Activeset

Activeset refers to the set of inequality constraints that are binding at a feasible point in constrained optimization. If a problem is to minimize f(x) subject to g_i(x) ≤ 0 for i in I and h_j(x) = 0 for j in J, then the active set at a point x is A(x) = { i ∈ I : g_i(x) = 0 }. In problems with bound constraints on variables, a bound is considered active when it is tight, that is, when the bound is attained.

Active-set methods are a family of iterative algorithms for constrained optimization that explicitly track a working set of constraints treated as equalities. At each iteration, the method solves a reduced, equality-constrained subproblem obtained by enforcing the constraints in the current working set to hold with equality. It then computes Lagrange multipliers associated with these active constraints and uses them to decide how to update the working set. If the current solution satisfies optimality conditions for the full problem, the method terminates; otherwise it adjusts the working set by adding an inactive constraint that is violated or removing an active constraint whose multiplier suggests it could be relaxed.

In practice, active-set methods are widely used for quadratic programming and are applicable to linear and nonlinear constrained optimization. They can be efficient when the active set changes little between iterations, since each step solves a problem with a relatively small set of active constraints. Their performance depends on the problem structure and the quality of the initial active set. They contrast with interior-point methods, which pursue feasibility and optimality through a path that remains strictly within the feasible region.