A survey: Semidefinite programming, its duality, complexity and applications

SDP (Semidefinite Programming), as a sort of ‘cone-LP’, optimizes a linear function over the intersection of an affine space and a cone that has the origin as its apex. SDP, however, has been developed in the process of searching for better solution methods for NP-hard combinatorial optimization problems. We surveyed the basic theories necessary to understand SDP researches. First, we examined SDP duality, comparing it to LP duality, which is essential for the interior point method. Second, we showed that SDP can be optimized from an interior solution in polynomial time with a desired error limit. Finally, we summarized several research papers that showed SDP can improve solution methods for some combinatorial optimization problems, and explained why SDP has become one of the most important research topics in optimization. We tried to integrate SDP theories, relatively diverse and complicated, to survey research papers with our own perspectives, and thus to help researchers to pursue their SDP researches in depth.