The Optimization Method And Applications In Portfolio Selection

We study su?cient descent directions in unconstrained optimization and theapplication of robust optimization in portfolio.For unconstrained optimization, we propose a way to construct su?cient de-scent directions by the use of a projection technique. Based on this projectiontechnique, the projected Newton direction, quasi-Newton directions and conjugategradient directions are all su?cient descent directions. It is important to notethat this property does not rely on the convexity of the objective function. It isalso independent with the line search used. Furthermore, by choosing appropriatesteplength, the above projected Newton method and quasi-Newton methods canmaintain superlinear convergence property. We then apply the technique to de-rive a PSB (Powell-Symmetric-Broyden) based method. The PSB based methodlocally reduces to the standard PSB method with unit steplength. Under appro-priate conditions, we show that the PSB based method with Armijo line searchis globally and superlinearly convergent for uniformly convex problems. We alsodo some numerical experiments. The results show that the PSB based method iscompetitive with the standard BFGS method.The past 50 years have witnessed a remarkable development in portfolio se-lection both in theory and real practice. Portfolio, for simply, is to invest moneyon several di?erent assets which can ensure the return meanwhile divert the risk.The numerical optimization has been an important tool for the portfolio selection.In the specific context of portfolio optimization, uncertainty arises, for in-stance, from imprecise knowledge of the expected returns and the covariance ma-trix of the risky assets. It has been well-known, these parameters are di?cult toestimate accurately. Di?erent techniques used in moment estimation may generatesignificantly di?erent point estimates of the input estimates which, in turn, leadto large variations in the composition of optimal portfolios. How to deal with theuncertainty of the parameters has become a hot topic in optimization and hasreceived much attention since the last two decades.Robust optimization, one of the most popular topics in the field of optimiza-tion and control since the late 1990s, deals with an optimization problem involvinguncertain parameters. The core of robust optimization is to approximate the un-certainty set by some relatively simple set (such as the box set or ellipsoid set). Itthen converts the original problem to a deterministic problem which can seek a solution simultaneously satisfies all possible constraint instances.In this thesis, we will study on the mean-variance model and the portfolioselection using CVaR strategy with data uncertainty. We show how to formulateand solve robust portfolio selection problems based on the recent progress in ro-bust optimization. By the use of statistics theory and time series techniques, weconstruct simple form uncertainty set which contain most possible realizations ofthe uncertain parameters. We then convert the original problem to a deterministicproblem which can obtain a solution that is guaranteed to be“good”for mostpossible realizations of the uncertainty parameters. To demonstrate our model andmethod, we do numerical experiments with real market data.We then study on indexing investment. We use the linear tracking error be-tween the returns of a portfolio and a benchmark as the risk. Then for a given levelof expected return, we investigate minimizing the tracking error. We show howto formulate and solve the robust model with some di?erent kinds of uncertaintysets. Real market simulation is presented to illustrate the method.For institution investor, it is necessary to concern the execution cost. Weconsider the execution of portfolio transactions with the aim of minimizing theexecution cost for a given level of volatility risk. In the model, uncertainty in theproblem arises, for instance, from imprecise knowledge of the permanent impactparameter, temporary market impact parameter and the volatility of the asset. Toget a reliable portfolio selection, in this thesis, we show how to apply the robustoptimization to solve the problem. Real market simulation is presented to illustratethe method.This thesis is supported by the National Natural Science Foundation of Chinagranted(10771057) and the Major Project of Ministry of Education of China granted(309023).