OptimizationEquality - Equalities and equality constraints - MATLAB (original) (raw)

Equalities and equality constraints

Description

An OptimizationEquality object contains equalities and equality constraints in terms of OptimizationVariable objects or OptimizationExpression objects. Each equality uses the comparison operator==.

A single statement can represent an array of equalities. For example, you can express the equalities that each row of a matrix variable x sums to one in this single statement:

constrsum = sum(x,2) == 1

Use OptimizationEquality objects as constraints in an OptimizationProblem, or as equations in an EquationProblem.

Creation

Create equalities using optimization expressions with the comparison operator==.

Include equalities in the Constraints property of an optimization problem, or the Equations property of an equation problem, by using dot notation.

prob = optimproblem; x = optimvar('x',4,6); SumToOne = sum(x,2) == 1; prob.Constraints.SumToOne = SumToOne; % Or for an equation problem: eqprob = eqnproblem; eqprob.Equations.SumToOne = SumToOne;

You can also create an empty optimization equality by using optimeq oroptimconstr. Typically, you then set the equalities in a loop. For an example, see Create Equalities in Loop. However, for the most efficient problem formulation, avoid setting equalities 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 equalities that each row of x sums to one.

constrsum = sum(x,2) == 1

constrsum = 4×1 Linear OptimizationEquality array with properties:

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

See equality formulation with show.

View the equalities.

(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 equalities 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.

Similarly, to include the equalities in an equation problem, set a Constraints property to constrsum by using dot notation.

eqnprob = eqnproblem; eqnprob.Equations.constrsum = constrsum

eqnprob = EquationProblem with properties:

Description: ''
  Variables: [1×1 struct] containing 1 OptimizationVariable
  Equations: [1×1 struct] containing 1 OptimizationEquality

See problem formulation with show.

Create an empty OptimizationEquality object.

Create a 5-by-5 optimization variable array named x.

Create the equalities that row i of x sums to i2.

for i = 1:size(x,1) eq1(i) = sum(x(i,:)) == i^2; end

View the resulting equalities.

(1, 1)

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

(1, 2)

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

(1, 3)

x(3, 1) + x(3, 2) + x(3, 3) + x(3, 4) + x(3, 5) == 9

(1, 4)

x(4, 1) + x(4, 2) + x(4, 3) + x(4, 4) + x(4, 5) == 16

(1, 5)

x(5, 1) + x(5, 2) + x(5, 3) + x(5, 4) + x(5, 5) == 25

To use eq1 as a constraint in an optimization problem, set eq1 as a Constraints property by using dot notation.

prob = optimproblem; prob.Constraints.eq1 = eq1;

Similarly, to use eq1 as a set of equations in an equation problem, set eq1 as an Equations property by using dot notation.

eqprob = eqnproblem; eqprob.Equations.eq1 = eq1;

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 OptimizationEquality 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.