Characterizing Shadow Price via Lagrangian Multiplier for Nonsmooth Problem
11 Characterizing Shadow Price via Lagrangian Multiplier for Nonsmooth Problem
Yan GaoSchool of Management, University of Shanghai for Science and Technology,Shanghai 200093, ChinaEmail:[email protected]
Abstract.
In this paper, a relation between shadow price and the Lagrangian multiplierfor nonsmooth problem is explored. It is shown that the Lagrangian Multiplier is the upperbound of shadow price for convex optimization and a class of Lipschtzian optimizations.This work can be used in shadow pricing for nonsmooth situation. The several nonsmoothfunctions involved in this class of Lipschtzian optimizations is listed. Finally, anapplication to electricity pricing is discussed.
Keywords: shadow price, Lagrangian multiplier, nonsmooth optimization, pricing.
1. Introduction
Shadow price is the estimated price of a good or service for which no market price exists. Itis denoted as the infinitesimal change in the social benefit or social cost arising from aninfinitesimal change in the resources constraint. In other words, shadow price is the marginalbenefit or the marginal cost. From the perspective of constrained optimization, shadow price isthe change, per infinitesimal unit of the constraint, in the optimal value of the objectivefunction of an optimization problem obtained by relaxing the constraint. If the objectivefunction is benefit (cost), it is the marginal benefit (cost) of relaxing the constraint.Shadow price was initially proposed by Tinbergen in 1930s in term of microeconomics, seeNicholson and Snyder (2011). Then, it was shown that shadow price is equal to the value of theLagrangian multiplier at the optimal solution by Kantorovich, see Kantorovich (1965). Therelation between shadow price and the Lagrangian multiplier is a significant milestone inshadow price topic. That paves a way for us to investigate shadow price both analytically andcomputationally in real applications.By the Lagrangian multiplier method, linear problem was initially investigated, dualitylinear programming was used to analyze and compute shadow price. Then, convex problemwas studied, since under some mild condition, for instance Slater’s condition, a convexoptimization is equivalent to its duality problem, thus shadow price can be analyzed andcomputed by a dual optimization problem, see Bertsekas (2003) and Boyd, Vandenberghe(2004).As a powerful technique, shadow pricing is widely used in cost-benefit analyses, forinstance, evaluating public projects, determine price of public goods. Since the importance ofenvironmental problems and resource allocation issues appears, recently shadow pricing hasbeen used in the area of energy economy by Lee and Zhou (2015), resource pricing by Col et al(2017) and decision-making analysis by Ke et al (2016).The social welfare maximization mode is widely used for energy pricing. In that model, theobjective function is usually the sum of utility functions of consumers minus the cost of the consumed energy. According to microeconomics, a utility function is increasing and concave.In some application, cost functions are increasing and convex. Therefore, the social welfaremaximization is usually a convex optimization problem. Thus, shadow pricing can beformulated as minimizing a convex function (as cost) or maximizing a concave function (asbenefit) with some constraints, which express resource limitation. The dual methods are usedto solve these optimization problems since dual methods can find not only decision variablesbut also the Lagrangian multipliers, see Boyd and Vandenberghe (2004).In the social welfare maximization for electricity pricing, utility functions and costfunctions are usually chosen as quadratic functions, see Deng et al (2015). As we know, theutility function is a measure of the requirements and desires of an individual based on theconsumption or leisure activities. Generally speaking, a quadratic function or a smoothfunction cannot characterize the benefit of an individual exactly. On the other hand, accordingto approximation theory, a given continuous function can be approximated by a piecewiselinear function as close as we want, therefore it is good way to take a piecewise linear functionas a utility function. Unfortunately, piecewise linear functions are not smooth and it is not surewhether the Lagrangian multiplier of social welfare maximization problem, which nonsmoothfunctions are involved in, is shadow price. Therefore, we cannot choose a piecewise linearfunction as well as some piecewise smooth function as a utility function.The relation between shadow price and the Lagrangian multiplier is mainly concerned withsmooth situation. How to characterize shadow price by the Lagrangian multiplier fornonsmooth situation, there exist no publications to deal with this problem prehensively. Asknown, shadow price is the limitation of the quotient of the difference of objective function atan optimal solution and constraint change of the optimization problem. For the smooth case,the quotient of the difference of objective function can be expressed by its gradient, thereforethe limitation of the quotient exists, and is just the Lagrangian multiplier. For the nonsmoothcase, the quotient of the difference of objective function at optimal solution and constraintchange has multiple limitations generally, this leads to that shadow price is not a number. Inthis paper, we explore the relation between shadow price and the Lagrangian multiplier fornonsmooth situation. We give an upper bound of shadow price by Lagrangian multiplier fornonsmooth situation. In the context of this study, we consider that objective functions are aconvex function and a class of Lipschitzian functions, respectively. While we consider that theconstraint functions to be smooth since constraints express the capacity of the resources, whichis usually smooth and sometimes are linear.The rest of the paper is organized as follows. In the next section, some notions onnonsmooth analysis and optimization are reviewed. In Section 3, shadow price characterizedby the Lagrangian multiplier is developed. In Section 4, some nonsmooth functions arediscussed. In Section 5, shadow pricing for electricity market is discussed.
2. Preliminaries
In this section, we review some notions of nonsmooth analysis and the Lagrangian multiplier for nonsmooth optimization.
Definition 1 [see Bertsekas (2003)] . Let : R R n f be convex. The subdifferential of f at x , denoted by )( xf , is defined as the following: T ( ) { R | ( ) ( ) ( ), R } n n f x f y f x y x y , )( xf is said to be an element of the subdifferential or a subgradient of f at x . A convex function is directionally differentiable, according to Bertsekas (2003), itssudifferential and directional derivative have the following relation:
T( ) ( ; ) max , R nf x f x d d d . : R R n f is said to be locally Lipschzian if there exists a neighbourhood ( ) N x of x and a constant L such that | ( ) ( ) | || ||, , ( ) f x f x L x x x x N x . Definition 2 [see Clarke et al (1998)]. Let : R R n f be locally Lipschzian. Thegeneralized directional derivative, in the sense of Clarke, of f at x with respective to thedirection d , denoted by ( ; ) f x d , is defined as the following: ( ) ( )( ; ) lim sup y xt f y td f yf x d t . Although the generalized directional derivative does not equal to the directional derivativegenerally, however it is greater than the directional derivative if the directional derivativeexists, i.e., ( ; ) ( ; ), R n f x d f x d d o . Definition 3 [see Clarke et al (1998)]. Let : R R n f be locally Lipschzian. The generalizedgradient, in the sense of Clarke, of f at x , denoted by )( xf or Cl ( ) f x , is defined asthe following: Cl ( ) { R | ( ; ) , R } n n f x f x d d d , Cl ( ) f x is said to be an element of the generalized gradient, also a generalized gradient,of f at x . Similarly to the convex situation, by Clarke et al (1998) the generalized directionalderivative can be reformulated as the following: Cl T( ) ( ; ) max , R nf x f x d d d o .If we treat a convex function as a locally Lipschzian function, its generalized directionalderivative and generalized gradient happen to be the directional derivative and thesubdifferential, respectively.Let us consider the optimization problem: min ( ) f x ( ) s.t. ( ) 0, 1,..., , i g x i m where f , , 1,..., i g i m are functions defined on R n , not necessary smooth . If f , , 1,..., i g i m are continuously differentiable and * x is a solution of the problem(1), under some conditions, for instance Slater’s condition, there exist scalars ,..., 0 m such that * *1 ( ) ( ) 0 m i ii f x g x , ( ) * ( ) 0 i i g x , ( ) where ,..., m are called the Lagrangian multipliers of (1), see Bazaraa et al (1993).In Problem (1), if the objective function f represents the cost of an economic system,and , 1,..., i g i m represent the constraints of the supplement of resources, then theLagrangian multipliers ,..., m , in the sense of microeconomic, equal to shadow prices.If f , , 1,..., i g i m are nonsmooth and * x is a solution of Problem (1), under somecondition, there exist scalars ,..., 0 m such that * *1 m i ii f x g x , ( ) * ( ) 0, 1,..., i i g x i m , ( ) where is a kind subdifferential of f , for instance the subdifferential in sense ofconvex analysis , see Bertsekas (2003), Rockafellar et al (1998) and the generalized gradient,see Clarke et al (1998), ,..., 0 m are said to be Lagrangian multipliers also. In thenonsmooth case, the above Karush-Kuhn-Tucker condition is an inclusion, not an equality.
3. Shadow Price Characterized by the Lagrangian Multiplier
In this section, we will interpret that the relation between shadow price and the Lagrangianmultiplier for nonsmooth situation.
We start with several propositions, which will be used later on.
Proposition 1 [see Demyanov et al (1995), Gao (2018)]. If : R R n f is locally Lipschitzianand directionally differentiable , it holds that ( ) ( ) ( ; ) (|| ||) f x d f x f x d o d . (4) Proposition 2.
Let : R R n f and : R R, 1,..., ni g i m be convex. If there exist scalars ,..., 0 m such that (3a) holds, then * *1 m ni ii f x d g x d d . (5) Proof.
Based on convex analysis and (3a), we have * *1
T( ) ( ) m i ii f x g x d , * * T T( ) ( )1 max max i m if x g xi d d * *1 ( ; ) ( ; ), R m ni ii f x d g x d d .This yields (5). Proposition 3.
Let : R R n f and : R R, 1,..., ni g i m be locally Lipschtzian. If thereexist scalars ,..., 0 m such that (3a ) holds, where denotes the generalized gradient of alocally Lipschitzian function, then * *1 m ni ii f x d g x d d o o . (6) Proof.
Noticing Cl T( ) ( ; ) max f x f x d d o , the proof is similar to the proof in Proposition 2. From the perspective of applications, to interpret the relation between shadow price andLagrangian multiplier for nonsmooth optimization, we consider the case where constraints aresmooth. Since only first-order rate of change for the constraint is involved, same to Bertsekas(2003), it is enough to consider the linearly constrained optimization problem.We will discuss the one constraint case and multiple constraints case separately. Let usconsider the nonsmooth optimization problems as follows: T min ( ) s.t. f xa x b (7)and T min ( ) s.t. , 1,..., , i i f xa x b i m (8)where f is function (not necessary smooth) defined on R n , , R , 1,..., ni a a i m , , , 1,..., i b b i m are numbers.We suppose that the function f is the cost of an economic system in both (7) and (8). T a x b denotes the resource limitation, in view of applications, we suppose that allcomponents of a are nonnegative and at least one component of a is nonzero in theproblem (7). T i i a x b denotes the limitation of th i resource, we suppose that all componentsof i a are nonnegative and at least one component of i a is nonzero in the problem (8). Suppose that the function f is convex, * x is a minimizer and is a Lagrangianmultiplier of Problem (7). Then, one has that * f x a , T * ( ) 0 a x b .If the level of the constraint b is changed to b b , then the minimizer will change to * x x . Since shadow price is just to evaluate scarce resource, we suppose that T * a x b ,otherwise shadow price is zero. By deducing, we have that T * T * T T ( ) b b a x x a x a x b a x ,thus T b a x . Since convex function is locally Lipschitzian, by virtue of Propositions 1 and2, setting T ( ) g x a x b in Proposition 2, we deduce the corresponding change of theobjective function: * * * = ( ) ( ) ( ; ) (|| ||) f f x x f x f x x o x * ( ; ) (|| ||) g x x o x T (|| ||) a x o x , (9)Noticing T b a x , it is obtained that (|| ||) f b o x .Since all components of a are nonnegative and at least one component of a is nonzero, wehave that (1) f ob . (10)The inequality (10) implies that the Lagrangian multiplier gives the maximum rate of optimalvalue decrease over the level of constraint increase. In other worlds, a Lagrangian multiplier isan upper bound of shadow price.Any Lagrangian multiplier is an upper bound of shadow price, it is a good way to take theminimum Lagrangian multiplier as an upper of shadow price.We next discuss the case with multiple constraints. Suppose that * x is a minimizer, ,..., 0 m is a set of Lagrangian multipliers for Problem (8). Then, one has that * T1 m i ii f x a , T * ( ) 0, 1,..., i i i a x b i m .Without loss of generality, we suppose that
T *
0, 1,..., i i a x b i m ,otherwise if there exists an index i such that T * i i a x b , then its corresponding Lagrangianmultiplier is zero, so is shadow price. We further suppose that ,..., m a a are linearlyindependent. Therefore, the linear system T , 1,..., i i i a x b b i m has a solution * x x for any set of changes i b of i b for i m . The function f islocally Lipschitzian, by virtue of Propositions 1 and 2, setting T ( ) i i i g x a x b in Proposition2, it follows that * * = ( ) ( ) f f x x f x * ( ; ) (|| ||) f x x o x *1 ( ; ) (|| ||) m i ii g x x o x T1 (|| ||) m i ii a x o x .Since T , 1,..., i i a x b i m , one have that (|| ||) m i ii f b o x . (11)This means that a set of Lagrangian multipliers gives a maximum rate of optimal valuedecrease over the level of constraints increase, specifically speaking i is a upper bound of shadow price for i th resource.Different from the smooth problem, the nonsmooth problems does not admit uniqueshadow price, this is due to the intrinsic feature of nonsmoothness. Example 1 . Let x xf x x xx
Evidently, * x is a solution of the above problem. Noticing (0) [1, 2] f , a Lagrangianmultiplier satisfies , this implies . Thus is the minimumLagrangian multiplier and is also an upper bound. For the nonconvex case, we consider a class of Lipschitzian functions, which includesmaximum functions and some other nonsmooth functions. In Problem (7), we suppose that f is Lipschitzian, directionally differentiable and its generalized directive and directive coincide,i.e. ( ; ) ( ; ), R n f x d f x d d o . Suppose that * x is a minimizer and is aLagrangian multiplier of the problem (7). Then, we have that *Cl f x a , T * ( ) 0 a x b .If the level of the constraint b is changed to b b , then the minimizer will change to * x x . Same to the argument on the convex case, it follows T b a x . According to ( ; ) ( ; ) f x d f x d o , Propositions 1 and 3, setting T ( ) g x a x b in Proposition 2, we deducethe corresponding change of the objective function: * * = ( ) ( ) f f x x f x * ( ; ) (|| ||) f x x o x * ( ; ) (|| ||) f x x o x o * ( ; ) (|| ||) g x x o x o * ( ; ) (|| ||) g x x o x T (|| ||) a x o x .According to T b a x , we obtain that (1) f ob . That is to say a Lagrangian multiplier is a maximum rate of optimal value decrease overthe level of constraint increase.If f is Lipschitzian, directionally differentiable and its generalized directive and directivecoincide, * x is a minimizer, ,..., 0 m is a set of Lagrangian multipliers for Problem (8),furthermore suppose that ,..., m a a are linearly independent, similar to the argument on theconvex case, we can obtain that (|| ||) m i ii f b o x . (12)The inequality (12) means that the Lagrangian multipliers give the maximum rate of optimalvalue decrease over the level of constraints increase, i is the upper bound of shadow pricefor the i th resource.
4. Some Nonsmooth Functions
In the above sections, a class of locally Lipschitzian functions is involved in the discussionon shadow price. In this section, we will list some nonsmooth functions which belong to thatclass. The discuss will show that the function class is rather broad in some sense.
Suppose that : R R, 1,..., ni f i m are continuously differentiable. Let us consider thefollowing maximum function: ( ) max ( ) ii m F x f x . (13)Evidently, the function F is locally Lipschitzian and piecewise smooth. The problem ofminimizing max ( ) ii m f x , i.e., min max ( ) n ii mx f x , which is called the minimax problem, plays animportant role in nonsmooth optimization.For a fixed point R n x , define the following index set: ( ) { {1,..., } | ( ) ( )} i I x i m f x f x .By virtue of Clarke et al (1998) and Gao (2018), the generalized gradient of F at x has ofthe form: Cl ( ) co{ ( ) | ( )} i F x f x i I x .Moreover, ( ; ) ( ; ), R n F x d F x d d o , i.e., the generalized directional directive and thedirective of F coincide. Let us consider smooth composition of maximum functions of the form: ( ) (max ( ), , max ( )) m j mjj J j J G x g f x f x , (14)where : R R, , 1,..., nij i f j J i m and : R R m g are continuously differentiable, , 1,..., i J i m are finite index sets. It is easy to see that the function G is locally Lipschitzianand piecewise smooth. The function F given in (13) is a special case of the function G . Fora fixed point R n x , denote the index sets: ( ) { | ( ) max ( )}, 1,..., i i i ij ikk J J x j J f x f x i m .We suppose that the function g is increasing with respect to each variable, in other words, ( , , ) 0, 1,..., mi g y y i my .From Demyanov et al (1995), it follows that the generalized gradient of G at x has of theform: ( , , )( ) { R | ( ), ( )} n m ij ii I x i g y yG x f x j J xy ,moreover ( ; ) ( ; ), R n G x d G x d d o , i.e., the generalized directional directive and thedirective of G coincide. We consider a piecewise smooth univariate function, which is useful in some pricingmechanisms based on social welfare maximization. Suppose that , 0, 1, 2,... i i a a i , : R R, 0, 1, 2,... i f i are continuously differentiable with ( ) ( ), 0, 1, 2,... i i i i f a f a i ,A univariate piecewise smooth function can be formulated as the following form: ( ) ( ), [ , ), 0, 1, 2,... i i i H x f x x a a i . (15)Evidently, the function H is continuously differentiable except at , 0, 1, 2,... i a i . Wesuppose that ( ) ( ), 0, 1, 2,... i i i i f a f a i . (16)It can be verified that ( ; ) ( ; ), 1 i i H a d H a d d o if (16) is satisfied. This entails that ( ; ) ( ; ), R i i H a d H a d d o since there exist just two directions d on R . It is easyto see that (16) holds if the function H is convex. The condition ( ) ( ) i i i i f a f a impliesthat H behaves like a convex function at points , 0, 1, 2,... i a i .
5. Pricing Mechanism Based on Social Welfare Maximization
The social welfare maximization is widely used for shadow pricing in some fields, forinstance energy and environment. Recently, the real-time pricing method based social welfaremaximization have been widely studied, see Deng et al (2015) and references therein.We consider an electric power system that consists of an energy provider and k loadusers. The energy provider and all users are connected with each other through an informationcommunication infrastructure. Suppose that i x denotes the power consumption, i U denotesthe utility function of the user i , C denotes the cost function for the provider and L denotes the generating capacity of the electricity. The social welfare maximization model is asthe following: max ( ) ( ) s.t. k ki i ii ik ii U x C xx L (17)where R , 1, , i x i k K are variables, see Deng et al (2015). The dual methods are used tosolve the problem (17), the dual variable is just the Lagrangian multiplier, i.e., shadow price.According to microeconomics, a utility function is increasing and concave. The cost function isincreasing and convex sometime, thus the problem (17) is a convex optimization. Generally,the electricity pricing can be formulated as minimizing a convex function or maximizing aconcave function with some constraints, which express resource limitation.Usually, quadratic functions are chosen as utility functions for the problem (17), see Deng,et al (2015). As we know, a utility is a measure of the requirements and desires for anindividual based on the consumption or leisure activities. Generally speaking, a quadraticfunction or a smooth function cannot characterize a utility perfectly. By the mathematicaltheory, piecewise linear function can approximate any function, therefore it is good way toexpress utility by a piecewise linear function, or a piecewise smooth function for more accurate.Noticing that a utility function is with a univariate, the function H given in (15) could betaken as a utility function.
6. Concluding Remarks
In this paper, the relation between shadow prices and the Lagrangian multipliers fornonsmooth problem is developed. Except theoretical meaning, based on this work, finding a upper bound of shadow price could be transformed into computing Lagrangian multipliers fornonsmooth situation. References
Bazaraa MS, Sherali HD, Shett CM (1993) Nonlinear Programming Theory and Algorithms(John Wiley and Sons, New York).Bertsekas DP (2003) Convex Analysis and Optimization (Athena Scientific, Belmont).Boyd S, Vandenberghe L (2004) Convex Optimization (Cambridge University Press,Cambridge).Clarke FH, Leda Yu S, Stern RJ, Wolenski PR (1998) Nonsmooth Analysis and ControlTheory (Springer-Verlag, New York).Col B, Durnev A, Molchanov A (2017) Foreign risk, domestic problem: capital allocation andfirm performance under political instability,
Management Science , doi.org/10.1287/mnsc.2016.2638.Demyanov VF, Rubinov AM (1995) Constructive Nonsmooth Analysis (Peterlang, Frankfurtam Main).Deng R, Yang Z, Chow MY, Chen J (2015) A survey on demand response in smart grids:mathematical models and approaches.
IEEE Transactions on Industrial Informatics
Mathematical Programming
Management Science
Mathematical Programming