Editing Active-set method (section)

{{redirect|Active set|the band|The Active Set}}

In mathematical [[Optimization (mathematics)|optimization]], the '''active-set method''' is an algorithm used to identify the active [[Constraint (mathematics)|constraints]] in a set of [[Inequality (mathematics)|inequality]] constraints. The active constraints are then expressed as equality constraints,  thereby transforming an inequality-constrained problem into a simpler equality-constrained subproblem.

An optimization problem is defined using an objective function to minimize or maximize, and a set of constraints

: <math>g_1(x) \ge 0, \dots, g_k(x) \ge 0</math>

that define the [[feasible region]], that is, the set of all ''x'' to search for the optimal solution. Given a point <math>x</math> in the feasible region, a constraint 

: <math>g_i(x) \ge 0</math>

is called '''active''' at <math>x_0</math> if <math>g_i(x_0) = 0</math>, and '''inactive''' at <math>x_0</math> if <math>g_i(x_0) > 0.</math> Equality constraints are always active. The '''active set''' at <math>x_0</math> is made up of those constraints <math>g_i(x_0)</math> that are active at the current point {{harv|Nocedal|Wright|2006|p=308}}.

The active set is particularly important in optimization theory, as it determines which constraints will influence the final result of optimization. For example, in solving the [[linear programming]] problem, the active set gives the [[hyperplane]]s that intersect at the solution point. In [[quadratic programming]], as the solution is not necessarily on one of the edges of the bounding polygon, an estimation of the active set gives us a subset of inequalities to watch while searching the solution, which reduces the complexity of the search.