constrained optimization problem

Finding the extrema of an objective function $f(x,y)$ restricted to a constraint set $g(x,y)=k$ .

A constrained optimization problem asks for the extrema of an objective function $f(x,y)$ restricted to a constraint set — the points $(x,y)$ satisfying $g(x,y)=k$ for some fixed constant $k$ . This is a fundamentally smaller search than the unconstrained problem: only the (typically one-dimensional) constraint curve is in play, and a point can be a perfectly ordinary, non-critical point of the unconstrained $f$ while still being the constrained extremum. Substitution works whenever $g(x,y)=k$ can be solved explicitly for one variable; when it can't, Lagrange multipliers take over.

Examples

Minimizing $f(x,y)=x^2+y^2$ subject to $x+y=10$ : substituting $y=10-x$ gives the minimum at $(5,5)$ , where the constraint line is tangent to the level curve $x^2+y^2=50$ .