Global Optimization And Its Application In Finance

In the thesis, we study the numerical methods for global optimization and its applications in portfolio selection. We also study the numerical method for solving convex minimization with singular solutions.First, we consider to find a global solution of minimizing a nonconvex function with box constrains. To solve the problem, we propose a new simplex mechanism for the local search stage of global optimization. We show that when the objective function is a concave quadratic function, the reflection of the simplex mechanism is done along a descent direction. If the objective function is concave but not necessary quadratic, the mechanism always reflect a "worse" vertex with respective to a "better" vertex. To avoid the algorithm being trapped in a local optimizer, we combined the simplex mechanism with simulated annealing and introduced the idea of non-monotone line search in our algorithm to propose a new SA-Simplex method. Numerical results show that as a local explorer in global optimization, our simplex mechanism is reasonable and practically effective.The portfolio selection theory was founded by Markowtiz in 1952. It is a new starting point of modern finance. Its most important view is to use the variance or the standard variance of the return of the portfolio to measure portfolio risk. Portfolio theory has made great progress in the past 50 years both in theory and practice, and the optimization theory has becomes an important tool in the solution of those models. There are two basic strategies in portfolio management:active and passive. Most fund managers inclines to passive strategies because they think that in an efficient market, it is hardly beat the market for a long period. Recently, passive strategies have received much attention. It is estimated that there are in total over 1012 dollars passively invested in index funds in the USA market [1]. Index tracking is a popular form of passive fund management, which consists in reproducing the performance of a stock market index by investing in a subset of stocks included in the index. There are some realistic constraints that should be posed on the tracking portfolio, for example, no short sell, constraints on portfolio positions, the number of assets that allowed to track the index, transaction cost constraints and so on. In this thesis, we consider index tracking with realistic constraints. The objective function of tracking model is very complex nonconvex minimization problem. To solve the problem, we propose a convex approximation to it, which is a mix-integer quadratic programming. Under certain conditions, we prove that the objective function value of the approximation problem can be sufficiently close to the value of the original objective function near the optimal portfolio. We then propose a simple heuristic to find its global optimizer. Numeri-cal results show that our approximation works well and the heuristic is simple and efficient.The most important thing in portfolio is how to diversify risk as well as possi-ble. In plain language, the investor should not put all eggs in one basket. Marginal risk can be used to judge the risk contribution of a single asset, and could be used as a measure of portfolio diversification. Recently, marginal risk control has at-tracted increasing attention. However, as portfolio selection with marginal risk control always leads to a non-convex optimization problem, most of the related work dwells on the stage of ex post analysis. As the non-systematic risk could be elimiated through diversification, we focus on the management of systematic risk. To get a formulation that could be useful for management of systematic risk, we introduce the idea of factor models in our portfolio selection model. Our portfolio model is a non-convex programming but with some special structures. By ex-ploiting the special structure of the problems, we propose an efficient branch and bound method for its global optimizer. Especially, our branching variables are only related to the number of assets that are restricted with marginal risk constraints (m) and the number of market factors (C). Our branch an bound method is very efficient when the number of assets (n) is much less than m+C. We compare our proposed portfolio model with mean variance model through empirical studies and test the efficient of our proposed branch and bound method through numerical experiments.Currently, most of risk control in portfolio management refers to the risk re-striction of certain assets in portfolio. In fact, by properly examining the market risk factors that derive the return of the portfolio, we could control the portfolio risk profile more efficiently according to the tolerance level for some specific risk. In general, factor risk constrained portfolio selection problems lead to nonconvex optimization problems and are generally very hard to solve. We consider the port-folio selection problems with factor risk constraints in another way. Specifically, we control the market factors'risk-reward ratio and derive a quadratic programming model, which could be efficiently solve by some standard software. We compare our proposed portfolio model with the mean variance model through empirical analysis. The results show that by property control the market factors'risk-reward ratio, we could make the distribution of risk more evenly, and improve the performance of our managed portfolio as well.Global optimization refers to the problem with multiplier local optimizer. Traditional optimization methods are easily trapped in a local optimizer and fails when dealing with global optimization problems. Global optimization is a kind of very hard optimization problems and there exists no efficient algorithm yet. In fact, there are some very hard problems in local optimizations too, for exam-ple, convex minimization with singular solutions. Traditional Newton's method is very efficient for unconstrained convex minimization, and has the attractive local quadratic convergence property. However, if the problem has singular solutions, the convergence rate of Newton's method may reduce to be linear. In the last part of this thesis, we propose a truncated regularized Newton method to minimize a convex function with singular solutions. Under appropriate conditions, we prove the global convergence and local quadratic convergence of the proposed method. Numerical results show that our proposed method has potential advantages for solving large scale problems with singular solutions.This dissertation is supported by the National Natural Science Foundation of China (10771057) and the Major Project of Ministry of Education of China granted (309023).