Chapter 13 - More on Numerical Methods for Constrained Optimum Design

Abstract

Advanced topics related to numerical methods for constrained optimum design are presented. The concept of a potential constraint strategy is introduced where only the currently violated, active and nearly active constraints are included in defining the search direction calculation subproblem. The idea of an inexact line search based on the Armijo-like procedure is introduced and illustrated. A descent function for the constrained optimization problem is defined and used to calculate the step size. A bound-constrained optimization problem that involves minimization of a function subject to only simple bounds on the design variables is defined. An algorithm is given to solve this class of problems. Then the general optimization problem is considered, and a sequential quadratic programming (SQP) algorithm that incorporates second-order information about the problem is presented. This is now a method of choice for solving nonlinear programming problems. The method is derived, and a detailed algorithm is given. A brief introduction to some first-order methods that were popular before development of the SQP method, such as the method of feasible directions, gradient projection method, and generalized reduced gradient method, is given. Many methods, including the SQP method, require solution of a quadratic programming (QP) subproblem to solve for the search direction. Two methods to solve the QP subproblem are given.