Sampling-based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures (Q2910873)

The authors deal with a multistage optimization problem where some of the variables involved are stochastic. The multistage nature of the problem fixes that they deal with a stochastic process. A risk averse decision maker is considered and the corresponding recourse function is defined. The framework matches with the approach used, currently, see, for example, [\textit{A. Ruszczynski} and \textit{A. Shapiro}, Math. Oper. Res. 31, No. 3, 433--452 (2006); corrigendum ibid. 32, No. 2, 496 (2006; Zbl 1278.90283); Math. Oper. Res. 31, No. 3, 544--561 (2006; Zbl 1278.90284)].NEWLINENEWLINESection 2 is concerned with considering a multiperiod risk function \(\rho\) which depends on a sequence of random variables, considered as some kind of revenues. The involved probabilistic framework is defined and the accumulation of the revenues is used. Three definitions fix necessary regularity conditions to \(\rho\), as a function of the revenues (monotonicity, invariance wrt to translations, convexity) and needed properties of the so-called multiperiod extended polyhedral. Another one is used for defining when the functional \(\rho\) is a convex risk measure. The main result is a theorem which establishes, under suitable condition, that \(\rho\) is finite, convex and continuous and a dual representation is obtained. Four corollaries are derived from it. They establish the coherence, monotonocity, invariance convexity, finiteness, and continuity of \(\rho\), as well as conditions for being coherent in the one period case. Proposition 2.12 derives a particular definition of \(\rho\). It is closely related with the results of the Corollary 2.9. Proposition 2.13 establishes sufficient conditions for identifying when \(\rho\), being an extended polyhedral risk-measure, is convex. A couple of propositions sustain particularities of interest for the sequel of the paper. A theoretical example is worked out, for illustrating the meaning of the results, using the optimized certainty equivalent as derived by \textit{A. Ben-Tal} and \textit{M. Teboulle} [Math. Finance 17, No. 3, 449--476 (2007; Zbl 1186.91116)].NEWLINENEWLINEThe third section is concerned with the problem of minimizing \(\rho\). The optimization problem is stated and a stochastic dual dynamic programming (SDDP) is derived. Its convergence, within the frame work developed, is considered, and the authors study the problem as established that it satisfies the conditions stated by \textit{A. B. Philpott} and \textit{Z. Guan} [Oper. Res. Lett. 36, No. 4, 450--455 (2008; Zbl 1155.90437)] who proved the convergence of SDDP. A decomposition algorithm is worked out and a theorem gives the needed cuts.