A new descent direction method for linearly constrained continuous optimization problems and its application in portfolio management

Abstract

The problem we consider in this thesis is a linearly constrained continuous optimization problem with lower and upper bounds on variables. To solve the problem, a logarithmic barrier function and a barrier parameter are introduced to incorporate the lower and upper bounds into the objective function and derive a barrier optimization problem. To solve the barrier problem, we have found a new descent direction, which is obtained by computing the analytical center of a linear programming problem. This descent direction has a nice feature that the lower and upper bounds are always satisfied automatically if the step length is a number between zero and one. Based on the new descent direction, a descent direction method is developed for approximating a solution of the linearly constrained optimization problem. For any given value of the barrier parameter, we prove that the method converges to a stationary point globally. A portfolio selection model with transaction cost is formulated in this thesis. The model is an extension of Harry Markowitz’s mean-variance portfolio selection model, which gives some numerical indications for investors to make their decisions. The literature shows that the transaction cost is a very important factor involved in every transaction. When the transaction cost is taken into consideration, the model becomes very complicated to solve. In this thesis, we have applied the new descent direction method to solve the portfolio selection model with transaction cost. Numerical experiments show that the method seems efficient. Keywords: Linear Constraints, Nonlinear Programming, Descent Direction, Interior-Point Method, Lagrange Multipliers, Portfolio Selection, Transaction Cost, Mean-Variance Analysis