Handing a list of constraint expressions to a C++ function with MathLink

Question

I need to solve an optimization problem, which is defined in a Mathematica notebook.

Using Mathematica's FindMinimum is not an option, because it is too slow. So, the idea is to use an external solver for quadratically constraint quadratic problems and use MathLink to get the constraints of the problem from Mathematica, calculate the solution, and return the solution to Mathematica.

And here is my problem: The constraints (quadratic and linear) are given to me as a list of expressions, e.g.

{ -1+(-0.363263+x)^2+(-0.329466+y)^2<=0,
-1+(-0.248721+x)^2+(0.451803 +y)^2<=0, 
-1+(0.33444 +x)^2+(-0.41341+y)^2<=0, 
-1+(0.414249 +x)^2+(0.384528 +y)^2<=0, 
-1+(-0.65488+x)^2+(0.242478 +y)^2<=0, 
-1+(-0.176244+x)^2+(0.30843 +y)^2<=0, 
-1.4+x<=t, -0.5+y<=t, 0.4 -x<=t, -0.5-y<=t }

How do I deal with such a list in my C++ function? Do I need to change its template signature so that it takes a SymbolList instead of a RealList (as in my current version)?

Or is there a way to extract all the numbers in my constraint list and put them in a list in Mathematica?

Answer 1

I don't know if this answer helps you but what I wanted to convey would be too lengthy for a comment. First of all quoting from your comment

"...the objective function (it is t, so fairly trivial)"

this problem actually renders to be so simple that any mechanism involving Compile or binding with external optimization routine (e.g CPLEX) involving Mathlink becomes completely unnecessary. Given that you agree with my conclusion of your comment you can see that MMA built-in optimization functions FindMinimum and NMinimize are already pretty efficient.

linearcons=
-1.4 + x <= t &&
-0.5 + y <= t &&
 0.4 - x <= t &&
-0.5 - y <= t;
quadraticcons=
-1+(-0.363263+x)^2+(-0.329466+y)^2<=0&&
-1+(-0.248721+x)^2+(0.451803+y)^2<=0&&
-1+(0.33444+x)^2+(-0.41341+y)^2<=0&&
-1+(0.414249+x)^2+(0.384528+y)^2<=0&&
-1+(-0.65488+x)^2+(0.242478+y)^2<=0&&
-1+(-0.176244+x)^2+(0.30843+y)^2<=0

Now call the optimization functions and see the Timing

res=NMinimize[{t,quadraticcons && linearcons},{x,y,t}]; // AbsoluteTiming
res1 = FindMinimum[{t,quadraticcons && linearcons},{x,y,t}]; // AbsoluteTiming

{0.2100003, Null}

{0.0300001, Null}

Is not it practically fast enough for your "fairly trivial" objective function? Added advantage is that MMA finds the unique global minimum here. You can visualize the minimum as in case of this too simple objective the intersection of the parameter space spanned by the quadratic and the linear constraint is a unique point. Lets see the two parameter spaces seperately using RegionPlot3D and RegionPlot.

Now using the visualization code from a past question we can see that the linear and the quadratic constrain agrees at a single point and MMA finds it really fast as we have seen above. The sought after point {x,y,t} is

FindArgMin[{t, quadraticcons && linearcons}, {x, y, t}]

{0.537203, -0.076731, -0.137203}

However I am using a high end desktop with core i7 extreme processor so on average machine the timing may turn out to be a bit worse..

BR