A fast convergent sequential linear equation method for inequality constrained optimization without strict complementarity

In this paper, a type of smooth nonlinear optimization problems with inequality constraints is considered, and a new sequential linear equation algorithm is proposed by introducing a new constructing technique for the system of linear equations (SLE), which depends on the perturbation of the constraints’ gradients. At each iteration of the proposed algorithm, one or more SLEs need to be solved. Specially, only one SLE is required to be solved when the iterates are sufficiently close to the solution (i.e., after a finite number of iterations), which decreases the amount of computations. The search technique is a combination of the line Armijo-type and Newton step size. Under mild assumptions without the strict complementarity, it is shown that the proposed algorithm enjoys the properties of global and superlinear convergence. Finally, some preliminary numerical tests are reported, and the numerical results show that the proposed algorithm is promising.