MATLAB Implementation of Optimization Methods: BFGS and SUMT Penalty Function Approach

This article explores optimization methods and their practical implementation in MATLAB assignments. Optimization methods comprise mathematical techniques designed to identify the maximum or minimum values of functions, including gradient descent, Newton's method, and quasi-Newton methods. We focus specifically on two key algorithms: BFGS and the SUMT penalty function method. The BFGS algorithm, a quasi-Newton approach, iteratively approximates the inverse Hessian matrix to update parameter estimates without computing second derivatives directly - in MATLAB this typically involves maintaining an approximation matrix updated using gradient differences. The SUMT (Sequential Unconstrained Minimization Technique) penalty function method transforms constrained optimization problems into unconstrained ones by incorporating constraint violations into the objective function through penalty terms; MATLAB implementations often use logarithmic or quadratic penalty functions with progressively increasing penalty parameters. These methodologies provide effective solutions for both unconstrained and constrained optimization scenarios, with MATLAB's fminunc function optionally implementing BFGS and custom penalty functions being implementable for SUMT. This content aims to deepen your understanding of optimization techniques through practical computational perspectives.