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issatisfied

Constraint satisfaction of an optimization problem at a set of points

Since R2024a

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Syntax

allsat = issatisfied(prob,pts)
allsat = issatisfied(prob,pts,tol)
[allsat,sat] = issatisfied(___)

Description

example

allsat = issatisfied(prob,pts) returns a logical vector whose length is the number of points in pts. An entry in allsat is true if all the constraints in prob are satisfied to within the default tolerance of 1e-6 for the corresponding point in pts. Otherwise, the entry is false.

example

allsat = issatisfied(prob,pts,tol) returns true when the constraints at a point are satisfied to within the value tol, and false otherwise.

example

[allsat,sat] = issatisfied(___), for any previous input argument combinations, also returns an OptimizationValues object sat. This object's Constraints properties are logical indications of the constraint satisfactions for the associated constraints at the corresponding point in pts.

Examples

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Open Live Script

Create an optimization problem with several linear and nonlinear constraints.

x = optimvar("x");
y = optimvar("y");
obj = (10*(y - x^2))^2 + (1 - x)^2;
cons1 = x^2 + y^2 <= 1;
cons2 = x + y >= 0;
cons3 = y <= sin(x);
cons4 = 2*x + 3*y <= 2.5;
prob = optimproblem(Objective=obj);
prob.Constraints.cons1 = cons1;
prob.Constraints.cons2 = cons2;
prob.Constraints.cons3 = cons3;
prob.Constraints.cons4 = cons4;

Create 100 test points randomly.

rng default % For reproducibility
xvals = randn(1,100);
yvals = randn(1,100);

Convert the points to an OptimizationValues object for the problem, and determine where all the constraints are satisfied among the points.

vals = optimvalues(prob,x=xvals,y=yvals);
allsat = issatisfied(prob,vals);

Plot the feasible (satisfied) points with green circles and the infeasible points with red x marks.

xsat = xvals(allsat);
ysat = yvals(allsat);
xnosat = xvals(~allsat);
ynosat = yvals(~allsat);
plot(xsat,ysat,"go",xnosat,ynosat,"rx")

Open Live Script

Create an optimization problem with several linear and nonlinear constraints.

x = optimvar("x");
y = optimvar("y");
obj = (10*(y - x^2))^2 + (1 - x)^2;
cons1 = x^2 + y^2 <= 1;
cons2 = x + y >= 0;
cons3 = y <= sin(x);
cons4 = 2*x + 3*y <= 2.5;
prob = optimproblem(Objective=obj);
prob.Constraints.cons1 = cons1;
prob.Constraints.cons2 = cons2;
prob.Constraints.cons3 = cons3;
prob.Constraints.cons4 = cons4;

Create 100 test points randomly.

rng default % For reproducibility
xvals = randn(1,100);
yvals = randn(1,100);

Convert the points to an OptimizationValues object for the problem, and determine where all the constraints are satisfied among the points.

vals = optimvalues(prob,x=xvals,y=yvals);
allsat = issatisfied(prob,vals);

Plot the feasible (satisfied) points with green circles and the infeasible points with red x marks.

xsat = xvals(allsat);
ysat = yvals(allsat);
xnosat = xvals(~allsat);
ynosat = yvals(~allsat);
plot(xsat,ysat,"go",xnosat,ynosat,"rx")

Determine which points are feasible with respect to a tolerance of 1 rather than the default 1e-6.

tol = 1;
allsat1 = issatisfied(prob,vals,tol);

Plot the feasible (satisfied) points with green circles and the infeasible points with red x marks.

xsat1 = xvals(allsat1);
ysat1 = yvals(allsat1);
xnosat1 = xvals(~allsat1);
ynosat1 = yvals(~allsat1);
plot(xsat1,ysat1,"go",xnosat1,ynosat1,"rx")

With a looser definition of constraint satisfaction, more points are feasible.

Open Live Script

Create an optimization problem with several linear and nonlinear constraints.

x = optimvar("x");
y = optimvar("y");
obj = (10*(y - x^2))^2 + (1 - x)^2;
cons1 = x^2 + y^2 <= 1;
cons2 = x + y >= 0;
cons3 = y <= sin(x);
cons4 = 2*x + 3*y <= 2.5;
prob = optimproblem(Objective=obj);
prob.Constraints.cons1 = cons1;
prob.Constraints.cons2 = cons2;
prob.Constraints.cons3 = cons3;
prob.Constraints.cons4 = cons4;

Create 100 test points randomly.

rng default % For reproducibility
xvals = randn(1,100);
yvals = randn(1,100);

Convert the points to an OptimizationValues object for the problem.

vals = optimvalues(prob,x=xvals,y=yvals);

Evaluate the constraints at the points. issatisfied evaluates all the constraint satisfactions at all the test points simultaneously.

[~,sat] = issatisfied(prob,vals);

Plot the feasible (satisfied) points for the first constraint with green circles and the infeasible points with red x marks.

x1sat = xvals(sat.cons1);
y1sat = yvals(sat.cons1);
x1nosat = xvals(~sat.cons1);
y1nosat = yvals(~sat.cons1);
plot(x1sat,y1sat,"go",x1nosat,y1nosat,"rx")
title("Constraint 1 Satisfaction")

Repeat this process for the other three constraints.

x2sat = xvals(sat.cons2);
y2sat = yvals(sat.cons2);
x2nosat = xvals(~sat.cons2);
y2nosat = yvals(~sat.cons2);
plot(x2sat,y2sat,"go",x2nosat,y2nosat,"rx")
title("Constraint 2 Satisfaction")

x3sat = xvals(sat.cons3);
y3sat = yvals(sat.cons3);
x3nosat = xvals(~sat.cons3);
y3nosat = yvals(~sat.cons3);
plot(x3sat,y3sat,"go",x3nosat,y3nosat,"rx")
title("Constraint 3 Satisfaction")

x4sat = xvals(sat.cons4);
y4sat = yvals(sat.cons4);
x4nosat = xvals(~sat.cons4);
y4nosat = yvals(~sat.cons4);
plot(x4sat,y4sat,"go",x4nosat,y4nosat,"rx")
title("Constraint 4 Satisfaction")

The plots show the different regions of satisfaction for each constraint.

Input Arguments

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Optimization problem, specified as an OptimizationProblem object. Create prob using optimproblem. The issatisfied function evaluates the objectives and constraints in the properties of prob at the points in pts.

Example: prob = optimproblem(Objective=obj,Constraints=constr)

Points to evaluate for prob, specified as a structure or an OptimizationValues object.

  • The field names in pts must match the corresponding variable names in the objective and constraint expressions in prob.

  • The values in pts must be numeric arrays of the same size as the corresponding variables in prob.

If you use a structure for pts, then pts can contain only one point. In other words, if you want to evaluate multiple points simultaneously, pts must be an OptimizationValues object.

Example: pts = optimvalues(prob,x=xval,y=yval)

Tolerance for constraint satisfaction, specified as a nonnegative scalar. A constraint is considered to be satisfied if it evaluates to no more than tol. Otherwise, the constraint is unsatisfied.

For example, if c(x) is a nonlinear inequality constraint function, then when c(pt) ≤ tol, issatisfied returns true for that constraint at the point pt. Similarly, if ceq(x) is a nonlinear equality constraint function, then when abs(ceq(pt)) ≤ tol, issatisfied returns true for that constraint at the point pt.

Data Types: double

Output Arguments

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Indication that all constraints are satisfied at the given points, returned as a logical vector. allsat(i) corresponds to the query point pts(i). The returned value is true if all values of the constraint functions are no more than tol.

Indication that a constraint is satisfied at the given points, returned as an OptimizationValues vector. Index into sat using the constraint names. For example, if a constraint is named mycons, then sat.mycons is a logical array with true values for satisfied constraints.

More About

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Constraint Expression Values

For a constraint expression at a point pt:

  • If the constraint is L <= R, the constraint value is evaluate(L,pt) – evaluate(R,pt).

  • If the constraint is L >= R, the constraint value is evaluate(R,pt) – evaluate(L,pt).

  • If the constraint is L == R, the constraint value is abs(evaluate(L,pt) – evaluate(R,pt)).

Generally, a constraint is considered to be satisfied (or feasible) at a point if the constraint value is less than or equal to a tolerance.

Version History

Introduced in R2024a

See Also

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