# Convex Optimization: $\min \left(\sum f(x_i)\right)$ s.t.: $\sum g(x_i) \le C$, where x has no closed-form expression

I am a newbie here and now facing to a hard convex optimization problem, sincerely wish anyone can help me : ) There is a variable vector $\textbf{x}$, and we wish to minimize the sum of $f()$: $$\min~ \left(\sum f(x_i)\right),$$ where $f()$ is a function with many log and exp, thus $\textbf{x}$ has no closed-form expression. Until here I know I can solve by gradient descent method to get numerical results, but we further have one constraint, that, $$\text{s.t. }\sum g(x_i) \le C,$$ where $C$ is a constant and $g()$ is also a function with many log and exp, thus $\textbf{x}$ still has no closed-form expression...

Note that $f$ and $g$ are all with 1st and 2nd derivatives, and we already proved the convexity of $f$ and $g$. What we need to do is just to numerically solve for the vector $\textbf{x}$.

I used Mathematica and Matlab by some of their eternal commands/tools, but failed; and also I manually tried Lagrange dual solution, but by the end, due to the non-closed-form of $\textbf{x}$, we still cannot solve the derivative equations. I guess this is related to non-closed-form convex optimization, or gradient descent with constraints, but I am quite weak at Math, so, I just sincerely wish someone can point me out on any effective methods or algorithms ; ) (Or maybe I missed some better eternal function tools by Mathematcia and Matlab, you can also feel free to figure out.. ^^)

I do really appreciate your help on this, maybe this is quite easy to you guys, but this can save my life =)

THANK YOU VERY MUCH!

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Can we take a look at the non-closed form expressions, if they are not too big (i.e. fit in one line)? Or at least the definitions of $f$ and $g$ so that we can deduce the latter. – Patrick Da Silva Aug 12 '11 at 12:38
Have you tried CVX (cvxr.com/cvx)? – Alejandro Aug 12 '11 at 15:03
Dears, $f$ and $g$ are complicated with many exp(x) and log(x) in denominator and exponent part, $g$ is a long equation ... Also I've tried CVX but failed. I just wish to know what kind of method we can deal with this kind of constrained minimization. Thanks a lot. ^^ – Dobby Aug 13 '11 at 2:02
Sorry but is there any people that can help me? Thanks a lot! – Dobby Aug 19 '11 at 9:31

Try a real solver like IPOPT or KNITRO: http://neos-server.org/neos If those fail, you'll need to go back to the drawing board and check the properties of your problem. For instance does there exist an $x$ such that $\sum g(x_i) < C$ (strictly)? If not your problem is degenerate.