# evaluate

Evaluate optimization expression or objectives and constraints in problem

## Description

Use `evaluate`

to find the numeric value of an optimization
expression at a point, or to find the values of objective and constraint expressions in
an optimization problem, equation problem, or optimization constraint at a set of
points.

**Tip**

For the full workflow, see Problem-Based Optimization Workflow or Problem-Based Workflow for Solving Equations.

## Examples

### Evaluate Optimization Expression at Point

Create an optimization expression of two variables.

x = optimvar("x",3,2); y = optimvar("y",1,2); expr = sum(x,1) - 2*y;

Evaluate the expression at a point.

xmat = [3,-1; 0,1; 2,6]; sol.x = xmat; sol.y = [4,-3]; val = evaluate(expr,sol)

`val = `*1×2*
-3 12

### Evaluate Constraint Expressions

Create two optimization variables `x`

and `y`

and a 3-by-2 constraint expression in those variables.

x = optimvar("x"); y = optimvar("y"); cons = optimconstr(3,2); cons(1,1) = x^2 + y^2/4 <= 2; cons(1,2) = x^4 - y^4 <= -x^2 - y^2; cons(2,1) = x^2*3 + y^2 <= 2; cons(2,2) = x + y <= 3; cons(3,1) = x*y + x^2 + y^2 <= 5; cons(3,2) = x^3 + y^3 <= 8;

Evaluate the constraint expressions at the point $$x=1$$, $$y=-1$$. The value of an expression `L <= R`

is `L - R`

.

x0.x = 1; x0.y = -1; val = evaluate(cons,x0)

`val = `*3×2*
-0.7500 2.0000
2.0000 -3.0000
-4.0000 -8.0000

### Evaluate Objective Function at Solution

Solve a linear programming problem.

x = optimvar('x'); y = optimvar('y'); prob = optimproblem; prob.Objective = -x -y/3; prob.Constraints.cons1 = x + y <= 2; prob.Constraints.cons2 = x + y/4 <= 1; prob.Constraints.cons3 = x - y <= 2; prob.Constraints.cons4 = x/4 + y >= -1; prob.Constraints.cons5 = x + y >= 1; prob.Constraints.cons6 = -x + y <= 2; sol = solve(prob)

Solving problem using linprog. Optimal solution found.

`sol = `*struct with fields:*
x: 0.6667
y: 1.3333

Find the value of the objective function at the solution.

val = evaluate(prob.Objective,sol)

val = -1.1111

### Evaluate Optimization Problem Values

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.

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

Evaluate the objective and constraint functions at the points `pts`

.

val = evaluate(prob,pts);

The objective function values are stored in `val.Objective`

, and the constraint function values are stored in `val.cons1`

through `val.cons4`

. Plot the log of 1 plus the objective function values.

```
figure
plot3(xvals,yvals,log(1 + val.Objective),"bo")
```

Plot the values of the constraints `cons1`

and `cons4`

. Recall that constraints are satisfied when they evaluate to a nonpositive number. Plot the nonpositive values with circles and the positive values with x marks.

neg1 = val.cons1 <= 0; pos1 = val.cons1 > 0; neg4 = val.cons4 <= 0; pos4 = val.cons4 > 0; figure plot3(xvals(neg1),yvals(neg1),val.cons1(neg1),"bo") hold on plot3(xvals(pos1),yvals(pos1),val.cons1(pos1),"rx") plot3(xvals(neg4),yvals(neg4),val.cons4(neg4),"ko") plot3(xvals(pos4),yvals(pos4),val.cons4(pos4),"gx") hold off

As the last figure shows, `evaluate`

enables you to calculate both the value and the feasibility of points. In contrast, `issatisfied`

calculates only the feasibility.

### Evaluate Expressions in Equation

Create a set of equations in two optimization variables.

x = optimvar("x"); y = optimvar("y"); prob = eqnproblem; prob.Equations.eq1 = x^2 + y^2/4 == 2; prob.Equations.eq2 = x^2/4 + 2*y^2 == 2;

Solve the system of equation starting from $$x=1,y=1/2$$.

x0.x = 1; x0.y = 1/2; sol = solve(prob,x0)

Solving problem using fsolve. Equation solved. fsolve completed because the vector of function values is near zero as measured by the value of the function tolerance, and the problem appears regular as measured by the gradient.

`sol = `*struct with fields:*
x: 1.3440
y: 0.8799

Evaluate the equations at the points `x0`

and `sol`

.

vars = optimvalues(prob,x=[x0.x sol.x],y=[x0.y sol.y]); vals = evaluate(prob,vars)

vals = 1x2 OptimizationValues vector with properties: Variables properties: x: [1 1.3440] y: [0.5000 0.8799] Equation properties: eq1: [0.9375 8.4322e-10] eq2: [1.2500 6.7431e-09]

The first point, `x0`

, has nonzero values for both equations `eq1`

and `eq2`

. The second point, `sol`

, has nearly zero values of these equations, as expected for a solution.

Find the degree of equation satisfaction using `issatisfied`

.

[satisfied details] = issatisfied(prob,vars)

`satisfied = `*1x2 logical array*
0 1

details = 1x2 OptimizationValues vector with properties: Variables properties: x: [1 1.3440] y: [0.5000 0.8799] Equation properties: eq1: [0 1] eq2: [0 1]

The first point, `x0`

, is not a solution, and `satisfied`

is `0`

for that point. The second point, `sol`

, is a solution, and `satisfied`

is `1`

for that point. The equation properties show that neither equation is satisfied at the first point, and both are satisfied at the second point.

## Input Arguments

`expr`

— Optimization expression

`OptimizationExpression`

object

Optimization expression, specified as an `OptimizationExpression`

object.

**Example: **`expr = 5*x+3`

, where `x`

is an
`OptimizationVariable`

`pt`

— Values of variables in expression

structure

Values of the variables in an expression, specified as a structure. The structure `pt`

has the following requirements:

All variables in

`expr`

must match field names in`pt`

.The values of the matching field names must be numeric.

The sizes of the fields in

`pt`

must match the sizes of the corresponding variables in`expr`

.

For example, `pt`

can be the solution to an optimization problem, as returned by `solve`

.

**Example: **`pt.x = 3, pt.y = -5`

**Data Types: **`struct`

`cons`

— Constraint

`OptimizationConstraint`

object | `OptimizationEquality`

object | `OptimizationInequality`

object

Constraint, specified as an `OptimizationConstraint`

object, an `OptimizationEquality`

object, or an `OptimizationInequality`

object. `evaluate`

applies to these constraint objects only for a point specified as a structure, not as an `OptimizationValues`

object.

**Example: **`cons = expr1 <= expr2`

, where `expr1`

and `expr2`

are optimization expressions

`prob`

— Object for evaluation

`OptimizationProblem`

object | `EquationProblem`

object

Object for evaluation, specified as an `OptimizationProblem`

object or an `EquationProblem`

object. The `evaluate`

function
evaluates the objectives and constraints in the properties of `prob`

at
the points in `pts`

.

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

`pts`

— Points to evaluate for `prob`

structure | `OptimizationValues`

object

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`

.

**Note**

Currently, `pts`

can be an `OptimizationValues`

object only when `prob`

is an `EquationProblem`

object or an `OptimizationProblem`

object.

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)`

## Output Arguments

`val`

— Evaluation result

double | `OptimizationValues`

object

Evaluation result, returned as a double or an `OptimizationValues`

object.

When the first input argument is an expression or constraint,

`val`

is returned as a double array of the same size as the expression or constraint, and contains its numeric values at`pt`

.When the first input argument is an

`OptimizationProblem`

object or`EquationProblem`

object,`val`

is an`OptimizationValues`

object.`val`

contains the values of the objective and constraints or equations in`prob`

evaluated at the points in`pts`

. If`pts`

contains`N`

points, then`val`

has size 1-by-`N`

. For example, if`prob`

contains a constraint`con`

of size 2-by-3, and`pts`

is an`OptimizationValues`

object with`N`

= 5 points, then`val`

has size 1-by-5, and`val.Constraints.con`

has size 2-by-3-by-5.

**Warning**

The problem-based approach does not support complex values in the following: an objective function, nonlinear equalities, and nonlinear inequalities. If a function calculation has a complex value, even as an intermediate value, the final result might be incorrect.

## More About

### 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 R2017b**

### R2024b: Evaluate expressions in more objects

The `evaluate`

function now applies to the following objects:

For an example, see Evaluate Expressions in Equation.

### R2024a: Evaluate objective and constraint expressions in optimization problems at multiple points

The `evaluate`

function now applies to objective and constraint
expressions in `OptimizationProblem`

objects. If the evaluation
points are passed as an `OptimizationValues`

object, then the
function evaluates the expressions at all points in the object. For an example, see
Evaluate Optimization Problem Values.

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