A Quadratically Constrained Quadratic Algorithm Model For Nonlinear MINIMAX Problems

It is well known that the sequential quadratic programming (SQP) method is one of the efficient methods for solving smooth constrained optimization problems. So, many authors have applied the idea of SQP method to present effective algorithms for solving the minimax problems. It is a key problem of various SQP methods to overcome the so called Maratos effect under suitable conditions, for example, solves one or more additional quadratic programs or systems of linear equations, or compute explicit correction direction. To overcome the shortcoming of SQP methods, several authors study the sequential quadratically constrained quadratic programming (SQCQP) methods for some special inequality constrained optimization problems.In this paper, a quadratically approximate algorithm framework for solving general constrained minimax problems is presented. The framework contains the idea of SQP method, SQCQP method, norm-relaxed method and strong sub-feasible method. The global convergence of the algorithm framework is obtained under the Mangasarian-Fromovitz constraint qualification (MFCQ), and the conditions for superlinear convergence of the algorithm framework are presented under the MFCQ, the constant rank constraint qualification (CRCQ) as well as the strong second-order sufficiency conditions (SSOSC). And quadratic convergence rate is obtained under the MFCQ and SSOSC.