Stability in stochastic programming with recourse ... - Semantic Scholar
Mathematical Programming 28 (1984) 72-83 North-Holland
STABILITY RECOURSE
IN STOCHASTIC PROGRAMMING - ESTIMATED PARAMETERS
WITH
J. D U P A C O V A Charles University, Prague, Czechoslooakia Received 5 May 1982 Revised manuscript received 14 March 1983 In this paper, stability of the optimal solution of stochastic programs with recourse with respect to parameters of the given distribution of random coefficients is studied. Provided that the set of admissible solutions is defined by equa[ity constraints only, asymptotical normality of the optimal solution follows by standard methods, If nonnegativity constraints are taken into account the problem is solved under assumption of strict complementarity known from the theory of nonlinear programming (Theorem 1). The general results are applied to the simple recourse problem with random right-hand sides under various assumptions on the underlying distribution (Theorems 2-4). Key words: Stochastic Programming, Estimation, Stability, Asymptotical Normality, Minimax Approach, Deterministic Equivalent, Simple Recourse Problem.
1. Introduction W h e n solving stochastic programs, complete knowledge of the distribution of r a n d o m coefficients is usually supposed. In real-life situations, however, this assumption is hardly acceptable and the c o m m o n p r o c e d u r e s should be at least supplem e n t e d by p r o p e r stability studies. Consider the following stochastic p r o g r a m with recourse: maximize
Ee{cTx-~(x;A,b)}
on t h e s e t ~
(1)
where ~ is a set of admissible solutions. A n example of (1) is when a linear p r o g r a m maximize
c Tx
subject to
A x ~ b,
x ~ O,
has s o m e of c o m p o n e n t s of the m - v e c t o r b, n-vector c or (m, n)-matrix A random. Assume (i) For fixed A, b, g~(x ; A , b) is a nonnegative convex function of x. (ii) F o r arbitrary x ~ ~f, r ; A , b) is a convex function of A, b. (iii) ~ c ~ is a n o n e m p t y closed convex set. Provided that the joint distribution F of r a n d o m coefficients is known, (1) is in principle reducible to a nonlinear deterministic program. Such p r o g r a m s have been studied by m a n y authors from m a n y different viewpoints (see e.g. [11, 14]). Their 72
Z Dupadovd / Stability in stochastic programming
73
explicit form as well as their optimal solution depend on the given distribution F. In this paper uncertainty with respect to the distribution F will be taken into account. A first idea could be to study stability of the optimal solution of program (1) with respect to the underlying distribution directly. To a certain extent, it can be done using empirical distributions [15] or the concept of e-contamination (see [6, 7, 8]). In this paper, stability of the optimal solution of program (1) with respect to the parameters of the distribution F will be studied. Two alternatives will be considered: I. The distribution F belongs to a given parametric family of distributions. II. The distribution F belongs to a specified set of distributions defined by prescribed values of certain moments. In Case I, stability of the optimal solution with respect to the parameters and related statistical problems were studied for the simple recourse problem with normally distributed right-hand sides b,, 1 0}
(1)
where P-(r, n) and p e N " are a given matrix and vector, r ( P ) = r For any y~ Y, let s denote the optimal solution of the program maximize
f(x;y) o n t h e s e t ~ .
(2)
For the optimal solution ~ (aT) of the program maximize
on the set ~o,
f(x; n)
(3)
denote
J={/:Yj(~)>O}, \
Oxj Oxt
cardf=s,
s163
P/=(Pkj)1~k-~. (4)
.
Theorem 1. A s s u m e : (i) For any y ~ Y, f(- ; y) is a concave function on ~" such that the second order derivatives
:f
axj ax~'
ozy
o.rj Oy~"
l {i(r~); n) >O,
l~i 0, I ~ i ~<m}, The objective function 111.5) has the form (see [5, 4, 10]) f ( x ; rt, cr2) =
rain EF[cT~ ~ ~ q d X i - b i ) +}
F~.~n.~ 2
= C X --
(
i~= l
~tI,, _ _ a~vr r _ ~ I=I
r=l
)
" _v zq. ~r + 71, ,-i
V aqxi I-I
Provided that the true mean values r~,, l ~ < i ~ m , have been estimated and that their (vector.) estimate y'4" is asymptotically normally distributed, the asymptotica] normality of the optimal solution s of the substitute program max f(.r ; yN)
82
I. D u p a ~ o v d / Stability in stochastic p r o g r a m m i n g
with
(
n
~
a~ix j
-,. ,~qi c r i + .Y"i=1
again follows directly from T h e o r e m 1. To summarize, we. have Theorem 4. A s s u m e
(i)
f(x;y)=minE,elcrx F ,a_.~%.
- ~ qi(Xr
!.
+}
~~-1
with n
Xi = V,. a~jxj,
l
STABILITY RECOURSE
IN STOCHASTIC PROGRAMMING - ESTIMATED PARAMETERS
WITH
J. D U P A C O V A Charles University, Prague, Czechoslooakia Received 5 May 1982 Revised manuscript received 14 March 1983 In this paper, stability of the optimal solution of stochastic programs with recourse with respect to parameters of the given distribution of random coefficients is studied. Provided that the set of admissible solutions is defined by equa[ity constraints only, asymptotical normality of the optimal solution follows by standard methods, If nonnegativity constraints are taken into account the problem is solved under assumption of strict complementarity known from the theory of nonlinear programming (Theorem 1). The general results are applied to the simple recourse problem with random right-hand sides under various assumptions on the underlying distribution (Theorems 2-4). Key words: Stochastic Programming, Estimation, Stability, Asymptotical Normality, Minimax Approach, Deterministic Equivalent, Simple Recourse Problem.
1. Introduction W h e n solving stochastic programs, complete knowledge of the distribution of r a n d o m coefficients is usually supposed. In real-life situations, however, this assumption is hardly acceptable and the c o m m o n p r o c e d u r e s should be at least supplem e n t e d by p r o p e r stability studies. Consider the following stochastic p r o g r a m with recourse: maximize
Ee{cTx-~(x;A,b)}
on t h e s e t ~
(1)
where ~ is a set of admissible solutions. A n example of (1) is when a linear p r o g r a m maximize
c Tx
subject to
A x ~ b,
x ~ O,
has s o m e of c o m p o n e n t s of the m - v e c t o r b, n-vector c or (m, n)-matrix A random. Assume (i) For fixed A, b, g~(x ; A , b) is a nonnegative convex function of x. (ii) F o r arbitrary x ~ ~f, r ; A , b) is a convex function of A, b. (iii) ~ c ~ is a n o n e m p t y closed convex set. Provided that the joint distribution F of r a n d o m coefficients is known, (1) is in principle reducible to a nonlinear deterministic program. Such p r o g r a m s have been studied by m a n y authors from m a n y different viewpoints (see e.g. [11, 14]). Their 72
Z Dupadovd / Stability in stochastic programming
73
explicit form as well as their optimal solution depend on the given distribution F. In this paper uncertainty with respect to the distribution F will be taken into account. A first idea could be to study stability of the optimal solution of program (1) with respect to the underlying distribution directly. To a certain extent, it can be done using empirical distributions [15] or the concept of e-contamination (see [6, 7, 8]). In this paper, stability of the optimal solution of program (1) with respect to the parameters of the distribution F will be studied. Two alternatives will be considered: I. The distribution F belongs to a given parametric family of distributions. II. The distribution F belongs to a specified set of distributions defined by prescribed values of certain moments. In Case I, stability of the optimal solution with respect to the parameters and related statistical problems were studied for the simple recourse problem with normally distributed right-hand sides b,, 1 0}
(1)
where P-(r, n) and p e N " are a given matrix and vector, r ( P ) = r For any y~ Y, let s denote the optimal solution of the program maximize
f(x;y) o n t h e s e t ~ .
(2)
For the optimal solution ~ (aT) of the program maximize
on the set ~o,
f(x; n)
(3)
denote
J={/:Yj(~)>O}, \
Oxj Oxt
cardf=s,
s163
P/=(Pkj)1~k-~. (4)
.
Theorem 1. A s s u m e : (i) For any y ~ Y, f(- ; y) is a concave function on ~" such that the second order derivatives
:f
axj ax~'
ozy
o.rj Oy~"
l {i(r~); n) >O,
l~i 0, I ~ i ~<m}, The objective function 111.5) has the form (see [5, 4, 10]) f ( x ; rt, cr2) =
rain EF[cT~ ~ ~ q d X i - b i ) +}
F~.~n.~ 2
= C X --
(
i~= l
~tI,, _ _ a~vr r _ ~ I=I
r=l
)
" _v zq. ~r + 71, ,-i
V aqxi I-I
Provided that the true mean values r~,, l ~ < i ~ m , have been estimated and that their (vector.) estimate y'4" is asymptotically normally distributed, the asymptotica] normality of the optimal solution s of the substitute program max f(.r ; yN)
82
I. D u p a ~ o v d / Stability in stochastic p r o g r a m m i n g
with
(
n
~
a~ix j
-,. ,~qi c r i + .Y"i=1
again follows directly from T h e o r e m 1. To summarize, we. have Theorem 4. A s s u m e
(i)
f(x;y)=minE,elcrx F ,a_.~%.
- ~ qi(Xr
!.
+}
~~-1
with n
Xi = V,. a~jxj,
l
