Masakazu Muramatsu Facial reduction algorithm reduces the size of the positive semidefinite cone in SDP. The elimination method for a sparse SOS polynomial ([3]) removes unnecessary monomials for an SOS representation. In this paper, we establish a relationship between a facial reduction algorithm and the elimination method for a sparse SOS polynomial. Citation Technical Report CS-09-02, Department … Read more
To obtain a primal-dual pair of conic programming problems having zero duality gap, two methods have been proposed: the facial reduction algorithm due to Borwein and Wolkowicz [1,2] and the conic expansion method due to Luo, Sturm, and Zhang [5]. We establish a clear relationship between them. Our results show that although the two methods … Read more
We observe that in a simple one-dimensional polynomial optimization problem (POP), the `optimal’ values of semidefinite programming (SDP) relaxation problems reported by the standard SDP solvers converge to the optimal value of the POP, while the true optimal values of SDP relaxation problems are strictly and significantly less than that value. Some pieces of circumstantial … Read more
This short note extends the sparse SOS (sum of squares) and SDP (semidefinite programming) relaxation proposed by Waki, Kim, Kojima and Muramatsu for normal POPs (polynomial optimization problems) to POPs over symmetric cones, and establishes its theoretical convergence based on the recent convergence result by Lasserre on the sparse SOS and SDP relaxation for normal … Read more
SparesPOP is a MATLAB implementation of a sparse semidefinite programming (SDP) relaxation method proposed for polynomial optimization problems (POPs) in the recent paper by Waki et al. The sparse SDP relaxation is based on a hierarchy of LMI relaxations of increasing dimensions by Lasserre, and exploits a sparsity structure of polynomials in POPs. The efficiency … Read more
Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite programming (SDP) … Read more
This paper is based on a recent work by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. Let ${\cal E}$ and ${\cal E}_+$ be a finite dimensional real vector space and a symmetric cone embedded in ${\cal E}$; examples of $\calE$ and $\calE_+$ include a pair of the … Read more
We propose a pivotting procedure for a class of Second-Order Cone Programming (SOCP) having one second-order cone. We introduce a dictionary, basic variables, nonbasic variables, and other necessary notions to define a pivot for the class of SOCP. In a pivot, two-dimensional SOCP subproblems are solved to decide which variables should be entering to or … Read more
We propose a new relaxation scheme for the MAX-CUT problem using second-order cone programming. We construct relaxation problems to reflect the structure of the original graph. Numerical experiments show that our relaxation approaches give better bounds than those based on the spectral decomposition proposed by Kim and Kojima, and that the efficiency of the branch-and-bound … Read more