A New Class Gradient Projection Algorithm Of Convex Quadratic Programming

Quadratic programming, as a natural transition from linear programming to nonlinear programming, is very important in practical application. Therefore, in aspect of the development of optimization theory, also from the practical point, research on quadratic programming is very significant. This thesis mainly studied on the solving methods of convex quadratic programming with inequality constraints and equality constraints, respectively.Firstly, the background and signification were introduced. Review of quadratic programming's present research situation, the main work and content arrangements of this paper were presented. And then, we introduced the basic knowledge and basic theory of quadratic programming, including fundamental concepts, theorems, optimization conditions and the basic format of iterative algorithm.Secondly, a new algorithm for solving inequality constrained convex quadratic programming problems was proposed based on the original algorithm. It combined Fisher function with gradient projection method. This algorithm can improve the convergency speed by changing the search direction, which is a feasible and descent direction not depending on any linear search. And each iteration only need compute the projection matrix once. It can greatly save computation. The convergence of this new algorithm was proved. Some numerical experiments were used to testify it. The results showed that the new algorithm had a faster convergency speed and less iterations.In the end, a conjugate gradient projection algorithm (CGPA) was presented to solve equality constrained convex quadratic programming problem. This algorithm combined gradient projection method with conjugate gradient method of unconstrained optimization. It used DCG method, a modified conjugate gradient method, to calculate the parameter ? k. DCG method is a method with sufficient descent and need no line search. It is superior to PRP method. It is proved that the search direction of CGPA is feasible-decline under the Wolfe line search and is conjugate under the precise linear search. The convergence of CGPA is analyzed and proved. Some numerical experiments showed that CGPA is feasible, effective and numerical stable.