I encountered an optimization problem which does not belong to any well-known category of optimization.
The system has $M$ (typically $M=120$) real variables and $N$ (typically $N=100$) constraints (all of them are equations), where $M>N$. Thus, all constraints can be satisfied.
The objective function is linear (weighted sum).
Some constraints are quartic equations (all terms are degree 4). Other constraints are quadratic equations (all terms are degree 2).
How to solve such optimization problem? I imagine there are two (possible) general approaches:
- mathematical programming approach, e.g. nonlinear programming. However, I find no specific program which fits the configurations above.
- meta-heuristics, e.g., hill-climbing or simulated annealing. I do not prefer this approach because it does not address the nature of under-determined system.