Editing Active-set method (section)

==Active-set methods==
In general an active-set algorithm has the following structure:

: Find a feasible starting point
: '''repeat until''' "optimal enough"
:: ''solve'' the equality problem defined by the active set (approximately)
:: ''compute'' the [[Lagrange multipliers]] of the active set
:: ''remove'' a subset of the constraints with negative Lagrange multipliers
:: ''search'' for infeasible constraints among the inactive constraints and add them to the problem
: '''end repeat'''

The motivations for this is that near the optimum usually only a small number of all constraints are binding and the solve step usually takes superlinear time in the amount of constraints. Thus repeated solving of a series equality constrained problem, which drop constraints which are not violated when improving but are in the way of improvement (negative lagrange multipliers) and adding of those constraints which the current solution violates can converge against the true solution. The optima of the last problem can often provide an initial guess in case the equality constrained problem solver needs an initial value.

Methods that can be described as '''active-set methods''' include:<ref>{{harvnb|Nocedal|Wright|2006|pp=467–480}}</ref>
* [[Successive linear programming]] (SLP) <!-- acc. to: Leyffer... - alt: acc. to "MPS glossary", http://glossary.computing.society.informs.org/ver2/mpgwiki/index.php/Main_Page: Successive approximation -->
* [[Sequential quadratic programming]] (SQP) <!-- acc. to: Leyffer... - alt: acc. to "MPS glossary", http://glossary.computing.society.informs.org/ver2/mpgwiki/index.php/Main_Page: Successive approximation  -->
* [[Sequential linear-quadratic programming]] (SLQP) <!-- acc. to: Leyffer... - alt: acc. to "MPS glossary", http://glossary.computing.society.informs.org/ver2/mpgwiki/index.php/Main_Page: Successive approximation  -->
* [[Frank–Wolfe algorithm|Reduced gradient method]] (RG) <!-- acc. to: MPS glossary, http://glossary.computing.society.informs.org/ver2/mpgwiki/index.php/Main_Page - alt: acc. to "Optimization - Theory and Practice" (Forst, Hoffmann): Projection method -->
* [[Generalized Reduced Gradient|Generalized reduced gradient method]] (GRG) <!-- acc. to: MPS glossary, http://glossary.computing.society.informs.org/ver2/mpgwiki/index.php/Main_Page - alt: acc. to "Optimization - Theory and Practice" (Forst, Hoffmann): Projection method -->
<!-- ? Wilson's Lagrange-newton method -->
<!-- ? Method of feasible directions (MFD) -->
<!-- ? Gradient projection method -  alt: acc. to "Optimization - Theory and Practice" (Forst, Hoffmann): Projection method -->