On The Global Convergence Of An SLQP Algorithm Without A Penalty Function Or A Filter

We usually use penalty-type methods to solve nonlinear constrained optimization problems, which depend on various penalty functions. But it is very hard to choose the penalty parameter in these penalty-type methods. Indeed, the penalty parameter may result in an ill-conditioned problem numerically. Therefore, it is very important to devise the new method without a penalty function which we called it as penalty-free method. Filter technique, which is designed by Fletcher et. al. in1997, is so far the most classical penalty-free method with satisfactory numerical effect. But, Filter methods also have the insufficient that it needs to retain a filter set at the every iterate point, which may cause a bigger storage. Therefore, the research without a penalty function or a filter is of important significance in theory and practice.In this paper, we propose an SLQP algorithm without a penalty function or a filter to solve nonlinear programming with nonlinear inequality constraints. The com-putation of the trial step is devided into two phases. Firstly, it solves a trust region LP subproblem to obtain an estimate of the active constraints of the NLP problem. Then an equality constrained subproblem with trust region framework (TREQP) is solved. If the TREQP step fails to make progress, then other alternatives such as a second order correction (SOC) step can be tried. If all else fails, then a step based on the LP solution is tried. The acceptable criteria of the algorithm use neither any penalty function nor a filter. For very large scale problems, QP solvers can be less efficient certainly when compared against the performance of linear programming (LP) solvers on problems of similar size. The trust region subproblem with smaller dimensions is also easily solved. Under usual assumptions, we analyse the well definedness and the global convergence of the algorithm.