On the Closedness of the Linear Image of a Closed Convex Cone

When is the linear image of a closed convex cone closed? We present very simple and intuitive necessary conditions that (1) unify, and generalize seemingly disparate, classical sufficientconditions such as polyhedrality of the cone, and Slater-type conditions; (2) are necessary and sufficient, when the dual cone belongs to a class that we call nice cones (nice cones subsume all cones amenable to treatment by efficient optimization algorithms, for instance, polyhedral, semidefinite, and p-cones); and (3) provide similarly attractive conditions for an equivalent problem: the closedness of the sum of two closed convex cones.

[1]  Motakuri V. Ramana,et al.  An exact duality theory for semidefinite programming and its complexity implications , 1997, Math. Program..

[2]  R. Freund.  Review of A mathematical view of interior-point methods in convex optimization, by James Renegar, SIAM, Philadelphia, PA , 2004 .

[3]  Heinz H. Bauschke,et al.  Conical open mapping theorems and regularity , 1999 .

[4]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[5]  Z. Waksman,et al.  On point classification in convex sets. , 1976 .

[6]  D. R. Morrison,et al.  THE THEORY OF CONES , 1965 .

[7]  G. P. Barker.  Faces and duality in convex cones , 1978 .

[8]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[9]  B. Tam.  On the duality operator of a convex cone , 1985 .

[10]  G. P. Barker.  The lattice of faces of a finite dimensional cone , 1973 .

[11]  J. Borwein,et al.  Regularizing the Abstract Convex Program , 1981 .

[12]  G. Pataki.  The Geometry of Semidefinite Programming , 2000 .

[13]  Alfred Auslender,et al.  Closedness criteria for the image of a closed set by a inear operator , 1996 .

[14]  Michael L. Overton,et al.  Complementarity and nondegeneracy in semidefinite programming , 1997, Math. Program..

[15]  A. Brøndsted.  An Introduction to Convex Polytopes , 1982 .

[16]  Henry Wolkowicz,et al.  Strong Duality for Semidefinite Programming , 1997, SIAM J. Optim..

[17]  Robert G. Jeroslow,et al.  Duality in Semi-Infinite Linear Programming , 1983 .

[18]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[19]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[20]  R. Holmes.  Geometric Functional Analysis and Its Applications , 1975 .

[21]  G. P. Barker,et al.  Cones of diagonally dominant matrices , 1975 .

[22]  A. Berman.  Cones, matrices and mathematical programming , 1973 .

[23]  James Renegar,et al.  A mathematical view of interior-point methods in convex optimization , 2001, MPS-SIAM series on optimization.