A competitive inexact nonmonotone filter SQP method: convergence analysis and numerical results

We propose an inexact nonmonotone successive quadratic programming (SQP) algorithm for solving nonlinear programming problems with equality constraints and bounded variables. Regarding the value of the current feasibility violation and the minimum value of its linear approximation over a trust region, several scenarios are envisaged. In one scenario, a possible infeasible stationary point is detected. In other scenarios, the search direction is computed using an inexact (truncated) solution of a feasible strictly convex quadratic program (QP). The search direction is shown to be a descent direction for the objective function or the feasibility violation in the feasible or infeasible iterations, respectively. A new penalty parameter updating formula is proposed to turn the search direction into a descent direction for an l(1)-penalty function. In certain iterations, an accelerator direction is developed to obtain a superlinear local convergence rate of the algorithm. Using a nonmonotone filter strategy, the global convergence of the algorithm and a superlinear local rate of convergence are guaranteed. The main advantage of the algorithm is that the global convergence of the algorithm is established using inexact solutions of the QPs. Furthermore, the use of inexact solutions instead of exact solutions of the subproblems enhances the robustness and efficiency of the algorithm. The algorithm is implemented using MATLAB and the program is tested on a wide range of test problems from the CUTEst library. Comparison of the obtained numerical results with those obtained by testing some similar SQP algorithms affirms the efficiency and robustness of the proposed algorithm.

Automated summary

The algorithm proposed in this study utilizes inexact solutions of QPs to ensure global convergence, guaranteeing descent direction in feasible and infeasible iterations. Additionally, the introduction of a nonmonotone filter strategy enhances the robustness and efficiency of the algorithm.