A non-monotone pattern search approach for systems of nonlinear equations

Area of Science:

• Numerical Analysis
• Optimization Theory

Background:

• Solving systems of nonlinear equations is a fundamental problem in applied mathematics and computational science.
• Existing pattern search methods often face challenges with convergence, particularly in complex optimization landscapes.

Purpose of the Study:

• To introduce a new pattern search algorithm for solving systems of nonlinear equations.
• To develop an adaptive non-monotone strategy that improves convergence properties.

Main Methods:

• A novel non-monotone strategy is proposed, combining preceding successful iterates and the current function.
• The strategy is designed to be stronger when iterates are far from the optimizer and weaker when near it.
• A medium strategy is employed for intermediate iterate positions, bridging existing methods.

Main Results:

• The proposed algorithm demonstrates improved performance in solving systems of nonlinear equations.
• Numerical results confirm the global convergence of the new pattern search method.
• The adaptive non-monotone strategy effectively handles different stages of the optimization process.

Conclusions:

• The new pattern search algorithm with its adaptive non-monotone strategy offers a robust solution for nonlinear equation systems.
• The method's global convergence is theoretically established and numerically validated.
• This work contributes to the advancement of numerical optimization techniques.