I have to find the best position of a point respecting some constraint.
The point should be “above” N plane.
So it should respect theses inequation:
a1x + b1y +c1z + d1 > 0
a2x + b2y +c2z + d2 > 0
a3x + b3y +c3z + d3 > 0
.
.
.
.
anx + bny + cnz + dn > 0
Thats the first constraint.
The plane are always defining an existing volume, but this volume is not necessary closed.
The second constraint is that the solution point have to be the closest point to another out of the volume.
ie: the distance between my reference point and the solution point should be the smaller possible.
LINDO API provides you with an arsenal of powerful solvers for linear, nonlinear (convex & nonconvex), quadratic, quadratically constrained, and integer optimization. All solvers incorporate numerous enhancements for maximum speed and robustness.
You can easily redefine the problem as multiple linear problems.
Some logic reasoning will show that the optimal solution has to be on at least one of the planes you used as constraints.
So a linear version of the code you supplied can be redefined as: