Let $X\subset\mathbb R^n$ be the convex domain of the problem, with relative interior taken in $\operatorname{aff}(X)$. Let $f_0,f_1,\dots,f_m:X\to\mathbb R$ be finite convex functions, and let $h_1,\dots,h_p:X\to\mathbb R$ be affine. Assume $p^*$ is finite and that there exists $\bar{x}\in\operatorname{ri}(X)$ with \begin{align*} f_i(\bar{x})<0 \quad (1\le i\le m), \qquad h_j(\bar{x})=0 \quad (1\le j\le p). \end{align*} Then strong duality holds: \begin{align*} d^*=p^*. \end{align*} If, in addition, the set \begin{align*} V=\{(u,v,t)\in\mathbb R^m\times\mathbb R^p\times\mathbb R: \exists x\in X\text{ with } f_i(x)\le u_i,\ h_j(x)=v_j,\ f_0(x)\le t\} \end{align*} is closed near $(0,0,p^*)$, then the dual optimum is attained. This last closedness assumption is a course-specific sufficient condition ensuring that the separating hyperplane can support $V$ at the actual value point rather than only at a limiting perturbation.