Code covered by the BSD License

# fminsearchbnd, fminsearchcon

### John D'Errico (view profile)

11 Aug 2005 (Updated )

Bound constrained optimization using fminsearch

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[x,fval,exitflag,output]=fminsearchcon(fun,x0,LB,UB,A,b,nonlcon,options,varargin)
```function [x,fval,exitflag,output]=fminsearchcon(fun,x0,LB,UB,A,b,nonlcon,options,varargin)
% FMINSEARCHCON: Extension of FMINSEARCHBND with general inequality constraints
% usage: x=FMINSEARCHCON(fun,x0)
% usage: x=FMINSEARCHCON(fun,x0,LB)
% usage: x=FMINSEARCHCON(fun,x0,LB,UB)
% usage: x=FMINSEARCHCON(fun,x0,LB,UB,A,b)
% usage: x=FMINSEARCHCON(fun,x0,LB,UB,A,b,nonlcon)
% usage: x=FMINSEARCHCON(fun,x0,LB,UB,A,b,nonlcon,options)
% usage: x=FMINSEARCHCON(fun,x0,LB,UB,A,b,nonlcon,options,p1,p2,...)
% usage: [x,fval,exitflag,output]=FMINSEARCHCON(fun,x0,...)
%
% arguments:
%  fun, x0, options - see the help for FMINSEARCH
%
%       x0 MUST be a feasible point for the linear and nonlinear
%       inequality constraints. If it is not inside the bounds
%       then it will be moved to the nearest bound. If x0 is
%       infeasible for the general constraints, then an error will
%       be returned.
%
%  LB - lower bound vector or array, must be the same size as x0
%
%       If no lower bounds exist for one of the variables, then
%       supply -inf for that variable.
%
%       If no lower bounds at all, then LB may be left empty.
%
%       Variables may be fixed in value by setting the corresponding
%       lower and upper bounds to exactly the same value.
%
%  UB - upper bound vector or array, must be the same size as x0
%
%       If no upper bounds exist for one of the variables, then
%       supply +inf for that variable.
%
%       If no upper bounds at all, then UB may be left empty.
%
%       Variables may be fixed in value by setting the corresponding
%       lower and upper bounds to exactly the same value.
%
%  A,b - (OPTIONAL) Linear inequality constraint array and right
%       hand side vector. (Note: these constraints were chosen to
%       be consistent with those of fmincon.)
%
%       A*x <= b
%
%  nonlcon - (OPTIONAL) general nonlinear inequality constraints
%       NONLCON must return a set of general inequality constraints.
%       These will be enforced such that nonlcon is always <= 0.
%
%       nonlcon(x) <= 0
%
%
% Notes:
%
%  If options is supplied, then TolX will apply to the transformed
%  variables. All other FMINSEARCH parameters should be unaffected.
%
%  Variables which are constrained by both a lower and an upper
%  bound will use a sin transformation. Those constrained by
%  only a lower or an upper bound will use a quadratic
%  transformation, and unconstrained variables will be left alone.
%
%  Variables may be fixed by setting their respective bounds equal.
%  In this case, the problem will be reduced in size for FMINSEARCH.
%
%  The bounds are inclusive inequalities, which admit the
%  boundary values themselves, but will not permit ANY function
%  evaluations outside the bounds. These constraints are strictly
%  followed.
%
%  If your problem has an EXCLUSIVE (strict) constraint which will
%  not admit evaluation at the bound itself, then you must provide
%  a slightly offset bound. An example of this is a function which
%  contains the log of one of its parameters. If you constrain the
%  variable to have a lower bound of zero, then FMINSEARCHCON may
%  try to evaluate the function exactly at zero.
%
%  Inequality constraints are enforced with an implicit penalty
%  function approach. But the constraints are tested before
%  any function evaluations are ever done, so the actual objective
%  function is NEVER evaluated outside of the feasible region.
%
%
% Example usage:
% rosen = @(x) (1-x(1)).^2 + 105*(x(2)-x(1).^2).^2;
%
% Fully unconstrained problem
% fminsearchcon(rosen,[3 3])
% ans =
%    1.0000    1.0000
%
% lower bound constrained
% fminsearchcon(rosen,[3 3],[2 2],[])
% ans =
%    2.0000    4.0000
%
% x(2) fixed at 3
% fminsearchcon(rosen,[3 3],[-inf 3],[inf,3])
% ans =
%    1.7314    3.0000
%
% simple linear inequality: x(1) + x(2) <= 1
% fminsearchcon(rosen,[0 0],[],[],[1 1],.5)
%
% ans =
%    0.6187    0.3813
%
% general nonlinear inequality: sqrt(x(1)^2 + x(2)^2) <= 1
% fminsearchcon(rosen,[0 0],[],[],[],[],@(x) norm(x)-1)
% ans =
%    0.78633   0.61778
%
% Of course, any combination of the above constraints is
% also possible.
%
% See test_main.m for other examples of use.
%
%
%
%
% Author: John D'Errico
% E-mail: woodchips@rochester.rr.com
% Release: 1.0
% Release date: 12/16/06

% size checks
xsize = size(x0);
x0 = x0(:);
n=length(x0);

if (nargin<3) || isempty(LB)
LB = repmat(-inf,n,1);
else
LB = LB(:);
end
if (nargin<4) || isempty(UB)
UB = repmat(inf,n,1);
else
UB = UB(:);
end

if (n~=length(LB)) || (n~=length(UB))
error 'x0 is incompatible in size with either LB or UB.'
end

% defaults for A,b
if (nargin<5) || isempty(A)
A = [];
end
if (nargin<6) || isempty(b)
b = [];
end
nA = [];
nb = [];
if (isempty(A)&&~isempty(b)) || (isempty(b)&&~isempty(A))
error 'Sizes of A and b are incompatible'
elseif ~isempty(A)
nA = size(A);
b = b(:);
nb = size(b,1);
if nA(1)~=nb
error 'Sizes of A and b are incompatible'
end
if nA(2)~=n
error 'A is incompatible in size with x0'
end
end

% defaults for nonlcon
if (nargin<7) || isempty(nonlcon)
nonlcon = [];
end

% test for feasibility of the initial value
% against any general inequality constraints
if ~isempty(A)
if any(A*x0>b)
error 'Infeasible starting values (linear inequalities failed).'
end
end
if ~isempty(nonlcon)
if any(feval(nonlcon,(reshape(x0,xsize)),varargin{:})>0)
error 'Infeasible starting values (nonlinear inequalities failed).'
end
end

% set default options if necessary
if (nargin<8) || isempty(options)
options = optimset('fminsearch');
end

% stuff into a struct to pass around
params.args = varargin;
params.LB = LB;
params.UB = UB;
params.fun = fun;
params.n = n;
params.xsize = xsize;

params.OutputFcn = [];

params.A = A;
params.b = b;
params.nonlcon = nonlcon;

% 0 --> unconstrained variable
% 1 --> lower bound only
% 2 --> upper bound only
% 3 --> dual finite bounds
% 4 --> fixed variable
params.BoundClass = zeros(n,1);
for i=1:n
k = isfinite(LB(i)) + 2*isfinite(UB(i));
params.BoundClass(i) = k;
if (k==3) && (LB(i)==UB(i))
params.BoundClass(i) = 4;
end
end

% transform starting values into their unconstrained
% surrogates. Check for infeasible starting guesses.
x0u = x0;
k=1;
for i = 1:n
switch params.BoundClass(i)
case 1
% lower bound only
if x0(i)<=LB(i)
% infeasible starting value. Use bound.
x0u(k) = 0;
else
x0u(k) = sqrt(x0(i) - LB(i));
end

% increment k
k=k+1;
case 2
% upper bound only
if x0(i)>=UB(i)
% infeasible starting value. use bound.
x0u(k) = 0;
else
x0u(k) = sqrt(UB(i) - x0(i));
end

% increment k
k=k+1;
case 3
% lower and upper bounds
if x0(i)<=LB(i)
% infeasible starting value
x0u(k) = -pi/2;
elseif x0(i)>=UB(i)
% infeasible starting value
x0u(k) = pi/2;
else
x0u(k) = 2*(x0(i) - LB(i))/(UB(i)-LB(i)) - 1;
% shift by 2*pi to avoid problems at zero in fminsearch
% otherwise, the initial simplex is vanishingly small
x0u(k) = 2*pi+asin(max(-1,min(1,x0u(k))));
end

% increment k
k=k+1;
case 0
% unconstrained variable. x0u(i) is set.
x0u(k) = x0(i);

% increment k
k=k+1;
case 4
% fixed variable. drop it before fminsearch sees it.
% k is not incremented for this variable.
end

end
% if any of the unknowns were fixed, then we need to shorten
% x0u now.
if k<=n
x0u(k:n) = [];
end

% were all the variables fixed?
if isempty(x0u)
% All variables were fixed. quit immediately, setting the
% appropriate parameters, then return.

% undo the variable transformations into the original space
x = xtransform(x0u,params);

% final reshape
x = reshape(x,xsize);

% stuff fval with the final value
fval = feval(params.fun,x,params.args{:});

% fminsearchbnd was not called
exitflag = 0;

output.iterations = 0;
output.funcCount = 1;
output.algorithm = 'fminsearch';
output.message = 'All variables were held fixed by the applied bounds';

% return with no call at all to fminsearch
return
end

% Check for an outputfcn. If there is any, then substitute my
% own wrapper function.
if ~isempty(options.OutputFcn)
params.OutputFcn = options.OutputFcn;
options.OutputFcn = @outfun_wrapper;
end

% now we can call fminsearch, but with our own
% intra-objective function.
[xu,fval,exitflag,output] = fminsearch(@intrafun,x0u,options,params);

% undo the variable transformations into the original space
x = xtransform(xu,params);

% final reshape
x = reshape(x,xsize);

% Use a nested function as the OutputFcn wrapper
function stop = outfun_wrapper(x,varargin);
% we need to transform x first
xtrans = xtransform(x,params);

% then call the user supplied OutputFcn
stop = params.OutputFcn(xtrans,varargin{1:(end-1)});

end

end % mainline end

% ======================================
% ========= begin subfunctions =========
% ======================================
function fval = intrafun(x,params)
% transform variables, test constraints, then call original function

% transform
xtrans = xtransform(x,params);

% test constraints before the function call

% First, do the linear inequality constraints, if any
if ~isempty(params.A)
% Required: A*xtrans <= b
if any(params.A*xtrans(:) > params.b)
% linear inequality constraints failed. Just return inf.
fval = inf;
return
end
end

% resize xtrans to be the correct size for the nonlcon
% and objective function calls
xtrans = reshape(xtrans,params.xsize);

% Next, do the nonlinear inequality constraints
if ~isempty(params.nonlcon)
% Required: nonlcon(xtrans) <= 0
cons = feval(params.nonlcon,xtrans,params.args{:});
if any(cons(:) > 0)
% nonlinear inequality constraints failed. Just return inf.
fval = inf;
return
end
end

% we survived the general inequality constraints. Only now
% do we evaluate the objective function.

% append any additional parameters to the argument list
fval = feval(params.fun,xtrans,params.args{:});

end % sub function intrafun end

% ======================================
function xtrans = xtransform(x,params)
% converts unconstrained variables into their original domains

xtrans = zeros(1,params.n);
% k allows some variables to be fixed, thus dropped from the
% optimization.
k=1;
for i = 1:params.n
switch params.BoundClass(i)
case 1
% lower bound only
xtrans(i) = params.LB(i) + x(k).^2;

k=k+1;
case 2
% upper bound only
xtrans(i) = params.UB(i) - x(k).^2;

k=k+1;
case 3
% lower and upper bounds
xtrans(i) = (sin(x(k))+1)/2;
xtrans(i) = xtrans(i)*(params.UB(i) - params.LB(i)) + params.LB(i);
% just in case of any floating point problems
xtrans(i) = max(params.LB(i),min(params.UB(i),xtrans(i)));

k=k+1;
case 4
% fixed variable, bounds are equal, set it at either bound
xtrans(i) = params.LB(i);
case 0
% unconstrained variable.
xtrans(i) = x(k);

k=k+1;
end
end

end % sub function xtransform end

```