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.(adsbygoogle = window.adsbygoogle || []).push({});

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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# Linear Programming Constraints

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