# Optimize an underdetermined system with quartic constraints

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.
• Oh, I see you also posted this on SciComp.SE: math.stackexchange.com/q/3117184/14578. Please do not post the same question on multiple sites. Each community should have an honest shot at answering without anybody's time being wasted. – D.W. Feb 18 '19 at 5:19
• I'm voting to close this question because it was also posted on SciComp.SE. – D.W. Feb 18 '19 at 5:20
• The question on SciComp has now been removed so I think we should reopen this. – David Richerby Feb 20 '19 at 17:13
• The copy on SciComp is gone, but the question on Math.SE is still there, and I answered this question on Math.SE: math.stackexchange.com/q/3117184/14578. Did that answer meet your needs? If not, please edit the question to clarify your requirements and identify which requirement(s) that answer doesn't meet. – D.W. Mar 12 '19 at 16:49
• Yes, the answer meet my needs. I also posted my research. – whitegreen Mar 14 '19 at 3:36