Active-set method: Difference between revisions

Revision as of 21:43, 15 May 2024

Revision as of 19:00, 27 August 2024

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: <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>xx_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.