Multi-start iterative multivariate constrained nonlinear programming

[lweunice]

Hi,

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.

 

[Future Bruno]

So I guess this is a kind of multi-start iterative multivariate constrained nonlinear programming problem. I have found some packages like NLopt, IPOPT, and Gnu Octave. Are there any more packages out there that can solve the problem? If so, please let me know. Thank you!

 

[quicknote]

I would suggest using a combination of numerical optimization techniques and machine learning algorithms to solve this problem. There are many packages available in C/C++ that can handle constrained nonlinear programming, such as NLopt, IPOPT, and Ceres Solver. These packages use different methods such as gradient-based optimization, evolutionary algorithms, and swarm intelligence to find the optimal solution.

In addition, you can also try using machine learning algorithms such as deep learning or reinforcement learning to learn the underlying patterns and relationships in your data and then use that knowledge to optimize the cost function. These algorithms have been proven to be effective in solving complex optimization problems with high-dimensional data.

Another approach would be to use a combination of both numerical optimization and machine learning techniques. You can use the optimization methods to find a good initial guess for M and then use machine learning to fine-tune the solution and improve its accuracy.

Overall, there is no one-size-fits-all solution for this type of problem, and it may require some experimentation and tweaking to find the best approach. I recommend consulting with a data scientist or a numerical optimization expert to help you choose the most suitable method for your specific problem.