Improvement of Sequential Quadratic Programming Algorithms for Optimization

Abstract

We consider the general nonlinear optimization problem defined as, minimize a nonlinear real-valued function of several variables, subject to a set of nonlinear equality and inequality constraints, this class of problems arise in many real life applications, for example in engineering design, chemical equilibrium, simulation and data fitting . In this research, we study the theory and analyze algorithms of optimization methods that use for solving problems of this nature, called sequential quadratic programming (SQP). First, we give a review of line search SQP which solves the nonlinear constrained program by solving a sequence of associating quadratic programs (QP’s). A QP is a constrained optimization problem in which the objective function is quadratic and the constraints are linear. The inequality QP is solved by use of the primal active set method. The primal active set method solves a QP with inequality constrained QP by solving a sequence of corresponding equality constrained QP’s. The equality constrained QP is solved by solving an indefinite symmetric linear system of equations, the so-called Karush-Kuhn-Tucker (KKT) system. When solving the KKT system, the range space procedure or the null space procedure is used. Also, we describe another class of SQP methods algorithm for solving the nonlinear equality and inequality constrained optimization the algorithms use the trust region SQP technique. We define a model subproblem which minimizes a quadratic approximation of the Lagrangian subject to modified relaxed linearization of the problem nonlinear constraints and a trust region constraint. We also reviewed a new algorithm which uses a filter technique and a trust-region method in order to enforce global convergence and to improve the efficiency of traditional approaches. We also analyze the effect of approximate first and second derivatives on the performance of the filter-trust-region algorithm. Also a modified version of the trust-region filter-SQP method for nonlinear programming introduced by Fletcher, Leyffer, and Toint is presented. Hereby, the original trust-region SQP-steps can be used without an additional second order correction. The main modification consists in using the Lagrangian function value instead of the objective function value in the filter together with an appropriate infeasibility measure. Moreover, it is shown that the modified trust-region filter-SQP method has the same globa