on reduction of the multistage problem of stochastic programming with quantile criterion to the...

ISSN 0005-1179, Automation and Remote Control, 2014, Vol. 75, No. 4, pp. 688–699. © Pleiades Publishing, Ltd., 2014.Original Russian Text © A.I. Kibzun, O.M. Khromova, 2014, published in Avtomatika i Telemekhanika, 2014, No. 4, pp. 120–133.

TWO-LEVEL PROGRAMMING PROBLEMS

On Reduction of the Multistage Problem of Stochastic

Programming with Quantile Criterion to the Problem

of Mixed Integer Linear Programming

A. I. Kibzun and O. M. Khromova

Moscow State Aviation Institute, Moscow, Russiae-mail: [email protected], [email protected]

Received November 14, 2013

Abstract—Consideration was given to the a priori formulation of the multistage problem ofstochastic programming with a quantile criterion which is reducible to the two-stage problem.Equivalence of the two-stage problems with the quantile criterion in the a priori and a posterioriformulations was proved for the general case. The a posteriori formulation of the two-stageproblem was in turn reduced to the equivalent problem of mixed integer linear programming.An example was considered.

DOI: 10.1134/S0005117914040092

1. INTRODUCTION

The problems of stochastic linear programming traditionally use the expectation as the opti-mization criterion [1]. The aviation and space engineering pay special attention to the issues ofsystem reliability [2]; therefore, quite often it is required to establish a probability-guaranteed so-lution. Formulations of the problem of stochastic programming with quantile criterion are mostadequate in this case. The paper [3] is devoted to the methods for solution of the quantile opti-mization problem. However, in many applied problems [1] the multistage problems of stochasticprogramming arise.

The multistage problems of stochastic programming turn out to be allied to the dynamic prob-lems of control allowing for the effect of random factors [2]. In these problems, a first-stage strategyis selected and corrected at the subsequent stages (steps) depending on realization of the randomfactors. The multistage problems of stochastic programming generalize the two-stage problems ofstochastic programming. The two-stage problem with a quantile criterion was considered in [4]where equivalence of its a priori and a posteriori formulations was established. Relying on thesolution of the linear programming problem, this paper also estimated from above the quantilecriterion for the two-stage problem.

The linear one-stage problems of quantile optimization with the discrete distribution reducibleto the problems of mixed integer linear programming were studied in [5, 6]. The general case ofthe two-stage problem of quantile optimization that was reduced to the large-dimension problemof mixed integer programming was examined in [7].

The present paper studies the multistage problem of stochastic programming with quantilecriterion. It is reduced with the use of equivalent transformations to the two-stage problem whichin turn is reduced to the problem of mixed integer linear programming. An example is considered.

688

ON REDUCTION OF THE MULTISTAGE PROBLEM 689

2. A PRIORI FORMULATION OF THE MULTI-STAGE PROBLEMOF STOCHASTIC PROGRAMMING

We pose an N -stage problem of stochastic programming in the a priori formulation and introducethe loss function

ΦN (u, y(·),X) � cT0 u+ aT11(X1)u+ cT1 y1(u,X1) + aT12(X1,X2)u+ cT2 y2(u,X1,X2) + . . .

= cT0 u+N−1∑

i=1

aT1i(Xi)u+

N−1∑

i=1

cTi yi(u,Xi),

(2.1)

where u ∈ IRm is the plan of the first stage; y(·) = col(y1(·), . . . , yN−1(·)) is the vector function ofthe plans of the subsequent N − 1 stage selected in the class of measurable functions with valuesin IRmi ; X = col(X1, . . . ,XN−1), X

i = col(X1, . . . ,Xi), i = 1, N − 1, are collections of random vec-tors Xi ∈ IRni ; a1i(X

i), i = 1, N − 1, are the given measurable vector functions of the correspondingdimensions; and ci, i = 0, N − 1, are the given determinate column vectors.

We assume that the random vector X has the distribution density p(x), where x ∈ IRn andn = n1 + n2 + . . .+ nN−1.

Let there be a collection of N − 1 constraint

Φi(u, yi(·),Xi) � A2i(X

i)u+Biyi(u,Xi) � a3i(X

i), i = 1, N − 1, (2.2)

where yi(·) = col(y1(·), . . . , yi(·)); A2i(Xi), i = 1, N − 1, are the given measurable (si ×m) matrix

functions; a3i(Xi), i = 1, N − 1, are the given measurable vector functions of dimension si, and

Bi is the given (si ×mi) matrix.

Let us consider the probability function

Pϕ(u, y(·)) =P{Φ(u, y(·),X)�ϕ,Φi(u, yi(·),Xi)� a3i(X

i), i=1, N −1} (2.3)

and also the quantile function

ϕα(u, y(·)) � min{ϕ : Pϕ(u, y(·)) � α}, α ∈ (0, 1). (2.4)

Let us formulate in the a priori terms the N -stage problem of stochastic programming

ϕα = minu∈U, y(·)∈Y

ϕα(u, y(·)), uα = argminu∈U

[miny(·)∈Y

ϕα(u, y(·))], (2.5)

where U is the given set of permissible strategies of the first stage and Y is the set of permissiblestrategies of the following stages:

Y � {y(·) : yi(u,Xi) � 0, i = 1, N − 1}.By solution is meant problem (2.5) understood as the pair (ϕα, uα). If uα does not exist, that is,the minimum of (2.5) is not reached, then it is assumed that there is no solution to problem (2.5).

We consider by way of example the three-stage problem of stochastic programming. The prob-ability function takes on form

Pϕ(u, y(·)) = P{Φ3(u, y(·),X) � ϕ, Φi(u, yi(·),Xi) � a3i(X

i), i = 1, 2}, (2.6)

where

Φ3(u, y(·),X) = cT0 u+aT11(X1)u+ c

T1 y1(u,X1)+a

T12(X1,X2)u+ c

T2 y2(u,X1,X2),

Φ2(u, y2(·),X2) = A22(X1,X2)u+B2y2(u,X1,X2) � a32(X1,X2), (2.7)

Φ1(u, y1(·),X1) = A21(X1)u+B1y1(u,X1) � a31(X1).

AUTOMATION AND REMOTE CONTROL Vol. 75 No. 4 2014

690 KIBZUN, KHROMOVA

Remark 1. In the multistage problems of stochastic programming, it is expectation that is mostoften considered as the optimization criterion [1]. The multistage problems of stochastic program-ming are usually considered in the a posteriori formulation where at the current stage the optimalplan is selected depending on the realization of the random factors of the preceding stages. Thismeans in essence that the method of dynamic programming is used to solve the stochastic program-ming problem. Yet, as is well known from [8], the method of dynamic programming is applicableto the problem of stochastic control only if the criterion features additive or Markovian propertieswhich are characteristic namely of the expectation of the random loss function. The quantile crite-rion has neither Markovian nor additive properties [2]. In consequence of this fact, it is impossiblein principle to set down the multistage stochastic programming problem with quantile criterion inthe traditional a posteriori formulation. Therefore, the problem at hand is set down below in thea priori formulation where at all stages, except for the first step, the optimal plans are taken fromthe class of functions depending on the entire preceding information. We notice that the attentionto the relation between the two-stage problem in the a priori and a posteriori formulations wasdrawn for the first time in [4].

3. REDUCTION OF THE MULTISTAGE PROBLEM IN THE A PRIORIFORMULATION TO THE TWO-STAGE PROBLEM

Assume that the probabilistic measure is absolutely continuous relative to the Lebesgue measureand there exists density p(x) of the random vector X. We discretize the probabilistic measure asfollows. Let xk, k = 1,K , be randomly generated points according to the density p(x). We definetheir measures as pk � P{X = xk} = 1/K, k = 1,K .

Let X̃ � col(X̃1, . . . , X̃N−1) be the random vector corresponding to these measures, that is,P{X̃ = xk} = pk, where the random subvectors X̃i have the same dimension as Xi, i = 1, N − 1.Let FK(x) be the distribution function of the random vector X̃ . We consider the sampled distri-bution function F̂K(x) corresponding to the random vector X and having realization FK(x). Ac-

cording to the Glivenko–Cantelli theorem [9], for K → ∞ the convergence F̂K(x)almost sure−−−−−−−−−→F (x)

takes place for all x, where F (x) is the distribution function of the random vector X. We provethe following lemma.

Lemma. Let the random vector X̃ = col(X̃1, . . . , X̃N−1) have discrete distribution FK(x). Then,there are determinate functions fi(X̃1) such that

P{X̃i = fi(X̃1)} = 1, i = 2, N − 1.

The proof is given in the Appendix.

We formulate a statement according to which the N -stage problem comes to the two-stageproblem and for that use the definition introduced in [6].

Definition. Two optimization problems are regarded as equivalent

1) if either both problems have permissible solutions (with finite values of the objective functions)or both have no such solutions;

2) if these problems have permissible solutions, then the finite or infinite optimal values of theirobjective functions coincide;

3) if the optimal values of their objective functions are finite, then these values either are reachedor not in both problems;

4) if the optimal values of the objective functions are reached, then the explicitly describedalgorithm indicates from the optimal solution of one problem the optimal solution from the otherproblem;



5) if the optimal values of the objective functions are finite but not reached, then the explic-itly described algorithm indicates from the optimizing sequence of strategies of one problem theoptimizing sequence for the other problem.

In what follows, we understand equivalence in the sense of definition.

Theorem 1. In the sense of definition, the N -stage problem in the a priori formulation (2.5)with the discrete distribution FK(x) of the random vector X̃ is equivalent to a special two-stagestochastic programming problem.

Theorem 1 is proved in the Appendix.

Remark 2. Solution of such problem in the space of functions turns out to be sufficiently compli-cated because it is the quantile of the loss function that is the criterion in the formulated problem.As was already noticed, the method of dynamic programming reducing the problem of optimizationin the space of functions to successive solution of the problems of mathematical programming in afinite-dimensional space is inapplicable to the quantile criterion [2]. Owing to the above scheme ofmeasure discretization, one can reduce the multistage problem of stochastic programming to thetwo-stage problem studied in [4], although for a more special case. The structure of constraintsof the resulting two-stage problem repeats that of the constraints of the original problem. Addi-tionally, the values of the loss functions of both problems coincide, which proves equivalence of themultistage problem in the a priori formulation and the resulting two-stage problem in the sense ofdefinition.

4. REDUCTION OF THE TWO-STAGE PROBLEM IN THE A PRIORI FORMULATIONTO THE TWO-STAGE PROBLEM IN THE A POSTERIORI FORMULATION

We rearrange the two-stage problem (2.5) in s simpler form assuming that y1(·) � y(·), X̃1 � X .Let us consider the loss function

Φ(u, y(·),X) = cT0 u+ aT1 (X)u+ cT1 y(u,X) (4.1)

with the constraint of the second stage

Φ1(u, y(·),X) � A2(X)u+By(u,X) � a3(X), (4.2)

where u ∈ IRm, y ∈ IRm1 , X ∈ IRn, a3(X) is the vector function of dimension s, and the matricesA2(X) and B have dimensions (s × m) and (s × m1), respectively. We consider the problem ofquantile optimization of the form of (2.5):

ϕα = minu∈U, y(·)∈Y

ϕα(u, y(·)),

uα = argminu∈U

[miny(·)∈Y

ϕα(u, y(·))] (4.3)

with the quantile function defined according to (2.4) on the basis of the probability function

Pϕ(u, y(·)) = P{Φ(u, y(·),X) � ϕ, Φ1(u, y(·),X) � a3(X)}. (4.4)

Consider also the two-stage problem in the following a posteriori formulation. Let the realizationx ∈ IRn of the random vector X be known, and the set of permissible plans of the second stagebe defined as Y � {y : y ∈ IRm1 , y � 0}. Consider the problem of second stage assuming thatu ∈ U ⊂ IRm and x ∈ IRn are known:

Φ(u, x) � miny∈Y

{cT1 y|A2(x)u+By(u, x) � a3(x)

}, (4.5)



where c1 is the vector of dimension m1. If there is no y ∈ Y satisfying (4.5) for some u ∈ U and xfrom the set of permissible realizations, then we assume that Φ(u, x) = +∞.

We define the probability function

Pϕ(u) � P{aT1 (X)u +Φ(u,X) � ϕ

}

and the quantile functionϕα(u) � min

ϕ{ϕ : Pϕ(u) � α}

and formulate the first-stage problem

ϕα = minu∈U

[cT0 u+ ϕα(u)

],

uα = argminu∈U

[cT0 u+ ϕα(u)

].

(4.6)

Theorem 2. The two stage problem in the a posteriori formulation (4.6) is equivalent in thesense of definition to the two-stage problem in the a priori formulation (4.3).

Theorem 2 is proved in the Appendix.

Remark 3. The present paper considers a more general case of the two-stage problem (4.1)–(4.4)than in [4]. In particular, the matrix A2(X) and the vector a3(X) may depend on the randomvector X. Yet the scheme of proof based on the confidence method according to which the problemsof quantile optimization in the a priori and a posteriori formulations in the sense of definition proveto be equivalent to some generalized minimax problem remains the same. Then in turn equivalencebetween these minimax problems is established. Therefore, using the property of transitivity ofthe notion of equivalence we establish equivalence between the two-stage problems in the a prioriand a posteriori formulations. We recall that the method of dynamic programming is generallyinapplicable to the multistage problem of quantile optimization, but in a special case the a prioriformulation of the two-stage problem of quantile optimization turns out to be equivalent to thea posteriori formulation, that is, in this case the method of dynamic programming is applicable.

5. REDUCTION OF THE PROBLEM IN THE A POSTERIORI FORMULATIONTO THE PROBLEM OF MIXED INTEGER LINEAR PROGRAMMING

According to the confidence method [3], in the sense of definition problem (4.6) is equivalent tothe problem

ϕα = minu∈U

minS∈Fα

{cT0 u+ sup

x∈S[aT1 (x)u+Φ(u, x)]

},

(uα, Sα) = arg minu∈U,S∈Fα

{cT0 u+ sup


},

(5.1)

where Φ(u, x) is defined according to (4.5), S ∈ Fα is the confidence set, Fα � {S ∈ F : P(S) � α},and the probabilistic measure P corresponds to the discrete distribution of the vector X̃.

We fix the confidence set S ∈ Fα and consider the subproblem from (5.1):

ψ(S) = minu∈U

{cT0 u+ sup


}, (5.2)

where Φ(u, x) is determined from the solution problem (4.5).



According to [10], for subproblem (4.5) we set down its equivalent dual subproblem as

Φ(u, x) = maxv∈V

{(a3(x)−A2(x)u)

Tv},

where v ∈ IRs is the vector of dual variables and V is a convex polyhedron given by

V ={v : BTv � c1, vi � 0, i = 1, s

}.

Therefore, subproblem (5.2) comes to

ψ(S) = minu∈U

{cT0 u+ sup

x∈S

[aT1 (x)u+max

v∈V{(a3(x)−A2(x)u)

Tv}]}

. (5.3)

According to the duality theory [10], V is a bounded set, and since the function (a3(x)−A2(x)u)Tv

is linear in v for all x and u, its maximum reached at the vertices vj , j = 1, J , of the polyhedron V .Therefore, problem (5.3) is rearranged in

ψ(S) = minu∈U

{cT0 u+ sup

x∈S

[aT1 (x)u+ max

j=1,J

{(a3(x)−A2(x)u)

Tvj}]}

. (5.4)

Owing to the fact that after the measure discretization the random vector X̃ takes on only a finitenumber of values xk, k = 1,K , problem (5.4) comes to the problem

ψ(S) = minu∈U

{cT0 u+ max

r=1,Rmaxj=1,J

[aT1 (x

r)u+ (a3(xr)−A2(x

r)u)Tvj]}

, (5.5)

where the set S consists of the vectors xr, r = 1, R.

Since the function under maximum is linear in u and the maximum is found from the finitecollection of vectors {xr}Rr=1, {vi}Jj=1, the function within the braces is piecewise-linear and convexin u ∈ U .

Let now U be a convex polyhedron. Then, problem (5.5) becomes that of linear programming

ψ → minu∈U,ψ�0

(5.6)

under the linear constraints

cT0 u+ aT1 (xr)u+ (a3(x

r)−A2(xr)u)Tvj � ψ, r = 1, R, j = 1, J . (5.7)

The set S was fixed above, P(S) � α. In problem (5.1), we take the optimal set Sα and for thatintroduce the Boolean variables characterizing the membership of the points xk to the confidenceset S according to the rule

δk �{

1 if xk ∈ S0, otherwise.

Denote pk � P{X̃ = xk} = 1/K, k = 1,K.

Let known be γ > −∞ estimating from below the functions

γ � aT1 (xk)u+ (a3(x

k)−A2(xk)u)Tvj , k = 1,K, j = 1, J .

Consider the problem of mixed integer linear programming

ψ → minu∈U,δ1,...,δK ,ψ�0

(5.8)



under the constraints⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

cT0 u+ γ + δk[aT1 (x

k)u+ (a3(xk)−A2(x

k)u)Tvj − γ] � ψ, k = 1,K, j = 1, J

K∑

k=1

δkpk � α

δk ∈ {0, 1}, k = 1,K.

(5.9)

According to [6, Lemma 2], problem (5.8), (5.9) is equivalent to problem (4.6) in the sense ofdefinition. By integrating the aforementioned, the following theorem can be formulated.

Theorem 3. The two-stage problem in the a posteriori formulation (4.6) is equivalent in thesense of definition to the problem of mixed integer linear programming (5.8), (5.9).

Remark 4. The technique of reducing the problem of stochastic programming in the quantileterms to the problem of mixed integer linear programming which is used in this paper was presentedin [6] where consideration was given to the problem of quantile optimization with the discretedistribution of the random vectors. It was suggested to solve the resulting problem of mixedinteger linear programming with the use of the standard methods. In particular, problem (5.8)and (5.9) can be solved using the optimization packets such as IBM ILOG CPLEX. There existvarious devices to reduce enumeration at finding the optimal set in problem (5.1), in particular,with the use of the measure kernel [3]. In the case of discrete distribution, the measure kernel αcoincides with the convex envelope of all, except for the extreme, points from the distribution ofthe random vector X̃. Therefore, in the case of quasiconvexity in x̃ of the objective function itsuffices to enumerate only the extreme points of the set of all possible values. It deserves notingthat under the fixed δk problem (5.8) and (5.9) is that of linear programming. If just the randomvector X̃ was considered in the criterion instead of the matrix A2(X̃), then problem (5.1) wouldbelong to the class of the portfolio problems considered in many publications including [3].

The following theorem can be formulated with regard for the aforementioned and all the previousequivalent transitions.

Theorem 4. For the discrete distribution FK(x) generated on the basis of density p(x), the mul-tistage problem of stochastic programming in the a priori formulation like (2.5) is equivalent in thesense of definition to the problem of mixed integer programming (5.8) and (5.9).

Remark 5. For the multistage problem in the a priori formulation (2.5), solution of problem (5.8)and (5.9) is approximated. Yet, according to the Glivenko–Cantelli theorem [9], the sample functionof distribution F̂K(x̃) converges almost sure to the considered distribution function F (x) with thedensity p(x). Therefore, the solution of problem (5.8) and (5.9) also shows promise to convergeto the solution of problem (2.5). However, this question remains outside the scope of the presentpaper and needs a special examination.

6. EXAMPLE

Consider the two-stage problem in the a priori formulation (4.3). Let x = col(x1, x2); n = m =m1 = s = 2; u = col(u1, u2); c0 = col(3, 4); c1 = col(3, 1); a1(x) = ax, a3(x) = bx, where a = 2.5;

b = 1.5, A2(x) =

(x1 x1 + x20 x2

), B =

(2 35 1

)and let the random vector X have normal distribu-

tion N(0; I), where I is the covariance identity matrix. We give α = 0.9.

To solve problem (4.3), we generate K points xk, k = 1,K , according to the considered densityof normal distribution and solve problem (5.8) and (5.9). Let us analyze the resulting approximate



Results of numerical calculations

K ϕα(K) u1(K) u2(K)

10 2.3417 0.3262 0.7266100 2.2231 0.3191 0.74691000 1.8102 0.3159 0.7581

10 000 1.8026 0.3123 0.7608

solution depending on the volume of the sample K, where K = 10, 100, 1000, and 10 000. As theresult, we obtain solution of problem (4.3) for each value in K. The results of numerical calculationsare compiled in the table from which one can see that the solution of the approximate problem(5.8) and (5.9) stabilizes with increasing volume of the sample. Therefore, we can assume that thissolution stabilized about that of the two-stage problem (4.3) in the a priori formulation.

7. CONCLUSIONS

The paper formulated the multistage problem of quantile optimization in the a priori formula-tion and proved its equivalence to the two-stage problem of quantile optimization in the a posterioriformulation. The question of existence of solution of the problems under consideration was disre-garded in view of the definition of problem equivalence which allows for the case of no problemsolution in both formulations. The paper considered the scheme of measure discretization enablingone to reduce the multistage problem of stochastic programming in the a priori formulation to thetwo-stage problem. Owing to the results of [6], it is possible to reduce the two-stage problem inthe a priori formulation to that of mixed integer linear programming solved using the standardoptimization packets. Therefore, by solving the problem of mixed integer linear programming onecan determine an approximate solution of the multistage problem of stochastic programming witha quantile criterion.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research, project no. 11-07-00315-a and the Federal Goal-oriented Program “Scientific and Educational Personnel of InnovativeRussia” (activity 1.2.2, state contract no. 14.740.11.1128).

APPENDIX

Proof of Lemma. Let xk, k = 1,K , be an a posteriori sample corresponding to the density p(x).Let the random vector X̃ = col(X̃1, . . . , X̃N−1) have the distribution FK(x) with measures P{X̃ =xk} = 1/K, k = 1,K. Determine the distribution of the subvector X̃1. Since the original vector Xhas the density p(x), for any x the probability P{X = x} = 0. Let Xk, k = 1,K, be an a priorisample corresponding to the density p(x). Since P{X = x} = 0 for all x, we get that P{Xk =Xj} = 0 for all k �= j, that is, Xk �= Xj almost sure for all k �= j. Yet, the components Xk

i ,i = 1, N − 1, of the vector Xk also have the densities pi(xi). Therefore, P{Xk

i = Xji } = 0 for all

k �= j, i = 1, N − 1, which implies that Xki �= Xk

j almost sure for all k �= j, i = 1, N − 1.

Let F1K(x̃1) be the distribution function of the subvector X̃1 corresponding to these measures.Let x̃ be a realization of the random vector X̃, and x̃1, the realization of the random subvector X̃1.In compliance with the construction x̃1 = xk1 for some k = 1,K . But since among the subvectors xki ,k = 1,K, i = 1, N − 1, at least there are no two identical subvectors, the rest of the subvectors x̃i,i = 2, N − 1, of the realization x̃ coincide with xki , i = 2, N − 1, where k is the same number as at thefirst subvector. Therefore, we established the one-to-one correspondence x̃i = fi(x̃1), i = 2, N − 1,



between the subvector x̃1 and subvectors x̃i, i = 2, N − 1, which is valid for any realization x̃ ofthe vector X̃. Therefore, P{X̃i = fi(X̃1)} = 1, i = 2, N − 1, which proves the lemma.

Proof of Theorem 1. First we consider the three-stage problem of stochastic programmingin the a priori formulation with regard for discretization of the measure. According to (2.7) andlemma, we establish

Φ3(u, y(·), X̃) = cT0 u+ aT11(X̃1)u+ cT1 y1(u, X̃1)

+ aT12(X̃1, f2(X̃1))u+ cT2 y2(u, X̃1, f2(X̃1))

= cT0 u+ (aT11(X̃1) + aT12(X̃1, f2(X̃1)))u+ cT1 y1(u, X̃1) + cT2 y2(u, X̃1, f2(X̃1)).

By denoting

y1(u, X̃1) � col(y1(u, X̃1), y2(u, X̃1, f2(X̃1))),

a11(X̃1) � a11(X̃1) + a12(X̃1, f2(X̃1)),

cT1 = (cT1 , cT2 ),

we get

Φ3(u, y(·), X̃) = Φ2(u, y1(·), X̃1) � cT0 u+ aT11(X̃1)u+ cT1 y1(u, X̃1)

and similarly

Φ1(u, y1(·), X̃1) = A21(X̃1)u+B1y1(u, X̃1) � a31(X̃1),

Φ2(u, y2(·), X̃2) = A22(X̃1, f2(X̃1))u+B2y2(u, X̃1, f2(X̃1)) � a32(X̃1, f2(X̃1)).

Therefore, by introducing the vectors

Φ1(u, y1(·), X̃1) � col(Φ1(u, y1(·), X̃1),Φ2(u, y

2(·), X̃2)),

a31(X̃1) � col(a31(X̃1), a32(X̃1, f2(X̃1))),

y1(u, X̃1) � col(y1(u, X̃1), y2(u, X̃1, f2(X̃1))) = y1(u, X̃1)

and matrices

A21(X̃1) �(

A21(X̃1)

A22(X̃1, f2(X̃1))

),

B1 = (B1, B2),

constraints (2.7) are representable as

Φ1(u, y1(·), X̃1) � A21(X̃1)u+B1y1(u, X̃1) � a31(X̃1).

Consider the two-stage problem (2.5) with the probability function

Pϕ(u, y1(·)) = P{Φ2(u, y1(·), X̃1) � ϕ, Φ1(u, y1(·), X̃1) � a31(X̃1)

}. (A.1)

Since the values of the loss function and constraints in problem (2.5), (A.1) coincide with theloss function and constraints in problem (2.5) and (2.6), both problems are equivalent in the senseof definition. Reduction of the N -stage problem to a two-stage problem is proved by induction andcompletes the proof of Theorem 1.



Proof of Theorem 2. We prove Theorem 2 using the confidence method [3]. According to it,problems (4.3) and (4.6) are equivalent to the following generalized minimax problem representedunder the condition of existence of solutions of problem (4.3) and (4.6). Let S be the confidenceset, that is, P(S) � α. Let Fα � {S : P(S) � α}. Then, according to [3]

ϕα � minS∈Fα

minu∈U

miny(·)∈Y

γ(S, u, y(·)), (Sα , uα, yα(·))

� arg minS∈Fα,u∈U,y(·)∈Y

γ(S, u, y(·))(A.2)

under the constraints

0 � supx∈S

[a3(x)−By(u, x)−A2(x)u], (A.3)

where sup in (A.3) is understood in the row-wise sense and

γ(S, u, y(·)) � cT0 u+ supx∈S

[aT1 (x)u+ cT1 y(u, x)], (A.4)

and also

ϕα � minS∈Fα

minu∈U

γ(S, u), (Sα, uα) � arg minS∈Fα,u∈U

γ(S, u), (A.5)

where

γ(S, u) � cT0 u+ supx∈S

[aT1 (x)u+min

y∈Y


}]. (A.6)

We notice that here (ϕα, uα) is the optimal solution of problem (4.3) and (ϕα, uα) is the op-timal solution of problem (4.6), at that in the sense of definition problem (4.3) is equivalent toproblem (A.2) and problem (4.6) to problem (A.5). Therefore, it suffices to demonstrate [3] thatthe problems (A.2) and (A.5) are equivalent. To prove this, it suffices to demonstrate that theequalities

γ(S, u, yα(·)) = γ(S, u, yα(·)) = γ(S, u) (A.7)

where

yα(u, x) = argminy∈Y


}, x ∈ IRn, u ∈ IRm, (A.8)

are satisfied for any S ∈ Fα and u ∈ U . We notice that γ(S, u) = γ(S, u, yα(·)) according to (A.6)and (A.4) and now prove that γ(S, u, yα(·)) = γ(S, u, yα(·)). For that, under constraints (A.3) weconsider three cases of the problem

γ(S, u, yα(·)) � miny(·)∈Y

γ(S, u, y(·));

yα(·) � arg miny(·)∈Y

γ(S, u, y(·)), (A.9)

where (i) γ(S, u, yα(·)) equals ∞, (ii) γ(S, u) = ∞, and (iii) both values are less than ∞.

Case 1. Let γ(S, u, yα(·)) = ∞ and there be no y(·) ∈ Y such that constraints (A.3) are satisfied.To prove that γ(S, u) = ∞, we assume that it is not the case. Then, under fixed u and S for allx ∈ S there exists in problem (A.8) the optimal plan of the second stage yα(u, x). Therefore,yα(·) ∈ Y. At that, γ(S, u) < ∞ and the constraints in problem (A.8) are satisfied for all x ∈ S.



This means that constraints (A.3) are satisfied for yα(·). Consequently, yα(·) is the permissibleplan in problem (A.2) and at that γ(S, u, yα(·)) < ∞, which contradicts the above assumption.Therefore, γ(S, u) = ∞.

Case 2. Assume that γ(S, u) = ∞ and there exists a realization x ∈ S such that there is noy ∈ Y satisfying the constraints in problem (A.8). We prove that then γ(S, u, yα(·)) = ∞. Letthis be not the case. Then, there exists a measurable function y ∈ Y with values in Y such thatconstraints (A.3) are satisfied. Consequently, for this function y(·) the constraints 0 � a3(x) −By(u, x)−A2(x)u are satisfied for all x ∈ S. Therefore, for all x ∈ S there exists a plan of the secondstage y � y(u, x) ∈ Y , which contradicts the above assumption that γ(S, u) = ∞. Consequently,γ(S, u, yα(·)) = ∞.

Case 3. Let for the fixed u ∈ U and S ∈ Fα in problems (A.9) and (A.8) there be optimal plansyα(·), yα(·) for each x ∈ S. Then, the inequalities

0 � supx∈S

[a3(x)−Byα(u, x) −A2(x)u]

and

A2(x)u+Byα(u, x) � a3(x) for all x ∈ S

are satisfied. Therefore, yα(·) is a permissible but not necessarily optimal plan for problem (A.9),that is, γ(S, u, yα(·)) � γ(S, u, yα(·)).

Assume that γ(S, u, yα(·)) < γ(S, u, yα(·)). Then,

supx∈S

[aT1 (x)u+

{cT1 yα(u, x)|A2(x)u+Byα(u, x) � a3(x)

}]

< supx∈S

[aT1 (x)u+

{cT1 yα(u, x)|A2(x)u+Byα(u, x) � a3(x)

}],

which contradicts the fact that for each x ∈ IRn the plan yα(u, x) is optimal in the second-stageproblem (A.8). Therefore, γ(S, u, yα(·)) � γ(S, u, yα(·)). By uniting the relations obtained, weobtain that they are consistent only in the case of the equalities γ(S, u, yα(·)) = γ(S, u, yα(·)). Itwas shown above that γ(S, u, yα(·)) = γ(S, u). Therefore, (A.7) is satisfied, which proves Theorem 2.

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This paper was recommended for publication by A.V. Nazin, a member of the Editorial Board


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