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I need to solve a multi-start iterative multivariate constrained nonlinear programming problem. But I can't seem to be find any package that will solve it for me. I have been wondering if anyone would be kind enough to tell me some packages that can solve the problem. (C/C++ packages are preferred, and Fortran packages are least preferred.)

I need to solve the following problem:

M is THE MATRIX I want to solve. None of its entries is known.

I have a lot of vectors. What I do is to use M to project them into a new space, and try to keep their relative distance sequences.

Let a, b, c denote the vectors.

The test condition is:

If distance(a, b) > distance(a, c) AND distance(Ma, Mb) < distance(Ma, Mc)

It is an anomaly, so it entails some cost.

The cost function is f(t), and is defined as:

S(t) = (Ma - Mc) (Ma - Mc)^{T}- (Ma - Mb) (Ma - Mb)^{T}

f(t) = summation(from 0 to t) exp( S(t) )

and the constraint would be the squared sum of each row of M has to be 1, or the global minimum simply occurs when all the entries of M are 0s.

However, the tricky part is that if I want to know the pairs in my vectors that satisfy the test condition, I need to know M, but M is precisely the variable I want to solve!

So I suppose one way to do this is to guess M, acquire the vectors satisfying the test condition, and thus got the cost function, and then optimize.

And repeat the procedure to get a local minimum.

And restart the whole thing to try to get a better local minimum.

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# Multi-start iterative multivariate constrained nonlinear programming

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