MATLAB Implementation Examples for Unconstrained Optimization, Linear Programming, and Nonlinear Programming Problems

This implementation example demonstrates how to solve various optimization problems using MATLAB, including unconstrained optimization, linear programming, and nonlinear programming. MATLAB's Optimization Toolbox provides a comprehensive set of functions and tools for optimization analysis and solution finding.

For unconstrained optimization problems, the fminunc function is commonly employed. This function accepts a function handle that defines the objective function to be minimized, along with optional gradient information. The implementation typically requires defining the objective function using an anonymous function or separate function file. fminunc utilizes quasi-Newton methods and line search algorithms to find the minimum point, returning the optimal solution and objective function value.

For linear programming problems, the linprog function is used. This function requires inputs including the linear objective function coefficients, linear inequality and equality constraints, and variable bounds. The implementation involves structuring the problem in the standard form: min c'x subject to A*x ≤ b, Aeq*x = beq, and lb ≤ x ≤ ub. linprog employs the dual-simplex algorithm or interior-point methods to efficiently solve linear programming problems.

For nonlinear programming problems, the fmincon function is utilized. This function handles constrained optimization by accepting an objective function handle along with nonlinear constraint functions. The implementation includes defining both the objective function and constraint functions, which can include nonlinear inequalities and equalities. fmincon uses sequential quadratic programming (SQP) or interior-point algorithms to find solutions that satisfy all constraints while minimizing the objective function.

These examples demonstrate MATLAB's powerful optimization capabilities through practical implementations. The Optimization Toolbox provides robust algorithms and flexible interfaces for solving diverse optimization problems, making it suitable for both academic and industrial applications.