I have a function $f(X,Y) = \sum_{i=1}^n g(x_i) + \sum_{j=1}^m h(y_j) + \sum_{i=1}^n\sum_{j=1}^mk(x_i,y_j) + c$ and the objective is to prove that $f$ is strongly convex. These are the known properties regarding to the function:
- $g$, $h$ and $k$ are allconvex. $c$ is a constant.
- $X = \{x_1,x_2,\dots,x_n\}$, $Y=\{y_1,y_2,\dots,y_m\}$. $n$ and $m$ are known.
- $x_i \ge 1 \text{ for } i=1,2,\dots,n$; $y_j > 0 \text{ for }j=1,2,\dots,m$. $x_i$ and $y_j$ are both finite set.
- $g$, $h$ and $k$ are all continuous differentiable over their domains.
A concrete example would be: $f(X,Y) = (a_1\ln x_1 + b_1x_1) + (a_2\ln x_2 + b_2x_2) + (c_1(y_1-d_1)^2) +(x_1(y_1-e_1)^2) + (x_2(y_2-e_2)^2)$.
In the function of $f$, $a, b, c, d, e$ are all constants, $X = \{x_1,x_2\}$, $Y = \{y_1\}$. $g(x) = a\ln x + bx$, $h(y) = c(y-d)^2$ and $k(x,y) = x(y-e)^2$.
Is it correct to show all $f''(X,Y) >0$ to prove this? i.e. each $\frac{\partial^2 f}{\partial x \partial x} > 0$, $\frac{\partial^2 f}{\partial y \partial y} > 0$ and $\frac{\partial^2 f}{\partial x \partial y} > 0$.
By the way, does strongly convex have any connection to Lipschitz continuity?
