Minimisation of Equations and Constraints with Trigonometric Functions in MATLAB

Question

I have an optimisation script (below), which gives me the error:

"Undefined function 'cos' for input arguments of type 'optim.problemdef.OptimizationExpression'."

Just to make sure it wasn't a problem with the cos function, I changed the cos in confn2 to a sin and then got:

"Undefined function 'sin' for input arguments of type 'optim.problemdef.OptimizationExpression'."

However, if I put cos(pi) in the Command Window, I do get -1.

I have successfully run this script before with the exception being that the sigma (which appears in the cos()) was not an optimisation variable and thus the trig functions were evaluated leaving something linear. Is this type of optimisation script not tractable for constraints with trigonometric functions? Are there alternatives that are within MATLAB?

k1 = optimvar('k1', 'LowerBound', -3, 'UpperBound', 3);
k2 = optimvar('k2', 'LowerBound', -3, 'UpperBound', 3);
f = optimvar('f', 'LowerBound', -3, 'UpperBound', 3);
sigma = optimvar('sigma', 'LowerBound', 0, 'UpperBound', 6.28318530718);

obj = fcn2optimexpr(@eq1, k1, k2, f, sigma);

confn1 = obj == 0;
confn2 = -0.9313129901097519*k1 - 1.4693755421886672*k2 - 0.18532000686683578*f*cos((1/2)*(-0.372795 + 2*sigma)) + 0.9826782255931369*f*sin((1/2)*(-0.372795 + 2*sigma)) <= 0;

prob = optimproblem('Objective', obj);

prob.Constraints.confn1 = confn1;
prob.Constraints.confn2 = confn2;

k0.k1 = 0;
k0.k2 = 0;
k0.f = 0;
k0.sigma = 0;

[sol, fval, exitflag, output] = solve(prob, k0)

function f1 = eq1(k1, k2, f, sigma)
f1 = 0.01308996938995749 - 0.3642198710296203*k1 - 0.6784053942919677*k2 + 0.37064001373367156*f*sin((1/2)*(-0.372795 + 2*sigma));
end

cos is a recognised function, but not for "input arguments of type optim.problemdef.OptimizationExpression". You're essentially doing cos(sigma), which isn't defined because sigma isn't a numeric variable, you created it using optimvar. — Wolfie
Hey @Wolfie ! Thanks so much -- that is what I figured as that is when the error occurred. Is there a way to rectify that problem within this optimisation script that allows for sigma to remain an optimvar, or will I need to write something quite different? — Cameron F.
I have no idea, that's a different question, I've never used optimproblem. — Wolfie

rinkert rinkert · Accepted Answer · 2019-06-03T14:12:57

You can use fmincon instead, in which you can also specify an objective and constraints. This way the cos function will be evaluated numerically each iteration, and will not be called with the optim.problemdef.OptimizationExpression type.

You do need to specify your variables as a vector (intial guess x0, lowerbound lb, upperbound ub), see comment in the code.

% x0 = [k1 k2 f sigma];
x0 = [0 0 0 0];
lb = [-3 -3 -3 0];
ub = [3 3 3 2*pi];

% options
options = optimoptions('fmincon','Display','iter', 'ConstraintTolerance', 1e-12);

% optimization
x = fmincon(@objfun,x0,[],[],[],[],lb,ub,@constrfun, options)

% check constraints to check fulfilment of constraints
[c, ceq] = constrfun(x)

Your object function must accept a single vector x, which can be 'unpacked' to the individual variables.

function f1 = objfun(x)
    k1 = x(1);
    k2 = x(2); 
    f = x(3);
    sigma = x(4);
    f1 = 0.01308996938995749 - 0.3642198710296203*k1 - 0.6784053942919677*k2 + 0.37064001373367156*f*sin((1/2)*(-0.372795 + 2*sigma));
end

Your constraints must be defined in a constraint function, that returns c and ceq, in which c is the inequality constraint c(x) <= 0 and ceq is the equality constraint ceq(x) = 0.

function [c,ceq] = constrfun(x)
    k1 = x(1);
    k2 = x(2); 
    f = x(3);
    sigma = x(4);
    c = -0.9313129901097519*k1 - 1.4693755421886672*k2 - 0.18532000686683578*f*cos((1/2)*(-0.372795 + 2*sigma)) + 0.9826782255931369*f*sin((1/2)*(-0.372795 + 2*sigma));
    ceq = objfun(x);
end

Minimisation of Equations and Constraints with Trigonometric Functions in MATLAB

2 Answers