I have an objective function given by
$\underset{a_{c,i}}{\max}\hspace{1mm}\underset{i,i=1,\cdots,M}{\min}\hspace{1mm}\frac{s_i(a_{c,i})}{d_i}$
$c=1,2,\cdots,N$, $i=1,2,\cdots,M$
$s_i=\sum_{c=1}^Na_{c,i}f_{c,i}$ with $0\le a_{c,i}\le1$
where, $f_{c,i}\ge0$ (are known)
What can we say about the convexity of this objective function
$d_i$'s are $>0$ and $s_i(a_{c,i})>0$
Are $\max \min$ optimization objective nonlinear/nonconvex?
If so, how can I linearize/convexify it?
Let $a=(a_{c,i})$ for $c \in \{1, ..., N\}, i \in \{1, ..., M\}$. You have concave functions $g_i(a)$ for $i \in \{1, ..., M\}$. You want to solve: $$ \max_{a \in [0,1]^{NM}} \min_{i\in \{1, ..., M\}} g_i(a)$$ Let $x \in \mathbb{R}$. Your problem is equivalent to choosing $(a,x)$ to solve: \begin{align} \mbox{Maximize:} & \quad x\\ \mbox{Subject to:} & \quad g_i(a) \geq x \quad \forall i \in \{1, ..., N\} \\ & \quad a_{c,i} \in [0,1] \quad \forall c \in \{1, ..., N\}, i \in \{1, ..., M\}\\ &\quad x \in \mathbb{R} \end{align} This is a convex optimization problem. If your $g_i(a)$ functions are linear then this is a linear program.
In fact your $g_i(a)$ functions are not only linear but have a very simple structure: $$ g_i(a) = \frac{1}{d_i}\sum_{c=1}^N a_{c,i}f_{c,i} $$ with given constants $d_i>0, f_{c,i}\geq 0$. So each $g_i(a)$ function uses its own variables $(a_{1,i}, a_{2,i}, ..., a_{N,i})$ and the solution is trivial: $$a^*_{c,i}=1 \quad \forall i,c $$ $$x^* = \min_{i \in \{1, ..., N\}} g_i(a^*)$$