  We consider the semi-infinite system of polynomial inequalities of the form
\[ \mathbf{K}:=\{x\in\mathbb{R}^m\mid p(x,y)\ge 0,\ \ \forall y\in
S\subseteq\mathbb{R}^n\}, \] where $p(x,y)$ is a real polynomial in the
variables $x$ and the parameters $y$, the index set $S$ is a basic
semialgebraic set in $\mathbb{R}^n$, $-p(x,y)$ is convex in $x$ for every $y\in
S$. We propose a procedure to construct approximate semidefinite
representations of $\mathbf{K}$. There are two indices to index these
approximate semidefinite representations. As two indices increase, these
semidefinite representation sets expand and contract, respectively, and can
approximate $\mathbf{K}$ as closely as possible under some assumptions. In some
special cases, we can fix one of the two indices or both. Then, we consider the
optimization problem of minimizing a convex polynomial over $\mathbf{K}$. We
present an SDP relaxation method for this optimization problem by similar
strategies used in constructing approximate semidefinite representations of
$\mathbf{K}$. Under certain assumptions, some approximate minimizers of the
optimization problem can also be obtained from the SDP relaxations. In some
special cases, we show that the SDP relaxation for the optimization problem is
exact and all minimizers can be extracted.
