Timeline for optimization of a separable function
Current License: CC BY-SA 3.0
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| Jul 31, 2012 at 12:40 | comment | added | Higgs88 | Hello Gilead, I have alos an idea which maybe costs so much. If we inherit the previous notations, then we denote $$f_i(x_i)=\left(1+\rho x_ik_j+\sqrt{(x_ik_j+\rho)^2+1-\rho^2}-\frac{2\alpha_i^j}{\theta_i}\right)^2$$ and define $$G_1(d,u)=\max_{d\leq x_1\leq \min(u,X_1)}f_1(x_1)$$ Then for $0<d\leq u$, $G_1(d,u)$ can be computed explicitly. We can then construct $G_i(d,u)$ by induction. Assume that we know $G_{i-1}(d,u)$, $\forall i\geq 2$ $$G_i(d,u)=\max_{d\leq x_i\leq \min(u,X_i)}G_{i-1}(x_i,\frac{\theta_{i-1}}{\theta_i}x_i)+f_i(x_i)$$ Could you give some comments? | |
| Jul 31, 2012 at 12:33 | history | edited | Higgs88 | CC BY-SA 3.0 |
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| Jul 31, 2012 at 12:13 | comment | added | Higgs88 | Thank you for your reply. Could you please specify the numerical methods that you mentioned? For example nonlinear programming, piecewise-linear approximation, SOS2, MIP? Thanks a lot! | |
| Jul 17, 2012 at 21:31 | comment | added | Gilead | You can either solve this to a local optima using nonlinear programming (fast), solve to global optima using global optimization code (potentially computationally expensive), or solve a separable programming problem using piecewise-linear approximation of the [possibly] nonconvex objective function using SOS2 constraints--which is designed for problems like this--or simply using MIP constraints. Your method of minimizing $x$'s one-by-one (essentially solving $n$ univariate problems) may not give you a global solution unless a global solver is used in each instance. | |
| Jul 17, 2012 at 15:01 | history | asked | Higgs88 | CC BY-SA 3.0 |