= Mathematical optimization
{wiki}
= Optimization
{synonym}
<Mathematical optimization> turns a preference into a precise question: among the allowed decisions, which gives the best outcome? A model separates the <objective function>, which measures quality, from the <optimization constraints>, which determine the <feasible set>. This language unifies finding a cheap plan for a <transportation problem>, efficient <scheduling>, or an accurate model in <linear regression>. The separation is useful because changing a preference and changing what is possible are different modeling decisions. See https://web.stanford.edu/~boyd/cvxbook/bv_cvxslides.pdf#page=4[Boyd and Vandenberghe, introductory optimization models]{ref}.
For an <objective function> $f:F\to\mathbb R$, a <maximization problem> asks for a <feasible point> $z_*$ satisfying $f(z_*)\geq f(z)$ for every $z\in F$; a <minimization problem> reverses the inequality. This is a global <extremum>. A <local maximum> or <local minimum> compares only nearby <feasible points>, so finding one does not settle the global question.
Consider allocating ten units between two complementary inputs:
$$
\max xy\qquad\text{subject to }x+y=10,\quad x,y\geq0.
$$
The <budget constraint> forces a tradeoff: an extra unit of $x$ costs a unit of $y$. The best allocation is $(5,5)$, with <objective function> value $25$. Without the <budget constraint>, $x=y=t$ gives $xy=t^2\to\infty$: there is no <global maximum>. <Constrained optimization> can therefore have a finite answer even when the corresponding problem with fewer <optimization constraints> does not. Its beauty is that geometry, <calculus>, and an <arithmetic-geometric mean inequality> can certify the same answer.
Back to article page
