Constrained Optimization Using Direct Methods - Complex Method

This article explores MATLAB programming applications for constrained optimization problems, with focus on direct search methods—specifically the Complex Method. We begin by examining the definition and theoretical background of these optimization techniques to establish a solid foundation for their practical implementation. The discussion then proceeds to detailed MATLAB implementation strategies, including code development for vertex generation, reflection operations, and constraint handling mechanisms. Key implementation aspects cover the Complex Method's iterative process where MATLAB functions manage geometric transformations and feasibility checks. The algorithm maintains a simplex-like structure (complex) that evolves through reflection, expansion, and contraction operations while respecting nonlinear constraints. Finally, we demonstrate practical application case studies illustrating how these methods solve engineering design problems, including parameter optimization with performance constraints and design variable bounds. Each case includes MATLAB code snippets showing constraint formulation and convergence monitoring techniques.