Centro-affine differential geometry, Lagrangian submanifolds PDF
Preview Centro-affine differential geometry, Lagrangian submanifolds
Conicoptimizationandbarriers Barriersandcentro-affinegeometry Lagrangiansubmanifoldsinpara-Kählerspace Openproblems Centro-affine differential geometry, Lagrangian submanifolds of the reduced paracomplex projective space, and conic optimization RolandHildebrand UniversitéGrenoble1/CNRS June5,2012/DifferentialGeometry2012,Be˛dlewo RolandHildebrand Centro-affinedifferentialgeometryandconicoptimization Conicoptimizationandbarriers Barriersandcentro-affinegeometry Lagrangiansubmanifoldsinpara-Kählerspace Openproblems Outline 1 Conicoptimizationandbarriers Convexprograms Barriers onconvex sets Conicprograms Logarithmicallyhomogeneousbarriers 2 Barriers andcentro-affinegeometry Splitting theorem Centro-affineequivalentsof barriers Applications 3 Lagrangiansubmanifoldsin para-Kählerspace Cross-ratio manifold Objectsdefinedbycones Barriers andLagrangiansubmanifolds Applications RolandHildebrand Centro-affinedifferentialgeometryandconicoptimization Conicoptimizationandbarriers Barriersandcentro-affinegeometry Lagrangiansubmanifoldsinpara-Kählerspace Openproblems Outline 1 Conicoptimizationandbarriers Convexprograms Barriers onconvex sets Conicprograms Logarithmicallyhomogeneousbarriers 2 Barriers andcentro-affinegeometry Splitting theorem Centro-affineequivalentsof barriers Applications 3 Lagrangiansubmanifoldsin para-Kählerspace Cross-ratio manifold Objectsdefinedbycones Barriers andLagrangiansubmanifolds Applications RolandHildebrand Centro-affinedifferentialgeometryandconicoptimization Conicoptimizationandbarriers Barriersandcentro-affinegeometry Lagrangiansubmanifoldsinpara-Kählerspace Openproblems Outline 1 Conicoptimizationandbarriers Convexprograms Barriers onconvex sets Conicprograms Logarithmicallyhomogeneousbarriers 2 Barriers andcentro-affinegeometry Splitting theorem Centro-affineequivalentsof barriers Applications 3 Lagrangiansubmanifoldsin para-Kählerspace Cross-ratio manifold Objectsdefinedbycones Barriers andLagrangiansubmanifolds Applications RolandHildebrand Centro-affinedifferentialgeometryandconicoptimization Conicoptimizationandbarriers Convexprograms Barriersandcentro-affinegeometry Barriersonconvexsets Lagrangiansubmanifoldsinpara-Kählerspace Conicprograms Openproblems Logarithmicallyhomogeneousbarriers Convex optimization problems minimize linearobjective function withrespect to convex constraints min f(x) x X ∈ f = c,x , X convex h i X Rn is called the feasible set ⊂ RolandHildebrand Centro-affinedifferentialgeometryandconicoptimization Conicoptimizationandbarriers Convexprograms Barriersandcentro-affinegeometry Barriersonconvexsets Lagrangiansubmanifoldsinpara-Kählerspace Conicprograms Openproblems Logarithmicallyhomogeneousbarriers Regular convex sets Definition A regularconvex setX Rn is a closed convex sethaving ⊂ nonemptyinteriorandcontainingnolines. can assume thefeasible set to beregular RolandHildebrand Centro-affinedifferentialgeometryandconicoptimization Conicoptimizationandbarriers Convexprograms Barriersandcentro-affinegeometry Barriersonconvexsets Lagrangiansubmanifoldsinpara-Kählerspace Conicprograms Openproblems Logarithmicallyhomogeneousbarriers Definition of barriers Definition LetX Rn bea regularconvex set. A ν-self-concordantbarrier ⊂ for X is a smoothfunction F :Xo Rsuch that → F (x) 0 (convexity) 00 (cid:31) lim F(x) = + (boundarybehaviour) x ∂X → ∞ F hi 2 νF hihj forall h T Rn (gradientinequality) ,i ,ij x | | ≤ ∈ F hihjhk 2(F hihj)3/2 for allh T Rn ,ijk ,ij x | | ≤ ∈ (self-concordance) F definesa Hessianmetric onXo 00 usesonlythe affine connectionon Rn affine invariance ⇒ RolandHildebrand Centro-affinedifferentialgeometryandconicoptimization Conicoptimizationandbarriers Convexprograms Barriersandcentro-affinegeometry Barriersonconvexsets Lagrangiansubmanifoldsinpara-Kählerspace Conicprograms Openproblems Logarithmicallyhomogeneousbarriers Interior-point methods using barriers min c,x x X h i ∈ constrainedconvex program letF(x) = + for allx Xo ∞ 6∈ min τ c,x +F(x) x h i unconstrainedprogram,τ > 0 a parameter by convexity andboundarybehaviourofF this programis convex the minimizerx ofthe unconstrainedprogramtendsto the τ∗ minimizerx oftheconstrained programasτ + ∗ → ∞ RolandHildebrand Centro-affinedifferentialgeometryandconicoptimization Conicoptimizationandbarriers Convexprograms Barriersandcentro-affinegeometry Barriersonconvexsets Lagrangiansubmanifoldsinpara-Kählerspace Conicprograms Openproblems Logarithmicallyhomogeneousbarriers Interior-point methods using barriers min c,x x X h i ∈ constrainedconvex program letF(x) = + for allx Xo ∞ 6∈ min τ c,x +F(x) x h i unconstrainedprogram,τ > 0 a parameter by convexity andboundarybehaviourofF this programis convex the minimizerx ofthe unconstrainedprogramtendsto the τ∗ minimizerx oftheconstrained programasτ + ∗ → ∞ RolandHildebrand Centro-affinedifferentialgeometryandconicoptimization Conicoptimizationandbarriers Convexprograms Barriersandcentro-affinegeometry Barriersonconvexsets Lagrangiansubmanifoldsinpara-Kählerspace Conicprograms Openproblems Logarithmicallyhomogeneousbarriers Interior-point methods using barriers min c,x x X h i ∈ constrainedconvex program letF(x) = + for allx Xo ∞ 6∈ min τ c,x +F(x) x h i unconstrainedprogram,τ > 0 a parameter by convexity andboundarybehaviourofF this programis convex the minimizerx ofthe unconstrainedprogramtendsto the τ∗ minimizerx oftheconstrained programasτ + ∗ → ∞ RolandHildebrand Centro-affinedifferentialgeometryandconicoptimization
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