I have a target function, I've computed its Hessian to check convexity, it has a positive-definite sub-matrix and small negative-definite sub-matrix and a kernel. Sometimes it is even better -- the the Hessian semi-definite.

I solve material placement optimisation problem. Each variable $w_i$ corresponds to material amount in the i-th node, so each variable is between [0,1] or [0.1, 1] and sum of material is constant, that is $\sum_i w_i = const$.

The current idea is to use convex solvers for convex part, with randomised or brute-force approach for the rest of variables. I also thought about convexifiying the given function, and then tracking backwards from this "fully convex" solution. How those kind of problems is treated in practice? What is state of the art?

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