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?
