Restricted-Recourse Bounds for Stochastic Linear Programming

We consider the problem of bounding the expected value of a linear program (LP) containing random coefficients, with applications to solving two-stage stochastic programs. An upper bound for minimizations is derived from a restriction of an equivalent, penalty-based formulation of the primal stochastic LP, and a lower bound is obtained from a restriction of a reformulation of the dual. Our "restricted- recourse bounds" are more general and more easily computed than most other bounds because random coefficients may appear anywhere in the LP, neither independence nor boundedness of the coefficients is needed, and the bound is computed by solving a single LP or nonlinear program. Analytical examples demonstrate that the new bounds can be stronger than complementary Jensen bounds. (An upper bound is "complementary" to a lower bound, and vice versa). In computational work, we apply the bounds to a two-stage stochastic program for semiconductor manufacturing with uncertain demand and production rates.

[1]  William T. Ziemba,et al.  Tight Bounds for Stochastic Convex Programs , 1992, Oper. Res..

[2]  J. R. Evans,et al.  Maximum flow in probabilistic graphs-the discrete case , 1976, Networks.

[3]  Richard D. Wollmer,et al.  Investments in stochastic maximum flow networks , 1991, Ann. Oper. Res..

[4]  Warren B. Powell,et al.  Restricted Recourse Strategies for Dynamic Networks with Random Arc Capacities , 1994, Transp. Sci..

[5]  David P. Morton,et al.  Stochastic Network Interdiction , 1998, Oper. Res..

[6]  P. Kall,et al.  Stochastric programming with recourse: upper bounds and moment problems: a review , 1988 .

[7]  Karl Frauendorfer,et al.  Stochastic Two-Stage Programming , 1992 .

[8]  José H. Dulá,et al.  Bounding separable recourse functions with limited distribution information , 1991, Ann. Oper. Res..

[9]  Stein W. Wallace.  Investing in arcs in a network to maximize the expected max flow , 1987, Networks.

[10]  Warren B. Powell,et al.  A Stochastic Model of the Dynamic Vehicle Allocation Problem , 1986, Transp. Sci..

[11]  Hercules Vladimirou,et al.  Stochastic linear programs with restricted recourse , 1997 .

[12]  Y. Ermoliev.  Stochastic quasigradient methods and their application to system optimization , 1983 .

[13]  B. WETSt,et al.  STOCHASTIC PROGRAMS WITH FIXED RECOURSE : THE EQUIVALENT DETERMINISTIC PROGRAM , 2022 .

[14]  Paul H. Zipkin,et al.  Bounds for Row-Aggregation in Linear Programming , 1980, Oper. Res..

[15]  Jery R. Stedinger,et al.  SOCRATES: A system for scheduling hydroelectric generation under uncertainty , 1995, Ann. Oper. Res..

[16]  David A. Kendrick,et al.  GAMS : a user's guide, Release 2.25 , 1992 .

[17]  N. C. P. Edirisinghe,et al.  Second-order scenario approximation and refinement in optimization under uncertainty , 1996, Ann. Oper. Res..

[18]  Roger J.-B. Wets,et al.  Programming under uncertainty: The complete problem , 1966 .

[19]  Roger J.-B. Wets,et al.  Sublinear upper bounds for stochastic programs with recourse , 1987, Math. Program..

[20]  A. Madansky.  Inequalities for Stochastic Linear Programming Problems , 1960 .

[21]  Jitka Dupačová,et al.  Applications of stochastic programming under incomplete information , 1994 .

[22]  Stein W. Wallace.  A piecewise linear upper bound on the network recourse function , 1987, Math. Program..

[23]  N. C. P. Edirisinghe,et al.  New Second-Order Bounds on the Expectation of Saddle Functions with Applications to Stochastic Linear Programming , 1996, Oper. Res..

[24]  Karl Frauendorfer,et al.  Solving SLP Recourse Problems with Arbitrary Multivariate Distributions - The Dependent Case , 1988, Math. Oper. Res..

[25]  Peter Kall,et al.  Approximation Techniques in Stochastic Programming , 1988 .

[26]  E. Beale.  ON MINIMIZING A CONVEX FUNCTION SUBJECT TO LINEAR INEQUALITIES , 1955 .

[27]  J. Mulvey,et al.  Stochastic network optimization models for investment planning , 1989 .

[28]  R. Wets.  Stochastic Programs with Fixed Recourse: The Equivalent Deterministic Program , 1974 .

[29]  Julia L. Higle,et al.  Stochastic Decomposition: An Algorithm for Two-Stage Linear Programs with Recourse , 1991, Math. Oper. Res..

[30]  William T. Ziemba,et al.  A Bank Asset and Liability Management Model , 1986, Oper. Res..

[31]  Robert D. Doverspike,et al.  Network planning with random demand , 1994, Telecommun. Syst..

[32]  R. Wets.  Solving stochastic programs with simple recourse , 1983 .

[33]  R. Wets,et al.  Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse , 1986 .

[34]  Pitu B. Mirchandani,et al.  Shortest distance and reliability of probabilistic networks , 1976, Comput. Oper. Res..

[35]  Roger J.-B. Wets,et al.  Computing Bounds for Stochastic Programming Problems by Means of a Generalized Moment Problem , 1987, Math. Oper. Res..

[36]  R. Wollmer.  Critical path planning under uncertainty , 1985 .

[37]  Peter Kall,et al.  Stochastic Linear Programming , 1975 .

[38]  George B. Dantzig,et al.  Parallel processors for planning under uncertainty , 1990 .

[39]  George B. Dantzig,et al.  Decomposition techniques for multi-area generation and transmission planning under uncertainty: Final report , 1989 .

[40]  Horand I. Gassmann,et al.  Optimal harvest of a forest in the presence of uncertainty , 1989 .

[41]  Matthew Sta.  A Product-Mix Capacity Planning Model , 1997 .

[42]  R. Wets,et al.  Epi‐consistency of convex stochastic programs , 1991 .

[43]  C. HuangC.,et al.  Bounds on the Expectation of a Convex Function of a Random Variable , 1977 .

[44]  R. Wets,et al.  Stochastic programming , 1989 .

[45]  W. Ziemba,et al.  The Russell-Yasuda Kasai Model: An Asset/Liability Model for a Japanese Insurance Company Using Multistage Stochastic Programming , 1994 .

[46]  William T. Ziemba,et al.  Bounding the Expectation of a Saddle Function with Application to Stochastic Programming , 1994, Math. Oper. Res..

[47]  J. Birge,et al.  A separable piecewise linear upper bound for stochastic linear programs , 1988 .

[48]  William T. Ziemba,et al.  Implementing bounds-based approximations in convex-concave two-stage stochastic programming , 1996, Math. Program..

[49]  Peter Kall,et al.  Stochastic Programming , 1995 .

[50]  E M Beale.  THE USE OF QUADRATIC PROGRAMMING IN STOCHASTIC LINEAR PROGRAMMING , 1962 .

[51]  H. Frank,et al.  Shortest Paths in Probabilistic Graphs , 1969, Oper. Res..

[52]  Julia L. Higle,et al.  Finite master programs in regularized stochastic decomposition , 1994, Math. Program..

[53]  John R. Birge,et al.  Aggregation bounds in stochastic linear programming , 1985, Math. Program..

[54]  W. Ziemba,et al.  A tight upper bound for the expectation of a convex function of a multivariate random variable , 1986 .

[55]  José H. Dulá,et al.  An upper bound on the expectation of simplicial functions of multivariate random variables , 1992, Math. Program..

[56]  W. Zangwill.  Non-Linear Programming Via Penalty Functions , 1967 .

[57]  William T. Ziemba,et al.  Bounds for Two-Stage Stochastic Programs with Fixed Recourse , 1994, Math. Oper. Res..

[58]  A. Madansky.  Bounds on the Expectation of a Convex Function of a Multivariate Random Variable , 1959 .