Doctoral Dissertation Robust Optimization In Game Theory And Portfolio

Optimization with uncertainty widely appears in engineering control, gametheory, supply chain management and portfolio etc. It is a hot topic in the fieldof mathematical programming. Robust optimization (RO) is one of the mostimportant tools to deal with uncertainty optimization. One important approachto RO is to appropriately estimate the uncertain set so that the solution of theRO is feasible for every element in this set. Morever, the solution is not tooconservative. This dissertation studies some applications of RO in game theoryand portfolio. The major contributions of the dissertation include the followingfour ideas.1. We study the existence of the robust optimization equilibrium. We willpay particular attention to the case of box uncertainty for payo? matrix or mixedstrategy. We show that in this situation, the corresponding robust optimizationequilibrium can be converted to a solution to a mixed complementarity problem(MCP). We also do some numerical experiments.2. On the basis of cardinality, we study robust optimization equilibrium forthe case where either the uncertain payo? matrix or the uncertain mixed strategy issymmetric. We show that in this situation, the corresponding robust optimizationequilibrium can be converted to a solution to a mixed complementarity problem(MCP) under l1∩l∞norm or a solution to a second order cone complementarityproblem (SOCCP) under l2 norm. We also estimate probability of constraintdeviation and do some numerical experiments.3. On the basis of cardinality, we study robust optimization equilibrium forthe case where either the uncertain payo? matrix or the uncertain mixed strat-egy is asymmetric. We show that that in this situation, the corresponding robustoptimization equilibrium can also be converted to a solution to a mixed comple-mentarity problem (MCP) under l1∩l∞norm or a solution to a second order conecomplementarity problem (SOCCP) under l2 norm. We also estimate probabilityof constraint deviation and do some numerical experiments.4. We study the risk measure of worst-case VaR with support informationand show that it is a coherent risk measure. We also investigate its applicationsin robust portfolio and take some performance analyses.This dissertation is supported by the National Natural Science Foundation ofChina (10771057) and the Major Project of the Ministry of Education of China (309023).This dissertation is typeset by software LATEX2ε.