MATLAB Optimization Algorithms: Mastery and Practice

# 1. Overview of MATLAB Optimization Algorithms

MATLAB optimization algorithms play a crucial role in various fields such as engineering, economics, and scientific research. These algorithms integrate a variety of mathematical programming tools, providing solutions from simple to complex problems. Their applications are not limited to academic research but are also widely used in solving practical engineering problems.

In this chapter, we will introduce the definition, basic functions, and significance of MATLAB optimization algorithms in solving real-world problems. The content will cover the types of optimization algorithms, application scenarios, and why MATLAB is chosen as the platform for solving optimization problems. By studying this chapter, readers will gain a preliminary understanding of optimization algorithms and lay a foundation for further in-depth learning in subsequent chapters.

# 2. Basic Optimization Theories in MATLAB

## 2.1 Mathematical Foundation of Optimization Problems

### 2.1.1 Linear Programming and Nonlinear Programming

Linear programming is one of the most fundamental and extensively studied mathematical branches in optimization theory. It involves finding the maximum or minimum of a linear function subject to a series of linear inequalities or equations. Linear programming problems are often represented in the following form:

\[
\begin{align*}
\text{minimize} \quad & c^T x \\
\text{subject to} \quad & A x \leq b \\
& A_{eq} x = b_{eq} \\
& l \leq x \leq u
\end{align*}
\]

Where \( c \) is the vector of objective function coefficients, \( A \) and \( b \) are the coefficient matrix and vector of constraints, \( A_{eq} \) and \( b_{eq} \) represent the equality constraints, and \( l \) and \( u \) are the lower and upper bounds of the variables, ***mon methods to solve linear programming problems include the Simplex Method and Interior-Point Methods.

Nonlinear programming involves optimization problems where the objective function and/or constraints contain nonlinear functions. Its general form is as follows:

\[
\begin{align*}
\text{minimize} \quad & f(x) \\
\text{subject to} \quad & g_i(x) \leq 0, \quad i = 1, \ldots, m \\
& h_j(x) = 0, \quad j = 1, \ldots, p \\
\end{align*}
\]

Where \( f(x) \) is the nonlinear objective function to be minimized, \( g_i(x) \) are the inequality constraints, and \( h_j(x) \) are the equality constraints. Solving nonlinear programming problems is more complex, often using algorithms such as Gradient Descent, Newton's Method, and Sequential Quadratic Programming (SQP).

% MATLAB linear programming example
f = [-1; -1]; % Objective function coefficients
A = [1, 2; 1, -1; -1, 2]; % Constraint matrix
b = [2; 2; 3]; % Constraint condition right-hand values
lb = zeros(2,1); % Variable lower bounds
[x, fval] = linprog(f, A, b, [], [], lb); % Calling linprog function to solve

In the above code, the `linprog` function is used to solve linear programming problems. `f` is the vector of objective function coefficients, `A` and `b` define the inequality constraints, `lb` specifies the variable lower bounds, `x` is the optimal solution, and `fval` is the value of the objective function at the optimal solution.

### 2.1.2 Integer Programming and Combinatorial Optimization

Integer programming is an extension of linear programming where the decision variables are required to be integers. This makes the problem more complex as integer programming problems are classified as NP-hard problems. The general form of an integer programming problem is as follows:

\[
\begin{align*}
\text{minimize} \quad & c^T x \\
\text{subject to} \quad & A x \leq b \\
& x \in \mathbb{Z}^n
\end{align*}
\]

Where \( x \in \mathbb{Z}^n \) indicates that all decision variables must be integers***

***binatorial optimization refers to finding the optimal solution in a set of a finite number of elements, commonly found in graph theory, scheduling, transportation, ***binatorial optimization problems typically have a discrete solution space, with common examples including the Traveling Salesman Problem (TSP) and the Knapsack Problem.

% MATLAB integer programming example
f = [-1; -1]; % Objective function coefficients
A = [1, 2; 1, -1; -1, 2]; % Constraint matrix
b = [2; 2; 3]; % Constraint condition right-hand values
intcon = 1:n; % Specify which variables are integer variables
[x, fval] = intlinprog(f, intcon, A, b); % Calling intlinprog function to solve

In the code above, the `intlinprog` function is used to solve integer linear programming problems. `f`, `A`, `b`, `intcon` represent the objective function coefficients, constraint matrix, right-hand side of constraint conditions, and integer variable indices, respectively. The result is the integer vector `x` and the minimum value of the objective function at this vector `fval`.

## 2.2 Overview of MATLAB Optimization Toolbox

### 2.2.1 Functions and Commands in the Toolbox

The MATLAB optimization toolbox provides a series of functions and commands for solving linear, nonlinear, integer, binary, quadratic, and goal programming problems. These functions and commands are designed to provide users with efficient optimization algorithms and help them quickly solve problems.

- `fmincon`: Used to solve constrained nonlinear optimization problems.
- `linprog`: Used to solve linear programming problems.
- `intlinprog`: Used to solve integer linear programming problems.
- `quadprog`: Used to solve quadratic programming problems.
- `bintprog`: Used to solve binary linear programming problems.

These tool functions typically require users to specify the objective function, nonlinear constraints (if any), linear inequalities and equalities constraints, and the upper and lower bounds of variables.

### 2.2.2 Working Principle and Architecture of the Toolbox

The working principle behind the MATLAB optimization toolbox is the use of advanced algorithms and mathematical modeling techniques to solve optimization problems. The toolbox encapsulates algorithm implementations into a series of function interfaces, making it convenient for users to call. Its architecture includes several key components:

- **Problem Modeling**: Allows users to express their optimization problems mathematically.
- **Solver Selection**: Chooses the appropriate solver based on the problem type (linear, nonlinear, etc.).
- **Parameter Optimization**: Allows users to customize algorithm parameters, such as convergence conditions, iteration numbers, etc.
- **Result Output**: Provides visualization and analysis tools for results to help users understand the solutions.

% MATLAB using quadprog function to solve a quadratic programming problem
H = [1, -1; -1, 2]; % Quadratic term coefficient matrix
f = [-7; -12]; % Linear term coefficient vector
A = [1, 1; -1, 2; 2, 1]; % Linear inequality constraints
b = [2; 2; 3]; % Constraint condition right-hand values
lb = zeros(2,1); % Variable lower bound
[x, fval, exitflag, output] = quadprog(H, f, A, b, [], [], lb); % Calling quadprog function to solve

In the above example, the `quadprog` function is used to solve a quadratic programming problem. `H` is the quadratic term coefficient matrix of the objective function, `f` is the linear term coefficient vector, `A` and `b` define the linear constraint conditions, and `lb` specifies the variable lower bound. After solving, `x` stores the optimal solution, `fval` stores the value of the objective function at the optimal solution, `exitflag` and `output` represent the algorithm exit flag and output information, respectively.

## 2.3 Understanding Objective Functions and Constraints

### 2.3.1 How to Define an Objective Function

In MATLAB, defining an objective function is usually done by writing a separate function file that calculates and returns the value of the objective function. For the optimization toolbox, the objective function should accept a vector `x` as input and return a scalar value as output.

function f = myObjectiveFunction(x)
     f = (x(1) - 1)^2 + (x(2) - 2)^2; % Example objective function
end

### 2.3.2 Setting and Processing Constraints

Constraints are typically defined in two forms: functional form and linear form. Nonlinear constraints are implemented by defining a function that returns a two-element vector, where the first element contains the inequality constraints \( g_i(x) \), and the second element contains the equality constraints \( h_j(x) \).

function [c, ceq] = myConstraints(x)
     c = [1.5 + x(1)*x(2) - x(1) - x(2); ... % Inequality constraints
          x(1)*x(2) - 10]; 
     ceq = [x(1) + x(2) - 10]; % Equality constraints
end

Linear inequalities and equalities can be directly defined in matrix form and passed to the optimization function as function parameters.

For mixed constraint problems, a combination of the above two methods can be used. In this way, users can flexibly define complex optimization problems and use MATLAB's optimization toolbox to solve them.

# 3. Practical Applications of MATLAB Optimization Algorithms

In this chapter, we will delve into the specific performance of MATLAB optimization algorithms in practical applications. By demonstrating real-world cases of linear and nonlinear optimization, exploring advanced optimization techniques such as multi-objective optimization and genetic algorithms, and presenting a specific engineering case to show how MATLAB is used for optimization in projects. Through hands-on operations, we will reveal how these optimization algorithms transform into powerful tools for solving real-world problems.

## 3.1 Demonstration: Linear and Nonlinear Optimization Examples

Linear and nonlinear optimization are the cornerstones of problem-solving in MATLAB. We will begin by understanding the mathematical foundations of these two types of optimization problems and demonstrate how to implement them in MATLAB through case studies.

### 3.1.1 Practical Application of Linear Optimization

Linear programming, as a method to optimize a linear objective function subject to a set of linear constraints, is widely used in production scheduling, logistics transportation, financial investment, and other fields. This section will present a simple linear optimization problem and introduce how to use MATLAB's `linprog` function to solve it.

**Case Background**: Suppose there is a factory producing two products A and B. Each unit of product A and B requires different resources, and the factory's resources are limited. The goal is to maximize profits.

f = [-2; -1]; % Objective function coefficients, indicating that profits need to be maximized, so negative signs are used
A = [1, 2; 1, 0; 0, 1]; % Constraint condition coefficient matrix
b = [100; 50; 30]; % Resource limits
lb = [0; 0]; % Variable lower bounds, indicating that the quantity of products cannot be negative

[x, fval] = linprog(f, A, b, [], [], lb)

**Parameter Explanation**:
- `f`: Objective function coefficients, representing the profit contributions of products A and B.
- `A`, `b`: Matrices and vectors defining the linear inequality constraints.
- `lb`: Lower bounds for variables,这里是0，表示产品数量不能为负。
- `x`: Optimization results, indicating the production quantity of each product.
- `fval`: Optimal value obtained during the optimization process, i.e., the maximum profit.

Through the above code, we obtain the optimal production quantities of products A and B and the maximum profit.

### 3.1.2 Case Analysis of Nonlinear Optimization

Nonlinear optimization problems involve a broader range of fields, ***pared to linear optimization, nonlinear optimization is often more complex and may have multiple local optimal solutions.

**Case Background**: In the financial market, how can we optimize the asset portfolio to minimize risk while achieving the desired return?

f = @(x) 0.5 * x' * P * x - r' * x; % Risk minimization objective function
A = ones(1, n); % Ensure the total allocation of funds is 1
Aeq = [];
beq = 1;
lb = zeros(n, 1); % Investment ratios cannot be less than 0
ub = ones(n, 1); % Investment ratios cannot be greater than 1

x0 = ones(n, 1) / n; % Initial investment ratio
options = optimoptions('fminunc', 'Algorithm', 'quasi-newton');
[x, fval] = fminunc(f, x0, options, A, b, Aeq, beq, lb, ub)

**Parameter Explanation**:
- `f`: Defines the risk minimization objective function.
- `A`, `b`, `Aeq`, `beq`: Defines linear equality and inequality constraints.
- `lb`, `ub`: Defines the lower and upper bounds of variables, i.e., the limits on investment ratios.
- `x0`: The initial solution for optimization.
- `x`: Optimization results, indicating the optimal investment ratios for the asset portfolio.
- `fval`: Optimal value obtained during the optimization process, i.e., risk minimization.

The above code demonstrates the solution process of a nonlinear optimization problem, where MATLAB's optimization function `fminunc` is used to find the asset portfolio configuration that minimizes risk.

## 3.2 Application of Advanced Optimization Techniques

In many complex practical applications, multiple objectives need to be considered simultaneously when solving optimization problems. This often goes beyond the scope of simple linear and nonlinear optimization and requires the use of more advanced optimization techniques.

### 3.2.1 Multi-objective Optimization Methods

Multi-objective optimization involves simultaneously optimizing multiple objective functions, which may be conflicting with each other. MATLAB provides some specialized tools to deal with such problems, such as the `gamultiobj` function.

**Case Background**: In the product design process, we want to minimize costs and maximize product performance, which are often conflicting objectives.

f = @(x) [x(1)^2 + x(2)^2; (1-x(1))^2 + (1-x(2))^2]; % Objective function
A = []; b = [];
Aeq = []; beq = 1;
lb = [0, 0]; ub = [1, 1];

options = optimoptions('gamultiobj', 'PlotFcn', @gaplotpareto);
[x, fval] = gamultiobj(f, 2, [], [], A, b, Aeq, beq, lb, ub, options);

**Parameter Explanation**:
- `f`: Defines two objective functions representing cost and performance.
- `options`: Uses `PlotFcn` to plot the Pareto front, helping users choose suitable solutions.

This method can find trade-off solutions between multiple objective functions, i.e., the set of Pareto optimal solutions.

### 3.2.2 Genetic Algorithms and Evolutionary Strategies

Genetic algorithms and evolutionary strategies are heuristic search algorithms inspired by the principle of natural selection, suitable for global optimization of complex problems. MATLAB's `ga` function provides a general framework for implementing genetic algorithms.

**Case Background**: Optimizing a complex engineering problem involving multiple design variables and complex constraints.

% The definition of the objective function and constraints is omitted; see the previous parts
options = optimoptions('ga', 'PopulationSize', 100, 'MaxGenerations', 200);
[x, fval] = ga(f, 2, A, b, Aeq, beq, lb, ub, [], options);

**Parameter Explanation**:
- `options`: Parameter settings include population size and maximum number of iterations.

Genetic algorithms evolve new populations in each generation and select better individuals based on fitness, ultimately converging to a global optimal solution.

## 3.3 Engineering Case: Using MATLAB for Project Optimization

In this section, we comprehensively demonstrate MATLAB's optimization applications in real projects through a specific engineering optimization problem, from modeling to solving, and then to result analysis.

### 3.3.1 Modeling of Engineering Optimization Problems

We take the factory production scheduling as an example to introduce how to build an optimization model in MATLAB.

% Assume a factory has n machines and m jobs, with specific constraints between jobs and machines
% Objective function: Minimize the time to complete all jobs

% Define variables
n = 5; % Number of machines
m = 10; % Number of jobs
x = sdpvar(m, n, 'nonnegative'); % Define the decision variable matrix

% Define objective function and constraints
obj = sum(max(x)) + sum(sum(x)); % Minimize completion time and waiting time
constraints = [sum(x, 2) <= ones(m, 1); x >= 0]; % Machine capacity and job time constraints

% Optimization solution
options = sdpsettings('verbose', 1, 'solver', 'glpk');
sol = optimize([obj, constraints], [], options);

% Output results
disp('Shortest completion time:');
disp(value(obj, sol));

**Parameter Explanation**:
- `x`: Decision variable matrix, representing the start times of jobs on different machines.
- `obj`: Objective function, assuming the minimization of the sum of completion time and waiting time.
- `constraints`: Defines the scheduling constraints of jobs on machines.
- `options`: Sets the solver and output options.

### 3.3.2 Algorithm Selection and Result Analysis

After modeling, selecting the appropriate optimization algorithm is crucial. In this case, we used the `optimize` function, which is a general SDP solver interface that can call various solvers for optimization calculations.

% Analyze results
job_order = find(value(x, sol)); % Determine the job scheduling order based on optimization results
plot(job_order); % Visualize the job scheduling results

By analyzing the solution results, we can draw out the job scheduling order, thereby optimizing the factory's production scheduling.

In this chapter, we first demonstrated linear and nonlinear optimization examples, then introduced how to use MATLAB for multi-objective optimization and apply heuristic algorithms, and finally comprehensively showcased MATLAB's applications in real project optimization through an engineering case. Through these specific case analyses, we demonstrate the powerful capabilities of MATLAB optimization toolboxes and their great potential in solving real-world problems.

# 4. Advanced Topics in MATLAB Optimization Algorithms

The advanced topics in MATLAB optimization algorithms involve algorithm design, parallel computing, performance evaluation, and more, which are crucial for advanced users who wish to deeply understand and expand the functionality of MATLAB optimization toolboxes.

## 4.1 Custom Optimization Algorithms

Custom optimization algorithms are the ability of MATLAB users to extend the optimization toolbox's functionality through programming when facing specific problems. This not only requires a thorough understanding of the MATLAB language but also a certain level of algorithm design capability.

### 4.1.1 Writing Custom Functions

Writing custom optimization functions first requires modeling the problem, clarifying the objective function and constraints, and transforming them into a form recognizable by MATLAB. Then, we can implement the algorithm using MATLAB's programming language.

function [x, fval] = myOptimizationFunction(x0, options)
% Custom optimization function, accepts initial guess solution x0 and options structure
% Outputs the solution x and objective function value fval

% Objective function definition (example)
objective = @(x) (x(1) - 1)^2 + (x(2) - 2)^2;

% Use fminunc function for optimization
options = optimoptions(options, 'Algorithm', 'quasi-newton');
[x, fval] = fminunc(objective, x0, options);
end

In the above code, we defined a custom optimization function named `myOptimizationFunction`, which uses the built-in `fminunc` function to find the minimum value of the given objective function. We also set the optimization algorithm to the quasi-Newton method. This code illustrates how to encapsulate optimization logic and accept input parameters to implement a custom optimization algorithm.

### 4.1.2 Algorithm Verification and Testing

The written custom optimization algorithm requires strict verification and testing. Verification is a test of the theoretical correctness of the algorithm, while testing is an examination of the algorithm's actual performance. In MATLAB, we can use the built-in testing framework or manually designed test cases to complete this process.

% Test case
x0 = [0, 0]; % Initial guess solution
options = optimoptions('fminunc', 'Display', 'iter'); % Set optimization options

[x, fval] = myOptimizationFunction(x0, options);

% Display results
fprintf('Solution: x = [%f, %f]\n', x(1), x(2));
fprintf('Objective function value: %f\n', fval);

In the above code, we created a test case that uses the custom optimization function `myOptimizationFunction` to find the minimum value of the objective function. The test results will display the solution and the objective function value, thus verifying the effectiveness of the algorithm.

## 4.2 Application of Parallel Computing in Optimization

With the prevalence of multi-core processors, improving algorithm performance through parallel computing has become possible. MATLAB provides a parallel computing toolbox that can help users implement parallel processing in optimization algorithms.

### 4.2.1 Basics of Parallel Computing

The basics of parallel computing include understanding how to identify and use multiple cores in MATLAB and how to write code that can utilize these cores.

### 4.2.2 Using MATLAB Parallel Toolbox

MATLAB's parallel toolbox allows users to extend their computing power through clustering and distributed computing.

function [results] = parallelOptimization(func, x0, options)
% Parallel optimization function, func is the optimization objective function, x0 is the initial guess solution, options is the options structure

% Create a parallel pool
pool = parpool;
parforIdx = 1 : pool.NumWorkers; % Allocate workload

% Use parfor for parallel computation
results = zeros(size(parforIdx));
parfor i = parforIdx
     % Run optimization on each worker
     results(i) = func(x0(i), options);
end

% Close the parallel pool
delete(pool);
end

In this function, we use `parpool` to create a parallel pool and use the `parfor` loop to execute optimization tasks on different workers. This can significantly reduce execution time, especially when dealing with large-scale optimization problems.

## 4.3 Performance Evaluation and Comparison of Optimization Algorithms

The performance evaluation and comparison of optimization algorithms are key factors in deciding which algorithm to choose. Performance evaluation typically involves multiple indicators, including convergence speed, accuracy, stability, and computational resource consumption.

### 4.3.1 Performance Evaluation Indicators

In performance evaluation, we need to pay attention to the following indicators:

- **Convergence Speed**: The number of iterations required for the algorithm to reach the optimal solution.
- **Accuracy**: The difference between the solution found by the algorithm and the global optimal solution.
- **Stability**: Whether the algorithm's performance is consistent under different initial conditions.
- **Computational Resource Consumption**: The memory and time required for algorithm execution.

### 4.3.2 Algorithm Comparison and Selection Strategy

By comparing the performance evaluation indicators of different algorithms, we can choose the most suitable algorithm to solve a specific problem. The selection strategy is usually based on the characteristics of the problem, such as the scale of the problem, the nature of the objective function, and the available computational resources.

% Compare the performance of different optimization algorithms

% Define two objective functions
objective1 = @(x) sum(x.^2);
objective2 = @(x) sum((x-1).^2);

% Use different optimization algorithms
methods = {'fminunc', 'fmincon'}; % List of optimization function names
results = struct();

for i = 1:length(methods)
     method = methods{i};
     % Optimize for each method
     % Omit the optimization process code...
     % Save the results
     results.(method) = optimResult;
end

% Analyze results
% Omit the analysis code...

This code demonstrates how to define different objective functions and use different optimization algorithms (such as `fminunc` and `fmincon`) to optimize these functions. Then, the results are stored in a structure and compared.

Through the above content, this chapter has thoroughly discussed the advanced topics of MATLAB optimization algorithms, covering the writing and testing of custom algorithms and the importance of parallel computing and performance evaluation. These advanced topics are essential for deeply understanding and expanding MATLAB's optimization capabilities.

# 5. Future Trends and Developments of MATLAB Optimization Algorithms

With technological advancements and in-depth research, the field of MATLAB optimization algorithms also presents new challenges and opportunities. This chapter will explore the challenges optimization algorithms face in the current environment, applications in emerging fields, and predictions and prospects for future development.

## 5.1 Current Challenges and Opportunities for Optimization Algorithms

In the era of big data, optimization problems are becoming more complex, and traditional optimization algorithms may struggle to adapt to large-scale and high-dimensional datasets. Therefore, seeking more effective optimization strategies is the primary challenge at present.

### 5.1.1 Optimization Problems in the Big Data Environment

In the context of big data, optimization algorithms need to process an increasing number of variables and constraints, which poses higher requirements for the computational efficiency and quality of results. At the same time, the noise and incompleteness that may exist in the data also challenge the robustness of optimization algorithms.

To address these challenges, researchers are developing algorithms capable of handling large datasets. For example, stochastic gradient descent (SGD) and its variants have been widely applied in the field of machine learning for large-scale optimization problems. These algorithms can more efficiently approximate the optimal solution by updating with subsampled data.

### 5.1.2 The Rise of Intelligent Optimization Algorithms

Intelligent optimization algorithms, including genetic algorithms, particle swarm optimization, ant colony algorithms, etc., are valued for their ability to effectively solve problems that traditional algorithms struggle with. These algorithms simulate natural evolution or social behavior, possessing a high degree of randomness and flexibility, and are particularly suitable for solving complex, multimodal, and nonlinear optimization problems.

MATLAB has integrated some intelligent optimization algorithms, but continuous research will bring improvements to these algorithms and new algorithm variants. By combining machine learning and other artificial intelligence technologies, the performance of future optimization algorithms is expected to be significantly improved.

## 5.2 Exploring MATLAB's Applications in Emerging Fields

MATLAB, as a powerful mathematical software, is not only widely used in the field of engineering optimization but also shows its potential in emerging interdisciplinary research areas.

### 5.2.1 The Intersection of Machine Learning and Deep Learning

Machine learning and deep learning are transforming the way data analysis and processing are conducted, and optimization algorithms play a vital role in these areas. MATLAB provides a series of machine learning and deep learning toolboxes, and the optimization algorithms embedded in these toolboxes can help build and train complex models.

A typical example is during the training of neural networks, where optimization algorithms adjust network weights to minimize the loss function. MATLAB supports automatic differentiation, greatly simplifying this process and enabling researchers and engineers to develop and apply deep learning models more efficiently.

### 5.2.2 MATLAB's Role in Interdisciplinary Research

MATLAB is not limited to the fields of engineering and computer science but also plays an important role in interdisciplinary research such as bioinformatics, financial analysis, and environmental science. These areas often need to solve complex optimization problems, such as finding the optimal sequence in gene sequencing, performing risk management in financial models, and optimizing resource allocation in environmental planning.

MATLAB's versatility and flexibility make it an ideal tool for interdisciplinary research. By integrating expertise from different fields, MATLAB can provide a one-stop solution from data processing to optimization algorithm implementation.

## 5.3 Predictions and Prospects

As technology advances and problems become more complex, MATLAB's optimization toolbox is also constantly evolving. This section will look forward to the potential update directions of MATLAB's optimization toolbox in the future and the development of community support.

### 5.3.1 Future Update Directions of MATLAB Optimization Toolbox

With the improvement of computer hardware performance and in-depth algorithm research, the future MATLAB optimization toolbox may integrate more efficient and intelligent algorithms. This includes optimizing existing algorithms to improve their performance on large-scale problems and introducing new algorithms to cover a broader range of application scenarios.

In addition, improving the user interface and enhancing the user experience are also key directions for the future. MATLAB may provide more visualization tools and interactive elements, allowing users to understand and use the optimization toolbox more intuitively.

### 5.3.2 Development of Learning Resources and Community Support

To help users better utilize MATLAB's optimization toolbox, future learning resources and community support will become richer. This includes updating official documentation, increasing online tutorials, and activating user forums and Q&A communities.

Open-source projects and case studies may become important learning resources, which will be created and maintained by the MATLAB user community. By sharing and discussing optimization cases from different fields, users can not only improve their skills but also contribute their own knowledge and experience.