= Feasible region
{title2=$F$}
{wiki}
= Feasible set
{synonym}
The <set> of all <feasible points> of an <optimization> problem. It depends on the <optimization constraints> and the decision domain; the <objective function> ranks its members.
Think of three successive questions. Which decisions can be represented in the model? Which satisfy every <optimization constraint>? Which of those receive the best <objective function> value? For decisions $z$ in a domain $D$, <inequalities> $g_i(z)\leq0$ and <equations> $h_j(z)=0$ define the <feasible region>
$$
F=\{z\in D:g_i(z)\leq0\text{ for every }i,\ h_j(z)=0\text{ for every }j\}.
$$
The domain must be contained in the <domain of a function> for every <function> being evaluated; choosing <integers> rather than <real numbers> as decisions can also change the <feasible set>. Constraint satisfaction determines membership of $F$. Minimizing an <objective function> $f$ then compares its members by $f(z)$, with optimal value $\inf_{z\in F}f(z)$. Changing the scores while keeping $D$ and the <optimization constraints> fixed leaves $F$ unchanged, provided the new <objective function> is defined throughout $D$. These distinctions separate an allowed decision from a best decision.<Convex Optimization by Stephen Boyd and Lieven Vandenberghe>[§4.1.1, printed pp. 127–128: problem domain, feasible set, optimal value and attainment]{ref}
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