fcn2optimexpr

Convert function to optimization expression

Syntax

[out1,out2,...,outN] = fcn2optimexpr(fcn,in1,in2,...,inK)

[out1,out2,...,outN] = fcn2optimexpr(fcn,in1,in2,...,inK,Name,Value)

Description

[out1,out2,...,outN] = fcn2optimexpr(fcn,in1,in2,...,inK) converts the function fcn(in1,in2,...,inK) to an optimization expression with N outputs.

[out1,out2,...,outN] = fcn2optimexpr(fcn,in1,in2,...,inK,Name,Value) specifies additional options using one or more name-value pair arguments. For example, you can save a function evaluation by passing OutputSize.

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 on 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 pair 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 specifying 'ReuseEvaluation'.

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

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]
             solver: 'fmincon'

toc

Elapsed time is 164.410724 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 Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [out1,out2] = fcn2optimexpr(@fun,x,y,'OutputSize',[1,1],'ReuseEvaluation',true) specifies that out1 and out2 are scalars that will be reused between objective and constraint functions without recalculation.

'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 out1, OutputSize specifies the size of out1. If the function has multiple outputs out1,…,outN, OutputSize specifies that all outputs have the same size.

• A cell array of integer vectors — The size of output outj 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).

'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) 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.

Tips

• When possible, create your objective or nonlinear constraint functions by using supported operations on optimization variables and expressions instead of fcn2optimexpr. Doing so has these advantages:

  • solve includes gradients calculated by automatic differentiation. See Effect of Automatic Differentiation in Problem-Based Optimization.

  • solve has a wider choice of available solvers. When using fcn2optimexpr, solve uses only fmincon or fminunc.

  For details, see Convert Nonlinear Function to Optimization Expression and Supported Operations on Optimization Variables and Expressions.

Algorithms

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'.