To use a nonlinear function as an objective or nonlinear constraint function in the problem-based approach, convert the function to an optimization expression using
fcn2optimexpr. This example shows how to convert the function using both a function file and an anonymous function.
To use a function file in the problem-based approach, you need to convert the file to an expression using
For example, the
expfn3.m file contains the following code:
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;
To use this function file as an optimization expression, first create optimization variables of the appropriate sizes.
u = optimvar('u',3,3,'LowerBound',-1,'UpperBound',1); % 3-by-3 variable v = optimvar('v',3,'LowerBound',-2,'UpperBound',2); % 3-by-1 variable
Convert the function file to an optimization expressions using
[f,g,mineval] = fcn2optimexpr(@expfn3,u,v);
Because all returned expressions are scalar, you can save computing time by specifying the expression sizes using the
'OutputSize' name-value pair argument. Also, because
expfn3 computes all of the outputs, you can save more computing time by using the
ReuseEvaluation name-value pair.
[f,g,mineval] = fcn2optimexpr(@expfn3,u,v,'OutputSize',[1,1],'ReuseEvaluation',true)
f = Nonlinear OptimizationExpression [argout,~,~] = expfn3(u, v)
g = Nonlinear OptimizationExpression [~,argout,~] = expfn3(u, v)
mineval = Nonlinear OptimizationExpression [~,~,argout] = expfn3(u, v)
To use a general nonlinear function handle in the problem-based approach, convert the handle to an optimization expression using
fcn2optimexpr. For example, write a function handle equivalent to
f and convert it.
fun = @(x,y)-exp(-y'*x*y); funexpr = fcn2optimexpr(fun,u,v,'OutputSize',[1,1])
funexpr = Nonlinear OptimizationExpression anonymousFunction1(u, v) where: anonymousFunction1 = @(x,y)-exp(-y'*x*y);
To use either expression as an objective function, create an optimization problem.
prob = optimproblem; prob.Objective = f; % Or, equivalently, prob.Objective = funexpr;
Define the constraint
g <= 0 in the optimization problem.
prob.Constraints.nlcons1 = g <= 0;
Also define the constraints that
u is symmetric and that .
prob.Constraints.sym = u == u.'; prob.Constraints.mineval = mineval >= -1/2;
View the problem.
OptimizationProblem : Solve for: u, v minimize : [argout,~,~] = expfn3(u, v) subject to nlcons1: arg_LHS <= 0 where: [~,arg_LHS,~] = expfn3(u, v); subject to sym: u(2, 1) - u(1, 2) == 0 u(3, 1) - u(1, 3) == 0 -u(2, 1) + u(1, 2) == 0 u(3, 2) - u(2, 3) == 0 -u(3, 1) + u(1, 3) == 0 -u(3, 2) + u(2, 3) == 0 subject to mineval: arg_LHS >= (-0.5) where: [~,~,arg_LHS] = expfn3(u, v); variable bounds: -1 <= u(1, 1) <= 1 -1 <= u(2, 1) <= 1 -1 <= u(3, 1) <= 1 -1 <= u(1, 2) <= 1 -1 <= u(2, 2) <= 1 -1 <= u(3, 2) <= 1 -1 <= u(1, 3) <= 1 -1 <= u(2, 3) <= 1 -1 <= u(3, 3) <= 1 -2 <= v(1) <= 2 -2 <= v(2) <= 2 -2 <= v(3) <= 2
To solve the problem, call
solve. Set an initial point
rng default % For reproducibility x0.u = 0.25*randn(3); x0.u = x0.u + x0.u.'; x0.v = 2*randn(3,1); [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.
sol = struct with fields: u: [3x3 double] v: [3x1 double]
fval = -403.4288
exitflag = OptimalSolution
output = struct with fields: iterations: 70 funcCount: 1070 constrviolation: 1.3317e-12 stepsize: 6.1260e-06 algorithm: 'interior-point' firstorderopt: 5.6090e-04 cgiterations: 126 message: '...' solver: 'fmincon'
View the solution.
0.9167 0.4393 -0.9774 0.4393 0.1531 0.2139 -0.9774 0.2139 0.6913
2.0000 -2.0000 2.0000
The solution matrix
u is symmetric. All values of
v are at the bounds.