The optimization module provides direct numerical solvers and a problem-based modelling layer. Use direct solvers when the coefficient matrices, residual function, or objective function are already available. Use the problem-based layer when the model is easier to read as variables, expressions, and constraints.
| Problem type | Direct solver | Typical inputs |
|---|---|---|
| Scalar bounded minimization | fminbnd | Objective function and finite interval. |
| Unconstrained minimization | fminunc, fminsearch | Objective function and initial point. |
| Zero finding | fzero | Function and bracket or initial point. |
| Linear programming | linprog | Linear objective, linear constraints, and bounds. |
| Mixed-integer linear programming | intlinprog | Linear objective, integer variable indices, constraints, and bounds. |
| Quadratic programming | quadprog | Quadratic objective, linear constraints, and bounds. |
| Constrained nonlinear minimization | fmincon | Nonlinear objective, constraints, and bounds. |
| Least squares | lsqnonneg, lsqnonlin | Linear or nonlinear residual model. |
| Nonlinear equations | fsolve | Vector function and initial point. |
Solver behavior is controlled with optimoptions or optimset. Use optimget to read an option with a fallback value. Linear and mixed-integer linear problems use the HiGHS backend when it is available. Problem-based compilation routes continuous linear, mixed-integer linear, quadratic and constrained nonlinear problems to the corresponding direct solver. Nonlinear problems with integer or binary variables are rejected explicitly.
The problem-based workflow starts with optimvar and optimproblem. Expressions and constraints are built with ordinary arithmetic. solve calls a supported direct solver, and prob2struct returns the direct-solver structure for inspection or lower-level execution.
Binary variables created with optimvar have default bounds 0 and 1. Integer and binary variables route linear models to intlinprog; continuous linear models route to linprog; continuous nonlinear constrained models route to fmincon. Maximization problems are converted internally to minimization and solve returns the objective value in the original problem sense.
When a direct solver proves a problem infeasible and returns no primal vector, solve returns a solution structure with the model variable names and empty values. This keeps diagnostic outputs such as exitflag and output.message available without failing during result unpacking.
opts = optimset('Display', 'off');
[x, fval] = fminbnd(@(x) (x - 1.5)^2 + 0.25, -2, 4, opts)fun = @(x) 3*x(1)^2 + 2*x(1)*x(2) + x(2)^2 - 4*x(1) + 5*x(2);
opts = optimoptions('fminunc', 'Display', 'off');
[x, fval] = fminunc(fun, [1, 1], opts)f = [-1; -1];
A = [1 2; 4 2];
b = [4; 12];
opts = optimoptions('linprog', 'Display', 'off');
[x, fval, exitflag] = linprog(f, A, b, [], [], [0; 0], [], opts)f = [-5; -4; -3];
intcon = 1:3;
A = [2 3 1; 4 1 2];
b = [5; 8];
lb = [0; 0; 0];
ub = [1; 1; 1];
opts = optimoptions('intlinprog', 'Display', 'off');
[x, fval, exitflag] = intlinprog(f, intcon, A, b, [], [], lb, ub, [], opts)x = optimvar('x', 2, 'LowerBound', 0);
prob = optimproblem('Objective', -x(1) - x(2));
prob.Constraints.capacity = [1 2; 4 2] * x <= [4; 12];
problem = prob2struct(prob);
[sol, fval, exitflag] = solve(prob);
sol.x| Version | Description |
|---|---|
| 2.0.0 | initial version |