Research On Sequential Quadratics Programming Method For Solving Constrained Optimization

The purpose of the thesis is to study the numerical method for solving nonlinear constrained optimization. We propose several sequential quadratic programming (SQP) algorithms, and establish their global convergence and superlinear convergence. We do numerical experiments to test the proposed algorithms.In Chapter 2, by the use of an active set estimate technique, we propose an active set SQP method for nonlinear constrained optimization. The method generates a sequence of feasible points. Major advantage of the proposed method lies in that the main search direction is determined by a lower dimension quadratic programs. To overcome Maratos effect, we calculate a higher-order correction direction by solving a reduced least squares problem. Under appropriate conditions, we show that the proposed method is globally and superlinearly convergent.In Chapter 3, we present a modified SQP method for the minimax problem. In the algorithm, the main search direction is obtained by solving a quadratic program which always has a solution. In order to avoid Maratos effect, different from the previous technique where a quadratic programs is solved, we solve a system of linear equations to obtain a higher-order correction direction. Under some mild conditions, we obtain the global and superlinear convergence.In Chapter 4, we have proposed a feasible point SQP algorithm for nonlinear inequality constrained optimization problems. At each iteration, we determined a descent and feasible direction by solving a reduced quadratic programming sub-problem and a reduced system of linear equations, respectively. We then device a feasible descent direction through a suitable combination of the descent direction and the feasible direction. To overcome Maratos effect, a higher-order correction direction is obtained by solving another reduced system of linear equations. The algorithm is proved to be globally and superlinearly convergent under some mild conditions. A good feature of the proposed algorithm is that the coefficient matrix for the system of linear equations do not involve multiplier estimate. Furthermore, the structure of coefficient matrix is simpler than the one in the previous algorithms.In Chapter 5, we propose a noninterior type feasible point QP-free algorithm for nonlinear inequality constrained optimization problems. The generated iterates are feasible but not necessary interior points of the feasible region. At each iteration, a search direction is obtained by solving four systems of linear equa- tions with the same coefficient matrix. The algorithm is proved to be globally and superlinearly convergent under some mild conditions.In Chapter 6, we propose an interior point type feasible QP-free algorithm for nonlinear inequality constrained optimization problems. At each iteration, by solving three systems of linear equations with the same coefficient matrix, a search direction is generated. The algorithm is proved to be globally and superlinearly convergent under some mild conditions. Advantages of the algorithm include: the uniformly nonsingularity of the coefficient matrices and the boundedness of the approximate Lagrange multipliers without the strictly complementarity are obtained. Moreover, the global convergence is achieved even if the number of the stationary points is infinite.