Parametric and Multiobjective Optimization with Applications in Finance

Abstract

<p> In this thesis parametric analysis for conic quadratic optimization problems is studied. In parametric analysis, which is often referred to as parametric optimization or parametric programming, a perturbation parameter is introduced into the optimization problem, which means that the coefficients in the objective function of the problem and in the right-hand-side of the constraints are perturbed. First, we describe linear, convex quadratic and second order cone optimization problems and their parametric versions. Second, the theory for finding solutions of the parametric problems is developed. We also present algorithms for solving such problems. Third, we demonstrate how to use parametric optimization techniques to solve multiobjective optimization problems and compute Pareto efficient surfaces. </p> <p> We implement our novel algorithm for hi-parametric quadratic optimization. It utilizes existing solvers to solve auxiliary problems. We present numerical results produced by our parametric optimization package on a number of practical financial and non-financial computational problems. In the latter we consider problems of drug design and beam intensity optimization for radiation therapy. </p> <p> In the financial applications part, two risk management optimization models are developed or extended. These two models are a portfolio replication framework and a credit risk optimization framework. We describe applications of multiobjective optimization to existing financial models and novel models that we have developed. We solve a number of examples of financial multiobjective optimization problems using our parametric optimization algorithms. </p>