1
$\begingroup$

Suppose two non-negative convex functions $f$ and $g$ be given. We want to solve the following optimization $$\max_{g\leq\epsilon}f.$$

Now suppose that both $f$ and $g$ can be upper-bounded by a convex combination of finite number of functions $\{f_i\}$, and $\{g_i\}$ i.e., $$f\leq \sum_{i}\alpha_i f_i,$$ and $$g\leq\sum_{i}\alpha_i g_i,$$

Can we show that $$\max_{g\leq \epsilon}f\leq \sum_i\alpha_i\max_{g_i\leq \epsilon_i}f_i$$ where $\sum_{i}\alpha_i\epsilon_i=\epsilon$?

If not, is there any established relations between these two?

$\endgroup$
1
  • $\begingroup$ What if $g(x)=0$ for all $x$, $g_1(x)=g_2(x)=10$ for all $x$, and $\epsilon=0$. There may not be any values $x$ for which $g_i(x) \leq \epsilon_i$. $\endgroup$
    – Michael
    Mar 21, 2016 at 21:56

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.