Ambiguous Probabilistic Optimization Problems Based On Modified X~2- Distance Divergence

Many practical problems with important values can be modeled as ambiguous probabilistic optimization problems, such as decision making problem, which often exist uncertainty distribution. Ambiguous probabilistic optimization problems can be classified as ambiguous probability minimization problems(PM) and ambiguous chance constrained programs(CCP). This paper aims at discussing efficient methods for solving ambiguous probabilistic programming based on modified X~2- distance divergence. Likelihood ratio(LR) is used to construct the ambiguity sets. Moreover, equivalent form of the ambiguous probabilistic optimization problems is established based on modified X~2- distance divergence function. We also discuss the transformation forms between the ambiguous probabilistic optimization problems and the original probabilistic optimization problems under a nominal distribution. The main contents of this paper are organized as follows:Chapter 1 reviews the research background of ambiguous probabilistic optimization problems, and introduces basic knowledge involved in the paper.Chapter 2 studies ambiguous probabilistic optimization problems based on modified X~2- distance divergence. First of all, ambiguous sets defined by Ф- divergence function are introduced, and the ambiguous set based on modified X~2- distance divergence is constructed. Second, the inner maximization problem of the models is solved. We study the worst-case probability function. Applying the change-of-measure technique, we convert an optimization problem with respect to distribution P to a convex optimization problem with respect to LR l(x). Moreover, equivalence of Lagrange dual problems is proved by using the duality theory of convex optimization problem. Consequently, we obtain the equivalent form of ambiguous probabilistic optimization problems.Chapter 3 discusses transformation forms of two classes of ambiguous probabilistic optimization problems based on modified X~2- distance divergence respectively. Firstly, when the ambiguous sets are defined by modified X~2- distance divergence, we prove that the ambiguous PM is essentially the same as the original probability minimization problem. Secondly, we discuss the ambiguous CCP may be reformulated as CCP with unique confidence level, which is an optimal value of one dimensional optimization problem. Finally, bisection search algorithm and 0.618 search algorithm are presented for solving new confidence level.Chapter 4 discusses application of ambiguous probabilistic optimization problems in the Value-at-Risk. Numerical example illustrates that the ambiguous CCP can be reformulated as CCP with unique confidence level based on modified X~2- distance divergence.