Approximation Schemes for Linear Two-stage Problems of Sto- Sto-cl1astic Programming

In this section we consider a special class of stochastic programming problems, so-called two-stage problems, and we describe the realization of the sequential approximation method in this case. In 2.2.1 we formulate the problem and review its basic properties and in Section2.2.2we consider the special case with a discretely distributed random vector. Section 2.2.3 is devoted to estimates of the accuracy of approximation, which are followed in 2.2.4by the analysis of refining strategies. The special case of so-called simple recourse is discussed separately in 2.2.5.

2.2.1 Basic properties of linear two-stage problems.

The linear two-stage problem of stochastic programming is defined as follows:

minimize ItjJ(x)

=

^cTx

+ 10

^Q(x,e(w))p(dw)]

subject to Ax

=

b, z ~0,

(2.11)

where c E Rnl, b E Rml and A of dimension ml x nl are defined as in a common linear programming problem. The function Q(x,e(w)) that appears in the additional part of the objective in(2.11)is defined as the optimal value of another linear programming problem which has x as a parametE'r and involves random coefficientse(w)

=

(q(w), h(w),T(w)):

minimize

qT(w)z

subject to Wy

=

h(w) - T(w)x, (2.12)

y ~

o.

The linear programming problem (2.12) is called the second stage problem, or the recourse problem; it consists in finding the best recourse decision yE R~2, when the first stage decision xE R~l and random realization of the parameters q(w) ERⁿ2,h(w) ER^m2andT(w) of dimensionm2 xnl are already established.

The m2 Xn2 matrix W is deterministic.

Since the expected value of the minimum recourse costQ(x,e(w)) modifies the objective of the first-stage problem (2.11), the whole model (2.11)-(2.12) has a certain internal dyna.mical structure: whE'n looking for an optimal first

Approximation Techniques 39 stage decision x we have to t.ake into account not only the direct first stage costc^Tx but also the expected value of the future recourse cost. Ifthere is no feasible solution to (2.12) we assumeQ(x, e(w)) = +00,and this should also be considered at the first stage.

We are especially interested in stochastic programming problems with reo course because of their wide application to modeling decision problems which involve random data. Ifsome constraints, e.g. Tx= h, in a linear programming problem include random coefficients in T or h and we have to take the deci·

sion before knowing the realizationsT(w) and h(w) ofT and h, it is generally impossible to require that the equality

T(w)x = h(w) (2.13)

be satisfied for each realization of the stochastic constraint parameters. The problem with recourse is a way of overcoming these modeling difficulties; the recourse decisionymay be interpreted as a correction in(2.13),and the recourse cost Q(x,e(w))-as a penalty for discrepancy in (2.13).

In a more general model the matrixW in (2.12) could be random too, but for the ease of exposition we assume that it is deterministic; such a model is called the problem with fixed recourse. Most of the theory and computational methods have been developed for this class of linear two· stage problems.

Let us review briefly basic properties of the problem (2.11)-(2.12). The feasible set of (2.11) is the intersection of the set given by the first stage con·

straints

K1 ={xERnl :Ax=b,x~O}

and of the induced feasible set

K2 = {x ERnj :Q(x,e(w))

<

⁰⁰ with probability I}.

(2.14)

(2.15) While Kj is described explicitly and easy to handle, the induced set K₂ is defined implicitly and hard to express analytically. However, ifthe matrix W in (2.12) is such that {Wy: y ~ O} = R^m2 (i.e. the corrections Wy in (2.12) can cancel any error), we have K₂ = Rnj. Problems with such a property are called problems with complete recourse. In the special case of W = [I, - I] we speak about simple recourse. Although generally the induced feasible set K2

need not contain Kj we still have the following property.

(a) The setsKj ,K2 and K = Kj

n

K2 are convex and closed.

As far as the recourse costQ(x, e(w)) is concerned, many interesting the·

oretical results are available. First, by the theory of duality in linear program·

ming we know thatQ(x, e(w))

>

-00 (i.e. the second stage problem is bounded from below) if and only if one can find 11 ER^m2 such thatW^T'U ~

q(w).

Since the case of unboundedness is of no interest for us, we shall from now on assume that the above condition is satisfied for each realization of the random vec·

torq(w). Under this assumption the recourse function possesses the following properties.

40 Stochastic Optimization Problems (b) For any 6xed xEK and any q the function (h, T) -. Q( x, e= (q, h, T)) is

piecewise linear and convex.

(c) For any 6xedx^E K andanyhand T the function q -. Q(x, e= (q, h,T)) is piecewise linearandconcave.

(d) For any 6xed e= (q, h, T) the function x -.Q(x,e) isa convex piecewise linear function onK.

Under the additional condition that the random variablee(w)

=

(q(w), h(w), T(w)) has finite second moments we finally obtain the following result.

(e) The function Q(x)=

1

₀ Q(x,e(w))p(dw) is finite and convex inK.

A detailed discussion of properties of linear two-stage stochastic program·

ming problems can be found in [12] and [35].

Properties (a)-(e) are of fundamental importance for the concepts and methods discussed in this chapter and will be frequently used in subsequent sections. We also assume that we deal with the case of complete recourse (no induced constraints). Motivation for the later assumption is rather obvious:

with K2 =1= Rⁿ¹ it would be extremely difficult to ensure that solutions to approximate problems are in the induced feasible set of the original problem.

2.2.2 The two-stage problem with a discrete random vector

Let us consider in more detail properties of stochastic programming problem with recourse in case of a discretely distributed random vector

e

^attaining

values:

e

^{= (q1 ,h1,T1)} with probability^P1

>

0,

e

^{= (q2,h2,T2)} with probabilityP2

>

0, e L= (qL,hL,T^L)with probability PL

>

0, where

LPI

L ^{= 1.}

t=1

In this case the two-stage problem (2.11 )-(2.12)takes on the form

L

minimize [J(x) = c^Tx

+ L

peQ(x, eel]

£=1

subject to Ax= b

x ~ 0,

where Q(x, eel is the minimum objective value in the recourse problem mllllIlllze (qe)Ty

subject to Wy = he - Tex, y ~0,

(2.16)

(2.17)

(2.18)

(2.19)

Appro:zimation Techniques 41

e

^{= 1,2, ...IL. If}we denote by

ye(x), e

^{= 1,2, ... ,L,}the solutions to problems (2.19) at a given x, we can express the first stage objective as

~(x)

^{= c}

^T

^z

+

Lpe(l)T ye(x).L i=l

(2.20)

Of course, the solutionsye(x) depend onxin a rather involved way, so that the products (qe)T yi(x) are piecewise linear (ef. property (d) in 2.2.1). However, instead of considering (2.18)-(2.19)as a two-level problem, we can put together the first stage problem (2.18) and all realizations of the second stage problem (2.19) into a large linear programming model:

minimize

cTx+pe(ql)T yl +p2(l) Ty2 + ... pL(qL)TyL

subject to

Ax

=

^b

T:r+Wyl =

^h

^l

_{(2.21 )}

T 2

x

+W y2 =

^h

²

TLz +WyL =

^h

^L

Z ~ 0, yl _~ 0,y2 _~ 0 _'" yL ~O.

Problems (2.18)-(2.19) and (2.21) are equivalent in the sense that they have the same set of solutions, as the first stage decision vector x is concerned, and the optimal values ofyl,

y2, .. . , yL

in (2.21) are solutions to the realizations of the second stage problem (2.19) at the optimalx.

Smpming up, a two stage problem with a discretely distributed random vector

e

turns out to be equivalent to a large-scale linear programming prob·

lem, which can be solved by powerful linear programming techniques, which take account of its special dual block angula.r structure. These techniques are discussed in detail in chapter 5 of this volume (see also [13], [28] and [33]).

42 StochaBt£c Optimization ProblemB 2.2.3 Error estimates

Let us now investigate relations between a two-stage problem with an arbitrary distribution of the random parameter

e

and its approximation resulting from the discretization of

e.

Recall that, according to the ideas sketched in Section 2.1.2, the discretely distributed approximation

e

^to

e

is constructed for a given partitionSL = (B1 ,B2 , ••• ,BL) of the supportBof

e

as follows:

pa ^{= ell =}

^Pe, l= 1,2, ... ,L, (2.22) where

e, e, ... , e'

are conditional expectations of

e

^{in Be,}

et

=

E{e/e

E

Bd,

l= 1,2, ...,L (2.23)

and

Pe=

p{e

EBe}, l=I,2, ...,L, (2.24 ) (2.25)

L

LPe= 1.

e=1

We expect (2.23) to be a good choice, since the conditional expectations mini-mizes

Ell e- ^el1

² with respect to all discrete distributions corresponding to our partition

[UJ.

After replacing

e

in (2.11)-(2.12) by the discrete variable

e

^{we obtain}

an approximating problem of the form (2.21). Obviously, this problem is much easier to solve than the original one, but now we need estimates of errors caused by the approximation. Such estimates can be derived from general properties

(a)-(d) of two-stage problems, discussed in Section 2.2.1.

(2.26) l= 1,2, ... ,L.

Lower Bounds

Let us assume that all the subsets Be, l = 1,2, ... ,L, are convex and the function

Q(z, e)

in (2.11) is convex in

e

for each z. By property (b), the latter condition is satisfiedifqin (2.12) is deterministic, and only

T(w)

and

h(w)

vary randomly.

Under this assumption, with

et

and perepresenting conditional exp ecta-tions and probabilities defined by (2.23)-(2.24)' for each blockBlfrom Jensen's inequality (see IU])we obtain

L e

Q(z,e(w))p(dw) ~ peQ(z, ell,

Im Dokument Stochastic quasigradient methods. Numerical techniques for stochastic optimization (Seite 56-60)