A generalized super-memory gradient projection method of strongly sub-feasible directions with strong convergence for nonlinear inequality constrained optimization

In this work, combining the properties of the generalized super-memory gradient projection methods with the ideas of the strongly sub-feasible directions methods, we present a new algorithm with strong convergence for nonlinear inequality constrained optimization. At each iteration, the proposed algorithm can sufficiently use the information of the previous tt steps’ iterations to generate a new iterative point. Particularly, the intervals of parameters in the super-memory gradient projection direction are adjustable. The main properties of the new algorithm are described as follows: (i) the improving super-memory gradient projection direction is a combination of the generalized gradient projection and the tt steps’ super-memory gradients, which include both the previous tt steps’ search directions and gradients; moreover, only the gradients associated with a generalized active constrained set are dealt with rather than the gradients of all constraints; (ii) the initial point can be chosen arbitrarily, and at each iteration, the number of the functions satisfying the inequality constraints is nondecreasing. Especially, once a feasible iteration is obtained, then the subsequent iterations are also feasible; (iii) under suitable assumptions, it possesses global and strong convergence. Finally, some preliminary numerical results show that the proposed algorithm is promising.