difference between fmincon and fminsearch in matlab.

The fundamental difference between fminsearch and fmincon in MATLAB comes down to constraints and derivative requirements.

• fminsearch is an unconstrained, derivative-free solver based on the Nelder-Mead simplex algorithm. It only evaluates function values and is best for noisy, non-smooth, or discontinuous functions without any constraints.
• fmincon is a constrained, gradient-based solver (using Interior-Point, SQP, or Active-Set algorithms). It solves continuous, smooth objective functions subject to bounds, linear constraints, and nonlinear constraints.

Key Differences at a Glance

1. When to Use fminsearch (Example)

Use fminsearch when your objective function is unconstrained, black-box, or contains noise/discontinuities where derivatives cannot be computed reliably:

% Minimize Rosenbrock Banana Function (Unconstrained)
objective = @(x) 100*(x(2) - x(1)^2)^2 + (1 - x(1))^2;
x0 = [-1.2, 1.0]; % Initial guess

[x_opt, fval] = fminsearch(objective, x0);

fprintf('Optimal x: [%.4f, %.4f]\n', x_opt(1), x_opt(2));
fprintf('Minimum value: %.6f\n', fval);

2. When to Use fmincon (Example)

Use fmincon when you have physical boundaries (e.g., temperatures > 0, voltages < 240V) or mathematical equality/inequality constraints:

% Minimize Rosenbrock function subject to bound constraints: 0 <= x1 <= 0.8, 0 <= x2 <= 2.0
objective = @(x) 100*(x(2) - x(1)^2)^2 + (1 - x(1))^2;
x0 = [0.2, 0.2];

% Bounds: lb <= x <= ub
lb = [0.0, 0.0];
ub = [0.8, 2.0];

% Optimization options
options = optimoptions('fmincon', 'Algorithm', 'interior-point', 'Display', 'off');

[x_opt, fval] = fmincon(objective, x0, [], [], [], [], lb, ub, [], options);

fprintf('Constrained Optimal x: [%.4f, %.4f]\n', x_opt(1), x_opt(2));
fprintf('Minimum value: %.6f\n', fval);

Decision Rule: Which One Should You Pick?

1. If your problem has any bounds or constraints (e.g., \(x_1 + x_2 \le 5\)), always use fmincon.
2. If your problem is smooth and unconstrained, use fminunc (gradient-based) rather than fminsearch for much faster convergence.
3. If your problem is unconstrained but noisy, discontinuous, or has non-differentiable step functions, use fminsearch.
4. If your function has multiple local minima (multimodal), use global optimization solvers like ga (Genetic Algorithm) or particleswarm.