Research On Generalized Gradient Projection And QP-free Algorithms For Inequality Constrained Minimax Problems

In this paper, generalized gradient projection and QP-free algorithms for inequality constrained minimax problems are discussed. The main works are illustrated as follows:First of all, combining the idea of generalized gradient projection and a new working set, a generalized gradient projection algorithm is proposed for solving inequality constrained minimax problems. The proposed algorithm starts with a feasible initial point. At each iteration, by means of the new working set, the improve direction, which is feasible and descent, is gener-ated by one generalized gradient projection explicit formula. What’s more, the mode of updating the new working set can ensure the projection matrix simple, which can further reduces the computational cost. Under some nec-essary assumptions, the algorithm possesses global convergence and strong convergence.In addition, it will increase the computational cost to get a feasible ini-tial point and the inverse of generalized projection matrixes. In chapter 4, to overcome the defects, motivated by the idea of quasi-strongly sub-feasible direction, a QP-free algorithm with arbitrary initial point is presented. At each iteration, the feasible and descent direction is obtained by solving two sequential systems of linear equations (SSLE) with the same coefficient ma-trix. After several iterations, the elements in the lower right corner of the regressive matrixes are all zero. This makes the coefficient matrix sparse and simple such that the computational cost is greatly reduced. In the line search, quasi-strongly sub-feasible direction algorithm is used to increase the feasi-bility of the iteration point sequence. Under some necessary assumptions, global convergence and strong convergence of the algorithm are proved.Finally, some numerical results are reported, which show that the pro-posed two algorithms are effective.