Linear Programming Constraints

[Kreizhn]

I'm trying to minimize a function over a rather complicated surface. I'm using an algorithm that takes an initial guess, finds the tangent plane at that point, minimizes using a linear programming algorithm, then (tries to) project back onto the complicated surface.

More specifically, if $\xi$ is a vector and $H(\xi)$ is the surface, I want to solve the linear programming problem
$$\min c^T \xi, \quad \text{subject to } \nabla H(\xi) (\xi - \bar \xi) = 0$$
Now my problem is that I'm having trouble setting up the constraints. This wouldn't normally be difficult, except that my surface is parameterized as
$$H(\xi) = X_f(\xi) - X_d$$
where $X_f, X_d$ are matrices.

Normally in these cases $H(\xi)$ is at worst vector-valued and so $\frac{\partial H}{\partial \xi_j}$ is a vector so that $\nabla H(\xi)$ is a matrix. However in this case $\frac{\partial H}{\partial \xi_j}$ is itself a matrix.

How do I handle this? Do I ``vectorize'' $\frac{\partial H}{\partial \xi_j}$? Do I break it into real and imaginary parts then vectorize? I'm not sure how to handle this.

 

[fresh_42]

If your surface consists of vectors or points shouldn't play a role. In the context they are points, however they look like. The condition is meant to constrain the set of all points, regardless its structure. The structure might play a role in the optimization process as it contains additional information which might be useful, e.g. the shape of the boundaries where the optimal solution is likely to be found.