Centro-affine differential geometry, Lagrangian submanifolds

66 Pages·2012·1.66 MB·English

by  Roland Hildebrand

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