Nonlinear Conjugate Gradient Method And Robust Optimal Portfolio

Nonlinear conjugate gradient methods form a class of welcome methods forsolving large-scale unconstrained optimization problem. In recent years, there hasbeen much new progress in nonlinear conjugate gradient methods. This thesisfurther studies nonlinear conjugate gradient methods. Combined with the con-jugate gradient method and projection gradient method, we propose a conjugategradient-type method for solving linear equality constrained optimization prob-lem, and apply it to portfolio problem. Finally we further study robust optimalportfolio problem.In Chapter2, based on the modified secant equation by Li and Fukushima,we propose two modified Hestenes-Stiefel (HS) conjugate gradient methods. Acommon nice property of the proposed methods is that they can generate sufcientdescent directions without any line search. Under mild conditions, we show that themethods with Armijio line search are globally convergent. Moreover, the R-linearconvergence rate of the modified HS methods is established. Preliminary numericalresults show that the proposed methods are promising, and are competitive withthe well-known CG-DESCENT method.In Chapter3, combining the idea of the modified Fletcher-Reeves conjugategradient method and the Rosen gradient projection method, we propose a conju-gate gradient-type method for linear equality constrained optimization problem.Search direction generated by the method is a feasible descent direction. Con-sequently the generated iterates are feasible points, moreover, the sequence offunction is decreasing. Under mild conditions, we show that the method withArmijio line search is globally convergent. Moreover, when the method with exactline search is used to solve a linear equality constrained quadratic programming,it will terminate at the solution of the problem within finite iterations We applythe proposed method to Markowitz mean-variance portfolio optimization problem.Numerical results show that the method is more efective than the Rosen gradientprojection method.In Chapter4, we study robust mean semi-absolute deviation models for port-folio optimization. We consider the case where the return of assets belongs to abox uncertainty set or ellipsoidal uncertainty set. We derive robust counterpartsfor both portfolio optimization. The first model is a Linear programming (LP) andthe last one is a second-order cone programming (SOCP), both can be computed efciently. The empirical analysis and comparisons from the real market data indi-cate that the robust models can obtain a portfolio strategy with the better wealthgrowth rate and more stable return.Andrew and Chen, Conine and Tamarkin found that the asymmetries in thedata reject the null hypothesis of multivariate normal distributions. Exiting robustCVaR method is under the box uncertainty set or the ellipsoidal uncertainty setwhich are symmetric. This may be overly conservative for capturing the deviationsof the asymmetric asset returns. In Chapter5, based on the robust optimizationtechniques by Chen et al, we study robust CVaR method in which the mean re-turn of the portfolio belongs a non-symmetric afne uncertainty set. We derive acomputationally tractable robust optimization method for minimizing the CVaRof a portfolio. A remarkable characteristic of the new method is that the robustoptimization model reserves the complexity of original portfolio optimization prob-lem, i.e., the robust counterpart problem is still a linear programming problem.Moreover, it takes into consideration asymmetries in the distributions of returns.We present some numerical experiments with simulated and real market data toillustrate the behavior of the robust optimization model.Conditional value-at-risk (CVaR) has become the most popular risk measuredue to its coherence property and tractability. However, recent study has indicatedthat optimal solutions to the CVaR minimization are highly susceptible to estima-tion error of the risk measure because the estimate depends on only a small portionof sampled scenarios. In Chapter6, considering the error of samples in conditionalvalue-at-risk (CVaR), we propose a Worst-Case CVaR risk (called AWCVaR). Weshow that the AWCVaR is a coherent risk measure, and the robust problem is stilla linear programming problem. The empirical analysis shows that AWCVaR ismore efective than the conditional value-at-risk (CVaR).