sample-path solution of stochastic variational inequalities

Math. Program. 84: 313–333 (1999) / DOI 10.1007/s10107980038a Springer-Verlag 1999

Gül Gürkan· A. Yonca Özge· Stephen M. Robinson

Sample-path solution of stochastic variationalinequalities

Received July 30, 1997 / Revised version received June 4, 1998Published online October 21, 1998

Abstract. Sample-path optimization is a simulation-based method for solving optimization problems thatarise in the study of complex stochastic systems. In this paper we broaden its applicability to include thesolution of stochastic variational inequalities. This formulation can model equilibrium phenomena in physics,economics, and operations research. We describe the method, provide general conditions for convergence,and present numerical results of an application of the method to a stochastic economic equilibrium model ofthe European natural gas market. We also point out some current limitations of the method and indicate areasin which research might help to remove those limitations.

Key words. variational inequality – sample-path optimization – simulation – coherent orientation – strongregularity

1. Introduction

In this paper we present a method for solving a variational inequality defined by a poly-hedral convex setC and a functionf that is an almost-sure limit of a computablesequence of random functionsfn. Such a situation commonly arises in simulationoptimization and stochastic economic equilibrium problems involving expectations orsteady-state functions. The method is a modification ofsample-path optimizationpro-posed by Plambecket al. [23,24] and analyzed in [28]. That method, in turn, is closelyrelated to M-estimation [15,16,38,39] and to the stochastic counterpart method [32];for additional references see [28]. As in the original version, the functionsfn may beoutputs of a simulation run of lengthn: from observing thefn and making appropriatecomputations with them, we hope to find an approximate solution of the variationalinequality defined byC and f . The approach we present here was outlined withoutproofs in the proceedings paper [9].

In our view, the present work makes the following contributions:

– To propose a modeling structure for stochastic variational inequalities in which theexpectation or limit functions must be estimated by simulation,

G. Gürkan: CentER for Economic Research, Tilburg University, P.O.Box 90153, 5000 LE Tilburg, TheNetherlands. Corresponding author.

A.Y. Özge: Information Technology Laboratory, General Electric Company, One Research Circle, Niskayuna,NY 12309, USA. The work of this author was done at the University of Wisconsin–Madison.

S.M. Robinson: Department of Industrial Engineering, University of Wisconsin-Madison, 1513 UniversityAvenue, Madison, WI 53706-1572, USA

Mathematics Subject Classification (1991):90C33, 93E30

314 Gül Gürkan et al.

– To present conditions under which the approximating problems can be shown tohave solutions with probability 1, provided the simulation run length is sufficientlylong,

– To provide bounds for the closeness of those solutions to solutions of the limitproblem, in terms of the goodness of approximation,

– To present numerical experiments on problems of moderate size to show that thetechnique can be implemented and to illustrate its performance.

In the process of establishing these points, we hope to show also that by using simulation– together with gradient estimation techniques – one may provide an effective alternativeto discrete scenario representations of uncertainty with their associated data managementproblems (see [20] for example).

The deterministic variational inequality problem has been much used since its originsin the middle 1960s, for which see e.g. [11,36]. Its application to the modeling ofeconomic equilibrium in the Project Independence study during the 1970s generatedgreat interest; see Hogan [14]. Harker and Pang [10] present an excellent survey of thedevelopments in the subject up to 1990, whereas Ferris and Pang [6] examine a largenumber of applications of complementarity problems in engineering and economics.

Josephy [17] introduced a variant of the Newton method for numerical solution ofgeneralized equations, which are reformulations of variational inequalities. Since then,much progress has been made in globalizing the Newton idea and making it more robust.One of the best current production software packages embodying this idea is the PATHsolver of Dirkse and Ferris [3], a stabilized Newton method based on a generalizationof the line search idea.

As far as we are aware, the idea of using stochastic variational inequalities to modeluncertainty in equilibrium phenomena has not been extensively used. Exceptions are theworks of Haurieet al. [12] and De Wolf and Smeers [2]. Haurieet al. used a dynamicstochastic model to find the equilibrium price and quantity of the natural gas to betraded in the European market. However, their model contained an affine price/demandrelation that resulted in anintegrablemodel: that is, one which reduces to an optimizationproblem. In Section 4, we extend their model to incorporate a different price/demandrelationship, resulting in a non-integrable model, and introduce a different stochasticstructure. In particular, while the model of [12] employed a scenario representation ofuncertainty with a scenario tree of only four branches, we propose using simulation toobserve a large number of instances of the random parameters, averaging their effects,and solving the resulting problem.

Of course, these ideas are not new in the case of optimization: see [23,24,28,34].Shapiro considers similar questions in [33], where the asymptotic behavior of solutionsof simulation optimization problems is studied. The conditions developed there differfrom those we provide here in two main respects; first, our conditions apply to generalvariational inequalities, not just those arising from optimization problems; second,we prove the existence and convergence of solutions of the approximating problems,whereas Shapiro assumes them. In order to obtain these stronger results we makestronger assumptions than does Shapiro: in particular, we use acoherent orientationcondition that is equivalent to the well-known condition of strong regularity, for whichsee e.g. [5,22,25].

Sample-path solution of stochastic variational inequalities 315

The remainder of this paper is organized in four sections. In Section 2, we formulatethe problem and describe the method that we propose. We also provide the necessarybackground on variational inequalities and continuous convergence. In Section 3, wefirst prove some auxiliary results, then explain some technical concepts including thenormal map and critical cone, and analyze the method using these concepts. Our analysisuses the notion of coherent orientation. Since this concept is quite technical we deal withit in the appendix following Section 5. In Section 4, we present the results of numericalexperiments applying the method to an energy planning problem in the European naturalgas market. We solve two versions of this problem, modeling uncertainty with discreteand with continuous probability distributions. Finally, in Section 5 we summarize thework we have presented and suggest some directions for future research.

2. Background and formulation

In this section we define the variational inequality formulation that we shall use, explainsome general aspects of the methodology, and present definitions of certain conceptsfrom analysis that will be needed in what follows.

For a closed convex setC (in general, a subset of a Hilbert space, but here a subsetof Rk) and a functionf from an open set2 ⊂ Rk to Rk, the variational inequalityproblem is to find a pointx0 ∈ C ∩2, if any exists, satisfying

for eachx ∈ C, 〈x− x0, f(x0)〉 ≥ 0, (1)

where〈y, z〉 denotes the inner product ofy andz. Geometrically, (1) means thatf(x0) isan inward normal toCatx0. Many problems from applications involve polyhedral setsC,and in this paper we restrict our attention to such sets. Two immediate examples area system ofk nonlinear equations ink unknowns, and the first-order necessary optima-lity conditions for a nonlinear-programming problem with continuously differentiableobjective and constraint functions.

Naturally, not all variational inequality problems arise from optimization. In someeconomic equilibrium models the lack of certain symmetry properties results in a modelthat is said to benon-integrable.In such a model, it is not possible to find the equili-brium prices and quantities by substituting an associated optimization problem for thevariational inequality. For discussion of an actual model of this type that was heavilyused in policy analysis, see [14]. The theory that we shall develop here does not requireany symmetry properties, so it applies to non-integrable models. In fact, the applicationto the European gas market given in Section 4 required the solution of a non-integrablestochastic economic equilibrium model.

Our method works with a vector-valued stochastic process{ fn(ω, x) | n = 1,2, . . . }and a vector-valued functionf(x) with x ∈ Rk. For alln ≥ 1 andx ∈ Rk, the randomvariablesfn(ω, x) are defined on a common probability space(�,F, P), and for almostall ω the fn(ω, · ) converge pointwise tof( · ). The method seeks a pointx0 solving (1)with the limit function f ; it works by fixing a largen and a sample pointω, solving thedeterministic variational inequality withf( · ) = fn(ω, · ), and taking the solutionxn(ω)

as an estimate ofx0. In Section 3 we give conditions ensuring that with probability 1,whenn is sufficiently large thexn(ω) exist and are close tox0.


One way of envisioning this setup is to regardfn(ω, x) as an estimate of the functionvalue f(x)obtained from a simulation run of lengthn, wherex is the decision variable andω represents the random element, i.e., the random number streams used in the simulation.One can use the method of common random numbers to evaluatefn(ω, x) for differentvalues ofx. Furthermore, exact values for the derivatives or directional derivatives ofthe fn can be obtained using well-established methods of gradient estimation suchas infinitesimal perturbation analysis; see [8,13,37]. The method proposed here thusinherits most of the advantages of sample-path optimization, such as the ability to usedeterministic techniques to solve the variational inequality.

In the particular case of an unconstrained optimization problem the method takesa special form. If we write the first order optimality conditions for the problem, weobtaink nonlinear equations ink unknowns. Then solving the associated variationalinequality would amount to finding a zero of the gradient of the objective function. Inthe stochastic context the approximate solution of this problem will be an estimate ofa critical point of this objective function. Such a point may not be an optimizer unlesscertain second-order conditions are satisfied. However, when the objective function isknown to be locally convex (or locally concave, depending on the sense of optimization),around the critical point, the solution point will be an optimizer. See [9] for an actualimplementation in that case.

At this point it may be helpful to correct a possible misconception about this formula-tion. One might think, because we use an expectation or limit function in the variationalinequality that we try to solve, that this method is some variant of the so-called “expectedvalue method” in stochastic optimization, in which a random variable is replaced by itsexpected value before solution (thereby producing an incorrect formulation). This is notso, as one can see by considering the simple example of minimizing (inx) the function8(x) := Eξφ(x, ξ), where

φ(x, ξ) ={

3α(ξ − x)2 if ξ ≤ x,

3β(ξ − x)2 if ξ ≥ x.

Here we suppose that 0< β ≤ α, and thatξ is uniformly distributed on [0,1]. This modelformulates the problem of selectingx to estimate an uncertain quantityξ, with a quadraticloss whose magnitude depends on whether one has overestimated or underestimatedξ.

A computation shows that

8(x) =

βx3+ β(1− x)3, if x ≤ 0,

αx3+ β(1− x)3, if x ∈ [0,1],αx3+ α(1− x)3, if x ≥ 1,

so that8 is convex andC2; its minimizer is

x̂ := (α− β)−1[(αβ)1/2− β].To illustrate our approach on this simple problem, we note thatφ satisfies the conditionsfor interchange of derivative and expectation, so that we can express the necessary andsufficient optimality condition

0= d

dx8(x) = d

dxEξφ(x, ξ)


as

0= Eξd

dxφ(x, ξ).

Our method would then proceed to approximate Eξd

dxφ(x, ξ) by simulation and tofind a zero of the resulting function. This procedure would approximate the correctminimizer. By contrast, if we replaced the random variableξ by its expectation of 1/2,we would obtain the function

8̃(x) = φ(x,Eξ) ={

3α(.5− x)2 if x ≥ .5,

3β(.5− x)2 if x ≤ .5,

whose minimizer is evidentlyx = .5, a point that does not minimize8(x) unlessα = β.Returning to the description of our method, we note that in order to guarantee

the closeness of the estimatexn(ω) to the true solutionx0 we need to impose certainfunctional convergence properties on the sequence{ f n}. The specific property we requireis calledcontinuous convergence; it is equivalent to uniform convergenceto a continuouslimit on compact sets [18]. For an elementary treatment of the relationship betweendifferent types of functional convergence, see [18], and for a comprehensive treatmentof continuous convergence and related issues, see [31]. In particular, Theorem 1 andCorollary 7 of [18], and Theorems 7.11, 7.17, and 7.18 of [31] provide other equivalentnotions of convergence or conditions that imply continuous convergence.

Definition 1. A sequencefn of extended-real-valued functions defined onRk converges

continuouslyto an extended-real-valued functionf defined onRk (written fnC−→ f )

if for any x ∈ Rk and any sequence{xn} converging tox, one has fn(xn) → f(x).A sequence of functions fromRk into Rm converges continuously if each of themcomponent functions does so.

To understand the rationale for requiring continuous convergence, consider a sequenceof functions fn and of pointsxn such that for eachn, xn solves the variational inequality

defined by fn andC, andxn → x asn→∞. Now if fnC−→ f then the limit pointx

is a solution of the limit variational inequality defined byf andC. Therefore we mightreasonably use solutions of the former as estimates of the limit problem. However,although this result is useful, it unfortunately guarantees neither the existence of thesolutionsxn nor their convergence.

To guarantee such existence and convergence we need to impose a certain generalizednonsingularity condition. Because this condition is technical and does not involve themain concepts introduced in this paper, we develop it in the appendix. The proof ofTheorem 2 in the next section uses this development.

3. Solution of stochastic variational inequalities

In this section we discuss the solution of stochastic variational inequalities via thesample-path method. The main result, Theorem 2, shows that under mild regularity


conditions on the limit functionf and on a solutionx0 of the original problem (1),for any continuous sample functionfn that is sufficiently close tof the variationalinequality

find x ∈ C such that〈 fn(x), c− x〉 ≥ 0 for all c ∈ C,

has a solution close tox0. We first prove a few auxiliary technical results, then proceedwith the main convergence result, and finally comment on its implications.

Theorem 1. Let f be a continuous function from an open set2 in Rk toRk having aninverse that is Lipschitzian onf(2) with modulusλ > 0. Let x∗ be a point of2 suchthat f(x∗) = 0. Then there is a positive numberα0 with B(x∗, α0) ⊂ 2 such that foranyα ∈ (0, α0] and any continuous functiong : B(x∗, α)→ Rk satisfying

γ := supx∈B(x∗,α)

‖ f(x)− g(x)‖ < λ−1α, (2)

the functiong has at least one zero, and each such zero lies inB(x∗, λγ).

Proof. f(2) is open by invariance of domain ( [4], Prop. 7.4); hence it is a neighborhoodof the origin inRk. Continuity of f then implies that there is a positiveα0 such thatB(x∗, α0) ⊂ 2 and

f(B(x∗, α0))+ λ−1α0B ⊂ f(2), (3)

whereB is the unit ball. Fixα ∈ (0, α0] and observe that (3) still holds if we substituteα for α0. For brevity writeB∗ = B(x∗, α). Let g be a function fromB∗ to Rk withthe properties listed in the statement of the theorem. Then for eachx ∈ B∗ and eacht ∈ [0,1] we have

(1− t)g(x)+ t f(x) = f(x)+ (1− t)[g(x)− f(x)]⊂ f(B∗)+ λ−1αB ⊂ f(2).

ThereforeH(x, t) = f −1[(1− t)g(x)+ t f(x)] is a well defined and continuous functionfrom B∗ × [0,1] intoRk.

Observe that for eachx on the boundary ofB∗ and anyt ∈ [0,1], H(x, t) 6= x∗.Otherwise, for some suchx andt we would have

f(x∗)− f(x) = (1− t)[g(x)− f(x)].But then

λ−1α = λ−1‖x∗ − x‖ ≤ ‖ f(x∗)− f(x)‖ ≤ γ < λ−1α,

and we reach a contradiction.Now let i be the identity ofB∗. Applying the homotopy invariance principle ( [21],

Th. 6.2.2), we find that

deg( f −1 ◦ g, int B∗, x∗) = deg(i , int B∗, x∗) = 1.

Accordingly, there must be at least one pointx̄ in B∗ such thatf −1(g(x̄)) = x∗: that is,such thatg(x̄) = f(x∗) = 0.


Next, letx belong toB∗ but with‖x− x∗‖ > λγ . Since

‖ f−1[(1− t)g(x)+ t f(x)] − f −1[ f(x)]‖ ≤ λ‖(1− t)[g(x)− f(x)]‖≤ (1− t)λγ,

we must have

‖H(x, t)− x∗‖ ≥ ‖x− x∗‖ − ‖ f−1[(1− t)g(x)+ t f(x)] − f −1[ f(x)]‖≥ ‖x− x∗‖ − (1− t)λγ. (4)

Taking t = 0 we find that f −1[g(x)] 6= x∗ and thereforeg(x) 6= 0. Hence each zeroof g lies in B(x∗, λγ).

utThe next result is a simple corollary of Theorem 1 that investigates the behavior of

approximate solutions of the equationg(x) = 0, for the functiong of that theorem.

Corollary 1. Let f ,2, λ, x∗, andα0 be as in Theorem 1, and letα ∈ (0, α0]. Leth bea continuous function fromB(x∗, α) toRk andy be a point ofRk such that

η := supx∈B(x∗,α)

‖ f(x)− h(x)‖, η+ ‖y‖ < λ−1α.

Then the functionh−yhas at least one zero, and each such zero lies inB(x∗, λ(η+‖y‖)).Proof. In Theorem 1 takeg= h− y, noting that thenγ ≤ η+ ‖y‖.

utWe next prove the main result of the paper. Roughly speaking, it says that if the

variational inequality defined by the limit function has a solutionx0 satisfying a gene-ralized nonsingularity condition, then for sufficiently good approximations of the limitfunction the approximating problems must have solutions close tox0.

In order to state and prove this theorem we need several definitions. We will onlygive a summary of relevant notions here; the interested reader can find a comprehensivetreatment in [27].

The first definition is that of thenormal mapassociated with (1). This map is usedto convert a variational inequality defined byf andC to a single-valued equation; it isdefined by

fC(z) = f(5C(z))+ z−5C(z), (5)

where5C is the Euclidean projector onC. If x0 solves (1) thenz0 = x0− f(x0) satisfiesfC(z0) = 0. Further, ifz0 is a zero of fC thenx0 = 5C(z0) solves (1); see [27] forexample.

Next, we define thenormal coneNC(x) of C at x to be the set

{y∗ | for eachc ∈ C, 〈y∗, c− x〉 ≤ 0}provided thatx ∈ C, and to be empty otherwise. An equivalent way of expressing (1) isthen thegeneralized equation

0 ∈ f(x0)+ NC(x0). (6)


If x ∈ C then thetangent coneof C at x, written TC(x), is the polar ofNC(x): thatis, the set of ally such that〈y∗, y〉 ≤ 0 for eachy∗ ∈ NC(x). Thecritical conedefinedby C together with a given pointz ∈ Rk (not necessarily inC) is then defined by

K(z) = TC(5C(z)) ∩ {y∗ ∈ Rk | 〈y∗, z−5C(z)〉 = 0}.We have now introduced all of the necessary definitions except that ofcoherent orien-tation,which generalizes the usual notion of nonsingularity for linear transformations.Because of the preliminary development required for that definition, we present it in theappendix.

Theorem 2. Let2 be an open subset ofRk and letC be a polyhedral convex set inRk.Let x0 be a point of2, and supposef is a function from2 to Rk. Let { fn | n =1,2, . . . } be random functions from2 to Rk such that for allx ∈ 2 and all finitenthe random variablesfn(x) are defined on a common probability space(�,F, P). Letz0 = x0− f(x0) and assume the following:

1. With probability one, eachfn for n = 1,2, . . . is continuous andfnC−→ f .

2. x0 solves the variational inequality defined byf andC.3. f has a strong Fréchet derivatived f(x0) at x0, andd f(x0)K(z0) is coherently orien-

ted.

Then the restriction offC to a neighborhood ofz0 has an inverse that is Lipschitzianwith some modulusλ. Further, there exist a compact subsetC0 ⊂ C∩2 containingx0,a neighborhood21 ⊂ 2 of x0, a scalarβ > 0, and a set1 ⊂ � of measure zero, withthe following properties: forn = 1,2, . . . , ω ∈ �, andy ∈ Rk with ‖y‖ ≤ β, let

ξn(ω) = supx∈C0

‖ fn(ω, x)− f(x)‖,

and

Xn(ω, y) := {x ∈ C ∩21 | for eachc ∈ C, 〈 fn(ω, x)− y, c− x〉 ≥ 0}.For eachω /∈ 1 there is then a finite integerNω such that for eachn ≥ Nω the setXn(ω, y) is a nonempty, compact subset ofB(x0, λ(ξn(ω)+ ‖y‖)).Proof. Determine1 having measure zero so that off1 the properties listed in hypo-thesis (1) hold for alln. Let ω /∈ 1. We suppressω from here on. Note that conti-nuous convergence is equivalent to uniform convergence on compacts to a continuous

limit [18], so f is continuous. We next show that( fn)CC−→ fC. Supposexn → x0.

Then

( fn)C(xn)− fC(x0) = [ fn(5C(xn))− f(5C(x0))]+ [(xn −5C(xn))− (x0−5C(x0))].

The first term in brackets approaches zero by the continuous convergence offn (hypo-thesis (1)) and the continuity of5C; the second approaches zero because of the continuity

of 5C. Therefore( fn)CC−→ fC. Next, use hypotheses (2) and (3) together with


Theorem 4 in the appendix to conclude that the restriction of the functionfC to someopen neighborhoodU ⊂ 5−1

C (2) of z0 has an inverse that is Lipschitzian onfC(U)with some modulusλ. Apply Theorem 1 tofC onU to produce anα0 with the propertieslisted in that theorem.

The Minty mapM(z) = (5C(z), z− 5C(z)) is a Lipschitzian homeomorphismfromRk onto the graph ofNC, and we know thatM(z0) = (x0,− f(x0)) =: w0. Choosepositive numbersξ, η, andε so that the balls21 = B(x0, ξ) andY1 = B(− f(x0), η)

satisfy the following conditions: (1)B0 := B(z0, α0) ⊃ M−1(W), whereW is theneighborhoodNC ∩ (21 × Y1) of w0 in NC; (2) − f(21) ⊂ Y1; (3) ξ + η + ε ≤ α0.Now choose a positiveα so thatα < min{α0, λε} andM−1(W) ⊃ B(z0, α) =: B1. LetC0 = 5C(B0). First note that for anyn, ( fn)C(z) − fC(z) = fn(5C(z)) − f(5C(z)),so that

ξn := supx∈C0

‖ fn(x)− f(x)‖ = supz∈B0

‖( fn)C(z)− fC(z)‖.

Chooseβ < λ−1α, y with ‖y‖ ≤ β, andN large enough so that ifn ≥ N then

ξn + ‖y‖ < λ−1α.

For n ≥ N, Corollary 1 tells us that( fn − y)C has a zero inB1. But the projectiononto C of any point ofB1 lies in C ∩ 21, so the projection of this zero ontoC liesin C ∩ 21 and solves the variational inequality forfn − y and C. ThereforeXn isnonempty; it is compact becausefn is continuous andXn ⊂ 21. Now suppose thatsome pointx′n ∈ 21 solves the variational inequality forfn − y andC. Thenx′n ∈ C.Let z′n = x′n − fn(x′n)+ y. Then( fn − y)C(z′n) = 0. Further,

‖z′n − z0‖ ≤ ‖x′n − x0‖ + ‖ f(x′n)− f(x0)‖ + ‖ fn(x′n)− f(x′n)‖ + ‖y‖

≤ ξ + η+ ξn + ‖y‖.But ξn+‖y‖ < λ−1α < ε, soz′n belongs toB0. But then Corollary 1 tells us that in fact

‖z′n − z0‖ ≤ λ(supz∈B0

‖( fn)C(z)− fC(z)‖ + ‖y‖) = λ(ξn + ‖y‖),

and nonexpansivity of the projection implies‖x′n − x0‖ ≤ ‖z′n − z0‖. ThereforeXn ⊂B(x0, λ(ξn + ‖y‖)).

utIn Theorem 2 we could have considered a random limit functionf instead of

a deterministic one, in which caseλ, C0, 21, and x0 would depend on the samplepointω. However, in most problems for which we think the method might be useful,f is a deterministic function, (for example, the gradient of a steady-state performancemeasure in a stochastic system). The form of the theorem is very general in that itallows us to work not only with thefn but also with small perturbations of thefn.This is important when using a numerical method having finite precision to solve thevariational inequality defined byfn andC.

Also note that the conditions derived in Theorem 2 are directly applicable to a specialcase of stochastic variational inequalities: namely, mixed complementarity problems in-volving expectations or steady-state performance functions. When the original problem


to be solved is a constrained stochastic optimization problem, the coherent orientationcondition reduces to strong regularity of the generalized equation expressing the firstorder necessary optimality conditions. In the unconstrained case, this condition is equi-valent to the nonsingularity of the Hessian at the solution point. For applications tosimulation optimization, see [22].

4. Numerical example

In this section we illustrate an application of the foregoing methodology to a model ofthe European natural gas market. We describe the problem, explain briefly the proce-dure used, and finally present numerical results. Theoretically, the method we proposeis capable of handling both discrete and continuous probability distributions; by consi-dering both types of uncertainties in the gas market problem we illustrate its practicalutility as well.

The problem is to find the market price and quantity of natural gas to be produced andshipped to various markets from various producers. We adopt the model and partiallythe notation of Haurieet al.[12], modifying several parameters and functional relations.In particular, we use a different price/demand relation that results in a non-integrablemodel. As was done in [12], we consider a decomposition of the European gas marketinto the producers and the consumers of natural gas. This is a simple preliminarymodel in the sense that it does not take into account the possibility of the transmissioncompanies’ and/or distributors’ acting as players as well.

The model hasm players (producers and exporters of natural gas), each player con-trolling a set of production units,n markets, andT time periods. LetUi be the set ofproduction units controlled by theith producer, fori = 1, . . . ,m. For each̀ ∈ Ui ,j = 1, . . . ,n, andt = 1, . . . , T, the variables are defined as follows:

Rt` = remaining reserves at beginning of periodt (R1

` is a given nonnegative number).

Kt` = available capacity at beginning of periodt (K1

` is a given nonnegative number).qt` j = annual production that is shipped from production unit` to market j during

periodt.Dt

j = annual domestic production of marketj during periodt (given).

Qtj = total annual quantity available on marketj during periodt: that is,

Qtj = Dt

j +m∑

i=1

∑`∈Ui

qt` j . (7)

I t` = physical capacity invested in during periodt for use in periodt + 1. Investment is

assumed to take place in one lump sum at the end of periodt.ct` j = constant marginal transportation cost for shipping from production unit` to

market j during periodt.yt = number of years in periodt. We usedys = 5 for s= 1,2,3 andy4 = 20.r = the (yearly) interest rate in effect; we usedr = 0.10.


During each period, each producer incurs a (constant) annual cash flow for everyyear in that period. To compute the present value of these cash flows we use a factorftexpressing the time value of money: for periodt,

ft = [(1+ r)yt − 1]/[r(1+ r)∑t

s=1 ys].The investment costs are linear with marginal cost0` ≥ 0, whereas marginal productioncosts follow a curve that has a reverse L shape in order to ensure that the unit productioncost is approximately constant when away from the production capacity but goes toinfinity at the capacity limit; see [19]. Therefore the production cost function is takento beG`(x) = a`x − b` ln(X` − x), wherea` andb` are parameters,x represents thequantity produced, andX` is the capacity limit. LetPt

j (Q) be the price of natural gas inmarket j during periodt when the amount available in that market isQ.

In contrast to the affine demand law used in [12], we use the following:

Ptj (Q) = p0t

j (Q/q0tj )

1/etj , (8)

wherep0tj andq0t

j are base prices and base demands respectively andetj is the price

elasticity of demand for natural gas in marketj during periodt.Unlike the affine demand law, (8) leads to a model that is generally non-integrable.

We mean by this that the vector function involved in the variational inequality (in thiscase, (16) below) is not the derivative of any scalar function (because its derivative isnot symmetric). Therefore the equilibrium condition cannot be written as the set ofnecessary conditions for an optimization problem.

We now develop the model itself, beginning with calculation of the net present value(NPV) of a producer’s profit. In each year of periodt, the revenue from shipments tomarket j from unit` is qt

` j Ptj (Q

tj ), and the corresponding shipping cost isct

` j qt` j , while

the yearly production cost for unit̀is G`(∑n

j=1 qt` j ). The investment cost paid at the

end of periodt for unit ` is0` I t`. Therefore the NPV of profit for produceri is

Fi (qt` j , I t

`) =T∑

t=1

∑`∈Ui

[ft

{ n∑j=1

[Ptj (Q

tj )− ct

` j ]qt` j − G`(

n∑j=1

qt` j )

}− 0` I t

`(1+ r)−∑t

s=1 ys].

(9)

For eachi and each̀ ∈ Ui , reserves satisfy

Rt+1` = Rt

` − yt

n∑j=1

qt` j , t = 1, . . . , T, (10)

and we require

Rt+1` ≥ 0, t = 1, . . . , T; (11)

recall thatR1` is given. Similarly, the capacities satisfy

Kt+1` = Kt

` + I t`, t = 1, . . . , T, (12)


and

Kt` −

n∑j=1

ytqt` j ≥ 0, t = 1, . . . , T. (13)

In addition to these, we have the quantity balance equation (7), and we require that thedecision variablesqt

` j and I t` be nonnegative.

The producers are the USSR (the former Soviet Union), the Netherlands, Norway,and Algeria. Among those Norway has two producing units, whereas all the othershave one each. There are six markets: UK, the Netherlands, FRGer (the former WestGermany), BelLux (Belgium/Luxembourg), Italy, and France.

Our next step is to formulate the first-order necessary conditions for an equilibriumas a variational inequality in the variablesqt

` j and I t`. We begin by discarding the

auxiliary variablesQtj , Rt

`, andKt` and rewriting the reserve depletion constraints by

replacing (10) and (11) by

R1` −

t∑s=1

ys

n∑j=1

qs` j ≥ 0, ` ∈ Ui , i = 1, . . .4, t = 1, . . . ,4, (14)

and replacing (12) and (13) by

K1` +

t−1∑s=1

I s` −

n∑j=1

ytqt` j ≥ 0, ` ∈ Ui , i = 1, . . .4, t = 1, . . . ,4, (15)

with the standard convention that∑0

1 is vacuous.The i th producer wishes to maximizeFi (qt

` j , I t`), given by (9) withQt

j replacedby the right-hand side of (7), subject to the constraints (14) and (15). These individualmaximization problems are coupled by the presence in each problem of the productionquantity decisions of all the other producers. For an equilibrium we wish to find values ofthe decision variablesqt

` j andI t` so that these maximizations take place simultaneously.

We therefore write the first-order necessary optimality conditions for each producer’sproblem of maximizing (9) subject to the constraints (14) and (15) as well as nonnega-tivity of the decision variablesqt

` j and I t`, and try to solve these simultaneously. Note

that with the demand law (8), the objective function is not generally convex, so thefirst-order necessary conditions may not always be sufficient.

To write these conditions we regard the left-hand sides of (14) and (15) as vectors ofdimensionγi , whereγi = 4 for i = 1,2,4 andγ3 = 8 (since Norway has two producingunits, whereas all the others have one each), and we denote these vectors by−g1

i and−g2

i respectively. We then associate dual variablesu1i andu2

i , also of dimensionγi ,with the constraints−g1

i ≥ 0 and−g2i ≥ 0 respectively. We write the quantitiesqt

` jfor produceri (that is, for` ∈ Ui and for allt and j ) as a vectorxi of dimensionαi ,whereαi = 4× 6 = 24 for i = 1,2,4 andα3 = 2× 4× 6 = 48. Similarly, we writethe physical capacity investmentsI t

` for ` ∈ Ui as a vectorzi of dimensionβi , whereβi = 3 for i = 1,2,4 andβ3 = 6.

As the constraints are affine, no constraint qualification is needed, and simultaneoussatisfaction of the first-order conditions for all producers can then be expressed by the


following generalized equation: fori = 1, . . . ,4 find nonnegative vectorsxi , zi , u1i , and

u2i such that

0 ∈ − ∂Fi∂xi+u1

i∂g1

i∂xi+u2

i∂g2

i∂xi+N

Rαi+(xi )

0 ∈ − ∂Fi∂zi

+u2i∂g2

i∂zi+N

Rβi+(zi )

0 ∈ −g1i +N

Rγi+(u1

i )

0 ∈ −g2i +N

Rγi+(u2

i )

(16)

Note again that the instances of (16) for differenti are coupled by the quantity deci-sions, so that (16) is actually a single variational inequality containing 175 variables:120 quantitiesqt

` j , 15 investment decisionsI t`, and a total of 40 dual variables.

The parameters used in the experiments are in Tables 1–3; subscripts` are omitted.Prices are in 1983 dollars and quantities are inmtoe(million tons of oil equivalent).

Table 1.Cost coefficients and parameters for production units

Producer a b X K1 R1 0 c1 c2 c3 c4

USSR 1.606 51 80 3750 3750 0.0 0.58 0.56 0.55 0.55Netherlands 1.212 67 80 1900 1900 0.0 0.14 0.13 0.13 0.12

Norway1 1.507 85 80 300 300 0.0 0.35 0.34 0.34 0.33Norway2 1.507 85 80 0 1936 0.5 0.35 0.34 0.34 0.33Algeria 2.102 96 80 3087 3087 0.0 0.70 0.69 0.64 0.62

Table 2.Base prices and base demands in markets,p0 andq0

Market Period 1 Period 2 Period 3 Period 4BelLux 5.12 7.8 2.56 9.4 3.41 9.4 5.12 9.5FRGer 5.27 40.7 2.64 46.2 3.52 46.5 5.27 44.6France 5.25 23.6 2.62 28.3 3.50 29.8 5.25 28.5Italy 5.15 25.3 2.57 34.9 3.43 37.5 5.15 37.2

Netherlands 5.16 28.9 2.58 29.9 3.44 32.2 5.16 29.7UK 4.54 43.8 2.27 50.3 3.03 56.4 4.54 53.7

Table 3.Price elasticities and domestic production in markets,e andD

Market Period 1 Period 2 Period 3 Period 4BelLux -1.07 0.00 -1.26 0.00 -1.34 0.00 -1.42 0.00FRGer -1.46 13.70 -1.58 13.80 -1.68 13.80 -1.79 13.80France -0.81 4.80 -1.19 2.90 -1.57 3.00 -2.01 3.00Italy -1.15 10.40 -1.36 10.00 -1.45 10.00 -1.54 10.40

Netherlands -0.94 22.93 -1.13 20.96 -1.29 24.11 -1.45 23.90UK -0.61 33.70 -0.87 35.00 -1.10 37.00 -1.30 38.00

We consider two versions of this problem. In the first one, in each period, weconsider a shutdown possibility in Algeria which results in an interruption of produc-tion/transportation of natural gas in the current period and in all future periods. Define


Table 4.Prices and quantities for Period 4 (when shutdown is uncertain)

Market n = 20 n = 200 n = 1000 n = 15000 n = 100000 n = ∞BelLux 5.19 9.31 5.26 9.15 5.29 9.07 5.27 9.11 5.27 9.13 5.27 9.12FRGer 4.77 53.28 4.83 52.17 4.86 51.64 4.84 51.94 4.84 52.02 4.84 52.01France 4.82 33.77 4.88 32.95 4.91 32.56 4.90 32.78 4.89 32.84 4.89 32.83Italy 4.87 40.50 4.93 39.77 4.96 39.42 4.94 39.62 4.94 39.67 4.94 39.66

Netherlands 5.99 23.90 5.99 23.90 5.99 23.90 5.99 23.90 5.99 23.90 5.99 23.90UK 4.53 53.79 4.58 53.06 4.61 52.71 4.59 52.91 4.59 52.96 4.59 52.95

ψt to be 0 if Algeria interrupts its production in periodt and 1 otherwise; then thisinduces a change in the reserve constraints for Algeria in the following fashion (Algeriahas one production unit; hence` = 1):

Rt+11 = Rt

1ψt −6∑

j=1

ytqt1 j . (17)

This should not be regarded as a physical change in the reserves; it is imposed only toguarantee that no natural gas is traded between Algeria and the rest of Europe. Notethat one does not need to recover the right value of the reserve in future periods after aninterruption occurs, since the shutdown stays in effect in all future periods.

As a consequence of the shutdown possibility, (14) becomes

R11

t∏s=1

ψs−t∑

s=1

ys(

t∏k=s+1

ψk)

6∑j=1

qs1 j ≥ 0, t = 1, . . . ,4, (18)

for Algeria, with the convention that∏t

s = 1 if s> t. Note that we want to solve thelimiting problem (16) withE[ψ] instead ofψ.

For each periodt, we assumedψt to be 0 with probability 0.40. After samplingfrom the uniform distribution and generating sequencesψn

t , we computed the samplefunctions fn by averaging the constraints (18). To compute the limiting distribution willgenerally not be possible for more complicated distributions; however in this case wecan solve the variational inequality with the limit functionf and compare the solutionwith the results of the simulations. Table 4 shows the results for different simulationlengths,n = 20, n = 200,n = 1000,n = 15000, andn = 100000, as well as for thelimit function, n = ∞. To model the problem we used GAMS [1], and in the solutionwe used the PATH solver [3] that is implemented in GAMS.

The solution is a very detailed report containing prices of natural gas in every marketand the amount each producer ships to each market. Since the values of these quantitiesin different periods exhibited similar convergence behavior, we report in Table 4 only theprice and quantity information for the last period. For eachn, the number of iterationsand the number of function evaluations required by PATH were 12 and 16 respectively.

Table 4, incidentally, shows that we get quite close to the true solution with a si-mulation run length as small as 200 random numbers. That table also shows that theprices in the Netherlands are the same for each value ofn. This is due to the fact thatthe Netherlands does not import natural gas and the annual domestic production is a pa-rameter in this model. As a consequence of (7) and (8), the total quantity available in


Table 5.Parameters for model with uncertain oil price

t ηt ort πLt πU

t1 -0.10 30 16 342 -0.12 15 12 183 -0.24 30 24 364 -0.36 35 28 42

Table 6.Prices and quantities for Period 4 (with demand law (19) in effect)

Market n = 20 n = 200 n = 1000 n = 20000 n =∞BelLux 5.17 9.28 5.18 9.28 5.19 9.31 5.19 9.31 5.19 9.30FRGer 4.75 53.03 4.76 53.07 4.77 53.27 4.77 53.24 4.77 53.23France 4.80 33.58 4.81 33.62 4.82 33.78 4.82 33.75 4.82 33.74Italy 4.85 40.34 4.86 40.36 4.87 40.48 4.87 40.46 4.87 40.46

Netherlands 5.95 23.90 5.97 23.90 5.99 23.90 5.99 23.90 5.99 23.90UK 4.51 53.65 4.52 53.66 4.53 53.77 4.53 53.75 4.53 53.75

the domestic Dutch market and its market price will not be affected by the shutdown inAlgeria.

We also considered a different version of this problem in which there is no shutdownin Algeria but in which the prices and demands for natural gas depend on the uncertainprice of oil, and therefore are themselves uncertain. In this model the demand law (8) isreplaced by

Ptj (Q) = p̂t

j (Q/q̂tj )

1/etj , (19)

where

p̂tj = p0t

j (opt/ort), q̂tj = q0t

j (opt/ort)ηt .

Hereopt is a random oil price taken to be uniformly distributed on[πLt , π

Ut ], ort is

a fixed reference price for oil, andηt is an elasticity relating the relative demand fornatural gas to the relative price of oil. The values ofηt , ort , πL

t , andπUt are given in

Table 5. As a consequence of uncertainty in oil prices,P(Q) in (9) becomes a randomquantity and we want to solve the limiting problem withE[P(Q)] instead ofP(Q).

For approximate solution of the limit variational inequality, we again sampled fromthe uniform distribution and generated sequencesopn

t . We then solved the approximatingvariational inequality withPn(Q), the average of the observedP(Q)’s. Table 6 presentsthe results for Period 4 for simulation lengthsn = 20, n = 200, n = 1000, andn = 20000.

For comparison purposes, we then computed the expectation of the quantityp̂tj /

(q̂tj )

1/etj and solved the variational inequality with the limit function; the solution is

given in the column corresponding ton = ∞ in Table 6. The PATH solver requiredthe same numbers of iterations and function evaluations as in the previous model. Inboth versions, we observed that the amount of time required to solve the variationalinequality did not vary much with the length of the simulation runs.


For illustrative purposes we formulated these two example problems so that wecould compute the limit functions exactly and could thereby verify the accuracy of thecomputed results. In practice, of course, this would not be possible (otherwise one wouldnot need to use simulation), and it is in no way necessary for application of the methodwe have described. Two application areas in which this method has been successfullyused and in which the limit function is not computable are option pricing [9] and networkdesign [22].

5. Conclusion

In this paper we have shown how to extend a simulation-based method, sample-pathoptimization, to solve stochastic variational inequalities. We have justified the methodby exhibiting sufficient conditions for convergence, and we have presented the results ofa small numerical experiment on a non-integrable economic equilibrium model of theEuropean natural gas market. These results indicate that the method is implementable.However, we have not presented any analysis of its speed of convergence, nor have weshown how to obtain confidence regions for the computed results. These questions arethe focus of current research.

One important practical aspect of the method concerns the approximating func-tions fn. For the theory these need only be continuous and converge continuously toa relatively nice limit f . However, for computational purposes more is needed. Forexample, in the European gas market problem of Section 4 we used the GAMS mode-ling language for computational solution. To be able to use GAMS we need a closedform expression for thefn; the automatic differentiation capability of GAMS then isused to evaluate gradients of these functions in order to apply an efficient solution al-gorithm. This means that for this example we needed thefn to be expressible in closedform and differentiable; in fact, the underlying theory for the PATH solver requires someadditional conditions on the derivatives [3].

The requirement for a closed-form expression arose because we wanted to use theautomatic differentiation capability of GAMS, so this is not really a basic require-ment. However, most efficient methods for solving variational inequalities will requirenumerical evaluation of both thefn and their derivatives. To avoid this, one might re-sort to techniques that do not require derivative values, such as the resolvent method(see [29,30] for example), but the performance of such methods is usually considera-bly worse than that of methods that utilize gradient information. In addition, except inspecial cases it may be difficult or impossible to compute the resolvent. An alternateapproach when evaluating exact gradients of thefn is not possible might be to use ap-proximations of these gradients in the solution of the variational inequality. Josephy [17]provides conditions for local convergence of such solutions. However, we are not awareof any systematic use of this idea for practical computation, and therefore we cannotassess its numerical effectiveness.

The model presented in Section 4 is static; uncertainty enters via one or moreparameters appearing in the model. In such situations it is often realistic to expect to beable to obtain closed-form expressions for thefn. There are other situations in which thiscan be considerably more difficult. One example of such a problem is the option-pricing


application of [9]. This was a problem of unconstrained optimization, sof was thegradient of a functionφ and the problem was to find a zero of dφ. Unfortunately, it wasnot possible to find well-behaved approximating functionsφn whose gradients couldbe used as thefn. Instead, thefn were obtained by applying methods of perturbationanalysis from [7] and [35] to obtain estimators that could then be averaged to constructthe fn.

A difficult case in general appears to be that in which one is modeling a dynamicsystem and thefn relate to performance of the system as observed over some timeinterval, say one of lengthn. If the underlying problem is one of optimization and weare solving a variational inequality derived from the first order optimality conditions,then the fn will contain gradients of the objective function, presumably a functionrelated to system performance, and possibly also gradients of nonlinear constraints(constraints that are entirely linear can be subsumed in the definition ofC). Althoughit is often possible to obtain numerical values for such gradients — and hence for thefn — by using general methods of perturbation analysis [8,13], at present we knowof no generally applicable method for obtaining derivatives of thefn in this case.However, methods are available for some special cases; for example, see [40] for resultsfor the system times of customers as a function of the parameters of the service timedistributions in a GI/GI/1 queue.

In summary, if the fn are differentiable and have closed-form expressions, thenone can conveniently apply modeling languages, such as GAMS, having automaticdifferentiation capabilities. When the underlying system is static, as in the option pricingproblem of [9], one may be able to use methods of perturbation analysis to find closed-form expressions for suitablefn and their gradients. When the underlying system isdynamic, thefn may not have closed forms, but again one may use perturbation analysisto evaluate them. How to evaluatetheir gradients in this case is, however, still largelyan open question; existing methods are tailor-made to specific systems. This seems tobe an important area for further research.

6. Appendix: Nonsingularity concepts for generalized equations

In this appendix we recall and explain some results from previous work that we use inthe statement and proof of Theorem 2. These center around the condition ofcoherentorientationand its consequences for the solution of variational inequalities.

We start with the critical coneK(z) associated with the polyhedral convex setC, andthe pointz ∈ Rk. This cone is defined in the discussion preceding Theorem 2. Let us fixanyz and writeK = K(z). As K is polyhedral it has only finitely many faces; it is nothard to show that for each nonempty faceF the normal cone ofK takes a constant value,sayNF , on the relative interior ofF. The following facts are demonstrated in [26]:

– For each nonempty faceF of K the setσF = F + NF is a nonempty polyhedralconvex set of dimensionk in Rk.

– The collectionNK of all theseσF for nonempty facesF of K , called thenormalmanifold of K , is a subdivided piecewise-linear manifold, and the union of itscellsσF is all ofRk.


– In each of the cellsσF the projector5K reduces to an affine map (generally differentfor differentσF).

If A is a linear transformation fromRk toRk, then we say that the normal mapAKis coherently orientedif the determinants of the affine maps obtained by restrictingAK to σF all have the same nonzero sign. This condition of coherent orientation is theappropriate extension of the concept of nonsingularity from equations to generalizedequations when the setC is polyhedral.

To understand this idea better, it may help to consider some examples. The simplestis the case in whichC = Rk: that is, in which the linear generalized equation is just anordinary linear equation. One sees immediately that then the critical coneK is alsoRk,so there is just one nonempty faceF, namelyRk itself. Then coherent orientation justmeans that the determinant ofA is nonzero: that is,A is nonsingular. Therefore this ideais indeed an extension of nonsingularity.

The next example is the case in whichK is a subspace ofRk. The coherent orientationrequirement then reduces to nonsingularity of thesectionof A in K : that is, the linearmap fromK to K given by5K ◦ (A|K), whereA|K denotes the restriction ofA to K .This case arises in particular when the normal-map equationfC(z) = 0 comes froma linear complementarity problem of whichx0 is a solution. For such a problem werequirex0 ≥ 0, Ax−a ≥ 0, and〈x0, Ax0−a〉 = 0. Then f(x) = Ax−a, C = Rk+, andz= x0 − (Ax0− a). If the pair(x0, Ax0− a) exhibitsstrict complementary slackness(that is, for each indexi exactly one of(x0)i and (Ax0 − a)i is zero), thenK is thesubspace consisting of allx ∈ Rk such thatxi = 0 if (x0)i = 0, and the section ofAin K is represented by the principal submatrix ofA corresponding to the indicesi forwhich (x0)i > 0. It is this principal submatrix that is then required to be nonsingular.

Of course, in generalK is not a subspace, the normal manifold may have manycells, and the coherent orientation condition can be difficult to check. Nonetheless, it isuseful because it involves only a finite set of matrices, each of which can in principlebe checked for the appropriate determinental sign. For more detailed explanations,additional examples in more general cases, and different characterizations of coherentorientation for a linear transformation, we refer to [5,26,27].

Because we require coherent orientation as a hypothesis in the inverse-functiontheorem below, it is worthwhile to note an equivalence between coherent orientationand another well known property. Suppose thatC is any polyhedral convex subset ofRk,A is a linear map fromRk toRk, anda ∈ Rk. Letx0 be a point satisfying the generalizedequation

0 ∈ Ax+ a+ NC(x). (20)

We say that (20) isstrongly regularat x0 if there are neighborhoodsU of x0 andV ofthe origin inRk such that solution of the generalized equation

v ∈ Ax(v)+ a+ NC(x(v)) (21)

defines a single-valued, Lipschitzian functionx(·) from V to U. The property of strongregularity was introduced in [25] and has been extensively used since then.

Dontchev and Rockafellar, in Remark 3 of [5], note an equivalence between strongregularity and coherent orientation. However, they give neither a precise statement nor


a proof. Because strong regularity is currently the better known of the two concepts, wegive here a statement of the equivalence result.

Theorem 3. LetC be a polyhedral convex subset ofRk, let x0 be a solution of(20)anddefinen0 = −(Ax0+ a) andz0 = x0 + n0. Then(20) is strongly regular atx0 if andonly if AK(z0) is coherently oriented.

Proof. Write K for K(z0). First suppose thatAK is coherently oriented. Theorem 5.2of [26] says thatAK is coherently oriented if and only ifAC is a local Lipschitzianhomeomorphism atz0. We note thatAC(z0) = −a, and thatAC is a piecewise affinemap. Letg be the local inverse ofAC near−a, defined from(V − a) to W, whereV isa neighborhood of the origin andW is a neighborhood ofz0. Theng is piecewise affine(hence Lipschitzian), and is a homeomorphism. Forv ∈ V let x(v) = 5C[g(v − a)].This mapx is Lipschitzian, and it is straightforward to show that it is the only solutionof (21) nearx0. Therefore (20) is strongly regular atx0.

Conversely, if (20) is strongly regular atx0 then there is a locally unique, Lipschitziansolution mapy(v) defined on a neighborhoodV of the origin, such that for eachv ∈ Vthe pointy(v) is the only solution of (21) nearx0. Let z(v) = y(v)+ v− [Ay(v)+ a].It is then easy to show thatz(v) is the only solution nearz0 of AC(z) = v. Hence byTheorem 5.2 of [26],AK must be coherently oriented.

ut

The generalized nonsingularity property provided by coherent orientation lets usobtain a simple inverse-function theorem for the normal maps defined in (5) above. Weuse this theorem in the proof of our main result, Theorem 2.

Theorem 4. Let C be a polyhedral convex set inRk, z0 be a point ofRk andx0 be theEuclidean projection ofz0 onC. Let f be a function from an open set2 containingx0to Rk. Suppose thatfC(z0) = 0, and letK be the critical coneK(z0). Assume thatfhas a strong Fréchet derivatived f(x0) at x0, and thatd f(x0)K is coherently oriented.Then there is an open setU containingz0 such that fC is well defined onU, fC(U) isopen, andfC|U has a Lipschitzian inverse onfC(U).

Proof. The normal mapfC is well defined on the open set5−1C (2), which containsz0.

Let g(x, y) = f(x) − y and takey0 = 0. It is easy to see thatg(·, y0)C(z0) =fC(z0) = 0, and that dxg(x0, y0) = d f(x0). By hypothesis df(x0) is strong, and asg(x, y) = f(x)− y the partial derivative dxg(x0, y0) is also strong. The hypotheses alsoshow that dxg(x0, y0)K is coherently oriented, and hence by Theorem 5.2 of [26] it isa homeomorphism.

Theorem 3 of [27] now shows that there is a positive numberλ0 such that for eachλ > λ0 there are neighborhoodsU0 of z0 andV0 of the origin, and a mapz : V0→ U0that is Lipschitzian with modulusλ, such that for eachy ∈ V0, z(y) is the uniquesolution inU0 of g(·, y)C(z) = 0 (i.e., of fC(z) = y). As z is continuous there is anopen subsetV of V0 such that 0∈ V andz(V) =: U ⊂ int U0. It is then straightforwardto show thatU is open, thatfC(U) = V, and thatz|V is the inverse offC|U.

ut


We refer to Theorem 3 of [27] for more information, in particular as to how theLipschitz modulus forf −1

C can be estimated via that of df(x0)−1K . If C = Rk (the case

of nonlinear equations, discussed above, in whichK = Rk and the requirement is thatd f(x0) be nonsingular), the situation is especially simple: thenλ can be taken to be anynumber greater than‖d f(x0)

−1‖.

Acknowledgements.The research reported here was sponsored by the Air Force Office of Scientific Research,Air Force Materiel Command, USAF, under grant numbers F49620-95-1-0222 and F49620-97-1-0283. TheU. S. Government has certain rights in this material, and is authorized to reproduce and distribute reprints forGovernmental purposes notwithstanding any copyright notation thereon. The views and conclusions containedherein are those of the authors and should not be interpreted as necessarily representing the official policiesor endorsements, either expressed or implied, of the sponsoring agency or the U. S. Government.

References

1. Brooke, A., Kendrick, D., Meeraus, A. (1992): GAMS: A User’s Guide, Release 2.25. Boyd & Fraser,Danvers, MA

2. De Wolf, D., Smeers, Y. (1997): A stochastic version of a Stackelberg-Nash-Cournot equilibrium model.Manage. Sci.43, 190–197

3. Dirkse, S.P., Ferris, M.C. (1995): The PATH solver: A non-monotone stabilization scheme for mixedcomplementarity problems. Optim. Methods Software5, 123–156

4. Dold, A. (1972): Lectures on Algebraic Topology. Springer, Berlin5. Dontchev, A.L., Rockafellar, R.T. (1996): Characterizations of strong regularity for variational inequali-

ties over polyhedral convex sets. SIAM J. Optim.6, 1087–11056. Ferris, M.C., Pang, J.S. (1997): Engineering and economic applications of complementarity problems.

SIAM Rev.39, 669–7137. Fu, M.C., Hu, J.-Q. (1995): Sensitivity analysis for Monte Carlo simulation of option pricing. Prob. Eng.

Inform. Sci.9, 417–4498. Glasserman, P. (1991): Gradient Estimation via Perturbation Analysis. Kluwer, Norwell, MA9. Gürkan, G., Özge, A.Y., Robinson, S.M. (1996): Sample-path solution of stochastic variational inequa-

lities, with applications to option pricing. In: Charnes, J.M., Morrice, D.M., Brunner, D.T., Swain, J.J.,eds., Proceedings of the 1996 Winter Simulation Conference, pp. 337–344

10. Harker, P.T., Pang, J.S. (1990): Finite-dimensional variational inequality and nonlinear complementarityproblems: A survey of theory, algorithms, and applications. Math. Program.48, 161–220

11. Hartman, P., Stampacchia, G. (1966): On some non-linear elliptic differential-functional equations. ActaMath.115, 271–310

12. Haurie, A., Zaccour, C., Legrand, J., Smeers, Y. (1987): A stochastic dynamic Nash-Cournot model forthe European gas market. Technical Report G-87-24. GERAD, École des Hautes Etudes Commerciales,Montréal, Québec, Canada

13. Ho, Y.-C., Cao, X.-R. (1991): Perturbation Analysis of Discrete Event Dynamic Systems. Kluwer,Norwell, MA

14. Hogan, W.W. (1975): Energy policy models for Project Independence. Comput. Oper. Res.2, 251–27115. Huber, P.J. (1964): Robust estimation of a location parameter. Ann. Math. Statistics35, 73–10116. Huber, P.J. (1967): The behavior of maximum likelihood estimates under nonstandard conditions. In:

LeCam, L.M., Neyman, J., eds., Proceedings of the Fifth Berkeley Symposium on Mathematical Statisticsand Probability, I: Statistics, pp. 221–233. University of California Press, Berkeley, CA

17. Josephy, N.H. (1979): Newton’s method for generalized equations and the PIES energy model. Ph.D. Dis-sertation. Department of Industrial Engineering, University of Wisconsin-Madison, Madison, Wisconsin

18. Kall, P. (1986): Approximation to optimization problems: An elementary review. Math. Oper. Res.11,9–18

19. Mathiesen, L., Roland, K., Thonstad, K. (1987): The European natural gas market: Degrees of marketpower on the selling side. In: Golombek, R., Hoel, M., Vislie, J., eds., Natural Gas Markets and Contracts,pp. 27–58. Elsevier, Amsterdam

20. Mulvey, J.M., Vladimirou, H., (1991): Applying the progressive hedging algorithm to stochastic genera-lized networks. Ann. Oper. Res.31, 399–424

21. Ortega, J.M., Rheinboldt, W.C. (1970): Iterative Solution of Nonlinear Equations in Several Variables.Academic Press, New York, London


22. Özge, A.Y. (1997): Sample-path solution of stochastic variational inequalities and simulation optimizationproblems. Ph.D. Dissertation. Department of Industrial Engineering, University of Wisconsin-Madison,Madison, Wisconsin

23. Plambeck, E.L., Fu, B.-R., Robinson, S.M., Suri, R. (1993): Throughput optimization in tandem produ-ction lines via nonsmooth programming. In: Schoen, J., ed., Proceedings of the 1993 Summer ComputerSimulation Conference, pp. 70–75. Society for Computer Simulation, San Diego, CA

24. Plambeck, E.L., Fu, B.-R., Robinson, S.M., Suri, R. (1996): Sample-path optimization of convex sto-chastic performance functions. Math. Program.75, 137–176

25. Robinson, S.M. (1980): Strongly regular generalized equations. Math. Oper. Res.5, 43–6226. Robinson, S.M. (1992): Normal maps induced by linear transformations. Math. Oper. Res.17, 691–71427. Robinson, S.M. (1995): Sensitivity analysis of variational inequalities by normal-map techniques. In:

Giannessi, F., Maugeri, A., eds., Variational Inequalities and Network Equilibrium Problems, pp. 257–269. Plenum Press, New York and London

28. Robinson, S.M. (1996): Analysis of sample-path optimization. Math. Oper. Res.21, 513–52829. Rockafellar, R.T. (1976): Augmented Lagrangians and applications of the proximal point algorithm in

convex programming. Math. Oper. Res.1, 97–11630. Rockafellar, R.T. (1976): Monotone operators and the proximal point algorithm. SIAM J. Control Optim.

14, 877–89831. Rockafellar, R.T., Wets, R.J.-B. (1997): Variational Analysis. Springer, Berlin32. Rubinstein, R.Y., Shapiro, A. (1993): Discrete Event Systems: Sensitivity Analysis and Stochastic Opti-

mization by the Score Function Method. Wiley, Chichester, New York33. Shapiro, A. (1993): Asymptotic behavior of optimal solutions in stochastic programming. Math. Oper.

Res.18, 829–84534. Shapiro, A., Homem-de-Mello, T. (1998): A simulation-based approach to two-stage stochastic program-

ming with recourse. Math. Program.81, 301–32535. Shi, L. (1996): Perturbation analysis with discontinuous sample performance functions. IEEE Trans.

Autom. Control41, 1676–168136. Stampacchia, G. (1969): Variational inequalities. In: Theory and Applications of Monotone Operators,

Proceedings of the NATO Advanced Study Institute held in Venice, Italy, June 17-30, 1968. Edizioni“Oderisi”, Gubbio, Italy

37. Suri, R. (1989): Perturbation analysis: the state of the art and research issues explained via the GI/G/1queue. Proc. IEEE77, 114–137

38. Wald, A. (1949): Note on the consistency of the maximum likelihood estimate. Ann. Math. Statistics20,595–601

39. Wolfowitz, J. (1949): On Wald’s proof of the consistency of the maximum likelihood estimate. Ann.Math. Statistics20, 601–602

40. Zazanis, M.A., Suri, R. (1994): Perturbation analysis of the GI/GI/1 queue. Queueing Syst.18, 199–248

sample-path solution of stochastic variational inequalities

Documents