MATLAB Implementation and Simulation of MMA Algorithm

Simulation Implementation and Performance Analysis of MMA Algorithm Based on MATLAB

The Method of Moving Asymptotes (MMA) is a numerical algorithm widely used for structural optimization problems, particularly suitable for nonlinear optimization problems with constraints. This algorithm progressively approaches the optimal solution by adjusting moving asymptotes, characterized by fast convergence speed and good stability.

Core Algorithm Concept MMA algorithm controls the update magnitude of design variables by introducing upper and lower asymptotes. During each iteration, the algorithm constructs a convex subproblem to approximate the original problem and updates design variables by solving this subproblem. The movement strategy of asymptotes is crucial as it directly affects the algorithm's convergence and efficiency.

MATLAB Implementation Workflow Parameter Initialization: Set initial values for design variables, initial positions of asymptotes, convergence tolerance, and other parameters using initialization functions. Subproblem Construction: Create an approximate convex optimization subproblem based on current design variables and asymptote positions through mathematical modeling. Subproblem Solution: Compute the optimal solution using built-in optimization tools (e.g., `fmincon`) or quadratic programming solvers with proper constraint handling. Asymptote Update: Adjust asymptote positions according to current solutions to ensure better approximation of the original problem through update functions. Convergence Check: Monitor changes in design variables or objective functions against convergence criteria using conditional statements. Return to step 2 if conditions are not met.

Simulation Results Demonstration Typical simulation experiments may include: Structural Topology Optimization: For example, optimizing material distribution of a cantilever beam for minimum compliance. Pre-simulation designs might show uniform material distribution, while post-simulation results clearly demonstrate material concentration in critical stress areas. Objective Function Convergence Curve: Plotting objective function values against iteration numbers visually demonstrates MMA's convergence characteristics using MATLAB's plotting functions.

Performance Comparison Illustration Initial Design Diagram: Displays unoptimized initial structures or parameter distributions. Optimized Result Diagram: Shows optimal designs after algorithm convergence, typically demonstrating significant performance improvements or more rational parameter distributions. Convergence Curve Diagram: Compares iteration processes of objective functions or constraints to verify algorithm stability and efficiency through graphical analysis.

MATLAB implementation of MMA algorithm not only facilitates understanding of mathematical principles but also visually demonstrates optimization effects through graphical results, making it highly suitable for research and educational applications in engineering optimization problems.