extremize

Extremize is a mathematical term meaning to determine the values that maximize or minimize a given quantity. In calculus or optimization, one seeks local or global extrema of a real-valued function f defined on a domain D. An extremum can be local (occurring in a neighborhood) or global (across the entire domain).

In unconstrained optimization, extrema are found by identifying critical points where the derivative (or gradient in multiple dimensions) is zero. For a single variable, f'(x) = 0 indicates a candidate; the second derivative test or higher-order conditions help classify a local maximum or minimum. In several variables, one examines the gradient and the Hessian matrix to determine the nature of critical points, while also considering function values at domain boundaries when the domain is bounded.

Constrained optimization involves finding extrema subject to one or more constraints. A standard method is the method of Lagrange multipliers, solving equations that set the gradient of the objective equal to a linear combination of constraint gradients, together with the constraint equations. For inequality constraints, optimality conditions are expressed via Karush–Kuhn–Tucker conditions.

Calculus of variations extends extremization to functionals, such as extremizing an integral S[y] = ∫ L(x, y, y') dx. This yields the Euler–Lagrange equations, governing the extremal functions with appropriate boundary conditions.

Numerical methods for extremization include gradient descent (minimization) and gradient ascent (maximization), as well as Newton-type methods and line-search or trust-region strategies. Convex problems guarantee global optima, while nonconvex problems may yield multiple local extrema.

Extremization has applications across physics (principle of least action), economics (cost or utility minimization/maximization), engineering, and machine learning.