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$Is there general theory for solving optimization problem of the following kind

\begin{align} &\max _{x_1,x_2,...,x_N} N \\ \text{ s.t.}& \sum_{i=1}^N f(x_i) \le a\\ &\sum_{i=1}^N g(x_i) \le b\\ & \sum_{i=1}^N x_i \le c \end{align}

where $f()$ and $g()$ are given convex, non-negative functions. The 'weird' property of this problem is that we have to optimize over the argument of the summation. At this stage I am looking at reference and examples.

Thank you in advance.

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  • $\begingroup$ Are the functions $f$ and $g$ convex? If not, please remove the "convex-optimization" tag. And is there any other information about $f$, $g$, and $x$ you can share? It matters. $\endgroup$ – Michael Grant Aug 9 '15 at 21:32
  • $\begingroup$ Yes, they are. I will add. But even then is it still a convex optimization problem? $\endgroup$ – Boby Aug 9 '15 at 21:44
  • $\begingroup$ Oh no, it's definitely not convex. But it might be quasiconvex. Again, is there anything else you can say about $f$, $g$, and $x$? For instance, are they nonnegative? Monotonic? $\endgroup$ – Michael Grant Aug 9 '15 at 21:51
  • $\begingroup$ I guess I can add non-negativity. Have you seen a problem like that in some context before? Also, for simplicity we can just look at the case when there is no $g$. $\endgroup$ – Boby Aug 9 '15 at 22:15
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    $\begingroup$ No, I haven't. But Boby, come on, details like convexity and non-negativity matter. They can spell the difference between being able to solve the problem efficiently and not being able to solve it at all. When you ask a question here you need to supply all the information about the problem you have. $\endgroup$ – Michael Grant Aug 9 '15 at 22:17
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The obvious guess is going to be that the optimal solution will take the form $x_1=x_2=\dots=x_N$ for some $N$. Thus, this suggests solving the following optimization problem:

$$\begin{align*} \text{maximize } \; &N\\ \text{subject to } \; &N f(x) \le a, N g(x) \le b, Nx \le c \end{align*}$$

where $x$ ranges over $\mathbb{R}$ and $N$ ranges over $\mathbb{N}$. Now you have only a two-dimensional optimization problem.

For instance, you could iterate over $N=1,2,3,\dots$ and for each fixed $N$, search for $x \in [-\infty,c/N]$ such that $f(x) \le a/N$ and $g(x) \le b/N$. Depending on the form of $f,g$, this might be feasible to solve.

If this is possible, you could speed it up by using bisection search to find the largest such $N$ (i.e., start by iterating over $N=1,2,4,8,16,\dots$; then when you find $N_0,N_1$ such that $N_0$ is feasible and $N_1$ isn't feasible, use binary search to find the largest $N$ such that $N_0 \le N < N_1$ and $N$ is feasible).

Better yet, define

$$h(x) = \max(f(x)/a, g(x)/b, x/c).$$

You can verify that $N,x$ is a feasible solution if and only if $h(x) \le 1/N$. Thus, to maximize $N$, it suffices to find $x^*$ that minimizes $h(x)$, and take $N=\lfloor 1/h(x^*) \rfloor$. Since $h:\mathbb{R} \to \mathbb{R}$ is a unidimensional function, minimizing $h(x)$ might be feasible via a number of methods, depending on the specific properties of of $f,g$.

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