Shapiro, A. and Homem-de-Mello, T. - H. Milton Stewart School of ...

RATE OF CONVERGENCE OF MONTE CARLO APPROXIMATIONS 73probabilities P {ω = ω k } = p k , k =1,...,K,wehave(2.1)K∑E P h(x, ω) = p k h(x, ω k ).k=1We assume that the feasible set Θ is closed **and** convex **and** that for every ω ∈ Ω, thefunction h(·,ω) is convex. This implies that the expected value function f(·) is alsoconvex, **and** hence the “true” problem (1.1) is convex. Also if P is discrete **and** thefunctions h(·,ω k ), k =1,...,K, are piecewise linear **and** convex, then f(·) is piecewiselinear **and** convex. That is what happens in two-stage stochastic programming witha finite number **of** scenarios.Let ω 1 ,...,ω N be an i.i.d. r**and**om sample in (Ω, F), generated according to thedistribution P , **and** consi**de**r the corresponding approximating program (1.2). Notethat, since the functions h(·,ω j ) are convex, the approximating (sample average)function ˆf N (·) is also convex, **and** hence the approximating program (1.2) is convex.We show in this section that, un**de**r some natural assumptions which hold, forinstance, in the case **of** two-stage stochastic programming with a finite number **of**scenarios, w.p.1 for N large enough any optimal solution **of** the approximating problem(1.2) belongs to the set **of** optimal solutions **of** the true problem (1.1). That is, problem(1.2) yields an exact optimal solution (w.p.1) when N is sufficiently large.The statement “w.p.1 for N large enough” should be un**de**rstood in the sensethat for P -almost every ω ∈ Ωthere exists N ∗ = N ∗ (ω), such that for any N ≥N ∗ the corresponding statement holds. The number N ∗ is a function **of** ω, i.e., it**de**pends on the r**and**om sample **and** therefore in itself is r**and**om. Note also that, sinceconvergence w.p.1 implies convergence in probability, the above statement impliesthat the probability **of** the corresponding event to happen tends to one as the samplesize N tends to infinity.We **de**note by A the set **of** optimal solutions **of** the true problem (1.1) **and** byf ′ (x, d) the directional **de**rivative **of** f at x in the direction d. Note that the set Ais convex **and** closed, **and** since f is a real valued convex function, the directional**de**rivative f ′ (x, d) exists, for all x **and** d, **and** is convex in d. We discuss initially thecase when A is a singleton; later we will consi**de**r the general setting.Assumption (A). The true problem (1.1) possesses unique optimal solution ¯x, i.e.,A = {¯x}, **and** there exists a positive constant c such that(2.2)f(x) ≥ f(¯x)+c‖x − ¯x‖ ∀x ∈ Θ.Of course condition (2.2), in itself, implies that ¯x is the unique optimal solution **of**(1.1). In the approximation theory optimal solutions satisfying (2.2) are called sharpminima. It is not difficult to show, since problem (1.1) is convex, that Assumption(A) holds iff(2.3)f ′ (¯x, d) > 0 ∀ d ∈ T Θ (¯x) \{0},where T Θ (¯x) **de**notes the tangent cone to Θ at ¯x. In particular, if f(x) is differentiableat ¯x, then Assumption (A) (or equivalently (2.3)) holds iff −∇f(¯x) belongs to theinterior **of** the normal cone to Θ at ¯x. Note that since f ′ (¯x, ·) is a positively homogeneousconvex real valued (**and** hence continuous) function, it follows from (2.3) thatf ′ (¯x, d) ≥ ε‖d‖ for some ε>0 **and** ∀ d ∈ T Θ (¯x). We refer to a recent paper [4], **and**references therein, for a discussion **of** that condition **and** some **of** its generalizations.

74 ALEXANDER SHAPIRO AND TITO HOMEM-DE-MELLOIf the function f(x) is piecewise linear **and** the set Θ is polyhedral, then problem(1.1) can be formulated as a linear programming problem, **and** the above assumption(A) always holds provi**de**d ¯x is the unique optimal solution **of** (1.1). This happens,for example, in the case **of** a two-stage linear stochastic programming problem witha finite number **of** scenarios provi**de**d it has a unique optimal solution. Note thatAssumption (A) is not restricted to such situations only. In fact, in some **of** ournumerical experiments sharp minima (i.e., Assumption (A)) happened to hold inthe case **of** continuous (normal) distributions. Furthermore, because the problem isassumed to be convex, sharp minima are equivalent to first or**de**r sufficient conditions.Un**de**r such conditions, first or**de**r (i.e., linear) growth (2.2) **of** f(x) holds globally, i.e.,for all x ∈ Θ.Theorem 2.1. Suppose that (i) for every ω ∈ Ω the function h(·,ω) is convex,(ii) the expected value function f(·) is well **de**fined **and** is finite valued, (iii) the set Θis closed **and** convex, (iv) Assumption (A) holds. Then w.p.1 for N large enough theapproximating problem (1.2) has a unique optimal solution ˆx N **and** ˆx N =¯x.Pro**of** **of** the above theorem is based on the following proposition. Results **of** thatproposition (perhaps not exactly in that form) are basically known, but since its pro**of**is simple we give it for the sake **of** completeness. Denote by h ′ ω(x, d) the directional**de**rivative **of** h(·,ω) at the point x in the direction d **and** by H(B,C) the Hausdorffdistance between sets B,C ⊂ R m , that is,{}(2.4)H(B,C) :=maxsupx∈Cdist(x, B), sup dist(x, C)x∈BProposition 2.2. Suppose that the assumptions (i) **and** (ii) **of** Theorem 2.1 aresatisfied. Then, for any x, d ∈ R m , the following holds:.(2.5)(2.6)limN→∞ ‖d‖≤1f ′ (x, d) =E P {h ′ ω(x, d)} ,sup ∣∣f ′ (x, d) − ˆf N(x, ′ d) ∣ =0w.p.1,(2.7)(lim H ∂ ˆf)N (x),∂f(x) =0 w.p.1.N→∞(2.8)Pro**of**. Since f(·) is convex we have thatf ′ (x, d) = inft>0f(x + td) − f(x),t**and** the ratio in the right-h**and** si**de** **of** (2.8) **de**creases monotonically as t **de**creasesto zero, **and** similarly for the functions h(·,ω). It follows then by the monotoneconvergence theorem that}f ′ h(x + td, ω) − h(x, ω)(2.9)(x, d) =E P{inf,t>0 t**and** hence the right-h**and** si**de** **of** (2.5) is well **de**fined **and** the equation follows.We have that(2.10)ˆf ′ N(x, d) =N −1N ∑j=1h ′ ωj (x, d).

RATE OF CONVERGENCE OF MONTE CARLO APPROXIMATIONS 75Therefore by the strong form **of** the law **of** large numbers it follows from (2.5) thatfor any d ∈ R m ′, ˆf N (x, d) converges to f ′ (x, d) w.p.1 as N →∞. Consequently forany countable set D ⊂ R m ′we have that the event “lim N→∞ ˆf N (x, d) =f ′ (x, d) ∀d ∈ D” happens w.p.1. Let us take a countable **and** **de**nse subset D **of** R m . Recallthat if a sequence **of** real valued convex functions converges pointwise on a **de**nsesubset **of** R m , then it converges uniformly on any compact subset **of** R m (e.g., [18,Theorem 10.8]). Therefore, since the functions ˆf N ′ (x, ·) are convex, it follows from thepointwise convergence **of** ˆf N ′ (x, ·) onD that the convergence is uniform on the unitball {d : ‖d‖ ≤1}. This proves (2.6).Recall that if g is a real valued convex function, then g ′ (x, ·) coinci**de**s with thesupport function **of** its subdifferential ∂g(x). Therefore the Hausdorff distance betweenthe subdifferentials **of** f **and** ˆf N ,atx, is equal to the supremum on the lefth**and**si**de** **of** (2.6) (see, e.g., [12, Theorem V.3.3.8]). Consequently (2.7) follows from(2.6).Pro**of** **of** Theorem 2.1. As we discussed earlier, Assumption (A) is equivalentto condition (2.3) which, in turn, implies that f ′ (¯x, d) ≥ ε for some ε>0 **and** alld ∈ T Θ (¯x) ∩ S m−1 , whereS m−1 := {d ∈ R m : ‖d‖ =1}.By (2.6) it follows that w.p.1 for N large enough(2.11)ˆf ′ N(¯x, d) > 0 ∀ d ∈ T Θ (¯x) ∩ S m−1 .Since the approximating problem is convex, this implies that ¯x is a sharp (**and**hence unique) optimal solution **of** the approximating problem. This completes thepro**of**.Let us consi**de**r now a situation where the true problem (1.1) may have multipleoptimal solutions, i.e., the set A is not necessarily a singleton. In that case Theorem2.1 can be generalized, un**de**r stronger assumptions, as follows.Theorem 2.3. Suppose that (i) the set Ω is finite, (ii) for every ω ∈ Ω thefunction h(·,ω) is piecewise linear **and** convex, (iii) the set Θ is closed, convex, **and**polyhedral, (iv) the true problem (1.1) has a nonempty boun**de**d set A **of** optimal solutions.Then the set A is compact, convex, **and** polyhedral, **and** w.p.1 for N largeenough the approximating problem (1.2) has a nonempty set A N **of** optimal solutions**and** A N is a face **of** the set A.Pro**of** **of** the above theorem is based on the following lemma which may have anin**de**pen**de**nt interest.Lemma 2.4. Suppose that the assumptions (i) **and** (ii) **of** Theorem 2.3 are satisfied.Then the following holds: (a) There exists a finite number **of** points z 1 ,...,z r(in**de**pen**de**nt **of** the sample) such that for every x ∈ R m , there is k ∈{1,...,r} suchthat ∂f(x) =∂f(z k ) **and** ∂ ˆf N (x) =∂ ˆf N (z k ) for any realization **of** the r**and**om sample.(b) W.p.1 the subdifferentials ∂ ˆf N (x) converge to ∂f(x) uniformly in x ∈ R m , i.e.,(lim sup H ∂ ˆf)(2.12)N (x),∂f(x) =0 w.p.1.N→∞ x∈R m(c) If, in addition, the assumptions (iii) **and** (iv) are satisfied, then there exists afinite number **of** points x 1 ,...,x q (in**de**pen**de**nt **of** the sample) such that the pointsx 1 ,...,x l , l

76 ALEXANDER SHAPIRO AND TITO HOMEM-DE-MELLOholds,(2.13)ˆf N (x i ) < ˆf N (x j ) for any i ∈{1,...,l} **and** j ∈{l +1,...,q},then the set A N is nonempty **and** forms a face **of** the set A.Pro**of**. It follows from the assumptions (i) **and** (ii) that the expected value functionf(x) is piecewise linear **and** convex, **and** hence f(x) can be represented as a maximum**of** a finite number **of** affine functions l i (x), i =1,...,n. Consequently the space R mcan be partitioned into a union **of** convex polyhedral sets C 1 ,...,C n such that f(x),restricted to C i , coinci**de**s with l i (x), i =1,...,n.Let us make the following observations. Suppose that f(x) is affine on a convexpolyhedral set C. Then function h(·,ω) is also affine on C for every ω ∈ Ω. In**de**ed,suppose for a moment that the set C has a nonempty interior **and** that for someω ∈ Ωthe corresponding function h(·,ω) is not affine on C. Since h(·,ω) is piecewiselinear **and** convex, this can happen only if there is a point ˆx in the interior **of** Csuch that ∂h(ˆx, ω) is not a singleton. By the Moreau–Rockafellar theorem (see [18,Theorem 23.8]) we have that ∂f(ˆx) = ∑ Kk=1 p k∂h(ˆx, ω k ). Therefore if ∂h(ˆx, ω) isnota singleton, then ∂f(ˆx) is also not a singleton. This, however, cannot happen sincef(x) is affine on C. In case the interior **of** C is empty, we can restrict the problemto the linear space generated by C **and** proceed as above. Now, since the sampleaverage function ˆf N (x) is a linear combination **of** the functions h(·,ω), ω ∈ Ω, withnonnegative coefficients, it follows that ˆf N (x) is also affine on C for any realization**of** the r**and**om sample.Our second observation is the following. Let g(x) be a convex function taking aconstant value over a convex set S. Then ∂g(x) is constant over the relative interior**of** S (e.g., [3, Lemma 1.115]). By adding an affine function to g(x), we obtain thatthe same property holds if g(x) is affine over S.By the above observations we can take points z i in the relative interior **of** eachface **of** the sets C 1 ,...,C n . Note that an extreme point **of** a set C i is viewed as itsface, **of** dimension zero, **and** its relative interior coinci**de**s with the consi**de**red extremepoint. Since each set C i is polyhedral, it has a finite number **of** faces, **and** hence thetotal number **of** such points will be finite. This completes the pro**of** **of** the assertion(a). Assertion (b) follows immediately from Proposition 2.2 **and** assertion (a).Let us prove (c). Since the function f(x) is piecewise linear, the set A is a convexpolyhedral set, **and** by assumption (iv), A is compact.Let us observe that by adding a barrier function **of** the form ψ(x) :=α dist(x, Θ)to the objective function f(x), for sufficiently large α>0, we can reduce the trueproblem to the unconstrained problem(2.14)Minx∈R m E P h ∗ (x, ω),where h ∗ (x, ω) :=h(x, ω)+ψ(x). It is well known that, for α large enough, the optimalsolutions **of** problems (1.1) **and** (2.14) coinci**de** (see, e.g., [2, Proposition 5.4.1]). SinceΘ is convex, the barrier function, **and** hence the functions h ∗ (·,ω), are also convex.Moreover, since by the assumption (iii) the set Θ is polyhedral, the barrier functionis also polyhedral if we take distance with respect to the l 1 norm in R m . Therefore,without loss **of** generality, we can assume in the subsequent analysis that Θ = R m ,i.e., that the problem un**de**r consi**de**ration is unconstrained.Let S be a sufficiently large convex compact polyhedral set (e.g., a cube) suchthat the set A is inclu**de**d in the interior **of** the set S. Such a set exists since A is

RATE OF CONVERGENCE OF MONTE CARLO APPROXIMATIONS 77boun**de**d. Consi**de**r the sets C i ′ := C i ∩ S, i =1,...,n. These sets are polyhedral **and**compact. We can assume that all these sets are different from each other **and** thatA coinci**de**s with the set C 1. ′ Now let {x 1 ,...,x q } be the set **of** all extreme points(vertices) **of** the sets C 1,...,C ′ n ′ such that, for some l0 such that the distancefrom the null vector 0 ∈ R m to ∂f(x) is greater than ε ∀ x ∉ A. Together with(2.12) this implies that w.p.1 for N large enough, 0 ∉ ∂ ˆf N (x) ∀ x ∉ A, **and** henceany x ∉ A cannot be an optimal solution **of** the approximating problem. This showsthat w.p.1 for N large enough the inclusion A N ⊂ A holds. Finally let us observethat since f(x), **and** hence ˆf N (x), are linear on A, **and** A N is the set **of** minimizers **of**ˆf N (x) over A, it follows that A N is a face **of** A.Let us give now the second pro**of**. Let {x 1 ,...,x q } be the set **of** points constructedin assertion (c) **of** Lemma 2.4. Since this set is finite **and** A is the set **of** minimizers **of**f(x), we have that there exists ε>0 such that f(x i )+ε

78 ALEXANDER SHAPIRO AND TITO HOMEM-DE-MELLOUn**de**r the assumptions **of** the above theorem, the set A N **of** optimal solutions **of**the approximating problem is convex **and** polyhedral. The above theorem shows thatw.p.1 for N large enough, every optimal solution **of** the approximating problem is anoptimal solution **of** the true problem **and** every vertex **of** the set **of** optimal solutions**of** the approximating problem is a vertex **of** the set **of** optimal solutions **of** the trueproblem.In or**de**r to see what may happen consi**de**r the following example. Let h(x, ω) :=|x 1 − ω|, where x =(x 1 ,x 2 ) ∈ R 2 **and** ω ∈ Ωwith Ω:= {−2, −1, 1, 2} ⊂R. Supposethat the probability **of** ω being equal to any **of** the points **of** Ωis 0.25 **and** let Θ :={x ∈ R 2 : |x 2 | ≤ 1}. Then the set A **of** optimal solutions **of** the correspondingtrue problem is A = {x : |x 1 |≤1, |x 2 |≤1}. On the other h**and**, for large N,the set **of** optimal solutions **of** the approximating problem is given either by the face{x : x 1 = −1, |x 2 |≤1} or the face {x : x 1 =1, |x 2 |≤1} **of** the set A.3. Exponential rate **of** convergence. In the previous section we showed that,un**de**r appropriate assumptions, the approximating problem (1.2) yields an exact optimalsolution **of** the true problem (1.1) w.p.1 for N large enough. Since convergencew.p.1 implies convergence in probability, it follows that the probability **of** this eventtends to one as N tends to infinity. That result, however, does not say how large thesample size N should be in or**de**r for the approximating problem to provi**de** such anexact solution.Similarly to the example presented in the introduction, it turns out that, in thecase un**de**r consi**de**ration (i.e., when Ωis finite **and** h(·,ω) are piecewise linear), theconvergence **of** the corresponding probability to one is exponentially fast. A consequence**of** this somewhat surprising fact is that one does not need a very large sampleto find the optimal solution **of** (1.1), which shows that Monte Carlo approximationstechniques can be an effective approach to solve such problems.In this section we formalize **and** prove this result. We begin by consi**de**ring againthe case where the true problem (1.1) has a unique optimal solution ¯x. Suppose thatAssumption (A) holds. Recall that S m−1 **de**notes the sphere in R m , **and** consi**de**r theBanach space Z := C(S m−1 ) **of** real valued continuous functions **de**fined on S m−1**and** equipped with the sup-norm. By restricting a positively homogeneous functionto S m−1 , we can i**de**ntify Z with the space **of** continuous positively homogeneousfunctions on R m . Denote by Z ∗ the dual space **of** Z, i.e., the space **of** continuouslinear functionals **de**fined on Z.Let B be the σ-algebra **of** Borel sets in Z. Consi**de**r the function(3.1)η(d, ω) :=h ′ ω(¯x, d), d ∈ R m ,ω∈ Ω.The function η(·,ω) is convex, **and** hence continuous, **and** is positively homogeneous.Therefore it can be consi**de**red as an element **of** Z. Moreover, the mapping ω ↦→ η(·,ω),from (Ω, F) into(Z, B), is measurable **and** hence η(·,ω) can be consi**de**red as a r**and**omelement **of** (Z, B). Let P be the probability measure on Z induced by the measure P .Note that E P η(d, ω) =f ′ (¯x, d) **and** that the measure P is concentrated on the subset**of** Z formed by convex positively homogeneous functions.Assumption (B). There exists a constant κ>0 such that‖η(·,ω)‖ Z ≤ κ, for P -almost every ω.This assumption clearly holds if the set Ωis finite. Note that‖η(·,ω)‖ Z =sup |h ′ ω(¯x, d)| .d∈S m−1

RATE OF CONVERGENCE OF MONTE CARLO APPROXIMATIONS 79Therefore Assumption (B) means that the subdifferentials ∂h(¯x, ω) are uniformlyboun**de**d for P -almost every ω. Notice that this is what happens in two-stage stochasticprogramming problems with complete recourse if only the right-h**and** si**de** is r**and**om,since in that case the dual feasibility set does not **de**pend on ω. Completerecourse implies that the dual feasibility set is also boun**de**d. Therefore, in such casesthe subdifferentials ∂h(¯x, ω) are uniformly boun**de**d for all ω.Let us recall now a few facts about r**and**om variables on Banach spaces. Letη 1 ,η 2 ,... be an i.i.d. sequence **of** r**and**om elements **of** (Z, B), with the common distributionP, **and** **de**fine ζ N := N −1 ∑ Nj=1 η j. Note that Assumption (B) implies that∫Z ‖z‖ ZP(dz) < ∞. Then, by the strong law **of** large numbers (for Banach spaces)we have that ζ N → ζ := E[η] w.p.1, where the convergence is in the norm **of** Z **and**the expectation operator corresponds to the so-called Bochner integral (see, e.g., Hiai[10]).Let∫M(z ∗ ):= e z∗ (z) P(dz), z ∗ ∈ Z ∗ ,be the moment generating function **of** P (i.e., **of** η(·,ω)). A version **of** Cramér’stheorem for Banach spaces (see, e.g., Deuschel **and** Stroock [8]) can be stated asfollows. If for any α ∈ [0, ∞) wehave∫(3.2)e α‖z‖ P(dz) < ∞,Zthen a large **de**viations principle (LDP) holds for {ζ N }, i.e., for any B-measurable setΓ ⊂ Z we have that− inf z∈ int(Γ) I(z)(3.3)≤ lim inf N→∞ N −1 log[P (ζ N ∈ Γ)]≤ lim sup N→∞ N −1 log[P (ζ N ∈ Γ)] ≤−inf z∈ cl(Γ) I(z).Here int(Γ) **and** cl(Γ) **de**note the interior **and** the topological closure, respectively, **of**the set Γ ⊂ Z, **and** I(z) is the large **de**viations rate function, which is given by(3.4)I(z) := sup {z ∗ (z) − log M(z ∗ )}.z ∗ ∈Z ∗Notice that (3.2) follows immediately from Assumption (B).For any d ∈ S m−1 we can **de**fine a functional zd ∗ ∈ Z∗ as zd ∗ (z) :=z(d). LetM d (t) :=M(tzd ∗ ). Note that we can also writeM d (t) =E P{e tη(d,ω)} ,so we recognize M d (t) as the moment generating function **of** the (one-dimensional)r**and**om variable X := η(d, ω). Note also that Assumption (B) implies that M d (t) < ∞∀ t ∈ R. Consi**de**r the rate function **of** η(d, ω), that is,(3.5)I d (α) :=sup[tα − log M d (t)] .t∈RBy taking z ∗ in the right-h**and** si**de** **of** (3.4) **of** the form z ∗ := tzd ∗ , we obtain that, forany z ∈ Z,(3.6)I(z) ≥sup sup [tz(d) − log M d (t)] =d∈S m−1 t∈Rsup I d (z(d)).d∈S m−1

80 ALEXANDER SHAPIRO AND TITO HOMEM-DE-MELLOLet A N be the set **of** optimal solutions **of** the approximating problem (1.2), **and**consi**de**r the event(3.7)E N := { the set A N is nonempty **and** A N = {¯x} } .The above event E N means that the approximating problem possesses a unique optimalsolution ˆx N **and** that ˆx N =¯x. Denote by EN c the complement **of** the event E N. Notethat the probability P (E N ), **of** the event E N , is equal to 1 − P (EN c ). The followingtheorem shows that the probability **of** the event ENc approaches zero exponentiallyfast.Theorem 3.1. Suppose that the assumptions **of** Theorem 2.1 are satisfied **and**that Assumption (B) holds. Then there exists a constant β>0 such that(3.8)lim supN→∞1N log[P (E c N)] ≤ −β.(3.9)Pro**of**. Consi**de**r ζ N (·) :=N −1 ∑ Nj=1 η(·,ωj )= ˆf N ′ (¯x, ·) **and** the set}F :={z ∈ Z :infd∈T Θ(¯x)∩S m−1 z(d) ≤ 0Since the topology on Z is that **of** uniform convergence, it follows that the min-functionφ(z) :=infd∈T Θ(¯x)∩S m−1 z(d)is continuous on the space Z, **and** hence the set F is closed in Z. By the **de**finition **of**the set F , we have that if ζ N ∉ F , then ζ N (d) > 0 ∀ d ∈ T Θ (¯x)∩S m−1 . Consequently,in that case, ˆx N =¯x is the unique optimal solution **of** the approximating problem.Therefore we have thatP (E c N) ≤ P (ζ N ∈ F ).It follows then by the last inequality **of** (3.3) that we need only to show that theconstant.(3.10)β := infz∈F I(z)is positive.Consi**de**r a fixed direction d ∈ T Θ (¯x) ∩ S m−1 , **and** let X **de**note the correspondingr**and**om variable η(d, ω). Let Λ(t) := logM d (t) = log E[e tX ] be the logarithmicmoment generating function **of** X. By the dominated convergence theorem wehave that M d (t) is differentiable ∀ t ∈ R **and** Md ′ (t) =E[XetX ]. It follows thatΛ ′ (t) =E[Xe tX ]/E[e tX ] **and** hence, since |X| ≤κ by Assumption (B),Similarly, we have|Λ ′′ (t)| =E[X 2 e tX ](3.11)∣ E[e tX ]|Λ ′ (t)| ≤ E[|X|etX ]E[e tX ]≤ κ ∀ t ∈ R.− (Λ ′ (t)) 2 ∣ ∣∣∣≤|κ 2 − (Λ ′ (t)) 2 |≤ κ 2 ∀ t ∈ R.

RATE OF CONVERGENCE OF MONTE CARLO APPROXIMATIONS 81By the mean value theorem, (3.11) implies that ∀ t, s ∈ R,(3.12)|Λ ′ (t) − Λ ′ (s)| ≤ κ 2 |t − s|.Since the function Λ(·) is convex, it follows from a result in convex analysis (e.g., [12,Theorem X.4.2.2]) that the conjugate function I d =Λ ∗ is strongly convex modulus1/κ 2 , that is,I d (α 2 ) ≥ I d (α 1 )+I d(α ′ 1 )(α 2 − α 1 )+ 12κ 2 |α 2 − α 1 | 2∀ α 1 ,α 2 ∈ R. Since at ᾱ d := E[X] =f ′ (¯x, d) we have that I d (ᾱ d )=Id ′ (ᾱ d)=0,itfollows that(3.13)I d (α) ≥ 12κ 2 |α − ᾱ d| 2 ∀ α ∈ R.By Assumption (A) we have that f ′ (¯x, d) ≥ c ∀ d ∈ T Θ (¯x) ∩ S m−1 , **and** hence weobtain that(3.14)I d (0) ≥c22κ 2 ∀ d ∈ T Θ (¯x) ∩ S m−1 .By the **de**finition **of** the set F we have that if z ∈ F , then there exists d ∈ T Θ (¯x)∩S m−1such that z(d) ≤ 0. It follows then by (3.6) **and** (3.14) that I(z) ≥ c 2 /(2κ 2 ) for anyz ∈ F . Consequently we obtain(3.15)β ≥c22κ 2 ,which completes the pro**of**.The inequality (3.8) means that the probability that the approximating problem(1.2) has a unique optimal solution which coinci**de**s with the optimal solution **of** thetrue problem (1.1) approaches one exponentially fast. The inequality (3.15) also givesan estimate **of** the corresponding exponential constant.Consi**de**r now a situation where the true problem (1.1) may have multiple solutions.As in the case **of** convergence w.p.1 presented in section 2, stronger assumptionsare nee**de**d. Let A N be the set **of** optimal solutions **of** the approximating problem (1.2),**and** consi**de**r the event(3.16)M N := { the set A N is nonempty **and** forms a face **of** the set A } .Theorem 3.2. Suppose that the assumptions **of** Theorem 2.3 hold. Then thereexists a constant β>0 such that(3.17)lim supN→∞1N log[P (Mc N)] ≤ −β.Pro**of**. It is possible to prove this theorem by using arguments **of** Theorem 3.1combined with assertions (a) **and** (b) **of** Lemma 2.4. The pro**of** becomes even simplerif we use assertion (c) **of** Lemma 2.4. Let {x 1 ,...,x q } be the set **of** points constructedin assertion (c) **of** Lemma 2.4. Recall that {x 1 ,...,x l } forms the set **of** extreme points**of** A, **and** that f(x i )

RATE OF CONVERGENCE OF MONTE CARLO APPROXIMATIONS 83the last equality arising from the fact that the function I(·) is increasing on the interval[p, ∞). From (4.1) we obtain that[ (p −1 − 1) 1/2 ](4.2)I(0.5) = log.2(1 − p)For p =1/3 wehaveI(0.5) = log( 3√ 24)=0.0589, **and** hence the probabilityP (X/N ≥ 1/2) converges to zero at the exponential rate e −0.0589N . Note that in theconsi**de**red (one-dimensional) case the upper bound **of** Cramér’s theorem holds for anyN (**and** not just in the limiting sense). It follows that the probability that the sampleestimate ˆx N is equal to the true optimal solution is greater than (1 − 2e −0.0589N ) m ,which for large N is approximately equal to 1−2me −0.0589N . Consequently the probabilitythat the sample estimate ˆx N is not equal to the true optimal solution **de**creasesexponentially fast with the sample size N **and** increases linearly with the number **of**variables m. For example, for N = 100 **and** m = 50 we have, by the above estimate,that the probability **of** the sample estimate ˆx N being equal to the true optimalsolution is at least (1 − 2e −5.89 ) 50 =0.76. This can be compared with the exact probability**of** that event, which is about 0.96. This is quite typical for the large **de**viationsestimates. For finite **and** not too “large” N, the large **de**viations estimates give poorapproximations **of** the corresponding probabilities. What the large **de**viations theoryprovi**de**s, **of** course, is the exponential rate at which the corresponding probabilitiesconverge to zero.Suppose now that each variable Y i has the following discrete distribution: it cantake values -1, -0.5, 0.5, **and** 1 with equal probabilities 0.25. In that case the set **of**optimal solutions **of** the true problem (1.3) is not a singleton **and** is given by the cube{x : −0.5 ≤ x i ≤ 0.5}. We have that the probability that the sample estimate ˆx iNbelongs to the interval [−0.5, 0.5] is at least 1−2P (X ≥ N/2), where X ∼ B(N,0.25).Again we obtain that the probability that ˆx N is an exact optimal solution **of** the trueproblem is approaching one exponentially fast with increasing N.Now let m = 1 **and** suppose that the distribution **of** Y is discrete with possiblevalues given by an odd number r =2l + 1 **of** points equally spaced on the interval[−1, 1] with equal probabilities **of** 1/r. For “large” r we can view this as a discretization**of** the uniform distribution on the interval [−1, 1]. Then by the same arguments asabove we obtain that the probability that ˆx N = 0 is at least 1 − 2P (X ≥ N/2), whereX ∼ B(N,p) with p = l/r.An estimate **of** how fast N grows as a function **of** the number **of** variables m **and**the number **of** discretization points r can be obtained using again large **de**viationstechniques. Suppose that m ≥ 1 **and** that each r**and**om variable Y i , i =1,...,m, hasa discrete distribution as above. From (4.2) we have that in this case the constantβ := I(0.5) is given byβ = 1 [ ] r2(4.3)2 log r 2 ,− 1**and** henceP (ˆx N =0)≥ (1 − 2e −βN ) m ∼ = 1 − 2me −βN .Consequently, for a fixed ε>0, a (conservative) estimate **of** the sample size N nee**de**dto obtain P (ˆx N =0)≥ 1 − ε is given byN = β −1 log(2m/ε) ∼ = (2r 2 − 1) log(2m/ε),

84 ALEXANDER SHAPIRO AND TITO HOMEM-DE-MELLOso we see that N grows quadratically with the number **of** discretization points **and**logarithmically with the number **of** r**and**om variables.4.2. A two-stage stochastic programming problem. We now present somenumerical results obtained for the capacity expansion problem CEP1 **de**scribed in [11],which can be mo**de**led as a two-stage stochastic programming problem with completerecourse. The problem has 8 **de**cision variables with 5 constraints (plus bound constraints)on the first stage, **and** 15 **de**cision variables with 7 constraints (plus lowerbound constraints) on the second stage. The r**and**om variables, which correspond to**de**m**and** in the mo**de**l, appear only on the right-h**and** si**de** **of** the second stage. Thereare 3 i.i.d. r**and**om variables, each taking 6 possible values with equal probability, sothe sample space has size 6 3 = 216.For the sake **of** verification, we initially solved the problem exactly by solving theequivalent **de**terministic LP, **and** we obtained the true minimizer ¯x. Notice that thisoptimal solution is unique. We then solved the corresponding Monte Carlo approximations,with sample sizes N =2, 5, 10, 15, 20, 35, 50. For each sample size, we solvedthe approximating problem 400 times **and** counted how many times the optimal solutionˆx N , **of** the approximating problem, coinci**de**d with the true solution ¯x. Thecorresponding proportion ˆp is then an estimate **of** the probability P (ˆx N =¯x). Sincethe generated replications are in**de**pen**de**nt, it follows that an unbiased estimator **of**the variance **of** ˆp is given by ˆp(1− ˆp)/399. From this value we obtain a 95% confi**de**nceinterval whose half-width is **de**noted by ∆. The results are displayed in Table 4.1.Table 4.1Estimated probabilities P (ˆx N =¯x).N ˆp ∆2 0.463 .0495 0.715 .04410 0.793 .04015 0.835 .03620 0.905 .02935 0.958 .02050 0.975 .015Notice again the exponential feature **of** the numbers on the table, i.e., how fastˆp gets close to one. It is interesting to notice that convergence in the CEP1 mo**de**lis even faster than in the median problem, even though the median problem is muchmore structured (in particular, the median problem is separable) with a smaller samplespace (27 points for 3 r**and**om variables, as opposed to 216 points in the CEP1mo**de**l). For instance, in the median problem a sample size **of** 20 gives the true optimalsolution with probability 0.544, whereas in the CEP1 problem that probabilityis approximately 0.9. These results corroborate the i**de**as presented in the previoussections, showing that convergence can be very fast if there is a sharp minimum suchas in the case **of** the CEP1 mo**de**l. The results also suggest that the separabilityinherent in the median problem was not a major factor in the speed **of** convergence,which encourages us to think that the numerical results reported here can be obtainedin more complex problems. Of course, more research is nee**de**d to draw any **de**finiteconclusions.5. Conclusions. We presented in this paper some results concerning convergence**of** Monte Carlo simulation-based approximations for a class **of** stochastic programmingproblems. As pointed out in the introduction, the usual approach to con-

RATE OF CONVERGENCE OF MONTE CARLO APPROXIMATIONS 85vergence analysis found in the literature consists in showing that optimal solutions **of**approximating problems converge, w.p.1, to optimal solutions **of** the original problemor in obtaining bounds for the rate **of** convergence via central limit theorem or large**de**viations type asymptotics. We show, un**de**r some specific assumptions (in particular,un**de**r the assumption that the sample space is finite), that the approximatingproblem provi**de**s an exact optimal solution w.p.1 for sample size N large enough **and**,moreover, that the probability **of** such an event approaches one at an exponential rate.This suggests that, in such cases, Monte Carlo simulation based algorithms could beefficient, since one may not need a large sample to find an exact optimal solution.The median problem presented in section 4 illustrates that point. For a problemwith 3 200 scenarios, an approximating problem which employs only N = 120 samples,**of** a vector **of** dimension m = 200, yields the exact optimal solution approximately95% **of** the time. Even more impressively, it is possible to show by the same type **of**calculations that N = 150 samples are enough to obtain the exact optimal solutionwith probability **of** about 95% for m = 1000 r**and**om variables, i.e., for 3 1000 scenarios.Estimates **of** the sample size N, which were obtained in section 4 by the large **de**viationsapproximations, give slightly bigger values **of** N (for example, they give N = 180instead **of** N = 150 for m = 1000). In either case the required sample size grows asa logarithm **of** the number m **of** r**and**om variables in that example. Of course, onemust take into account the fact that this is a very structured problem, **and** in a moregeneral case one may not get such drastically fast convergence; in fact, the flatterthe objective function is around the optimal solution, the slower the convergence willbe. Nevertheless, the CEP1 mo**de**l studied in section 4 seems to indicate that fastconvergence is obtained in more general problems, even in the absence **of** separability.One should, however, be cautious about these results, especially with respect tothe following aspect. The fact that the convergence is exponential does not necessarilyimply that a small sample suffices. In**de**ed, the constant β in the correspondingexponential rate e −βN can be so small that one would need a large sample size N inor**de**r to achieve a reasonable precision. The lower bound (3.15) gives us an i**de**a aboutthe exponential constant β. In the median example, with r discretization points foreach r**and**om variable Y i , i =1,...,m, we have that we can take c =1/r **and** κ =1,if we use l 1 norm in the space R m . This gives us the lower bound β ≥ 1/(2r 2 ), whichcan be compared with the exact value **of** β = 1 2 log[r2 /(r 2 − 1)] ∼ = 1/(2r 2 − 1). Notethat the estimate β ≥ 1/(2r 2 ) does not **de**pend on the number m **of** r**and**om variables.This happens since any multiplicative constant before e −βN can be absorbed into theexponential rate as N tends to infinity.Another remark concerns the assumption **of** Monte Carlo sampling in our analysis.By doing so, we were able to exploit properties **of** i.i.d. samples, which we used to**de**rive our results. In practice, however, one might think **of** implementing variancereduction techniques in or**de**r to reduce even more the nee**de**d sample sizes. Theincorporation **of** such techniques into stochastic optimization algorithms has beenshown to be very effective in practice (see, e.g., [1], [5], [21]). Research on specificapplications **of** variance reduction techniques to the type **of** problems discussed in thispaper is un**de**rway.Acknowledgment. We thank the referees for constructive comments whichhelped to improve the presentation **of** the paper.

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