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Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem
 Mathematical Programming, Series A
"... An interior point method (IPM) defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the ..."
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An interior point method (IPM) defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the feasible region, the unique solution of the ODE system is a curve passing through the point, with tangents parallel to the search directions along the curve. We call such curves offcentral paths. We study offcentral paths for the monotone semidefinite linear complementarity problem (SDLCP). We show that each offcentral path is a welldefined analytic curve with parameter µ ranging over (0,∞) and any accumulation point of the offcentral path is a solution to SDLCP. Through a simple example we show that the offcentral paths are not analytic as a function of µ and have first derivatives which are unbounded as a function of µ at µ = 0 in general. On the other hand, for the same example, we can find a subset of offcentral paths which are analytic at µ = 0. These “nice ” paths are characterized by some algebraic equations.
in a wide neighborhood of the central path
"... Copyright information to be inserted by the Publishers Predictorcorrector methods for sufficient linear complementarity problems ..."
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Copyright information to be inserted by the Publishers Predictorcorrector methods for sufficient linear complementarity problems
Asymptotic Behavior of HKM Paths in Interior Point Methods for Monotone Semidefinite Linear Complementarity Problems: General Theory
, 2006
"... An interior point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as the ..."
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An interior point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as the solutions of the system of ODEs. In [9], these offcentral paths are shown to be welldefined analytic curves and any of their accumulation points is a solution to the given monotone semidefinite linear complementarity problem (SDLCP). Offcentral paths of a simple example are also studied in [9] whose asymptotic behavior near the solution of the example is analyzed. In this paper, which is an extension of [9], we study the asymptotic behavior of offcentral paths for general SDLCPs (using the dual HKM direction), instead of for a given example. We give a necessary and sufficient condition for when an offcentral path is analytic as a function of µ at a solution of the SDLCP. Then we show that if the given SDLCP has a unique solution, the first derivative of its offcentral path, as a function of