A new filter QP-free method for the nonlinear inequality constrained optimization problem

Area of Science:

• Optimization Theory
• Mathematical Programming

Background:

• Minimizing smooth optimization problems with inequality constraints is a fundamental challenge.
• Existing methods often require feasibility or struggle with incompatibility issues.

Purpose of the Study:

• To propose a new filter QP-free infeasible method for constrained optimization.
• To address the incompatibility issue in optimization algorithms.

Main Methods:

• The method is based on solving nonsmooth equations derived from Karush-Kuhn-Tucker (KKT) conditions using Lagrangian multipliers and nonlinear complementarity functions.
• It employs a filter technique incorporating the nonlinear complementarity problem function to avoid incompatibility.
• Each iteration perturbs Newton or quasi-Newton iterations for primal and dual variables.

Main Results:

• The proposed method demonstrates global convergence.
• Under mild conditions, superlinear convergence rates are achievable.
• Preliminary numerical results indicate the method's promise.

Conclusions:

• The developed filter QP-free infeasible method is a viable and promising approach for constrained optimization.
• The integration of nonlinear complementarity functions within the filter enhances robustness.