**Course Title:** MATLAB Programming: Applications in Engineering, Data Science, and Simulation **Section Title:** Optimization and Nonlinear Systems **Topic:** Solve real-world optimization problems using MATLAB's optimization toolbox. (Lab topic) **Objective:** In this lab, you will learn how to solve real-world optimization problems using MATLAB's optimization toolbox. You will explore various techniques and tools to model and solve optimization problems, and apply them to practical examples. **Pre-requisites:** * Review of optimization concepts: unconstrained and constrained optimization, multi-variable and multi-objective optimization (covered in the previous topics) * Familiarity with MATLAB's optimization toolbox **Materials Needed:** * MATLAB software ( Release 2022a or later) * Optimization toolbox * Example files and datasets (available for download) **Introduction:** Optimization is the process of finding the best solution among a set of possible solutions, given a set of constraints and objectives. In engineering, data science, and many other fields, optimization problems are ubiquitous. MATLAB's optimization toolbox provides a robust set of tools to model and solve optimization problems. **Step 1: Defining the Optimization Problem** The first step in solving an optimization problem is to define the problem mathematically. This involves specifying the objective function, constraints, and any other relevant parameters. Example: Suppose we want to optimize the design of a cantilever beam to minimize the stress at the point of attachment. The beam is subject to a load of 100 N and has a length of 1 m. The objective function is the stress at the point of attachment, and the constraint is that the beam's width must be greater than or equal to 0.1 m. **Step 2: Choosing the Optimization Algorithm** MATLAB's optimization toolbox provides a range of algorithms to solve optimization problems, including: * `linprog`: Linear programming * `quadprog`: Quadratic programming * `fmincon`: Constrained nonlinear optimization * `fminsearch`: Unconstrained nonlinear optimization For this example, we will use `fmincon` to solve the constrained nonlinear optimization problem. **Step 3: Implementing the Optimization Model** Once we have defined the optimization problem and chosen the algorithm, we can implement the optimization model in MATLAB. This involves writing a function that computes the objective function and constraints. Example: ```matlab % Define the objective function function stress = objfun(x) % Define the beam's properties L = 1; % length (m) P = 100; % load (N) w = x(1); % width (m) h = x(2); % height (m) % Compute the stress at the point of attachment stress = P / (w * h); end % Define the constraint function function [c, ceq] = confun(x) % Define the minimum width constraint c = x(1) - 0.1; % width must be greater than or equal to 0.1 m ceq = []; % no equality constraints end ``` **Step 4: Running the Optimization** Once we have implemented the optimization model, we can run the optimization using the `fmincon` function. Example: ```matlab % Define the optimization problem objfun = @objfun; confun = @confun; x0 = [0.1; 0.1]; % initial guess A = [1 0]; % constraint matrix b = 0.1; % constraint vector % Run the optimization opts = optimoptions(@fmincon, 'Display', 'iter'); [x, fval] = fmincon(objfun, x0, A, b, [], [], [], [], confun, opts); ``` **Conclusion:** In this lab, you learned how to solve real-world optimization problems using MATLAB's optimization toolbox. You explored the process of defining the optimization problem, choosing the optimization algorithm, implementing the optimization model, and running the optimization. **Practice Problem:** * Download the example files and datasets for this lab. * Modify the optimization model to optimize the design of a rectangular beam instead of a cantilever beam. * Use `fmincon` to solve the constrained nonlinear optimization problem. **Resources:** * MATLAB Optimization Toolbox documentation: <https://www.mathworks.com/help/optim/> * Optimization examples: <https://www.mathworks.com/help/optim/examples.html> **Leave a Comment/Ask for Help:** If you have any questions or need help with this lab, please leave a comment below. If you have completed the lab and would like to share your results or feedback, please also leave a comment. What's Next? In the next topic, we will introduce digital image processing with MATLAB. You will learn how to read and write images, apply filters and transformations, and perform feature extraction and analysis. Topic: Introduction to Digital Image Processing with MATLAB (From: Image Processing and Signal Processing)