A new ɛ-generalized projection method of strongly sub-feasible directions for inequality constrained optimization

Abstract

In this paper, the nonlinear optimization problems with inequality constraints are discussed. Combining the ideas of the strongly sub-feasible directions method and the ε-generalized projection technique, a new algorithm starting with an arbitrary initial iteration point for the discussed problems is presented. At each iteration, the search direction is generated by a new ε-generalized projection explicit formula, and the step length is yielded by a new Armijo line search. Under some necessary assumptions, not only the algorithm possesses global and strong convergence, but also the iterative points always get into the feasible set after finite iterations. Finally, some preliminary numerical results are reported.

An improved sequential quadratic programming algorithm for solving general nonlinear programming problems

“…Step 5. Solve system of linear equations: Solve (8) to get a solution (d k 1 , h k 1 ), and compute dk by (10) and (11). Step 6.…”

“…The detailed proof of this lemma can be found in [10]. Due to Lemma 3.1, we assume that c k ≡ c for all k in the rest of this paper.…”

“…Note that problem (2) only has inequality constraints, so the corresponding QP subproblem has not equality constraints, and SQP algorithms will be always successful under some suitable conditions. More advantages and further applications of this technique can be seen in [9,10,11].…”

“…The superlinear convergence is derived under the strong second-order sufficient conditions (SSOSC) without the strict complementarity. Moreover, for further comparing the performance of the method of strongly sub-feasible directions (MSSFD) [3,10,12] with the method of quasi-strongly sub-feasible directions (MQSSFD) [14], the technical of MQSSFD is adopted in our new algorithm. Finally, some comparative numerical results are reported to show that our new algorithm is promising.…”

“…By Lemma 2.1 and (4), the relationship between problems(1)and(2)is shown in the following Lemma 2.2. Its proof can be referred to the one of Lemma 2.1(v) in[10]. If c > |π i (x)| for all i ∈ I 2 , then (x, µ) is the KKT point of problem(1) if and only if (x, λ) is the KKT point of problem(2), where µ and λ satisfy…”