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From: Stuart Nettleton on 8 Jul 2010 07:41
Hi, I am experimenting with external numerical constraint functions in
FindMinimum. Would anyone understand how the second FindMinimum might be
made to work in the way of the first?

FindMinimum[{Sin[x ] Sin[2 y], {x^2 + y^2 < 3, -x < 0, -y < 0}}, {x,
y}]
Clear[obj, con];
vars = {x, y};
lenvars = Length[vars];
obj[vars1_] := Module[{subst = Thread[Array[m, lenvars] -> vars1]},
ReplaceAll[Sin[m[1]] Sin[m[2]], subst]
] /; VectorQ[vars1, NumberQ];
con[vars2_] := Module[{subst = Thread[Array[n, lenvars] -> vars2]},
ReplaceAll[{n[1]^2 + n[2]^2, -n[1], -n[2]}, subst]
] /; VectorQ[vars2, NumberQ];
FindMinimum[{obj, Thread[con[vars] < {3, 0, 0}]}, vars]


Thanks,
Stuart

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From: Stuart Nettleton on 8 Jul 2010 20:34
Hi Daniel,
I am pleased that FindMinimum is happy with constraint return values like
{True, False, False} etc.
This gives me a whole new perspective.
Thanks once again for a magnificent insight!
Best regards,
Stuart

> obj[vars_] := Sin[vars[[1]]]*Sin[2*vars[[2]]] /; VectorQ[vars, NumberQ]
>
> con[vars_List, vals_] :=
> Thread[{vars[[1]]^2 + vars[[2]]^2, -vars[[1]], -vars[[2]]} <= vals] /;
> VectorQ[vals, NumberQ] && Length[vals]===3
>
> FindMinimum[{obj[vars], Thread[con[vars,{3,0,0}]]}, vars]

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From: Stuart Nettleton on 8 Jul 2010 20:34
Hi Daniel,
Looking deeper, I was perhaps too happy to see a solution. You have the
condition "/;VectorQ[vals, NumberQ]" in the constraint function using
"vals" instead of "vars". Is this still symbolic execution? If I add "&&
VectorQ[vars, NumberQ]" there is an execution error.
Stuart


On Fri, 09 Jul 2010 08:30:15 +1000, Stuart Nettleton
<Stuart.Nettleton(a)uts.edu.au> wrote:

> Hi Daniel,
> I am pleased that FindMinimum is happy with constraint return values
> like {True, False, False} etc.
> This gives me a whole new perspective.
> Thanks once again for a magnificent insight!
> Best regards,
> Stuart
>
>> obj[vars_] := Sin[vars[[1]]]*Sin[2*vars[[2]]] /; VectorQ[vars, NumberQ]
>>
>> con[vars_List, vals_] :=
>> Thread[{vars[[1]]^2 + vars[[2]]^2, -vars[[1]], -vars[[2]]} <= vals]
>> /;
>> VectorQ[vals, NumberQ] && Length[vals]===3
>>
>> FindMinimum[{obj[vars], Thread[con[vars,{3,0,0}]]}, vars]


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Senior Lecturer
School of Management & Systems
Faculty of Engineering & IT
Energy Policy Research Centre

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From: Bill Rowe on 8 Jul 2010 20:35
On 7/8/10 at 7:41 AM, Stuart.Nettleton(a)uts.edu.au (Stuart Nettleton)
wrote:

>Hi, I am experimenting with external numerical constraint functions
>in FindMinimum. Would anyone understand how the second FindMinimum
>might be made to work in the way of the first?

>FindMinimum[{Sin[x ] Sin[2 y], {x^2 + y^2 < 3, -x < 0, -y < 0}}, {x,
>y}]
>Clear[obj, con]; vars = {x, y}; lenvars = Length[vars];

>obj[vars1_]:= Module[{subst = Thread[Array[m, lenvars] -> vars1]},
>ReplaceAll[Sin[m[1]] Sin[m[2]], subst]]/; VectorQ[vars1, NumberQ];

>con[vars2_] := Module[{subst = Thread[Array[n, lenvars] -> vars2]},
>ReplaceAll[{n[1]^2 + n[2]^2, -n[1], -n[2]}, subst]
>]/; VectorQ[vars2, NumberQ];

>FindMinimum[{obj, Thread[con[vars] < {3, 0, 0}]}, vars]

You have made both functions con and obj needlessly complex.
Additionally, you are requiring con to work for an array of
numbers. So, you will not get a list of constraints when this
function gets evaluated.

First, obj can be simplified to:

obj[vars_]:=Times@@Sin[vars]

and con can be simplified to:

con[vars_] := Flatten@{Plus@@vars^2, -vars}

Since both of these are defined using SetDelayed a terminating
semicolon is not needed. And since they both localize the
supplied argument, it is not necessary to define them with
uniquely named arguments.

With these definitions, the FindMinimum problem can be done as follows:

In[95]:= FindMinimum[{obj[vars], Thread[con[vars] < {3, 0, 0}]}, vars]

Out[95]= {3.41113*10^-12,{x->0.00256066,y->1.33213*10^-9}}

and to verify this is the same answer as you would have gotten
by explicitly typing everything:

In[96]:= FindMinimum[{Sin[x] Sin[ y], {x^2 + y^2 < 3, -x < 0, -y <
0}}, {x, y}]

Out[96]= {3.41113*10^-12,{x->0.00256066,y->1.33213*10^-9}}


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