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For IEOR 6611 Semidefinite and Second-Order cone programming Fall 2001
Topic 1: Introductory remarks, a quick tour of SDP, the notion of Cone-LP, Convex cones, Duality theory, Extended Farkas lemma, Strong duality theorem, extension of complementary slackness theorem. (2 weeks)
Topic 2: The notion of degeneracy in cone-LP and in semidefinite programming, detailed analysis of degeneracy in SDP and in SOCP. The notion of strict complementarity (one-two weeks)
Topic 3: Introduction to Euclidean Jordan algebras and symmetric cones. How are LP, SOCP and SDP related? How can one set of proofs and analyses be used for all of them (two-three weeks)
Topic 4: Interior point methods for SDP and SOCP and symmetric cones, notions of logarithmic barrier, potential functions, central path, notion of proximity to central path, algorithms with polynomial-time iteration complexity. Notion of feasibility and boundedness and how the situation is different from linear programming. Primal-dual methods, the XS, SX, Nesterov-Todd, and XS+SX methods, Moneiro-Zhang family of classes, the commutative class of algorithms, polynomial-time path-following methods (two-three weeks)
Topic 5: Problems that can be formulated as SDP and related optimization models. Various Eigenvalue optimization problems, the class of problems, functions, and inequalities that can be formulated as SDP or SOCP (one-two week)
Topic 6: The Lovasz-Schrijver Lift-and-project method for SDP relaxations of 0-1 programs (one week)
Topic 7: Lovasz theta function and SDP, Goemans and Williamson's approximation algorithm for MAXCUT and related problems. (one week)
Topic 8: Nonnegative polynomials, moment problems and connection to semidefinite programming, applications to approximation theory, regression analysis, and other areas of statistics (one-two weeks)
Topic 9: Engineering and other applications (one week)