Abstract
In this paper, we study the two-stage stochastic linear semi-infinite programming with recourse to handle uncertainty in data defining (deterministic) linear semi-infinite programming. We develop and analyze volumetric barrier cutting plane interior point methods for solving this class of optimization problems, and present a complexity analysis of the proposed algorithms. We establish our convergence analysis by showing that the volumetric barrier associated with the recourse function of stochastic linear semi-infinite programs is a strongly self-concordant barrier and forms a self-concordant family on the first-stage solutions. The dominant terms in the complexity expressions obtained in this paper are given in terms of the problem dimension and the number of realizations. The novelty of our algorithms lies in their ability to kill the effect of the radii of the largest Euclidean balls contained in the feasibility sets on the dominant complexity terms.
Highlights
It can be seen that the SLSIP generalizes the ordinary stochastic linear programming by allowing infinite number of constraints on one hand, and the deterministic linear semiinfinite programming by allowing uncertainty in data on the other hand
AND COMPLEXITY Based on the self-concordance analysis established in Section IV, we develop a volumetric barrier cutting plane algorithm for SLSIPs, which is formally stated in Algorithm 1
We have studied the two-stage stochastic linear semi-infinite programming problem with discrete support
Summary
The purpose of this paper is to introduce and analytically study the two-stage stochastic linear semi-infinite programming (SLSIP in brief) with recourse in the dual standard form max cTx + E [Q(x, ω)]. In 2007, Ariyawansa and Zhu [27] derived a volumetric barrier decomposition interior point algorithm for two-stage stochastic (convex) quadratic linear programming. In 2011, Ariyawansa and Zhu [28] generalized their work in [27] to derive a volumetric barrier decomposition interior point algorithm for two-stage stochastic semidefinite programming. In 2015, Alzalg [29] exploited the work of Ariyawansa and Zhu in [27], [28] to derive a volumetric barrier decomposition interior point algorithm for two-stage stochastic second-order cone programming. We utilize the work of Luo et al [22] for deterministic linear semi-infinite programming on one hand and the work of Ariyawansa and Zhu [27] for ordinary stochastic quadratic linear programming on the other hand to derive volumetric barrier cutting plane decomposition algorithms for two-stage SLSIP problem with recourse. X := diag (x1, . . . , xn) to denote the n × n diagonal matrix whose diagonal entries are x1, . . . , xn
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