Timeline for Least sum squares given constraints on subcomponents
Current License: CC BY-SA 2.5
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| Feb 20, 2011 at 19:18 | history | edited | Brian Borchers | CC BY-SA 2.5 |
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| Feb 20, 2011 at 4:18 | comment | added | Gilead | @Brian, you're right, a high-order multivariate system of polynomials doesn't always have necessarily closed form solution, and if you understand "closed-form" in an engineering sense (i.e. "analytical solution"), then it definitely isn't that, even if it is decomposed into its Groebner basis. As for the second part of your assertion, my comment is in fact agreeing with yours in that I believe global optimization software is necessary at some level for solving this problem because it is inherently nonconvex owing to the equality constraints. | |
| Feb 20, 2011 at 2:48 | comment | added | Brian Borchers | @Gilead, if you write out the KKT conditions, you'll get a polynomial system of equations which also can't be solved by local numerical methods. I'd hardly call a system of polynomial equations a "closed form" solution to the problem. You could apply Groebner basis methods to that system of equations, but in my opinion you'd be better off using the polynomial optimization approach of Gloptipoly or one of its competitors to solve the minimization problem. | |
| Feb 19, 2011 at 22:42 | comment | added | Gilead | @Tony, there is a closed form solution if you write out the KKT conditions. There are no inequality constraints, so you don't have to worry about the complementarity conditions. That said, your equality constraints are either non-smooth or nonlinear (depending on how you formulate them), so your problem is definitely nonconvex. You will need to use global optimization software if you desire the global minimum; local methods will not be able to guarantee true optimality. | |
| Feb 19, 2011 at 5:48 | comment | added | Tony | Thanks for your reply. x is a small vector, about 9 to 15, divided into 3-5 subcomponents. And n is small too, about 9 to 20. I think it is feasible. I hope there's a closed-form solution. But I'll look into your suggested software. | |
| Feb 19, 2011 at 4:20 | history | answered | Brian Borchers | CC BY-SA 2.5 |