Semidefinite and Second Order Cone Programming Seminar Fall 2012 Lecture 2

1 Overview We had a general overview of semidefinite programming(SDP) in lecture 1, starting from this lecture we will be jumping into the theory. Some topics we will discuss in the next few lectures include: the duality theory, notion of complementary slackness, at least one polynomial time algorithm of solving SDP and application in integer programming and combinatorial optimization. Throughout the whole semester we only consider vectors in finite dimensional space R n unless otherwise explicitly point out. Also, all vectors are considered column vectors and represented by lower case bold letters such as a, b, etc. Definitions: Let S ⊆ R n be a set, then • S is an open set if for each x ∈ S, there is a sufficently small ball centered at x and contained in S, that is: ∀x ∈ S, ∃ > 0 such that {y ∈ R n : y − x < } ⊂ S.