Method

FindMinimum allows setting bounds on variables (box-constrained minimization) but does not find the minimum, instead giving a warning that the boundary was reached.

In[6]:= AbsoluteTiming @ 
 FindMinimum[(x - 1.5)^2 + (y + 1.3)^2, {{x, 0, -1, 1}, {y, 0, -1, 1}}]

During evaluation of In[6]:= FindMinimum::reged: The point {1.,-0.866667} is at the edge of the search region {-1.,1.} in coordinate 1 and the computed search direction points outside the region.

Out[6]= {0.00137144, {0.437778, {x -> 1., y -> -0.866667}}}

Rerunning the optimization, eliminating the variable at its bound finds the actual minimum.

In[11]:= AbsoluteTiming @ 
 FindMinimum[(x - 1.5)^2 + (y + 1.3)^2 /. x -> 1, {{y, 0, -1, 1}}]

Out[11]= {0.000507734, {0.34, {y -> -1.}}}

This is much faster than putting in the bounds in as constraints.

In[5]:= AbsoluteTiming @ 
 FindMinimum[{(x - 1.5)^2 + (y + 1.3)^2, -1 <= x <= 1, -1 <= y <= 
    1}, {{x, 0}, {y, 0}}]

Out[5]= {0.0341419, {0.34, {x -> 1., y -> -1.}}}

The function in the attached file automates this process, repeatedly elimating variables which reach their bounds, and is much faster than putting in constraints.

In[10]:= AbsoluteTiming @ 
 fmbox[(x - 1.5)^2 + (y + 1.3)^2, {{x, 0, -1, 1}, {y, 0, -1, 1}}]

Out[10]= {0.000847333, {0.34, {x -> 1, y -> -1.}}}

Automated function

fmbox[obj_,vwbi_List]:=
Block[{nvars,newobj,starts,var,init,lb,ub,testlist,resrls,fmres,allrls,test,objrl,val,bndvars},
Off[FindMinimum::reged];
nvars = Length[vwbi];
newobj = obj;
starts = vwbi;
testlist = Flatten[vwbi /. {var_,lb_?NumberQ, init_?NumberQ, ub_?NumberQ}-> {var == lb, var==ub}];
resrls = {};
allrls=Do[
fmres = FindMinimum[newobj,starts];
test = Select[testlist,#/. fmres[[2]]&];
If[Length[test]==0,
Return[Join[resrls,fmres[[2]]]],
objrl = test /. Equal[var_,val_?NumberQ] -> (var -> val);
newobj = newobj /. objrl;
bndvars = objrl[[All,1]];
starts = Select[starts,!MemberQ[bndvars,#[[1]]]&];
resrls = Join[resrls,objrl];
If[nvars==Length[resrls],Return[resrls]];
]; (*End If *)
,{i,nvars}]; (* End Do *)
{obj /.allrls,allrls}
]

Benchmarking

AbsoluteTiming @ fmbox[(x - .5)^2 + (y + .3)^2, {{x, 0, -1, 1}, {y, 0, -1, 1}}]

{0.000608449, {0., {x -> 0.5, y -> -0.3}}}

AbsoluteTiming @ fmbox[(x - 1.5)^2 + (y + .3)^2, {{x, 0, -1, 1}, {y, 0, -1, 1}}]

{0.00110092, {0.25, {x -> 1, y -> -0.3}}}

AbsoluteTiming @ fmbox[(x - 1.5)^2 + (y + 1.3)^2, {{x, 0, -1, 1}, {y, 0, -1, 1}}]

{0.000847333, {0.34, {x -> 1, y -> -1.}}}

Attachments: