constrained optimum

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The solution of a constrained optimization problem at which one or more constraints are binding. Also referred to as constrained maximum (minimum) when optimization involves maximizing (minimizing) the objective function. Assume f(x) is to be maximized subject to the constraints gi(x) ≥ 0, i = 1,…, m, where x = (x1,…, xn). Then the maximum occurs at a saddle point of the Lagrangian function

Lf(x) + Σiλigi(x)

with L maximized for each xi and minimized for λi. The optimum is constrained if, at the saddle point, at least one of the constraints holds as an equality. Examples of constrained optima are the maximization of utility subject to a budget constraint for a consumer who is never satiated, and the minimization of cost subject to a production constraint for a firm employing factors of production with strictly positive prices.

Subjects: Economics.

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