# grsm

Find the constrained minima of a real multi-objective function.

## Syntax

x = grsm(@func,x0)

x = grsm(@func,x0,A,b)

x = grsm(@func,x0,A,b,Aeq,beq)

x = grsm(@func,x0,A,b,Aeq,beq,lb,ub)

x = grsm(@func,x0,A,b,Aeq,beq,lb,ub,nonlcon)

x = grsm(@func,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

[x,fval,info,output] = grsm(...)

## Inputs

func
The function to minimize.
x0
An estimate of the location of a minimum.
A
A matrix used to compute A*x for inequality contraints.
Use [ ] if unneeded.
b
The upper bound of the inequality constraints A*x<=b.
Use [ ] if unneeded.
Aeq
A matrix used to compute Aeq*x for equality contraints.
Use [ ] if unneeded.
beq
The upper bound of the equality constraints Aeq*x=beq.
Use [ ] if unneeded.
lb
The design variable lower bounds.
Use [ ] if unbounded. Support for this option is limited. See Comments.
ub
The design variable upper bounds.
Use [ ] if unbounded. Support for this option is limited. See Comments.
nonlcon
The non-linear constraints function.
The function signature is as follows:
function [c, ceq] = ConFunc(x)
where c and ceq contain inequality and equality contraints, respectively. The inequality constraints are assumed to have upper bounds of 0.
The function can return 1 or 2 outputs.
options
A struct containing options settings.
See grsmoptimset for details.

## Outputs

x
The locations of the multi-objective minima.
fval
The multi-objective function minima.
info
The convergence status flag.
• info = 3: A constraint violation within TolCon occurred.
• info = 0: Reached maximum number of iterations.
output
A struct containing iteration details. The members are as follows.
Pareto
A logical matrix indicating which samples belong to the Pareto Front. Each column contains the front information for an iteration.
nfev
The number of function evaluations.
xiter
The candidate solution at each iteration.
fvaliter
The objective function values at each iteration.
coniter
The constraint values at each iteration. The columns will contain the constraint function values in the following order:
1. linear inequality contraints
2. linear equality constraints
3. nonlinear inequality contraints
4. nonlinear equality constraints

## Example

Plot the iterations and Pareto Front for the function ObjFunc.
function obj = ObjFunc(x)
obj = zeros(2,1);
obj(1) = 2*(x(1)-3)^2 + 4*(x(2)-2)^2 + 6;
obj(2) = 2*(x(1)-3)^2 + 4*(x(2)+2)^2 + 6;
end

init = [2; 0];
lowerBound = [1, -5];
upperBound = [5, 5];

options = grsmoptimset('MaxIter', 50);
[x,fval,info,output] = grsm(@ObjFunc,init,[],[],[],[],lowerBound,upperBound,[],options);

obj1 = output.fvaliter(:,1);
obj2 = output.fvaliter(:,2);
scatter(obj1, obj2);
hold on;

obj1P = fval(:,1);
obj2P = fval(:,2);
scatter(obj1P, obj2P);
legend('Iteration History','Pareto Front');

grsm uses a Global Response Surface Method.

To pass additional parameters to a function argument, use an anonymous function.

See the fmincon optimization tutorial, Activate-4030：OMLの最適化アルゴリズム, for an example with nonlinear constraints.

Options are specified with grsmoptimset. The defaults are as follows:
• Display: 'off'
• InitSamPnts: min(20,n+2)
• MaxFail: 20000
• MaxIter: 50
• PntsPerIter: 2
• Seed: 0
• StopNoImpr: 1000
• TolCon: 0.5 (%)