Research interests of our group lie in conic programming, global optimization, polynomial optimization, combinatorial optimization, and nonlinear programming.
Conic relaxations have been widely studied for modeling real-world applications. We study semidefinite relaxations, second-order cone relaxations, doubly nonnegative relaxations, copositive relaxations, and completely positive relaxations from the theoretical and computational viewpoints. - Our strengths have been exploiting the sparsity in the conic relaxations of the problems, especially the chordal sparsity by combining the result from graph theory. The sparsity technique is employed to increase the efficiency of solving the problems. -
Based on theoretical studies of modeling problems, software packages have been developed in Matlab. The released software packages are 'SparsePOP' for polynomial problems, 'SFSDP' for sensor network problems, 'SparseCoLO' for conic problems with sparsity exploitation, 'BBCPOP' for polynomial problems in binary and box variables, and 'NewtBracket' for conic optimization problems.
For combinatorial optimization problems, finding the solutions of fundamental problems such as the max-cut problems, the max-clique problems, multiple-knapsack problems, and quadratic assignment problems are the main subjects of our research. These problems are NP-hard problems. Solutions of NP-hard problems are sought with approximating schemes or branch and bound methods. Our focus is first on developing efficient and effective approximating schemes, more precisely, tight convex relaxations and providing their theoretical convergence results. Second, the approximation schemes are incorporated into the branch and bound methods to find the exact solutions of the problems. Various branching rules and cutting methods are studied and developed.
We actively participate in international conferences and collaborate with internationally renowned scientists.