Solving Optimization Problems with MATLAB | Master Class with Loren Shure

Lauren Shore welcomes attendees to the masterclass on optimization techniques using MATLAB, noting it is the final webinar in a series. She introduces herself and mentions her extensive experience with MathWorks. Lauren highlights the importance of optimization in various fields and provides everyday examples. She explains the design process for optimization and discusses the challenge of optimizing systems with numerous variables.

Lauren discusses gradient-based optimization techniques, emphasizing the importance of automatic techniques for efficiency. She explains how computers can systematically perform calculations and gradient estimates to find optimal solutions faster than manual methods. She touches on sensitivity and trade-off analysis and the possibility of finding non-intuitive designs, using NASA's antenna design as an example.

Lauren introduces the solver-based approach for optimization in MATLAB, applicable to all optimization solvers in the Optimization Toolbox and Global Optimization Toolbox. She explains the process of specifying objective functions, constraints, and bounds, and how MATLAB returns function values and solution locations. Lauren uses a simple data fitting problem to demonstrate setting up and solving a system of linear equations in MATLAB.

Lauren transitions to the problem-based approach for optimization, initially available for Optimization Toolbox solvers and later for global optimization techniques. She explains the use of optimization variables and constraints and demonstrates solving an example problem using the problem-based approach. Lauren compares solutions from the solver-based and problem-based methods, highlighting the advantages of each.

Lauren delves into data fitting with linear constraints, emphasizing the importance of accurately fitting data points. She shows how to add constraints to ensure certain points fit precisely through the curve. Lauren compares solutions obtained with and without constraints and demonstrates using the problem-based approach to incorporate constraints and achieve optimal solutions.

Lauren discusses non-linear optimization, noting its prevalence in real-world problems. She presents a non-linear function and explains the use of anonymous functions in MATLAB. Lauren demonstrates setting up and solving a non-linear optimization problem using the problem-based approach, highlighting the importance of accurate gradient calculations through automatic differentiation.

Lauren continues with non-linear optimization, showcasing MATLAB's automatic differentiation feature for accurate gradient evaluations. She compares solutions obtained using automatic differentiation and finite differences, illustrating the efficiency and accuracy of automatic differentiation. Lauren also touches on optimization tasks in the live editor and their advantages.

Lauren introduces mixed integer programming, a technique useful for problems requiring some variables to be integers. She explains the traveling salesperson problem as an example and demonstrates setting up and solving the problem using MATLAB. Lauren discusses constraints and iterative methods to eliminate subtours, ensuring an optimal global solution.

Lauren transitions to global optimization, explaining its goal of finding the lowest or highest value of a non-linear function with multiple local minima. She outlines various global optimization solvers, including multi-start, global search, pattern search, genetic algorithms, surrogate optimization, particle swarm, and simulated annealing. Lauren provides a high-level overview of each method and their applications.

Lauren details multi-start and global search techniques, which involve running local solvers from multiple start points to find global minima. She explains how multi-start evenly distributes starting points, while global search filters non-promising points. Lauren demonstrates solving an example problem with multi-start and global search, highlighting their effectiveness in finding optimal solutions.

Lauren explains pattern search, a derivative-free optimization method that evaluates function points around existing points and adjusts search patterns based on success. She demonstrates how pattern search expands and contracts the search grid to focus on optimal areas. Lauren shows the method's effectiveness in sampling the search space and finding global solutions.

Lauren introduces genetic algorithms, which use concepts from evolutionary biology to create and optimize generations of candidate solutions. She explains selection, crossover, and mutation processes in detail and demonstrates solving an optimization problem with a genetic algorithm. Lauren emphasizes the method's ability to work with both smooth and non-smooth functions.

Lauren discusses surrogate optimization, a method suitable for expensive function evaluations. She explains how surrogate functions approximate the actual function locally, allowing for efficient optimization. Lauren demonstrates solving an optimization problem with surrogate optimization, highlighting its ability to explore and refine solutions adaptively.

Lauren introduces particle swarm optimization, which simulates the behavior of flocks of birds or schools of fish. She explains how particles influence each other's velocities and positions to find optimal solutions. Lauren demonstrates the method, showing how particles explore the search space and converge on the global minimum.

Lauren provides an overview of various global optimization solvers, comparing their suitability for different types of problems. She summarizes the key features of each method, including gradient-based solvers, derivative-free solvers, and specialized solvers for multi-objective optimization. Lauren emphasizes the importance of choosing the right solver for the problem at hand.

Lauren discusses the benefits of parallel computing for speeding up optimization problems. She explains how parallel computing can evaluate multiple points simultaneously, improving efficiency. Lauren highlights the use of MATLAB's parallel computing tools and the potential for scaling up to larger clusters or cloud-based solutions.

Lauren covers the deployment of optimization algorithms using MATLAB Coder for generating C and C++ code. She explains how this allows optimization solutions to be applied to various hardware platforms. Lauren encourages users to provide feedback and suggestions for expanding MATLAB's capabilities in this area.

Lauren concludes the masterclass by summarizing key takeaways, including the variety of optimization problems that can be solved with MATLAB, the use of global optimization techniques, and the advantages of parallel computing and code generation. She provides resources for further learning and encourages participants to ask final questions. Lauren thanks the attendees and highlights upcoming events and challenges.