Linear Programming Constraints

  • Thread starter Kreizhn
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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 [itex] \xi [/itex] is a vector and [itex] H(\xi) [/itex] is the surface, I want to solve the linear programming problem
[tex] \min c^T \xi, \quad \text{subject to } \nabla H(\xi) (\xi - \bar \xi) = 0 [/tex]
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
[tex] H(\xi) = X_f(\xi) - X_d [/tex]
where [itex] X_f, X_d [/itex] are matrices.

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

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


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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.

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