Generalized bounds for convex multistage stochastic programs.

*(English)*Zbl 1103.90069
Lecture Notes in Economics and Mathematical Systems 548. Berlin: Springer (ISBN 3-540-22540-4/pbk). xi, 190 p. (2005).

In this book, the author elaborates on the technique of barycentric bounds that was initiated by K. Frauendorfer [Stochastic two-stage programming. Lecture Notes in Economics and Mathematical Systems. 392. Berlin: Springer-Verlag (1992; Zbl 0794.90042)] for two-stage stochastic programs and developed in the subsequent years also to multistage problems.

Selected basic definitions and results valid for general multistage stochastic programs are presented in Chapter 2. New regularity conditions which relax standard assumptions and guarantee subdifferentiability and (local) saddle property of the recourse functions are derived in Chapter 3. In Chapter 4, barycentric bounds are presented for multistage convex stochastic programs with nonlinear dependence of the involved functions both on decisions and random parameters. Besides of convexity, the basic assumption introduced here is that the random data, parameters in the objective functions and those in constraints, follow a block-diagonal autoregressive process. This assumption is substantial, e.g., for monotonicity of the barycentric bounds obtained by a gradually refined partitioning scheme. Under related regularity conditions, bounding sets for the optimal first-stage solutions are derived.

Various further extensions in Chapter 5 deal with relaxation of convexity of the objective and of the constraints with respect to random parameters. One of important results applies to multistage stochastic linear programs: They may be approached by the barycentric approximation and bounding techniques only if the recourse and technological matrices are non-random and the right-hand sides have a form of differences of convex functions of the random data. The data should follow a block-diagonal autoregressive process such that the objective functions depend on other random parameters than the right-hand sides.

In Chapter 6, the technique is applied to real-life hydropower industry problems.

Appendices include selected items in optimization which have been exploited to derive the results.

Selected basic definitions and results valid for general multistage stochastic programs are presented in Chapter 2. New regularity conditions which relax standard assumptions and guarantee subdifferentiability and (local) saddle property of the recourse functions are derived in Chapter 3. In Chapter 4, barycentric bounds are presented for multistage convex stochastic programs with nonlinear dependence of the involved functions both on decisions and random parameters. Besides of convexity, the basic assumption introduced here is that the random data, parameters in the objective functions and those in constraints, follow a block-diagonal autoregressive process. This assumption is substantial, e.g., for monotonicity of the barycentric bounds obtained by a gradually refined partitioning scheme. Under related regularity conditions, bounding sets for the optimal first-stage solutions are derived.

Various further extensions in Chapter 5 deal with relaxation of convexity of the objective and of the constraints with respect to random parameters. One of important results applies to multistage stochastic linear programs: They may be approached by the barycentric approximation and bounding techniques only if the recourse and technological matrices are non-random and the right-hand sides have a form of differences of convex functions of the random data. The data should follow a block-diagonal autoregressive process such that the objective functions depend on other random parameters than the right-hand sides.

In Chapter 6, the technique is applied to real-life hydropower industry problems.

Appendices include selected items in optimization which have been exploited to derive the results.

Reviewer: Jitka Dupačová (Praha)