Compact-box global minimization problems

A box in ${\mathbb{R}}^n$ is $D_0={\prod }_{j=1}^n[a_j,b_j]$ with finite real endpoints $a_j\le b_j$. Here $n\ge 1$. The box is closed and bounded, hence compact by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line. An optimization instance on a compact box consists of this $D_0$, a continuous objective $f:D_0\to \mathbb{R}$ (continuity in the inherited Euclidean metric, as in Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form), and a feasible set $C\subseteq D_0$. The unconstrained instance has $C=D_0$. A constrained instance may specify $C=\{x\in D_0:g_i(x)\le 0,1\le i\le m\}$ for a finite list of real functions $g_i$.

When $C\mathrm{\ne }\mathrm{\varnothing }$, the set $f[C]$ is nonempty and bounded below: $D_0$ is compact by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, its continuous image $f[D_0]$ is compact in the usual metric on $\mathbb{R}$ by The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset and therefore bounded by A compact subset of a metric space is closed and bounded. Thus $f[D_0]$ lies in a real interval $(c-r,c+r)$ for some $c\in \mathbb{R}$ and $r>0$, by Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space and The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded; hence $c-r$ is a lower bound of $f[C]\subseteq f[D_0]$. Therefore Every nonempty set bounded below has an infimum gives the real number $f^{\ast }=\inf f[C]$, using Greatest lower bound (infimum). No attainment is assumed for a general $C$. Statements about finite empty-set detection require a separate constraint certificate or a stated separation margin; objective continuity alone does not certify infeasibility.