# fcn2optimexpr

## Syntax

## Description

## Examples

### Convert Objective Function to Expression

To use a MATLAB® function in the problem-based approach when it is not composed of supported functions, first convert it to an optimization expression. See Supported Operations for Optimization Variables and Expressions and Convert Nonlinear Function to Optimization Expression.

To use the objective function `gamma`

(the mathematical function $$\Gamma (x)$$, an extension of the factorial function), create an optimization variable `x`

and use it in a converted anonymous function.

x = optimvar('x'); obj = fcn2optimexpr(@gamma,x); prob = optimproblem('Objective',obj); show(prob)

OptimizationProblem : Solve for: x minimize : gamma(x)

To solve the resulting problem, give an initial point structure and call `solve`

.

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

Solving problem using fminunc. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance.

`sol = `*struct with fields:*
x: 1.4616

For more complex functions, convert a function file. The function file `gammabrock.m`

computes an objective of two optimization variables.

`type gammabrock`

function f = gammabrock(x,y) f = (10*(y - gamma(x)))^2 + (1 - x)^2;

Include this objective in a problem.

x = optimvar('x','LowerBound',0); y = optimvar('y'); obj = fcn2optimexpr(@gammabrock,x,y); prob = optimproblem('Objective',obj); show(prob)

OptimizationProblem : Solve for: x, y minimize : gammabrock(x, y) variable bounds: 0 <= x

The `gammabrock`

function is a sum of squares. You get a more efficient problem formulation by expressing the function as an explicit sum of squares of optimization expressions.

```
f = fcn2optimexpr(@(x,y)y - gamma(x),x,y);
obj2 = (10*f)^2 + (1-x)^2;
prob2 = optimproblem('Objective',obj2);
```

To see the difference in efficiency, solve `prob`

and `prob2`

and examine the difference in the number of iterations.

x0.x = 1/2; x0.y = 1/2; [sol,fval,~,output] = solve(prob,x0);

Solving problem using fmincon. Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.

[sol2,fval2,~,output2] = solve(prob2,x0);

Solving problem using lsqnonlin. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance.

`fprintf('prob took %d iterations, but prob2 took %d iterations\n',output.iterations,output2.iterations)`

prob took 21 iterations, but prob2 took 2 iterations

If your function has several outputs, you can use them as elements of the objective function. In this case, `u`

is a 2-by-2 variable, `v`

is a 2-by-1 variable, and `expfn3`

has three outputs.

`type expfn3`

function [f,g,mineval] = expfn3(u,v) mineval = min(eig(u)); f = v'*u*v; f = -exp(-f); t = u*v; g = t'*t + sum(t) - 3;

Create appropriately sized optimization variables, and create an objective function from the first two outputs.

u = optimvar('u',2,2); v = optimvar('v',2); [f,g,mineval] = fcn2optimexpr(@expfn3,u,v); prob = optimproblem; prob.Objective = f*g/(1 + f^2); show(prob)

OptimizationProblem : Solve for: u, v minimize : ((arg2 .* arg3) ./ (1 + arg1.^2)) where: [arg1,~,~] = expfn3(u, v); [arg2,~,~] = expfn3(u, v); [~,arg3,~] = expfn3(u, v);

You can use the `mineval`

output in a subsequent constraint expression.

### Create Nonlinear Constraints from Function

In problem-based optimization, constraints are two optimization expressions with a comparison operator (`==`

, `<=`

, or `>=`

) between them. You can use `fcn2optimexpr`

to create one or both optimization expressions. See Convert Nonlinear Function to Optimization Expression.

Create the nonlinear constraint that `gammafn2`

is less than or equal to –1/2. This function of two variables is in the `gammafn2.m`

file.

`type gammafn2`

function f = gammafn2(x,y) f = -gamma(x)*(y/(1+y^2));

Create optimization variables, convert the function file to an optimization expression, and then express the constraint as `confn`

.

x = optimvar('x','LowerBound',0); y = optimvar('y','LowerBound',0); expr1 = fcn2optimexpr(@gammafn2,x,y); confn = expr1 <= -1/2; show(confn)

gammafn2(x, y) <= -0.5

Create another constraint that `gammafn2`

is greater than or equal to `x + y`

.

confn2 = expr1 >= x + y;

Create an optimization problem and place the constraints in the problem.

prob = optimproblem; prob.Constraints.confn = confn; prob.Constraints.confn2 = confn2; show(prob)

OptimizationProblem : Solve for: x, y minimize : subject to confn: gammafn2(x, y) <= -0.5 subject to confn2: gammafn2(x, y) >= (x + y) variable bounds: 0 <= x 0 <= y

### Compute Common Objective and Constraint Efficiently

If your problem involves a common, time-consuming function to compute the objective and nonlinear constraint, you can save time by using the `ReuseEvaluation`

name-value argument. The `rosenbrocknorm`

function computes both the Rosenbrock objective function and the norm of the argument for use in the constraint $$\Vert x{\Vert}^{2}\le 4$$.

`type rosenbrocknorm`

function [f,c] = rosenbrocknorm(x) pause(1) % Simulates time-consuming function c = dot(x,x); f = 100*(x(2) - x(1)^2)^2 + (1 - x(1))^2;

Create a 2-D optimization variable `x`

. Then convert `rosenbrocknorm`

to an optimization expression by using `fcn2optimexpr`

and set the `ReuseEvaluation`

name-value argument to `true`

. To ensure that `fcn2optimexpr`

keeps the `pause`

statement, set the `Analysis`

name-value argument to '`off'`

.

x = optimvar('x',2); [f,c] = fcn2optimexpr(@rosenbrocknorm,x,... 'ReuseEvaluation',true,'Analysis','off');

Create objective and constraint expressions from the returned expressions. Include the objective and constraint expressions in an optimization problem. Review the problem using `show`

.

```
prob = optimproblem('Objective',f);
prob.Constraints.cineq = c <= 4;
show(prob)
```

OptimizationProblem : Solve for: x minimize : [argout,~] = rosenbrocknorm(x) subject to cineq: arg_LHS <= 4 where: [~,arg_LHS] = rosenbrocknorm(x);

Solve the problem starting from the initial point `x0.x = [-1;1]`

, timing the result.

x0.x = [-1;1]; tic [sol,fval,exitflag,output] = solve(prob,x0)

Solving problem using fmincon. Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. <stopping criteria details>

`sol = `*struct with fields:*
x: [2×1 double]

fval = 4.5793e-11

exitflag = OptimalSolution

`output = `*struct with fields:*
iterations: 44
funcCount: 164
constrviolation: 0
stepsize: 4.3124e-08
algorithm: 'interior-point'
firstorderopt: 5.1691e-07
cgiterations: 10
message: 'Local minimum found that satisfies the constraints.↵↵Optimization completed because the objective function is non-decreasing in ↵feasible directions, to within the value of the optimality tolerance,↵and constraints are satisfied to within the value of the constraint tolerance.↵↵<stopping criteria details>↵↵Optimization completed: The relative first-order optimality measure, 5.169074e-07,↵is less than options.OptimalityTolerance = 1.000000e-06, and the relative maximum constraint↵violation, 0.000000e+00, is less than options.ConstraintTolerance = 1.000000e-06.'
bestfeasible: [1×1 struct]
objectivederivative: "finite-differences"
constraintderivative: "finite-differences"
solver: 'fmincon'

toc

Elapsed time is 165.623157 seconds.

The solution time in seconds is nearly the same as the number of function evaluations. This result indicates that the solver reused function values, and did not waste time by reevaluating the same point twice.

For a more extensive example, see Objective and Constraints Having a Common Function in Serial or Parallel, Problem-Based. For more information on using `fcn2optimexpr`

, see Convert Nonlinear Function to Optimization Expression.

## Input Arguments

`fcn`

— Function to convert

function handle

Function to convert, specified as a function handle.

**Example: **`@sin`

specifies the sine function.

**Data Types: **`function_handle`

`in`

— Input argument

MATLAB^{®} variable

Input argument, specified as a MATLAB variable. The input can have any data type and any size. You can include
any problem variables or data in the input argument `in`

; see Pass Extra Parameters in Problem-Based Approach.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

| `logical`

| `char`

| `string`

| `struct`

| `table`

| `cell`

| `function_handle`

| `categorical`

| `datetime`

| `duration`

| `calendarDuration`

| `fi`

**Complex Number Support: **Yes

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **```
[out1,out2] =
fcn2optimexpr(@fun,x,y,'OutputSize',[1,1],'ReuseEvaluation',true)
```

specifies that
`out1`

and `out2`

are scalars that a solver will reuse
between objective and constraint functions without recalculation.

`Analysis`

— Indication to analyze function

`"on"`

(default) | `"off"`

Indication to analyze the function `fcn`

to determine whether
it consists entirely of supported operations (see Supported Operations for Optimization Variables and Expressions), specified as
`"on"`

or `"off"`

.

If you want

`fcn2optimexpr`

to analyze`fcn`

and, if possible, use the supported operations to implement`fcn`

, specify`"on"`

. This specification enables`fcn`

to use automatic differentiation and to choose an appropriate solver as described in`Solver`

.If you do not want

`fcn2optimexpr`

to analyze`fcn`

and, therefore, to treat`fcn`

as a black box without automatic differentiation, specify`"off"`

. In this case,`solve`

uses only`fmincon`

,`fminunc`

, or`lsqnonlin`

as the solver.

For more information about the effects of `Analysis`

, see Limitations.

**Example: **```
[out1,out2] =
fcn2optimexpr(@fun,x,"Analysis","off")
```

**Data Types: **`char`

| `string`

`Display`

— Report function analysis details

`"off"`

(default) | `"on"`

Report function analysis details, specified as `"off"`

(do not
report) or `"on"`

(report). If `Analysis`

is
`"off"`

, there is nothing to report.

**Example: **```
[out1,out2] =
fcn2optimexpr(@fun,x,"Display","on")
```

**Data Types: **`char`

| `string`

`OutputSize`

— Size of output expressions

integer vector | cell array of integer vectors

Size of the output expressions, specified as:

An integer vector — If the function has one output

`out`

1,`OutputSize`

specifies the size of`out`

1. If the function has multiple outputs`out`

1,…,`out`

N,`OutputSize`

specifies that all outputs have the same size.A cell array of integer vectors — The size of output

`out`

j is the jth element of`OutputSize`

.

**Note**

A scalar has size `[1,1]`

.

If you do not specify the `'OutputSize'`

name-value pair
argument, then `fcn2optimexpr`

passes data to
`fcn`

in order to determine the size of the outputs (see Algorithms). By specifying
`'OutputSize'`

, you enable `fcn2optimexpr`

to
skip this step, which saves time. Also, if you do not specify
`'OutputSize'`

and the evaluation of `fcn`

fails
for any reason, then `fcn2optimexpr`

fails as well.

**Example: **```
[out1,out2,out3] =
fcn2optimexpr(@fun,x,'OutputSize',[1,1])
```

specifies that the three outputs
`[out1,out2,out3]`

are scalars.

**Example: **```
[out1,out2] =
fcn2optimexpr(@fun,x,'OutputSize',{[4,4],[3,5]})
```

specifies that
`out1`

has size 4-by-4 and `out2`

has size
3-by-5.

**Data Types: **`double`

| `cell`

`ReuseEvaluation`

— Indicator to reuse values

`false`

(default) | `true`

Indicator to reuse values, specified as `false`

(do not reuse) or
`true`

(reuse).

**Note**

`ReuseEvaluation`

may not have an effect when `Analysis`

=`"on"`

.

`ReuseEvaluation`

is not supported in a thread-based parallel
pool.

`ReuseEvaluation`

can make your problem run faster when, for
example, the objective and some nonlinear constraints rely on a common calculation. In
this case, the solver stores the value for reuse wherever needed and avoids
recalculating the value.

Reusable values involve some overhead, so it is best to enable reusable values only for expressions that share a value.

**Example: **```
[out1,out2,out3] =
fcn2optimexpr(@fun,x,"ReuseEvaluation",true,"Analysis","off")
```

allows
`out1`

, `out2`

, and `out3`

to be
used in multiple computations, with the outputs being calculated only once per
evaluation point.

**Data Types: **`logical`

## Output Arguments

`out`

— Output argument

`OptimizationExpression`

Output argument, returned as an `OptimizationExpression`

. The size of the expression depends on the input
function.

## Limitations

`Analysis`

Can Ignore Noncomputational Functions

The

`Analysis`

algorithm might not include noncomputational functions. This aspect of the algorithm can result in the following:`pause`

statements are ignored.A global variable that does not affect the results can be ignored. For example, if you use a global variable to count how many times the function runs, then you might obtain a misleading count.

If the function contains a call to

`rand`

or`rng`

, the function might execute the first call only, and future calls do not set the random number stream.A

`plot`

call might not update a figure at all iterations.Saving data to a

`mat`

file or text file might not occur at every iteration.

To ensure that noncomputational functions operate as you expect, set the

`Analysis`

name-value argument to`"off"`

.

For more information, see Limitations of Static Analysis.

## Algorithms

### Analysis

When the `Analysis`

argument is at the default setting of
`"on"`

, `fcn2optimexpr`

performs several steps in an
attempt to create the most efficient optimization expression. See the algorithm description
in fcn2optimexpr Algorithm Description.

You have several choices when including an objective or nonlinear constraint function in a problem object.

Use overloads. If all operations in a function are Supported Operations for Optimization Variables and Expressions, you can call the function directly on the optimization variables. For example,

prob.Objective = sin(3*x)*exp(-x-y);

Use

`fcn2optimexpr`

on an unmodified function. If at least one operation in a function is not supported, you must call`fcn2optimexpr`

. For example, the`besselh`

function is not supported, so to include it in an objective function, you must use`fcn2optimexpr`

.prob.Objective = fcn2optimexpr(@(z)besselh(3,z),x);

Modify a function so that its internal

`for`

loops appear in separate functions. Doing so enables static analysis to accelerate the loops. See Create for Loop for Static Analysis and Static Analysis of Optimization Expressions.Set the

`Analysis`

argument to`"off"`

in`fcn2optimexpr`

. Doing so causes`fcn2optimexpr`

to wrap the function as a black box, which is a fast operation. The resulting expression cannot take advantage of automatic differentiation (see Automatic Differentiation Background), so can cause a solver to use more function evaluations for finite difference gradient estimation.

### Output Size

To find the output size of each returned expression when you do not specify
`OutputSize`

, `fcn2optimexpr`

evaluates the function
at the following point for each element of the problem variables.

Variable Characteristics | Evaluation Point |
---|---|

Finite upper bound `ub` and finite lower bound
`lb` | `(lb + ub)/2 + ((ub - lb)/2)*eps` |

Finite lower bound and no upper bound | `lb + max(1,abs(lb))*eps` |

Finite upper bound and no lower bound | `ub - max(1,abs(ub))*eps` |

No bounds | `1 + eps` |

Variable is specified as an integer | `floor` of the point given previously |

An evaluation point might lead to an error in function evaluation. To avoid this error,
specify `OutputSize`

.

## Version History

**Introduced in R2019a**

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