Research On Some Algorithms For Constrained Optimization Problems

In this paper, some nonlinear programming theory and algorithms are discussed. The whole paper is divided into four parts, and it is organized as follows:In Chapter One, we briefly introduce the origin and development of the nonlinear optimization problems, and then we propose the problems that will be discussed in the furtherresearch.In Chapter Two, we analyze several constraint qualifications in constraint optimizationsuch as LICQ, SMFCQ, MFCQ, CRCQ, CPLD etc in detail. Their definitions are listed and the property of lagrange multipliers under these constraint qualifications are discussed for equality and inequality constraint optimization. The relationship among all these constraint qualifications is concluded and a relation graph is given. Moreover, we also show that the "WMFCQ" condition is not a constraint qualification by a counterexample.In Chapter Three, first, we analyze the generalized gradient projection method for nonlinear constrained optimization problems under non-degeneracy and degeneracy. Then we give a generalized gradient projection algorithm for nonlinear inequality and equality constrained optimization problems under MFCQ.In the fourth chapter, by the accurate active set identification technique we establish a new SQP feasible algorithm for nonlinear inequality constrained optimization problems, while the strict complementary assumption is never invoked. Per iteration, only one quadratic problem on a smaller scale and a generalized gradient projection need to be solved and computed. Under the assumptions, which don not need the strict complementary and are weaker than strong second order condition, the algorithm is proved to be globally and locally superlinearly convergent.