Stochastic Separated Continuous Conic Programming: Strong Duality and a Solution Method

Mathematical Problems in Engineering, Jan 2014

We study a new class of optimization problems called stochastic separated continuous conic programming (SSCCP). SSCCP is an extension to the optimization model called separated continuous conic programming (SCCP) which has applications in robust optimization and sign-constrained linear-quadratic control. Based on the relationship among SSCCP, its dual, and their discretization counterparts, we develop a strong duality theory for the SSCCP. We also suggest a polynomial-time approximation algorithm that solves the SSCCP to any predefined accuracy.

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Stochastic Separated Continuous Conic Programming: Strong Duality and a Solution Method

Stochastic Separated Continuous Conic Programming: Strong Duality and a Solution Method Xiaoqing Wang Lingnan (University) College, Sun Yat-sen University, Guangzhou, Guangdong 510275, China Received 3 November 2013; Accepted 29 November 2013; Published 9 January 2014 Academic Editor: Dongdong Ge Copyright © 2014 Xiaoqing Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We study a new class of optimization problems called stochastic separated continuous conic programming (SSCCP). SSCCP is an extension to the optimization model called separated continuous conic programming (SCCP) which has applications in robust optimization and sign-constrained linear-quadratic control. Based on the relationship among SSCCP, its dual, and their discretization counterparts, we develop a strong duality theory for the SSCCP. We also suggest a polynomial-time approximation algorithm that solves the SSCCP to any predefined accuracy. 1. Introduction Stochastic programming is one of the branches of optimization which enjoys a fast development in recent years. It tries to find optimal decisions in problems involving uncertain data, so it is also called “optimization under uncertainty” [1]. Since the problems in reality often involve uncertain data, stochastic programming has a lot of applications. Many deterministic optimization models have their stochastic counterpart; for example, the stochastic counterpart of linear programming is stochastic linear programming. In this paper, we consider the stochastic counterpart of a kind of optimization model called separated continuous conic programming () which has the following form: Here the control and state variables (both are decision variables), and , are vectors of bounded measurable functions of time . , , are closed convex cones in the Euclidean space with appropriate dimensions, are vectors, are matrices, and the superscript denotes the transpose operation. was first studied by Wang et al. [2]. They developed a strong duality theory for under some mild and verifiable conditions and suggested an approximation algorithm to solve with predefined precision. has a variety of applications in robust optimization and sign-constrained linear-quadratic control. However, many applications of are stochastic in nature in the sense that the values of some parameters in the resulted models may change over time with some probability distribution. To incorporate this kind of randomness into the model, we introduce the following stochastic counterpart of which we call stochastic separated continuous conic programming () problem: where is a random variable. is formulated with the similar idea as that of the stochastic linear programming [1, 3]. There are two stages in this problem; the values of some parameters in the second stage depend on the value of a random variable . Our goal in this paper is developing the strong duality for and suggesting a solution method to solve it approximately with predefined precision. Here is a summary of our main results. Through discretization, we connect and its dual to two ordinary conic programs, and we show that strong duality holds for and its dual under some mild (and verifiable) conditions on these two ordinary conic programs. Furthermore, the optimal values of those two conic programs provide an explicit bound on the duality gap between and its dual, based on which we suggest a polynomial-time approximation algorithm that solves to any predefined accuracy. According to our knowledge, we are the first to raise the model and there have been no other results on besides those in this paper. The paper is organized as follows. In Section 2, we present an overview on the related literature. We also give a concrete example to show the application of . In Section 3, we construct a dual for . We also discretize and its dual into two ordinary conic programs, and bring out their relations. In Section 4, we discuss the strong feasibility for , its dual, and their discretizations. We then establish the strong duality result for and its dual in Section 5. This leads to a polynomial-time approximation algorithm with an explicit error bound, detailed in Section 6. In Section 7, we summarize what we get for and point out some future research directions. For simpler presentation, in the remainder of this paper, we will concentrate on the following problem, which is the corresponding when is a discrete variable and only takes two different values with probability and , that is, there are only two scenarios in the second stage of : where the first-stage control and state variables are and , , and the second-stage control and state variables are , , and , . Also , , , , , , , . Note that although (3) is a deterministic optimization problem, it is not an . T (...truncated)


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Xiaoqing Wang. Stochastic Separated Continuous Conic Programming: Strong Duality and a Solution Method, Mathematical Problems in Engineering, 2014, 2014, DOI: 10.1155/2014/896591