v2006.03.09 - Convex Optimization

CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY 152.9 Positive semidefinite (PSD) cone . . . . . . . . . . . . . . . . . 1092.9.0.1 Membership . . . . . . . . . . . . . . . . . . . 1102.9.1 PSD cone is convex . . . . . . . . . . . . . . . . . . . . 1102.9.2 PSD cone boundary . . . . . . . . . . . . . . . . . . . 1142.9.2.1 rank ρ subset of the PSD cone . . . . . . . . 1162.9.2.2 Faces of PSD cone . . . . . . . . . . . . . . . 1162.9.2.3 Extreme directions of the PSD cone . . . . . 1172.9.2.4 PSD cone is not circular . . . . . . . . . . . . 1182.9.2.5 Boundary constituents of the PSD cone . . . 1242.9.2.6 Projection on S M +(ρ) . . . . . . . . . . . . . . 1262.9.2.7 Uniting constituents . . . . . . . . . . . . . . 1262.9.2.8 Peeling constituents . . . . . . . . . . . . . . 1272.9.3 Barvinok’s proposition . . . . . . . . . . . . . . . . . . 1282.10 Conic independence (c.i.) . . . . . . . . . . . . . . . . . . . . . 1292.10.1 Preservation of conic independence . . . . . . . . . . . 1312.10.1.1 linear maps of cones . . . . . . . . . . . . . . 1332.10.2 Pointed closed convex K & conic independence . . . . . 1332.10.3 Utility of conic independence . . . . . . . . . . . . . . 1352.11 When extreme means exposed . . . . . . . . . . . . . . . . . . 1352.12 Convex polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . 1362.12.1 Polyhedral cone . . . . . . . . . . . . . . . . . . . . . . 1372.12.2 Vertices of convex polyhedra . . . . . . . . . . . . . . . 1382.12.2.1 Vertex-description of polyhedral cone . . . . . 1392.12.2.2 Pointedness . . . . . . . . . . . . . . . . . . . 1402.12.3 Unit simplex . . . . . . . . . . . . . . . . . . . . . . . 1402.12.3.1 Simplex . . . . . . . . . . . . . . . . . . . . . 1422.12.4 Converting between descriptions . . . . . . . . . . . . . 1422.13 Dual cone & generalized inequality . . . . . . . . . . . . . . . 1452.13.1 Dual cone . . . . . . . . . . . . . . . . . . . . . . . . . 1452.13.1.1 Key properties of dual cone . . . . . . . . . . 1502.13.1.2 Examples of dual cone . . . . . . . . . . . . . 1522.13.2 Abstractions of Farkas’ lemma . . . . . . . . . . . . . . 1552.13.2.1 Null certificate, Theorem of the alternative . . 1572.13.3 Optimality condition . . . . . . . . . . . . . . . . . . . 1592.13.4 Discretization of membership relation . . . . . . . . . . 1602.13.4.1 Dual halfspace-description . . . . . . . . . . . 1602.13.4.2 First dual-cone formula . . . . . . . . . . . . 1612.13.5 Dual PSD cone and generalized inequality . . . . . . . 162