Robust Solution Analysis On Several Mathematical Programming Problems

In this paper,we first put forward a new concept on robust analysis and establish a unified model.Different from the fixed uncertainty set in robust optimization,the uncer-tainty set in this model is changeable and controlled by the variable "perturbation radius".The model aims to maximize the perturbation radius on the condition that the robust fea-sible region satisfies some requirements,which are described by a pre-decided decision set.The requirements could be that a given robust solution is still feasible or at least one robust feasible solution satisfies the properties required.In the first case,the decision set is a singleton and the maximal perturbation radius of the given solution is obtained.In the second case,the decision set is of a general form and the most perturbation-immune robust feasible solution with the properties satisfied is obtained.This new concept could be regarded as an extension of robust optimization and provides a new assessment tool for the robustness of the solution.Most models obtained could be equivalently reformed into tractable ones,such as linear program,second-order cone program and semi-definite program.For those intractable problems,a binary search algorithm is designed.Moreover,we study the single-period unit commitment(UC)problem in power sys-tem,which belongs to mixed-integer programming.Dealing with the parameter uncer-tainty,we establish a robust model by punishing the mismatch between the power supply and demand in the objective.In fact,both the deterministic problem and its robust coun-terpart are NP-hard,and the former is a degenerate case of the latter.For the deterministic one,based on conjugate function and Lagrangian dual,we propose a polynomial-time ap-proximation algorithm with error bounds.And the algorithm is then extended to solve the robust counterpart with the computational complexity and error bounds remained.Com-putational results support the effectiveness and efficiency of the algorithms.Chapter 1 summarizes some related work in the literature and introduces how the new concept emerges.Chapter 2 and Chapter 3 apply the new concept of robust analy-sis to linear programming and convex quadratically constrained quadratic programming,respectively,and give a detailed discussion.Chapter 4 provides the analysis and algo-rithm for the deterministic single-period UC problem and its robust counterpart.Chapter 5 includes the conclusions.