-linearly on x and ω , e.g., x ∈ P ( ω ) if and only if f ( x, ω ) ≥ , where18unction f ( x, ω ) is concave in x and quasiconvex in ω . For example, Yang and Xu [241, Corollary 2] recast (21b)as a linear matrix inequality if r ( ω ) , as well as each entry of T ( ω ) , is either convex quadratic or linear in ω .It is also possible to extend El Ghaoui et al. [68] by incorporating shape information into the ambiguity set F ( µ, Σ) .For example, Calafiore and El Ghaoui [31, Lemma 3.1] strengthens F ( µ, Σ) by additionally requiring P to be centrally symmetric (that is, P [ A ] = P [ − A ] for any Borel set A ⊆ R d ) and derives a conservative approximationof constraint (21b). Hanasusanto [86] considers a similar ambiguity set and allows the true covariance matrixto be bounded from above by Σ (instead of matching it exactly as in F ( µ, Σ) ). Consequently, Hanasusanto[86, Theorem 3.4.3] recasts (21b) as a set of conic constraints. Different from [31], Li et al. [124, Theorem 1]strengthens F ( µ, Σ) by requiring that P is α -unimodal (a generalized notion of unimodality; see Dharmadhikariand Joag-Dev [62] for definition). They show that constraint (21b) is equivalent to a set of second-order conicconstraints. Hanasusanto [86, Example 3.4.4] considers a similar ambiguity set, which bound the true covariancematrix from above by Σ , and recasts (21b) as linear matrix inequalities. Stellato [209] also considers a similarambiguity set as in Li et al. [124] but requires P to be centered around µ . In that case, Stellato [209, Section 4.1.1]recasts (21b) as a single second-order conic constraint. There are works that consider other shape information andprovide tractable conservative approximations of (21b) (i.e., maintaining computational tractability at a potentialcost of exactness). For example, Chen et al. [42] replace the covariance information in F ( µ, σ ) with bounds onforward and backward deviations, which capture the asymmetry of P , and derive a conservative approximationof (21b) via second-order conic constraints. Li et al. [123] drop the covariance restriction from F ( µ, Σ) whileadding in that P is log-concave and supported on an ellipsoid centered at µ . For this case, Li et al. [123] deriveconservative and relaxing approximations of (21b), all via second-order conic constraints. Postek et al. [180] replacethe covariance information in F ( µ, Σ) with the mean absolute deviation (MAD) from the mean and further requirethat ω is componentwise independent . For that case, Postek et al. [180] derive a conservative approximation of(21b) based on second-order conic constraints.The special case of combinatorial DRCCPs with individual chance constraints is in general intractable because ofthe binary decision variables. Nevertheless, various formulation strengthening and algorithmic techniques can beapplied to solve these problems more effectively. For example, Ahmed and Papageorgiou [3] exploit supermodularityof their distributionally robust set covering problem to derive a stronger and compact reformulation. Zhang et al.[248] derive a submodular relaxation of their DRCCP reformulation for a general binary packing problem and applyextended polymatroid inequalities. Zhang et al. [252] integrate various algorithmic techniques, including coefficientstrengthening and structure-aware reformulation, into a branch-and-price algorithm to solve a bin packing problem.Tractable reformulations for distributionally robust joint chance constraints, i.e., constraint (21b) with m ≥ , aremuch scarcer than for individual chance constraints. Indeed, Hanasusanto et al. [88, Section 2.3] show that DRCCPbecomes NP-hard if the ambiguity set involves any non-homogeneous dispersion measure (e.g., covariance as in F ( µ, Σ) ) or any non-conic support (e.g., a hyperrectangle), or if T ( ω ) involves any uncertainty (i.e., if T ( ω ) = T for some data matrix T ∈ R m × n ). Nevertheless, tractable reformulations do exist for ambiguity sets differentfrom F ( µ, Σ) or for chance constraints less general than (21b). For example, Hanasusanto et al. [88, Theorem 2]characterize an ambiguity set by the mean value, a positively homogeneous dispersion measure (e.g., MAD), and aconic support of ω , and derive a second-order conic reformulation of constraint (21b), in which T ( ω ) = T . Xie19 -3 -2 -1 0 1 2 3 O p t i m a l V a l ue ( $10 ) ED-F( , ' )ED-F( , ' , , ) (a) Optimal Value vs. φ , O p t i m a l V a l ue ( $10 ) ED-F( , ' )ED-F( , ' , , ) (b) Optimal Value vs. α Figure 2: Optimal values of ED- F ( µ, Σ) and ED- F ( µ, Σ , α ) with various φ and α and Ahmed [234, Theorem 2] consider a two-sided variant of (21b) with m = 2 and T ( ω ) = − T ( ω ) and derive asecond-order conic reformulation of constraint (21b) with regard to ambiguity set F ( µ, Σ) . Xie and Ahmed [233]derive exact and tractable reformulations of (21b) with regard to multiple types ambiguity sets, e.g., when F ( β ) involves linear moment constraints only (i.e., on the mean value of ω ) or when F ( β ) consists of a single (possiblynonlinear) moment constraint. Xie et al. [237] consider a subclass of constraints (21b) with separable uncertaintiesacross individual inequalities, i.e., each row of [ T ( ω ); r ( ω )] involves a different set of uncertain parameters and,correspondingly, a different ambiguity set. They show that, if either T ( ω ) or r ( ω ) involves no uncertainty, then(21b) admits an exact and tractable reformulation by applying the Bonferroni approximation (or union bound;see Bonferroni [28]).Various conservative approximations for distributionally robust joint chance constraints have been proposed. Chenet al. [41] propose to approximate the chance constraint in (21b) by using CVaR and subsequently approximate theresulting distributionally robust CVaR (DR-CVaR) constraint via a classical inequality of order statistics. These twolayers of approximation lead to a set of second-order conic constraints. Later, Zymler et al. [254] show that thesecond-layer approximation can be circumvented by deriving an exact reformulation of the DR-CVaR constraint,yielding a linear matrix inequality approximation of (21b). The approximations of [41] and [254] can both be furtherimproved by tuning certain scaling parameters. Unfortunately, it appears to be difficult to simultaneously optimizesuch scaling parameters and the decision x in DRCCP. Cheng et al. [45] obtain a different approximation from thatof [254] when different rows of T ( ω ) are independent.In Figs. 2a–2b, we summarize a case study of a distributionally robust chance-constrained economic dispatch (ED)problem that appears in Li et al. [124] to demonstrate the difference between F ( µ, Σ) and an alternative ambiguityset that incorporates α -unimodality into F ( µ, Σ) , denoted by F ( µ, Σ , α ) . Their case study uses the IEEE 30-bussystem and incorporates two uncertain parameters, representing prediction errors of the forecast power outputs attwo wind farms. The formulation and parameters of this problem can be found in [124, Section 5.1]. In particular,we assume that the uncertainties are α -unimodal with a mode at [0 , > and have a mean value µ = φ [1 , > with20 ∈ {− , − , . . . , } . In Fig. 2a, we compare the optimal values of ED with regard to F ( µ, Σ) and that of EDwith regard to F ( µ, Σ , α ) with α = 1 and various φ values. From this figure, we observe that the optimal valueof ED- F ( µ, Σ) is consistently higher than that of ED- F ( µ, Σ , α ) . This confirms that incorporating unimodalityinto the ambiguity set makes DRCCP less conservative. In Fig. 2b, we compare the optimal values of ED- F ( µ, Σ) and ED- F ( µ, Σ , α ) with φ = 0 and various α values. From this figure, we observe that, although the discrepancybetween ED- F ( µ, Σ) and ED- F ( µ, Σ , α ) declines as α increases, the convergence is sub-linear (in fact, it takesplace when α exceeds ). This demonstrates the significant influence of unimodality upon the ambiguity set andthe corresponding DRCCP.The case study just described highlights the utility of available distribution information in reducing the degree ofconservatism. In this regard, moment ambiguity sets are known to be more conservative than their counterpartsbased on discrepancy measures (e.g., a Wasserstein ambiguity set) when more data samples are available. On theother hand, there is a trade-off between conservatism and tractability—unlike with moment-based ambiguity sets,DRCCP with a Wasserstein ambiguity set is not polynomially solvable in general [236]. However, there have beenrecent developments in MIP formulations for DRCCP under Wasserstein ambiguity, which we describe in the nextsection. Due to its desirable statistical properties, the so-called
Wasserstein ambiguity set has witnessed an explosion ofinterest. Wasserstein ambiguity set F ( N, θ ) is defined as the θ -radius Wasserstein ball of distributions on R d aroundthe empirical distribution P N . This is defined as d W ( P , P ) := inf Π (cid:8) E ( ω,ω ) ∼ Π [ k ω − ω k ] : Π has marginal distributions P , P (cid:9) , where the , based on a norm k · k , between two distributions P and P is used. The Wassersteinambiguity set is then defined as F ( P N , θ ) := { P : d W ( P N , P ) ≤ θ } . Given a decision x ∈ X and randomrealization ω ∈ R d , we first define a safety set, S ( x ) , of outcomes such that S ( x ) = { ω ∈ Ω : x ∈ P ( w ) } . Thedistance from ω to the unsafe set is dist( ω, S ( x )) := inf ω ∈ R d {k ω − ω k : ω
6∈ S ( x ) } . (23)Chen et al. [43, Theorem 3] and Xie [232, Proposition 1] show that the formulation min x,v,u c > xx ∈ X , v ≥ , u i ≥ , i ∈ [ N ] , (24a) dist( ω i , S ( x )) ≥ v − u i , i ∈ [ N ] , (24b) (cid:15) v ≥ θ + 1 N X i ∈ [ N ] u i (24c)is an equivalent formulation of (21), by using the dual representation for the worst-case probability P [ x
6∈ P ( ω )] under the Wasserstein ambiguity set P ∈ F ( P N , θ ) provided in [27, 76, 153]. (See also Hota et al. [96] for adeterministic non-convex reformulation of (21) and CVaR-based inner approximation of (21) for certain safety sets.)21ote that formulation (21) is non-convex due to constraint (24b). However, for certain safety sets S ( · ) , MIPreformulations are possible [43, 99, 232]. Therefore, we can once again formulate a deterministic equivalent modeland solve it using off-the-shelf optimization software, thereby enabling the usage of these models by practitioners. In this section, we consider joint chance constraints with RHS uncertainty under certain common form of a safetyset. In particular, let S ( x ) := { ω : T x ≥ r ( ω ) } , (25)where r ( ω ) := Bω + e , for a given an m × d data matrix B , e ∈ R m , and T is a given m × n data matrix. For m = 1 (resp. m > ), we say that the problem is an individual (resp. joint) chance-constrained problem with RHSuncertainty. Let T j and B j be a row vector of appropriate dimension corresponding to the j th row of T and B ,respectively. In this case, the distance function is evaluated as [43] dist( ω, S ( x )) = max (cid:26) , min j ∈ [ m ] T j x − B j ω − e j k B j k ∗ (cid:27) , (26)where k · k ∗ is the dual norm. We can then introduce binary variables, z , to capture the non-convex constraint (24b)to arrive at the mixed-integer linear program [43, Proposition 2] min z,u,v,x c > x (27a)s.t. z ∈ { , } N , v ≥ , u i ≥ , i ∈ [ N ] , x ∈ X , (27b) (cid:15) v ≥ θ + 1 N X i ∈ [ N ] u i , (27c) M (1 − z i ) ≥ v − u i , i ∈ [ N ] , (27d) T j x − B j ω i − e j k B j k ∗ + M i z i ≥ v − u i , i ∈ [ N ] , j ∈ [ m ] , (27e)where M i , i ∈ [ N ] is a sufficiently large Big-M coefficient.A few remarks are in order. The computational studies of [43, 232] indicate that this MIP reformulation is difficultto solve in certain cases—state-of-the-art solvers terminate with large optimality gaps after an hour time limit.To address this challenge, Ho-Nguyen et al. [91] propose a number of results that make an order of magnitudeimprovement in the solution times. Note that formulation (30) is not immediately amenable to the improvementswe described for the SAA counterpart. For example, constraints (30e) do not have the mixing structure that theSAA counterpart benefited greatly from. In particular, the continuous variables u i are not shared across scenarios,whereas the mixing set requires common continuous variables. On the other hand, as argued in [91], the SAAcounterpart is a relaxation of (30). By making a key observation that relates the nominal SAA problem for P N toformulation (30), Ho-Nguyen et al. [91] give a stronger formulation and valid inequalities based on the same set ofbinary variables z . Furthermore, this strengthening does have the mixing structure. They also use pre-processingtechniques to reduce the formulation size drastically. On a related note, Ji and Lejeune [99] give a different MIPformulation of (21) under Wasserstein ambiguity under additional assumptions on the support of ω .22 .2.2 LHS uncertainty In this section, we consider joint chance constraints with RHS uncertainty under certain common form of a safetyset. In particular, let S ( x ) := { ω : T ( ω ) x ≥ r ( ω ) } , (28)where r j ( ω ) := b > ω j + e j , j ∈ [ m ] , for a given vector b ∈ R κ , ω j , j ∈ [ m ] is a projection of ω to a κ -dimensionalvector, and e ∈ R m . Also, let the j th row of T ( ω ) be given by T j ( ω ) := ω > A + T j for some n × κ data matrix A > and T ∈ R m × n . In this case, the distance function is measured by dist( ω, S ( x )) = max (cid:26) , min p ∈ [ P ] T j ( ω ) x − r j ( ω ) k A > x − b k ∗ (cid:27) , (29)We can then introduce binary variables, z to represent the non-convex constraint (24b) and make a transformation ofvariables to arrive at the mixed-integer conic program ([232, Theorem 2] and [44, Proposition 1 (for m = 1 )] min z,u,v,x c > x (30a)s.t. z ∈ { , } N , v ≥ , u i ≥ , i ∈ [ N ] , x ∈ X , (30b) (cid:15) v ≥ θ k A > x − b k ∗ + 1 N X i ∈ [ N ] u i , (30c) M i (1 − z i ) ≥ v − u i , i ∈ [ N ] , (30d) T j ( ω i ) x − r j ( ω i ) + M i z i ≥ v − u i , i ∈ [ N ] , j ∈ [ m ] , (30e)where M i , i ∈ [ N ] is a sufficiently large Big-M coefficient, under the assumption that A > x = b for any x ∈ X .This assumption can be relaxed with appropriate safeguards as described in [44, 92, 232].As in the case of SAA, the computational studies show that the LHS uncertainty case is a more challenging casethan the RHS uncertainty only. First, the resulting formulation is no longer linear, but conic. Furthermore, thecoefficients of the common variables x are scenario-dependent unlike the RHS uncertainty case. So it is not clear ifsimilar enhancements that Ho-Nguyen et al. [91] performed for the RHS uncertainty case can be done here. To thisend, Ho-Nguyen et al. [92] establish the link between the DRCCP and its SAA counterpart for the LHS case toidentify mixing-type valid inequalities and strengthen the formulation. This results in significant improvements inthe performance of the resulting MIP formulation. Distributionally robust variants of the resource planning problem(described in Section 2.4) with N = 100 that are unsolvable or terminate with high end gaps (40-80%) with theoriginal formulation are now solvable or have much small end gaps (<15%) with the enhancements proposed in[92].For combinatorial DRCCPs , for which the decision variables are pure binary, further strengthening is possible. Xie[232] observe the submodularity of the norm and the terms in the distance operator, and propose the use ofpolymatroid inequalities to strengthen the formulation. They report significant improvements in the performanceof the resulting algorithm. Kılınç-Karzan et al. [107] show how the polymatroid inequalities derived from theconic constraint can be generalized to the case of mixed-binary decisions. In addition, Shen and Jiang [203] derivepolymatroid inequalities when the random parameters are binary-valued and show how these inequalities can23e further strengthened via mixing and lifting schemes. In a related line of work, Wang et al. [224] consider anassignment problem and derive lifted cover inequalities based on a bilinear reformulation of their DRCCP. Conservative approximations for DRCCP with Wasserstein ambiguity are related to their SAA counterparts describedin Section 2.5. The approach of Erdo˘gan and Iyengar [71] may be seen as a (robust) scenario approximationcounterpart of [29, 33] with similar sample complexity results when the uncertainty set is defined by a Prohorovmetric, which is related to a Wasserstein metric. Furthermore, for distributionally robust CCPs under Wassersteinambiguity [96] give an approximation based on a CVaR interpretation of the reformulation [see, also, 232, for thisand two other approximations based on the scenario approximation and VaR approximation].
CCP is used to model risk-averse decision-making problems in a plethora of applications, ranging from chemicalprocesses [89, 90] to water quality management [211]. In this section, we review a few recent and active applicationdomains—this is not meant to be an exhaustive list.
Finance.
Chance constraints (or equivalently, VaR as defined in (3)) have been applied in finance to control risks.Linsmeier and Pearson [129] provide motivation of using VaR as a risk measure in significant volatile financialmarkets. VaR has been widely adopted (e.g., by the US Securities and Exchange Commission) as a method ofquantifying risks. Lemus Rodriguez [122], El Ghaoui et al. [68], Natarajan et al. [159], Zymler et al. [255], Huangand Zhao [97], Yao et al. [242], Çetinkaya and Thiele [37], Barrieu and Scandolo [18], Lotfi and Zenios [138],Li et al. [126], and Ji and Lejeune [100] apply VaR and worst-case VaR (analogous to the distributionally robustchance constraints) in finance via mathematical optimization. In addition, Rujeerapaiboon et al. [195] and Choiet al. [47] apply chance constraints in multi-period portfolio optimization.
Healthcare.
Chance constraints find applications in appointment scheduling (e.g., Deng and Shen [57]), surgeryplanning (e.g., Deng et al. [58], Wang et al. [223], and Zhang et al. [249]), operating room planning (e.g., Wanget al. [225], Wang et al. [224], and Najjarbashi and Lim [158]), vaccine allocation (e.g., Tanner and Ntaimo [212]),and social distancing during a pandemic (e.g., Duque et al. [67]), among others.
Power Systems.
Zhang and Li [244], Bienstock et al. [24], Zhang et al. [247], Duan et al. [66], Lubin et al.[141, 142] Dall’Anese et al. [52], Xie and Ahmed [234], Li et al. [123], and Li et al. [125] study chance-constrainedvariants of the optimal power flow problem. Ozturk et al. [171], Pozo and Contreras [181], and Wang et al. [222]consider chance constraints in the unit commitment problem. Vrakopoulou et al. [218], Pozo and Contreras[181], and Wu et al. [229] apply chance constraints to schedule electricity systems in face of random outages andcontingencies. Liu et al. [134], Liu et al. [135], Ravichandran et al. [191], and Zhang et al. [251] employ chanceconstraints to model an integrated system of power grid and electric vehicles. Other power system applicationsinclude coordinated load control (e.g., Zhang et al. [247] and Zhang et al. [250]), power grid topology control(e.g., Qiu and Wang [188] and Mazadi et al. [149]), and hydro power plant scheduling (e.g., Wu et al. [230] and Lodi24t al. [137]). We refer the reader to a recent survey [214] and references therein for a more detailed review of CCPin energy management.
Transportation and Routing.
Dinh et al. [63], Moser et al. [155], Pelletier et al. [174], Du et al. [64], Wu et al.[231], Muraleedharan et al. [156], Ghosal and Wiesemann [79], and Florio et al. [74] study chance constraints inthe optimal route design for vehicles (also see Cordeau et al. [49]). Blackmore et al. [26], Farrokhsiar and Najjaran[72], Banerjee et al. [16], Du Toit and Burdick [65], d. S. Arantes et al. [51], Castillo-Lopez et al. [36], and Oh et al.[170] study chance constraints to find paths for robots while avoiding obstacles.
Supply Chain, Logistics, and Scheduling.
Wang [220], Song and Luedtke [206], Hong et al. [95], Elçi andNoyan [69], Elçi et al. [70], and Noyan et al. [166] employ chance constraints in the design of networks for logisticsand humanitarian relief. Lejeune and Ruszczy´nski [120], Murr and Prékopa [157], Zhang et al. [246], and Liuand Küçükyavuz [130] apply chance constraints in logistics. Gurvich et al. [84] study chance constraints in thestaffing of call centers. Cohen et al. [48] apply chance constraints to cloud computing. Lu et al. [140] apply chanceconstraints in non-profit resource allocation.
Wireless Communication.
Li et al. [128], Soltani et al. [205], Mokari et al. [154], and Xu and Nallanathan[239] apply chance-constrained programming to accommodate the data rate requirement in orthogonal frequencydivision multiple access (OFDMA) systems. Ma and Sun [147] and Li et al. [127] apply chance constraints on thebeamforming problem in communication networks.
In this survey, we reviewed mixed-integer conic formulations of CCPs under various distributional assumptions. Wedescribed the trade-offs between tractability and conservatism of the corresponding optimization models, as well asthe trade-offs between the amount of distributional information used and over-fitting. There is some theoreticalguidance on selecting sample sizes or other design parameters, such as the Wasserstein ball radius. However, thisguidance is conservative, and instead the parameter choices are made and statistically verified using out-of-sampletests and cross-validation, in practice. There are many opportunities that arise from the recent developments in CCPmodels. As we outlined, these models often lead to mixed-integer conic formulations, which optimization softwareis now able to handle in modest sizes. The novel mixed-integer conic CCP models when coupled with paralleldevelopments in strengthening mixed-integer conic formulations [11–13, 107, 232, 248] are likely to enable thesolution of large-scale problems before resorting to conservative approximations. Such strengthening approachesoften exploit hidden submodularity—a recurring structure in many reformulations we discussed. Approximationscontinue to play an important role in applications where faster solution times are needed. In such cases, it is ofinterest to be able to provide some performance guarantees. In this regard, recent research in deriving strongrelaxations and approximation algorithms for structured problems is promising.We have primarily discussed single- or two-stage problems in this survey. Conceptually, one can also envisionCCPs with multiple decision epochs. Zhang et al. [246] consider multi-stage CCPs and give valid inequalities25or the SAA reformulation. Lulli and Sen [146] consider a multi-stage problem under a finite discrete demanddistribution, and propose a model wherein non-anticipativity is enforced only for the scenarios that meet the desiredservice constraint. The authors propose a branch-and-price algorithm, for the resulting formulation. Andrieu et al.[8], González Grandón et al. [80], and references therein, consider problems with dynamic chance constraints, andpropose solution methods under certain continuous distributions. Meraklı and Küçükyavuz [151] consider the riskassociated with parameter uncertainty in infinite-horizon Markov decision processes, and formulate this problemusing a chance-constrained optimization framework. Models and methods for multi-stage CCPs are sparser due totheir inherent difficulty not only in modeling, by taking into account the time consistency of solutions, but also indesigning scalable solution methods. This is an area of further research.In closing, we believe that the developments in easy-to-implement reformulations will usher in new and excitingapplications of CCPs, given the increasingly uncertain conditions of operations in various sectors (extreme weather,autonomous devices, renewable power, pandemics, political unrest, etc.).
Acknowledgments
Simge Küçükyavuz is supported, in part, by ONR grant N00014-19-1-2321 and NSF grant 2007814. Ruiwei Jiangis supported, in part, by NSF grant ECCS-1845980.
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