Linear Constraints

What Are Linear Constraints?

Several optimization solvers accept linear constraints, which are restrictions on the solution x to satisfy linear equalities or inequalities. Solvers that accept linear constraints include fmincon, intlinprog, linprog, lsqlin, quadprog, multiobjective solvers, and some Global Optimization Toolbox solvers.

Linear Inequality Constraints

Linear inequality constraints have the form A·x ≤ b. When A is m-by-n, there are m constraints on a variable x with n components. You supply the m-by-n matrix A and the m-component vector b.

Pass linear inequality constraints in the A and b arguments.

For example, suppose that you have the following linear inequalities as constraints:

x₁ + x₃ ≤ 4,
2x₂ – x₃ ≥ –2,
x₁ – x₂ + x₃ – x₄ ≥ 9.

Here, m = 3 and n = 4.

Write these constraints using the following matrix A and vector b:

$\begin{array}{l} A = [\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & - 2 & 1 & 0 \\ - 1 & 1 & - 1 & 1 \end{matrix}], \\ b = [\begin{matrix} 4 \\ 2 \\ - 9 \end{matrix}] . \end{array}$

Notice that the “greater than” inequalities are first multiplied by –1 to put them in “less than” inequality form. In MATLAB^® syntax:

A = [1 0 1 0;
    0 -2 1 0;
    -1 1 -1 1];
b = [4;2;-9];

You do not need to give gradients for linear constraints; solvers calculate them automatically. Linear constraints do not affect Hessians.

Even if you pass an initial point x0 as a matrix, solvers pass the current point x as a column vector to linear constraints. See Matrix Arguments.

For a more complex example of linear constraints, see Set Up a Linear Program, Solver-Based.

Intermediate iterations can violate linear constraints. See Iterations Can Violate Constraints.

Linear Equality Constraints

Linear equalities have the form Aeq·x = beq, which represents m equations with n-component vector x. You supply the m-by-n matrix Aeq and the m-component vector beq.

Pass linear equality constraints in the Aeq and beq arguments in the same way as described for the A and b arguments in Linear Inequality Constraints.