On the analyticity of underlying HKM Paths for monotone semidefinite linear complementarity problems

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An interior point method (IPM) defines a search direction at an interior point of the feasible region. These search directions form a direction field, which in turn defines a system of ordinary differential equations (ODEs). The solutions of the system of ODEs are called off-central paths, underlying paths lying in the interior of the feasible region. It is known that not all off-central paths are analytic, whether w.r.t. μ or μ √ , where μ represents the duality gap, at a solution of a given semidefinite linear complementarity problem, SDLCP (Sim and Zhao, Math. Program. 110:475–499, 2007). In Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008), we give a necessary and sufficient condition for when an off-central path is analytic as a function of μ √ at a solution of a general SDLCP. It is then natural to ask about the analyticity of a SDLCP off-central path at a solution, as a function of μ. We investigate this in the current paper. Again, we work under the assumption that the given SDLCP satisfies strict complementarity condition.
Original languageEnglish
Pages (from-to)193-215
JournalJournal of Optimization Theory and Applications
Issue number1
Publication statusPublished - Apr 2009


  • Semidefinite linear complementarity problems
  • Ordinary differential equations
  • Off-central paths
  • HKM directions
  • Analyticity

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