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… (More)
SparsePOP is a Matlab implementation of the sparse semidefinite programming (SDP) relaxation method for approximating a global optimal solution of a polynomial optimization problem (POP) proposed by Waki et al. . The sparse SDP relaxation exploits a sparse structure of polynomials in POPs when applying “a hierarchy of LMI relaxations of… (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 . We establish a clear relationship between them. Our results show that although the two methods can be regarded… (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 E and E + be a finite dimensional real vector space and a symmetric cone embedded in E; examples of E and E + include a pair of the N-dimensional Euclidean space and its nonnegative orthant, a… (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… (More)
In this paper, we introduce an aane scaling algorithm for semideenite programming, and give an example of a semideenite program such that the aane scaling algorithm converges to a non-optimal point. Both our program and its dual have interior feasible solutions, and unique optimal solutions which satisfy strict complementarity, and they are nondegenerate… (More)
Facial reduction algorithm reduces the size of the positive semi-definite cone in SDP. The elimination method for a sparse SOS polynomial () 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.