# 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

Mentor
2018 Award
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.

"Linear Programming Constraints"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving