A Gaussian Upper Bound for Gaussian Multi-Stage ... - Semantic Scholar

A Gaussian Upper Bound for Gaussian Multi-Stage Stochastic Linear Programs Eithan Schweitzer and Mordecai Avriel

Faculty of Industrial Engineering and Management Technion - Israel Institute of Technology Haifa 32000, Israel Operations Research Statistics and Economics Mimeograph Series No. 408 May 1995 Abstract This paper deals with two-stage and multi-stage stochastic programs in which the right-hand sides of the constraints are Gaussian random variables. Such problems are of interest since the use of Gaussian estimators of random variables is widespread. We introduce algorithms to nd upper bounds on the optimal value of two-stage and multi-stage stochastic (minimization) programs with Gaussian right-hand sides. The upper bounds are obtained by solving deterministic mathematical programming problems with dimensions that do not depend on the sample space size. The algorithm for the two-stage problem involves the solution of a deterministic linear program and a simple semide nite program. The algorithm for the multi-stage problem involves the solution of a quadratically constrained convex programming problem.

Key words: Stochastic Programming, Semide nite Programming, Multi-

Stage Linear Programs.

1 Introduction Multi-stage stochastic linear programs, introduced by Dantzig [7], [8] and Beale [1] are mathematical programming problems in which some of the parameters (usually those of future stages) are not known with certainty when decisions with regard to the problem variables must be taken. If the number of future scenarios is nite, an exact deterministic equivalent of the stochastic program can be formulated and solved by conventional methods. In a practical multi-period planning problem, the size of the resulting linear program, representing the deterministic equivalent, is often so large that obtaining an exact solution is unrealistic. As the number of future scenarios approaches in nity, solution of the stochastic program can be obtained by approximation schemes that bound the optimal value of the program. In recent works sampling techniques were proposed to approximate the deterministic equivalent problem, see e.g., Infanger [18]. However, when the number of stages is greater than two, the sampling approach may also become inadequate, because of the involved structure of the probability space that has to be taken into account. This paper deals with two-stage and multi-stage stochastic programs in which the right-hand sides of the constraints are Gaussian random variables. Such problems are of interest since the use of Gaussian estimators of random variables is widespread. We introduce algorithms to nd upper bounds on the optimal value of two-stage and multi-stage stochastic (minimization) programs with Gaussian right-hand sides. The upper bounds are obtained by solving deterministic mathematical programming problems with dimensions that do not depend on the sample space size. The algorithm for the two-stage problem involves the solution of a deterministic linear program and a simple semide nite program. The algorithm for the multi-stage problem involves the solution of a quadratically constrained convex programming problem. Since the dimensions of the deterministic linear and quadratically constrained convex programmming problems, that have to be solved for the computation of the upper bounds, do not depend on any probability space, the programs are small, relative to the possible size of the stochastic programming problem. Therefore, a small eort is needed for the computation of these upper bounds.

1

Although the upper bound is not always tight, as will be shown in Section 8, its tightness can be estimated. Upper bounds, depending on the probability space, were derived by Birge [4], Birge and Wallace [5], Birge and Wets [6], Huang, Ziemba and Ben-Tal [16], Madansky [21] and Wallace and Yan [27]. The upper bounds obtained by the algorithms proposed in this paper, can be calculated, in the case of Gaussian right-hand-sides, faster than the bounds found in the above works, as explained at the end of Section 2 In Section 2, the Gaussian multi-stage stochastic linear program, together with the structure of the associated probability space, is formulated and the de nitions of strong and weak solutions are introduced. Gaussian solutions and upper bounds are de ned in Section 3. Section 4 deals with nonnegativity constraints for Gaussian solutions and in Sections 5 and 6, ecient algorithms for calculating upper bounds for Gaussian stochastic two-stage and multistage linear programs are respectively developed. A discussion of the relative error of the Gaussian upper bound is given in Section 7. Section 8 shows some numerical results. In Section 9, some more applications and extensions relevant to Gaussian upper bounds are discussed.

2 Problem Formulation In order to formulate a multi-stage stochastic linear problem, the probability space of the uncertain parameters is introduced rst. Let ( ; FT ; P) be a probability space. Let fFt; t = 1; : : :; T g be a ltration on such that F1 = f;; g; where t = 1; : : :; T is the stage number. Let fbt; t = 1; : : :; T g be a sequence of mt-dimensional random vectors adapted to fFt; t = 1; : : :; T g; where mt; t = 1; : : :; T; are constants. Each bt is the right-hand-side vector of the multi-stage stochastic linear problem at stage t. Since b1 2 F1 = f;; g; it is a deterministic vector and de nes the certain parameters of the present. The random vectors b2; : : :; bT de ne the uncertain parameters of the future. De ne ( ; FT ; P) to be the probability space associated with the multi-stage stochastic linear problem. In a general formulation of multi-stage stochastic linear problems, the objective is to nd a deterministic decision (or strategy) x1 2 F1 and a 2

stochastic \plan" xt 2 Ft for each period t = 2; : : :; T: The stochastic process fxt ; t = 1 : : :; T g should optimize the expected value of a linear function PTt=1 c0txt ; where ct 2 Rnt and c0t is the transpose of ct. The vectors fxt; t = 1 : : :; T g are real valued nt-dimensional nonnegative vectors where nt; t = 1 : : :; T are constants. The stochastic process fxt ; t = 2; : : :; T g must satisfy the constraints: A1 x1 = b1 (2.1) ?Bt?1 xt?1 + Atxt = bt; t = 2; : : :; T (2.2) where At 2 Rmt nt and Bt?1 2 Rmt nt?1 ; t = 1; : : :; T . The general structure of the multi-stage stochastic linear program with random right-hand side will then be, (MSSLP) min z = c01 x1 + E(c02x2 + : : :: : : + c0T xT ) s:t: A1 x1 = b1 ?B1 x1 + A2x2 = b2 .. ... ... (2:3) . ?BT ?1xT ?1 + AT xT = bT x1 ; x2; : : :; xT  0 xt 2 Ft; t = 1; : : :T: Since fct; t = 1; : : :; T g are deterministic, the random vectors fxt; t = 1; : : :; T g are adapted to the natural ltration of fbt; t = 1; : : :; T g: The requirement of xt to be adapted is also known as non-anticipativity. In the formulation of (2.3), it means that for each realization of fbs ; s = 1; : : :; T g, the value of xt is a (measurable) function of solely fbs ; s = 1; : : :; tg but not fbs; s = t + 1; : : :; T g: When the stochastic process fbt; t = 1; : : :; T g is a multi-dimensional Gaussian process, problem (MSSLP) is called a Gaussian multi-stage stochastic linear program. The solution of (MSSLP) can be divided into two parts. The rst part is the \here-and-now" decision x1 and the second part is the \wait-and-see" solution xt; t = 2; : : :; T which is a plan for the future. If this solution exists, it de nes a unique z  , the optimal value of the objective function. This solution is called a strong solution and is de ned as follows: 3

De nition 1 A of (MSSLP) on a given probability space ( ; FT ; P) is a stochastic process X = fxt ; t = 1; : : :; T g with the following strong solution

properties:

(i) X is adapted to the ltration fFtg. (ii) x1 satis es (2.1) and x1  0. (iii) X satis es (2.2) and is nonnegative P-a.s. Computing a strong solution for multi-stage stochastic linear programs is the subject of a large number of research papers, see e.g., Birge [3], Birge and Wallace [5], Dantzig and Infanger [9], Entriken and Infanger [11], Gassman [12], Higle and Sen [13], Infanger [17], Rockafellar and Wets [24], Ruszczynski [25] and Wets [28]. Unfortunately, when the number of possible realizations of the uncertain parameters is large, ecient numerical solutions have not yet been found. Another solution, that has the same optimal value z  but is not a \wait-and-see" solution, exists. This solution is called a weak solution and is de ned as follows:

De nition 2 A where

weak solution

of (MSSLP) is a triple X , ( ; FT ; P), fFtg,

(i) ( ; FT ; P) is a probability space and fFtgTt=1 is a ltration with F1 = f;; g: (ii) X = fxt ; t = 1; : : :; T g is a real value process adapted to fFtg. (iii) x1 satis es (2.1) and x1  0. (iv) X satis es (2.2) in distribution and is nonnegative P-a.s. The main dierence between the strong solution and the weak solution becomes clear by their alternative de nitions:

For the strong solution: Given a probability space ( ; FT ; P); a ltration fFt; t = 1; : : :; T g with F1 = f;; g and a stochastic process fbt; t = 1; : : :; T g; determine a stochastic process X that satis es (i)-(iii) of Definition 1.

4

For the weak solution: Given a joint distribution of b = (b1; b2; : : :; bT ); nd a probability space ( 0; FT0 ; P0); a stochastic process b0 = (b01; b02; : : :; b0T ) on it that has the same distribution as b, a ltration fFt0; t = 1; : : :; T g on 0 and a stochastic process X that is a strong solution of (MSSLP) given ( 0 ; FT0 ; P0); fFt0 g and fb0t 2 Ft0; t = 1; : : :; T g:

For convenience de ne

0 BB A1 B ?B1 A2 A=B BB ... B@

...

?BT ?1 AT

0 1 BB b1 CC B b2 CC b=B BB .. CC B@ . CA

1 CC CC CC CA

bT

0 1 BB c1 CC B c2 CC c=B BB .. CC : B@ . CA

(2:4)

cT

Then problem (MSSLP) can be written as the stochastic linear program (SP)

min E(c0 x) s:t: Ax = b x0 xt 2 Ft

(2:5)

where A 2 Rmn is assumed to be of rank m, PTt=1 mt = m and PTt=1 nt = n: Note that the format of (SP) is more general than (MSSLP). For a multi-stage stochastic linear program de ned in the format of (SP), additional information about the probability space and the ltration (if needed) must be given. The algorithms presented in this paper require the solution of deterministic mathematical programming problems with dimensions that depend only on m and m, and do not depend on the size of the probability space. The computational eort required by the proposed algorithms does not depend on the number (in nitely many in the Gaussian case) of possible realizations of the future. The computational eort required by other algorithms, such as those referred to in Section 1, also depend, in one way or another, on the (practically large) size of the sample space. The rst rough estimation of the optimal value of a multistage stochastic linear program can be eciently calculated by the proposed algorithms. 5

3 Gaussian Solutions Consider the stochastic linear program (SP), as was de ned in (2.5). Suppose that b is a random vector from a multivariate normal distribution. Let b be its expectation vector and let  be its covariance matrix, i.e., bi = E(bi );

i;j = Cov(bi; bj ):

(3:1)

Suppose that A and c are deterministic. For the existence of a solution for (SP), when the right hand side is Gaussian, suppose that rank(A)=m and that ft j 9x 2 Rn+ ; Ax = tg = Rm (3:2) i.e., Rm is spanned by nonnegative linear combinations of the columns of A. In the multi-stage case, assumption (3.2) should not involve the deterministic part (A1; x1) of the problem. Problem (SP), with the nonnegativity constraints, has no feasible Gaussian solution since x  0 should occur almost surely. However, there is an , 0 <  < 1; for which there exists a feasible, with respect to the constraints Ax = b and xt 2 Ft; Gaussian solution that satis es P(x  0)  : For that case, consider the problem, (SGP)

min s:t:

E(c0x)

Ax = b P(x  0)   xt 2 F t

(3:3)

where 0 <  < 1 and is as close to 1 as possible, b  Nm(b; ) and A 2 Rmn with rank(A)=m. Let x be a feasible solution of (SGP) and de ne

xi = E(xi );

i;j = Cov(xi ; xj );

i2 = i;i:

(3:4)

Proposition 1 If (SP) has a feasible solution then (SGP) has a feasible solution.

2

Proof. Take  = 1:

6

De nition 3 The

de ned by

Gaussian

(SGPG)

-solution of (SP) is a feasible solution of (SGP)

E(c0x)

min s:t:

Ax = b P(x  0)   xt 2 Ft

(3:5)

x is a Gaussian r :v : Proposition 2 If for some ~ 2 (0; 1); (SGP) has a Gaussian -solution, then it has a Gaussian -solution for each  2 (0; ~ ]: Proof. Let x~  Nn (; ) be a Gaussian -solution of (SGP), then Ax~ =

b; xt 2 Ft and P(~x  0)  ~ : Let 0 <  < ~; then P(~x  0)  ; hence x~ is a feasible Gaussian -solution of (SGP) for each  2 (0; ~ ]: Therefore, (SGP) has a Gaussian -solution for each  2 (0; ~ ]: 2

Proposition 3 If (SGP) is solvable then the optimal value of (SGPG) is an

upper bound for the optimal value of (SGP).

Proof. To get a Gaussian solution for (SGP), it is solved with an additional

constraint \x is Gaussian" as in De nition 3. Since (SGP) is feasible and is a minimization problem, its solution with an additional constraint is an upper bound for its optimal value. 2 From the propositions above, it can be seen that an upper bound for the optimal value of (SP) can usually be achieved by a Gaussian solution of (SGP). However, in multi-stage stochastic problems, the \here and now" solution x1 is important because it gives the immediate decision. Consider (SGP) to be a multi-stage stochastic linear program by applying (2.4). The Gaussian solution of the multi-stage (SGP) includes the decision vector x1. The question is whether this decision is also feasible for (SP) or not. The following theorem answers this question in the armative.

Theorem 1 Suppose that assumption (3.2) holds.

Let X~ = fx~t ; t = 1; : : :; T g be a Gaussian solution of (MSSLP) with Gaussian right-hand-side and deterministic coecients. Then there exists a feasible strong solution for (MSSLP), X = fxt ; t = 1; : : :; T g; where x1 = x~1 :

7

Proof. Let

s = B1 x1 ; s~ = B1x~1: (3:6) X~ is a Gaussian solution of the Gaussian (MSSLP), therefore it satis es A1x~1 = b1;

x~1  0

(3:7)

among the other conditions. Every strong solution, X , of the Gaussian (MSSLP) satis es, for each ! 2 ; A1 x 1 = b 1 ; x 1  0 A2x2 (!) = b2(!) + s (3:8) Atxt(!) = bt(!) + Bt?1 xt?1 (!) xt (!)  0; xt 2 Ft: Since condition (3.2) holds, the system

A1x~1 = b1; x~1  0 A2 x2 (!) = b2(!) + s~ Atxt(!) = bt(!) + Bt?1 xt?1 (!) xt (!)  0; xt 2 Ft: has a solution X = fxt; t = 1; : : :; T g; where x1 = x~1 :

(3:9)

2

4 Linear Conditions for Positive Orthant Probability The stochastic linear program (SGP) includes the probabilistic constraint that x should be in the positive orthant with probability which is at least  Notice that this constraint is not a chance constraint in the usual sense where x is deterministic and satis es a probabilistic constraint but in the sense of stochastic programming. That constraint can be written as

Pf! j x(!)  0g   and that means that the random vector x is nonnegative with probability greater than or equal to . It was shown in Section 3 that a Gaussian solution 8

of (SGP) gives an upper bound for its optimal solution. In order to describe an ecient algorithm for nding a Gaussian solution, a discussion of positive orthant probabilities for multivariate Gaussian random variables is needed. This section will present linear relations between the expected value and the standard deviation vectors of multivariate Gaussian random variables that ensure that the positive orthant probability will be greater than or equal to a given .

Proposition 4 Let x  Nn(; D) where D is a nonsingular diagonal matrix with Di;i = i2: A sucient condition for P(x  0)   is that i  i?1 ( n1 ); i = 1; : : :; n

(4:1)

where  is the standard normal distribution function. Proof.

P(x  0) = =

n  Y i  P(xi  0) = P xi ? i  ? i i i=1 i=1   n n Y Y i n Y

  i i=1

 =

i=1

 n1 :

Since  is monotone and nondecreasing, (4.2) implies that i  ?1 ( n1 ); i or i  i?1 ( n1 ):

(4.2)

(4:3) (4:4)

2

Proposition 5 Let x  Nn(; ) where  is symmetric and positive de nite. A sucient condition for P(x  0)   is that q ?2 i  i n (); i = 1; : : :; n

(4:5)

where i2 = i;i and 2n is the Chi-square distribution function with n degrees of freedom.

9

Proof. For each  such that

fx j (x ? )0?1(x ? )  g  Rn+;

(4:6)

i.e., a translation of the ellipsoid fx j x0?1 x  g to the positive orthant,

P(x  0)  P((x ? )0?1(x ? )  ):

(4:7)

Since (x ? )0 ?1 (x ? ) has the Chi-square distribution with n degrees of freedom, P((x ? )0?1(x ? )  ) = 2n ( ): (4:8) Therefore, whenever

= ?n 2 (); (4:9) the vector x is inside the ellipsoid fx j (x ? )0 ?1 (x ? )  g with probability . Thus, if  satis es (4.6) with given by (4.9), then P(x  0)  : The minimal translation is when all the hyperplanes fx 2 Rn j xi = 0g; i = 1; : : :; n are tangents to the ellipsoid, which is in the positive orthant. The elements of the vector  which is the minimal translation are (CPi)

i = ? min xi s:t: x0 ?1 x  :

(4:10)

Let i be the vector on the i-th axis with unit norm. Using the Lagrange multipliers technique, the rst order necessary and sucient optimality conditions for the convex programming problem (CPi), solved separately for each i = 1; : : :; n; are

i + 2?1x = 0 (x0?1 x ? ) = 0: From (4.11) it follows that  6= 0; therefore x = ? 21 i : Substituting into (4.12) gives that  1 2 0 2 i i = : 10

(4.11) (4.12) (4:13) (4:14)

s

Hence

1 = 2 i;i

and

s

i : i;i So the optimal values of the elements of  are x = ?

q

i = ?xi = i;i = ip :

(4:15) (4:16) (4:17)

From (4.17) and (4.9), sucient conditions for the positive orthant probability of multivariate Nn (; ) vector to be greater than or equal to  are

q

i  i ?n 2 ();

i = 1; : : :; n:

(4:18)

2

5 A Gaussian Upper Bound for the Two-Stage Problem In this section, an ecient algorithm for calculating an upper bound for the optimal value and a Gaussian solution for (SGP) is will be introduced. The upper bound will be a bound for the optimal of (SGP) when (SGP) is a two-stage problem, i.e., when

1 0 A 0 A=B @ 1 CA : ?B1 A2

(5:1)

This upper bound, which is the objective function value of (SGP) calculated at its Gaussian -solution, will be called the Gaussian upper bound of (SGP). For the rest of this section, let A be de ned by (5.1). For a multi-stage problem, the same algorithm can be used to produce an upper bound for the optimal value of (SGP) given by a weak solution. In order to nd a Gaussian solution of (SGP), it is sucient to nd the parameters x and  of the Gaussian solution. For that, de ne the expected

11

value problem

min c0x (5:2) s:t: Ax = b x   where  = (1; : : :; n) consists of the square roots of the elements on the diagonal of  and is calculated by solving another problem, (CP), described below, before solving (EP). The parameter is a positive number that satis es (EP)

q

0 <  ?n 2 ():

(5:3)

The choice of should be such that the positive orthant probability will be greater or equal to the required . In Section 4 it was shown that q than ? 2

= r (); where r is theq rank of , is a proper choice. However, in most cases, the choice of < ?r 2 () is good enough. The choice of , that satis es (5.3) and that provides the required positive orthant probability, can be done by trial and error. From numerical experience, it is recommended to start with = ?1 ( n1 ) (see Proposition 4) or with a smaller . If the positive orthant probability constraint is not satis ed by this , then should be increased. The numerical experiments show that the desired is found after a very small number of such iterations. The matrix , from where the vector  is derived, should satisfy the equation AA0 =  and should be a covariance matrix. When the matrix  is all zeros, the vector  is also a zero vector and the optimal value of the linear program (JLB) min c0x (5:4) s:t: Ax = b x  0 is a lower bound of the solution of (SP) by Jensen's inequality (see Madansky [21]). Let x be the optimal solution of (JLB). If it is possible to nd a covariance 2 matrix  that also satis es i;i  x2i ; then the upper bound and the lower bound on (SGP) will be the same and will be equal to the optimal value

12

of (SGP). Therefore, it is desired to nd the covariance matrix  from the semide nite programming problem

  2 min Pi ri i2 ? x2i

(5:5) AA0 =  0 where   0 denotes that  is positive semide nite and the ri are \penalty" coecients. The choice of the penalty coecients will be such that if ci , the i-th coecient in the objective function of (EP), is large, then the associated i will be as small as possible. Therefore, the penalties will be chosen to be ri = maxfci; 0g: If all the coecients of the objective function of (EP) are nonpositive, it is not desired to choose the penalty coecients to be equal to zero. Since a covariance matrix with minimal variances is required, it is possible to set all penalty coecients to 1. From the above, the covariance matrix  can be calculated by solving the s:t:

covariance problem

min tr(C ) (5:6) s:t: AA0 =  0 where C = diag (c+1 ; : : :; c+n ) or C = I if all the ci are nonpositive, and tr(
) denotes the trace of a matrix. The linear program (EP) is deterministic and its size does not depend on any probability space, so it is not as large as the stochastic problem. However, the vector  must be calculated rst by solving the semide nite program (CP). Fortunately, Vanderbei and Yang [26] have shown that (CP) has an explicit solution. Let  = QFQ0 (5:7) where Q is the orthogonal matrix obtained from the Q-R factorization of (A0 j 0) 2 Rnn ; i.e., (CP)

QR = (A0 j 0);

0 1 0 P 0C R=B @ A 0 0

13

(5:8)

and P is a lower triangular m  m matrix. Let

G = P ?1 P ?10 and let

(5:9)

0 1 H H12 C H = Q0CQ = B @ 110 A

(5:10) H12 H22 where H11 2 Rmm ; H12 2 Rm(n?m) and H22 2 R(n?m)(n?m) : Then, by [26], 0 1 ?1 ? GH H G 12 22 CA : (5:11) F =B @ ? 1 ? 1 ? 1 0 0 0 ?(H12H22 ) G (H12H22 ) GH12H22 When the diagonal matrix C does not have full rank, H22?1 may not exist. Suppose that all the nonzero elements of c are the last r elements of the diagonal of C . Partitioning C into

1 0 0 0 C=B @ ~ CA 0 C

(5:12)

where C~ has full rank and is a diagonal matrix and rank(C~ ) = r < n, shows that H22 = (Q0CQ)22 may not be invertible. Let

0 1 0 C^ = B @ ~ 1 CA

(5:13)

C2

where C~ 21 2 Rrr ; then C^ C^ 0 = C and

H = Q0CQ = Q0C^(Q0 C^)0 = uu0 where

0 1 w u = Q0C^ = B A; @ C

w 2 Rmr ; v 2 R(n?m)r :

(5:15)

1 0 1 0 0 0 ww wv C B H11 H12 C H = uu0 = B A=@ 0 A @ 0 0

(5:16)

v

Hence,

(5:14)

vw

vv

14

H12 H22

and



0

(5:17) H12H22?1 = wv 0(vv 0)?1 = w (vv 0)?1 v : Also, (vv 0)?1 must satisfy (vv 0)?1 vv 0v = v: Therefore, whenever v 0 v has full rank (and is invertible), (vv 0)?1v exists and is equal to v (v 0v )?1: Equation (5.17) shows that the product H12H22?1 may exist even if H22 is not invertible. Therefore, solving (CP), in order to obtain the best possible , is easy and that  can be used in the solution of the linear program (EP) which gives the Gaussian upper bound of (SGP). The value calculated by the above algorithm can usually be used as an upper bound for (SP). However, it is obtained after extending the feasible set of the problem by changing the constraint x  0 P ? a:s: to P(x  0)  : Thus, it may happen that it will not be an upper bound for (SP) (only for (SGP)). That can be easily corrected in the following way: Let x = (x1; x2(! )) be the Gaussian -solution of (SP). De ne

? = f! 2 j x2 (! ) 62 Rn+2 g; and let

f

= c 0 x1 + 1

Z

c x2(!) dP(!):

n ? 2 0

(5:18) (5:19)

The following proposition gives the relation between this new value, f , and the actual optimal value of (SP).

Proposition 6 Assume that the feasible set fx1 j A1x1 = b1; x1  0g of the

rst stage is bounded. Then, the actual optimal value, f  , of the two stage problem (SP) satis es the estimate

p f   f + k 1 ? ;

(5:20)

where k is some constant (that can be extracted from the proof). Proof. Let x1 = x1 and

8 > < x (!); x2 (!)  0 x2 (!) = > 2 : argminu0fkuk j ?B1x1 + A2u = b2(!)g; x2(!) 6 0: 15

(5:21)

Then, using Assumption (3.2),

kx2(!)k  k1(kb2(!)k + kB1x1k) P ? a:s: on ?; or

kx2(!)k  k2 + k3kb2(!)k P ? a:s: on ? : The constructed (x1; x2(! )) is feasible to (SP) and

(5:22) (5:23)

0 f   c01x1 + E Z (c2x2) Z 0 0 = c 1 x1 + c x (!) dP(!) + ? c02x2 dP(!) ? 2 2

=f+

Z

?

n



c02x2 dP(!)

 f + kc2k  f + kc2k

Z

Z ?

?

kx2(!)k dP(!) (k2 + k3 kb2(! )k) dP(! )

= f + k2kc2kP( ? ) + k3kc2k

Z

?

Z

(by (5.23))

kb2(!)k dP(!) kb2(!)k2 dP(!)

 12 Z

 f + k2kc2k(1 ? ) + k2kc2k ?

 f + k2kc2k(1 ? ) + k2kc2k(1 ? ) 21 (tr + kbk2) 21 : p Since 1 ?   (1 ? ); the proposition is proved.

?

dP(!)

 21

(5.24)

2 In Section 9 it will be shown how changing the objective function of (CP) can allow a variety of stochastic programming problems. One example will be a stochastic programming problem that minimizes the variance and that can be bounded by applying the algorithm that was presented in this section.

6 A Gaussian Upper Bound for the Multi-Stage Problem In the case of multi-stage stochastic linear programs, the Gaussian upper bound calculated by the algorithm of Section 5 is an upper bound for the optimal value of (SGP) given by a weak solution. Although that upper bound is usually also an upper bound for the actual optimal value, it may happen that it will be less than the actual optimal value. This can happen because when 16

passing from strong solutions to weak solutions, the set of feasible solutions is extended and the optimal value may be decreased. For this reason, a more accurate algorithm is developed in this section. It requires more computational eort than the one of Section 5, since has to solve a single deterministic quadratically constrained convex programming problem. The size of that quadratically constrained program does not depend on any probability space. Let Dt;s 2 Rnt ms ; t = 1; : : :; T; s = 1; : : :; t; be a set of matrices. De ne

xt =

t X s=1

Dt;s bs;

t = 1; : : :; T:

(6:1)

To simplify notations, de ne

0 1 D 0 1 ; 1 B CC B B CC D2;1 D2;2 D=B B CC : .. ... B . B CA @ DT;1 DT;2


DT;T

(6:2)

Then, x = Db: Since b  Nm (b; );

x  Nn (Db; DD0); (6:3) and bt 2 Ft implies that xt 2 Ft : Therefore, in order to nd a Gaussian solution for (SGP), which is the Gaussian upper bound for (SGP), it is enough to nd a block lower diagonal matrix D, such that x = Db will satisfy the constraints of (SGP). The constraint A1 x1 = b1 is satis ed if and only if A1 D1;1b1 = b1:

(6:4)

The constraint ?B1 x1 + A2 x2 = b2 P ? a:s: is satis ed if and only if

? B1D1;1b1 + A2D2;1b1 + A2D2;2b2 = b2 8! 2 ;

(6:5)

that holds if and only if

A2 D2;2 = Im2 ; ?B1 D1;1b1 + A2D2;1b1 = 0: 17

(6.6) (6.7)

For t = 2; : : :; T; the constraint ?Bt?1 xt?1 + At xt = bt P ? a:s: is satis ed if and only if

? Bt?1

t?1 X s=1

Dt?1;s b2 + At

t X s=1

Dt;sbs = bt 8! 2 ;

(6:8)

that holds if and only if

At Dt;t = Imt ; ?Bt?1 Dt?1;s + AtDt;s = 0; s = 2; : : :; t ? 1; ?Bt?1 Dt?1;1b1 + AtDt;1b1 = 0:

(6.9) (6.10) (6.11)

From Section 4, a sucient condition for P(x  0)   is

Db  diag(DD0);

(6:12)

where the parameter is chosen the same way as was explained in Section 5. Finally t T T X X X (6:13) c0tDt;sbs: E( c0txt) = t=1 s=1

t=1

Therefore, the Gaussian upper bound for (SGP) can be calculated by solving the following program

P P min Tt=1 ts=1 c0t Dt;sbs s:t: A1 D1;1b1 = b1 AtDt;t = Imt ; t = 2; : : :; T ?Bt?1 Dt?1;s + AtDt;s = 0; s = 2; : : :; t ? 1 ?Bt?1 Dt?1;1b1 + AtDt;1b1 = 0 Db  diag(DD0);

(6:14)

that can be written, using matrix notations, as (CQP )

min s:t:

c0Db ADIb1 = Ib1 Db  diag(DD0);

18

(6:15)

where

0 BB b1 BB 0 Im2 B Ib1 = B BB 0 Im 3 BB .. ... B@ .

1 CC CC CC CC : CC CA

(6:16)

0 ImT Since the covariance matrix  is positive semide nite, (6.12) is a set of convex quadratic constraints with respect to the entries of D. All other constraints of (CQP) are linear constraints. Therefore, (CQP) is a convex quadratic programming problem.

7 Error Estimation for the Upper Bound As was shown in Sections 5 and 6, the calculation of the Gaussian upper bound is relatively easy. However, since the calculation, or even the estimation, of the optimal value of the multi-stage stochastic linear program is dicult, it is also hard to see if the upper bound is tight. Madansky [21] has shown, by Jensen's inequality, that the optimal value of the linear program (JLB) is a lower bound for the optimal value of the stochastic linear program with a random right-hand-side. Let b = (b1; b2; : : :; bT ) be the random vector comprised of all the righthand-side vectors of the multi-stage stochastic linear program (MSSLP), where bt is Ft-measurable. Let C (b) be the optimal value of (MSSLP) when the right-hand-side is b. Let G(b) be the Gaussian upper bound of the Gaussian (MSSLP). Hence, whenever b is Gaussian,

C (E(b))  C (b)  G(b) (7:1) by the results of Section 5 and by Jensen's inequality. If the distance between G(b) and C (E(b)) is small, then they give a good estimate of C (b). If the distance is large, it is not possible to determine which of the bounds is closer to the optimal value. However, by Jensen's inequality it can be shown that C (E(b))  C (E(bjF2))  : : :  C (E(bjFT ?1))  C (b)  G(b): (7:2) 19

The complexity of the calculation of C (E(bjFt)) is growing rapidly as t is increasing and the choice of t for solving the t-stage stochastic linear program depends on the computational capabilities of the solver. As the chosen t becomes larger, the interval in which the optimal value is bounded will become smaller. The tightness of the Gaussian upper bound depends on the parameters of the stochastic linear program, (A; c; b; ): There are cases where the gap between Jensen's lower bound and the Gaussian upper bound of the stochastic linear program can be large. As an example of a Gaussian stochastic linear problem with a large gap between Jensen's lower bound and the Gaussian upper bound, consider the following problem min c(x1 + x2) (7:3) s:t: x1 ? x2 = b; b  N (0; 1) x1; x2  0: When c > 0 it is easy to see that Jensen's lower bound is 0. For positive orthant probability greater than 0.999 it is sucient to take = 3 and to obtain the Gaussian upper bound, 3c, with

00 1 0 1 B@ x1 CA  N2 B@ B@ 1:5 CA ; 0:25 ?0:25 ?0:25 0:25 1:5 x2

!1 CA :

(7:4)

In this case the error is bounded by 3c. The dierence between the lower bound and the upper bound increases with c although the optimal solution and the Gaussian solution do not change. By xing c and changing the variance of b from 1 to v , the covariance matrix of the Gaussian solution will be multiplied by v and the right-handside of the lower bound constraints for x1 and x2 will be multiplied by pv. The dierence between the lower bound and the upper bound will increase p with v . Multiplying the coecients 1 and ?1 in the constraint by a is the same as dividing the variance of b by a2 . In that case, the dierence between the lower bound and the upper bound is changed by jaj?1 . 20

8 Numerical Results The algorithm for calculating the Gaussian upper bound was applied to some test problems. The test problems are well-known staircase linear programs that can be found in [14], [15] and [20]. The problems were modi ed to be Gaussian stochastic multi-stage linear problems so that assumption (3.2) holds. The names and sizes of the problems tested appear in Table 1. Problem CEP1 PGP2 SC105 SC205 SCTAP1 SCAGR7 SC50A SC50B SCSD1 SCSD6 SCAGR25

Two-Stage Size 9  8 ; 7  29 2  4 ; 7  30 90  91 ; 15  42 190  191 ; 15  42 270  432 ; 30  108 110  120 ; 19  58 34  36 ; 15  42 35  36 ; 15  42

Multi-Stage Total Total Size Stages 2 2 105  287 9 205  587 18 300  1020 10 129  368 7 49  122 4 50  122 4 73  874 3 147  1604 7 471  1412 25

Table 1: Multi-Stage Test Problems The modi cation of the staircase deterministic test problems into twostage and multi-stage stochastic linear programs was done separately. Therefore, for the same problem, the two-stage version and the multi-stage version may not have the same number of variables. The right-hand-side vector of the staircase deterministic test problems was changed to be Gaussian (stochastic) with the original deterministic values as its expectations and with a randomly generated covariance matrix, in which the variances of the right-hand-side vector increase with the number of stage. The Gaussian upper bound was rst computed on a set of two-stage Gaus21

Problem CEP1 PGP2 SC105 SC205 SCTAP1 SCAGR7 SC50A SC50B

()

2.7 2.9 3.2 3.2 3.3 3.2 3.2 3.0

Jensen's Estimated Gaussian Lower Optimal Upper Maximal Estimated Bound Value Bound Rel. Error Rel. Error 90200 90700 98800 0.0953 0.0893 429.509 450.01 1270.79 1.9587 1.8239 -52.202 -51.702 -50.948 0.0240 0.0146 -52.202 -52.129 -52.024 0.0034 0.0020 1412.25 3018.31 19704.45 12.9525 5.5283 6 6 6 -2.3314
10 -2.3313
10 -2.3277
10 0.0017 0.0016 -64.5751 -59.79 -50.9196 0.2115 0.1414 -70.0 -66.4327 -62.127 0.1125 0.0648

(*) For positive orthant probability greater than 0.99

Table 2: Bounds on Two-Stage Gaussian Stochastic Linear Programs sian stochastic linear problems. It was compared with Jensen's lower bound and with an estimated optimal value of the problem. The results appear in Table 2. The maximal relative error is the relative dierence between the Gaussian upper bound and Jensen's lower bound. The estimated relative error is the relative dierence between the Gaussian upper bound and the estimated optimal value. Table 3 contains the results for multi-stage Gaussian stochastic linear programs and the Gaussian upper bound is compared with Jensen's lower bound. The Gaussian upper bound is the one of Section 5, i.e., the upper bound of Table 3 are, theoretically, the upper bounds for the solution of the multi-stage problems given by weak solutions. The computation of the optimal value of a multi-stage problem is dicult, and therefore was not computed, but the bounds were computed in a few seconds using MINOS 5.3 and Matlab 4 on a Sun-4 workstation. The optimal value is positioned somewhere between the upper and lower bounds. Numerical results for the Gaussian upper bound on the strong solution of the multi-stage problem, as was described in Section 6, will appear in a forthcoming paper that deals with the solution of quadrati22

Problem Stages () SC105 9 3.4 SC205 18 3.8 SCTAP1 10 3.8 SCAGR7 7 3.8 SC50A 4 3.2 SC50B 4 3.4 SCSD1 3 4.0 SCSD6 7 3.8 SCAGR25 25 4.1

Jensen's Gaussian Lower Upper Maximal Bound Bound Rel. Error -52.202 -39.563 0.2421 -52.202 -48.607 0.0689 1412.25 17594.67 11.4586 -2.33139
106 -2.23365
106 0.0419 -64.5751 -55.8523 0.1351 -70.0 -61.033 0.1281 8.6667 676.857 77.096 50.5 346.11 5.8537 -1.67259
107 -1.65229
107 0.00121

(*) For positive orthant probability greater than 0.99

Table 3: Bounds on Multi-Stage Gaussian Stochastic Linear Programs cally constrained convex programming problems in the format of (CQP). Both Table 2 and Table 3 contain some examples with a large gap between Jensen's lower bound and the Gaussian upper bound in the sense of the example from Section 7. Test problems PGP2, SCTAP1, SCSD1 and SCSD6 are such examples. For those test problems, the Gaussian upper bound is indeed an upper bound, but it is not as tight as the upper bounds of the other test problems.

9 Extensions and Applications In many applications, like in portfolio management (see e.g., Markowitz [22], Dantzig and Infanger [9] and Mulvey and Vladimirou [23]), a minimization of the variance is needed, as in the problems whose objective functions is

E(c0x) + Var(d0x)

(9:1)

where  is a penalty parameter. When the objective is in the form of (9.1), a Gaussian upper bound can be easily calculated. 23

In the two-stage case, the Gaussian upper bound can be calculated by solving the mixed-variables program (a program in which some of the variables appear linearly and some other nonlinearly), min c0x + d0d s:t: Ax = b AA0 =  x ?   0   0:

(DSLP)

(9:2)

This problem is nonlinear, but can be bounded from above by solving the expected value problem (EP) after solving the covariance problem (CP) with d0d as its objective function. Since

d0d = tr(dd0);

(9:3)

the covariance problem is a semide nite program in the format of (CP) where C = dd0: The matrix dd0 is a rank-one matrix so H22?1 does not exist. However H12H22?1 exists and, as in (5.17), it can be determined by

H12H22?1 =

0 1 B@ w CA = Q0d:

wv 0 ; v0v

v

(9:4)

For a  calculated by the covariance problem, problem (DSLP) becomes a linear program and can be easily solved. In that case the covariance problem is bounded since d0d  0 for all d. In the multi-stage case, the Gaussian upper bound can be calculated by solving the quadratically constrained convex program min c0Db + d0DD0d s:t: ADIb1 = Ib1 Db  diag(DD0):

(9:5)

Another possible application for Gaussian multi-stage stochastic linear programs is the long-period investment problem (see [9], [23]). Long-period investment problems can be formulated in the format of (MSSLP) where the 24

random parameters are in the objective function. If the parameters of the objective function are multivariate Gaussian random vectors, the dual problem is a multi-stage stochastic linear program with Gaussian right-hand-side. An upper bound for the investment problem can be found using the Gaussian upper bound of Sections 5 and 6. A lower bound for the investment problem can be found using Jensen's inequality, i.e., solving the linear program where the expected values of the parameters are taken, see [21].

Acknowledgments

The authors are thankful to Professors Avi Mandelbaum and Arkadi Nemirovski of the Faculty of Industrial Engineering and Management at the Technion for valuable comments. Partial support was provided by the Fund for the Promotion of Research at the Technion.

References [1] Beale E.M.L., (1955), On Minimizing a Convex Function Subject to Linear Inequalities, Journal of the Royal Statistical Society, Series B 17, 173{184. [2] Billingsley P., (1986), Probability and Measure, Wiley, NY. [3] Birge J.R., (1985), Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs, Operations Research 33, 989{1007. [4] Birge J.R., (1985), Aggregation Bounds in Stochastic Linear Programming, Mathematical Programming 31, 25{41. [5] Birge J.R. and Wallace S.W., (1988), A Separable Piecewise Linear Upper Bound for Stochastic Linear programs, SIAM Journal of Control and Optimization 26, 3, 725{739. [6] Birge J.R. and Wets R.J.-B., (1986), Designing Approximation

Schemes for Stochastic Optimization Problems, in Particular for Stochastic Programs with Recourse, Mathematical Programming Study 27, 54{

102.

25

[7] Dantzig G.B., (1955), Linear Programming under Uncertainty, Management Science 1, 197{206. [8] Dantzig G.B., (1963), Linear Programming and Extensions, Princeton University Press, Princeton, NJ. [9] Dantzig G.B. and Infanger G., (1991), Multi-Stage Stochastic Linear Programs for Portfolio Optimization, Technical Report SOL 91-11, Department of Operations Research, Stanford University, Stanford, CA. [10] Durret R., (1991), Probability: Theory and Examples, Wadsworth, Paci c Grove, CA. [11] Entriken R. and Infanger G., (1990), Decomposition and Importance Sampling for Stochastic Linear Models, Energy, The International Journal 15, 7/8, 645{659. [12] Gassman H.I., (1990), MSLiP: A Computer Code for the Multistage Stochastic Linear Programming Problem, Mathematical Programming 47, 407{423. [13] Higle J.L. and Sen S., (1991), Stochastic Decomposition: An Algorithm for Two-Stage Linear Programs with Recourse, Mathematics of Operations Research 16, 650{669. [14] Higle J., Sen S. and Yakowitz D.S., (1990), Finite Master Programs in Stochastic Decomposition, Technical Report, Dept. of Systems and Industrial Engineering, The University of Arizona,Tucson, AZ. [15] Ho J.K. and Loute E., (1981), A Set of Staircase Linear Programming Test Problems, Mathematical Programming 20, 245{250. [16] Huang C., Ziemba W. and Ben-Tal A., (1977), Bounds on the Ex-

pectation of a Convex Function of a Random Variable: With Applications to Stochastic Programming, Operations Research 25, 315{325.

[17] Infanger G., (1991), Monte Carlo (Importance) Sampling within a Ben-

ders' Decomposition Algorithm for Stochastic Linear Programs, Extended

26

Version: Including Results of Large-Scale Problems, Annals of Operations

Research 39, 69{95.

[18] Infanger G., (1994), Planning Under Uncertainty: Solving Large-Scale Stochastic Linear Programs, Boyd and Fraser, Danvers, Ma. [19] Karatzas I. and Shreve S.E., (1991), Brownian Motion and Stochastic Calculus, Springer, NY. [20] Louveaux F. and Smeers N., (1988), Optimal Investment for Electricity Generation: A Stochastic Model and a Test Problem, in: Ermoliev Y. and Wets R. J.-B., Numerical Techniques for Stochastic Optimization, Springer-Verlag, Berlin. [21] Madansky A., (1960), Inequalities for Stochastic Linear Programming Problems, Management Science 6, 2, 197{204. [22] Markowitz H.M., (1991), Portfolio Selection: Ecient Diversi cation of Investments, Balckwell, Cambridge, Mass. [23] Mulvey J.M. and Vladimirou H., (1989), Stochastic Network Optimization Models for Investment Planning, Annals of Operations Research 20, 187{217. [24] Rockafellar R.T. and Wets R.J.-B., (1991), Scenarios and Policy Aggregation in Optimization Under Uncertainty, Mathematics of Operations Reaserch 16, 119{147. [25] Ruszczynski A., (1993), Parallel Decomposition of Multistage Stochastic Programming Problems, Mathematical Programming 58, 201{228. [26] Vanderbei R.J. and Yang B., (1993), The Simplest Semide nite Programs are Trivial, Technical Report SOR-93-12, Program in Statistics and Operations Research, Princeton University, Princeton, NJ. [27] Wallace S.W. and Yan T., (1993), Bounding Multi-Stage Stochastic Programs from Above, Mathematical Programming 61, 111{129.

27

[28] Wets R.J.-B., (1989), Stochastic Programming, in: Nemhauser G.L., Kan A.H.G. and Todd M.J. (eds.), Handbook in Operations Research and Management Science 1, Elsevier Science Publishers (North-Holland), Amsterdam, 573{629.

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