Prerequisite: Strong background in Linear Algebra and Real Analysis; Instructor’s consent.
This course aims at providing fundamental grounding in convex analysis to PhD students. Convex analysis is the study of properties of convex functions and convex sets, which are fundamental in studying convex optimization problems. In addition to covering basics of finite dimensional convex analysis, which will form the most part of this course, we will also introduce convex analysis in infinite dimensional spaces. In addition, we will also introduce Convex Geometry, a subject attracting lot of interest currently.
- Convex sets and functions. Basic definitions, epigraphs and level sets of convex functions, relative interiors, closures.
- Projection and Separation. The projector operator and its uniqueness, Separating hyperplanes, supporting hyperplanes, and their consequences.
- Structure of Convex sets. Faces, Extreme points, recession cones, and their properties. Exposed and non-exposed faces. Decomposition of convex sets.
- Conical Approximations. Tangent cones, Normal cones, and their properties.
- Support functions. sublinear functions, support functions of a nonempty set, correspondence between sub- linear functions and convex sets.
- Conjugacy in convex analysis. The convex conjugate of a function, calculus rules, conjugate duality and its applications in optimization.
- Infinite dimensional Convex Analysis. Analogues of some of the above results for general vector spaces, including the Hahn-Banach Theorem. 8
- Introduction to Convex Geometry. The space of convex bodies, mixed volumes. Blaschke selection theorem and the Brunn-Minkowski inequality.
- R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.
- R. T. Rockafellar, Conjugate Duality and Optimization, SIAM, 1974.
- J. B. Hiriart-Urruty and C. Lemar ́chal, Fundamentals of Convex Analysis, Springer-Verlag, 2001.
- D. P. Bertsekas, Convex Optimization Theory, Athena Scientific, 2009
- R. T. Rockafellar and R. J-b. Wets, Variational Analysis, Springer-Verlag, 1997.
- C. Zˇlinescu, Convex Analysis in General Vector Spaces, World Scientific, 2002.
- P. M. Gruber, Convex and Discrete Geometry, Springer-Verlag, 2007.
- Open Literature.