OptimizationInequality - Inequality constraints - MATLAB (original) (raw)

Description

An OptimizationInequality object contains an inequality constraint in terms of OptimizationVariable objects or OptimizationExpression objects. An inequality constraint uses the comparison operator <= or >=.

A single statement can represent an array of inequalities. For example, you can express the inequalities that each row of a matrix variable x sums to no more than one in this single statement:

constrsum = sum(x,2) <= 1

Use OptimizationInequality objects as constraints in an OptimizationProblem.

Creation

Create an inequality using optimization expressions with the comparison operator<= or >=.

Include inequalities in the Constraints property of an optimization problem by using dot notation.

prob = optimproblem; x = optimvar('x',4,6); SumLessThanOne = sum(x,2) <= 1; prob.Constraints.SumLessThanOne = SumLessThanOne;

You can also create an empty optimization inequality by using optimineq oroptimconstr. Typically, you then set the inequalities in a loop. For an example, see Create Inequalities in Loop. However, for the most efficient problem formulation, avoid setting inequalities in loops. See Create Efficient Optimization Problems.

Properties

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This property is read-only.

Optimization variables in the object, specified as a structure of OptimizationVariable objects.

Data Types: struct

Object Functions

Examples

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Create a 4-by-6 optimization variable matrix named x.

Create the inequalities that each row of x sums to no more than one.

constrsum = sum(x,2) <= 1

constrsum = 4×1 Linear OptimizationInequality array with properties:

IndexNames: {{}  {}}
 Variables: [1×1 struct] containing 1 OptimizationVariable

See inequality formulation with show.

View the inequalities.

(1, 1)

x(1, 1) + x(1, 2) + x(1, 3) + x(1, 4) + x(1, 5) + x(1, 6) <= 1

(2, 1)

x(2, 1) + x(2, 2) + x(2, 3) + x(2, 4) + x(2, 5) + x(2, 6) <= 1

(3, 1)

x(3, 1) + x(3, 2) + x(3, 3) + x(3, 4) + x(3, 5) + x(3, 6) <= 1

(4, 1)

x(4, 1) + x(4, 2) + x(4, 3) + x(4, 4) + x(4, 5) + x(4, 6) <= 1

To include the inequalities in an optimization problem, set a Constraints property to constrsum by using dot notation.

prob = optimproblem; prob.Constraints.constrsum = constrsum

prob = OptimizationProblem with properties:

   Description: ''
ObjectiveSense: 'minimize'
     Variables: [1×1 struct] containing 1 OptimizationVariable
     Objective: [0×0 OptimizationExpression]
   Constraints: [1×1 struct] containing 1 OptimizationConstraint

See problem formulation with show.

Create the constraint that a two-element variable x must lie in the intersections of a number of disks whose centers and radii are in the arrays centers and radii.

x = optimvar('x',1,2); centers = [1 -2;3 -4;-2 3]; radii = [6 7 8]; constr = optimineq(length(radii)); for i = 1:length(constr) constr(i) = sum((x - centers(i,:)).^2) <= radii(i)^2; end

View the inequality expressions.

arg_LHS <= arg_RHS

where:

    arg1 = zeros(3, 1);
    arg1(1) = sum((x - extraParams{1}).^2);
    arg1(2) = sum((x - extraParams{2}).^2);
    arg1(3) = sum((x - extraParams{3}).^2);
    arg_LHS = arg1(:);
    arg1 = zeros(3, 1);
    arg1(1) = 36;
    arg1(2) = 49;
    arg1(3) = 64;
    arg_RHS = arg1(:);

extraParams

Instead of using a loop, you can create the same constraints by using matrix operations on the variables.

constr2 = sum(([x;x;x] - centers).^2,2) <= radii'.^2;

Creating inequalities in a loop can be more time consuming than creating inequalities by using matrix operations.

Check the value of an inequality constraint and satisfaction of an equality constraint at a point specified by a structure.

x = optimvar("x"); y = optimvar("y"); ineq = x^2 + y^2/4 <= 2; eq = 4*x^2 + y^2 == 5; x0.x = 1; x0.y = 1; evaluate(ineq,x0) % Value = L - R, where L = x^2 + y^2/4 and R = 2

issatisfied(eq,x0) % Checks whether 4*x^2 + y^2 = 5

Version History

Introduced in R2019b

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You can now evaluate optimization expressions and constraints using evaluate and issatisfied for OptimizationInequality objects.

The value of a constraint depends on the constraint type. An equation is equivalent to an == constraint. For expressions L and R:

For details, see the evaluate andissatisfied reference pages.