Linear programs with strict inequalities and supremum objectives
Suppose that the supremum of a continuous function $f(x)$ subject to $Ax &lt; b$ is $c$, and assume furthermore that the constraints imply a bound on $\|x\|_\infty$. Thus there is a sequence of feasible points $x_n$ such that $f(x_n) \to c$. Since all points in the sequence have bounded norm, it follows that some subsequence of $x_n$ converges to a limit point $y$ satisfying $f(y) = c$ and $Ay \leq b$. This implies that in the bounded case, your problem is equivalent to classical linear programming.

In the unbounded case this argument doesn't work. For example, the infimum of $x-y$ subject to $x-y &gt; 0$ is $0$, but the sequence $(n+1/n,n)$ has no limit points. Nevertheless, the answer should be the same.