joint work with Julio González Díaz, Ángel M. González Rueda, Joaquín Ossorio Castillo, David Rodríguez Penas, Diego Rodríguez Martínez
We will present a new implementation of the reformulation linearization techniques, RLT, in the context of polynomial programming problems, originally introduced in Sherali and Tuncbilek (1991). The implementation incorporates most of the features of the RLT algorithm discussed in past literature. Moreover, additional enhancements have been introduced, such as parallelization and warm start features at various levels of the branching process. The current version of the algorithm has proven to be very efficient, and comparisons with other global optimization solvers will be presented.
joint work with Ya-Juan Wang
We are interested in solving higher-order moment portfolio optimization via DC (Difference of Convex) programming approach. The problem can be rewritten as a polynomial optimization which is equivalent to a DC program based on the Difference-of-Convex-Sums-of-Squares (DCSOS) decomposition. The later problem can be solved by DCA. We also propose an improved DC algorithm called Boosted DCA (BDCA). The main idea of BDCA is to introduce a line search based on Armijo typo rule to accelerate the convergence of DCA. Numerical simulation results show good performance of our methods.
joint work with Martin S. Andersen
We study the semidefinite relaxation of homogeneous nonconvex quadratically constrained quadratic programs where the matrices have a specific sparsity pattern. In particular, we consider problems where the graph of the aggregate nonzero structure of the matrices is a tree. We present a priori sufficient conditions for the semidefinite relaxation to be exact, which implies that the original problem is solvable in polynomial time. Checking these conditions requires solving a number of second order cone programs based on the data in the matrices, which can be done in polynomial time.