Comparison of fmincon and Genetic Algorithm: Performance Differences in Different Problems

# 1. Introduction to Optimization Algorithms

Optimization algorithms are mathematical tools used to find the optimal values of a given function. They are extensively applied in various fields, such as engineering, finance, and data science. There are numerous types of optimization algorithms, each with its unique principles and applications. This chapter will introduce the fundamental concepts of optimization algorithms, including their classification, principles, and applications.

# 2. The fmincon Algorithm

### 2.1 Principles of the fmincon Algorithm

The fmincon algorithm is a solver in MATLAB designed for solving nonlinear constrained optimization problems. It employs the interior-point method to solve the objective function. The interior-point method is an iterative algorithm that approximates the optimal solution by searching within the feasible domain.

Specifically, the fmincon algorithm first converts the constraints into a set of inequality constraints. It then approximates the optimal solution by solving a series of quadratic programming problems. During each iteration, the algorithm updates a function known as the barrier function, which penalizes points on the feasible domain boundary to infinity. By minimizing the barrier function, the algorithm gradually guides the search point into the interior of the feasible domain.

### 2.2 Setting Parameters for the fmincon Algorithm

The fmincon algorithm offers a variety of parameters for users to set, controlling the operation of the algorithm. Here are some commonly used parameters:

- **Algorithm:** Specifies the algorithm used by the solver, including interior-point, active-set, and trust-region methods.
- **Display:** Specifies the level of output during the algorithm's operation, including "off," "iter," "final," and "notify."
- **MaxFunEvals:** Specifies the maximum number of function evaluations allowed by the algorithm.
- **MaxIter:** Specifies the maximum number of iterations allowed by the algorithm.
- **TolFun:** Specifies the relative tolerance for the objective function value at which the algorithm stops iterating.
- **TolX:** Specifies the relative tolerance for the decision variables at which the algorithm stops iterating.

### 2.3 Example Application of the fmincon Algorithm

Below is an example of using the fmincon algorithm to solve a nonlinear constrained optimization problem:

% Objective function
fun = @(x) x(1)^2 + x(2)^2;

% Constraints
confun = @(x) [x(1) - 2; -x(2) + 1];

% Solver options
options = optimset('Display', 'iter', 'MaxIter', 1000, 'TolFun', 1e-6, 'TolX', 1e-6);

% Solve
x = fmincon(fun, [0, 0], [], [], [], [], [-inf, -inf], [2, 1], [], options);

% Output results
disp('Optimal solution:');
disp(x);
disp('Objective function value:');
disp(fun(x));

In this example, the objective function is a quadratic function, and the constraints are two linear inequalities. The fmincon algorithm successfully found the optimal solution and output the results.

**Code logic analysis:**

- `fun`: Defines the objective function, an anonymous function that takes a vector `x` as input and returns a scalar value.
- `confun`: Defines the constraints, another anonymous function that takes a vector `x` as input and returns a vector, where each element represents a constraint.
- `options`: Sets the solver options, including output level, maximum iterations, tolerances, etc.
- `fmincon`: Calls the fmincon solver, specifying the objective function, initial point, constraints, solver options, and other parameters.
- `x`: Stores the obtained optimal sol