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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)

- Sunyoung Kim, Masakazu Kojima, Hayato Waki
- SIAM Journal on Optimization
- 2009

A sensor network localization problem can be formulated as a quadratic optimization problem (QOP). For quadratic optimization problems, semidefinite programming (SDP) relaxation by Lasserre with relaxation order 1 for general polynomial optimization problems (POPs) is known to be equivalent to the sparse SDP relaxation by Waki et al. with relaxation order… (More)

- Hayato Waki, Sunyoung Kim, Masakazu Kojima, Masakazu Muramatsu
- SIAM Journal on Optimization
- 2006

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… (More)

- Hayato Waki, Sunyoung Kim, Masakazu Kojima, Masakazu Muramatsu, Hiroshi Sugimoto
- ACM Trans. Math. Softw.
- 2008

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. [2006]. The sparse SDP relaxation exploits a sparse structure of polynomials in POPs when applying “a hierarchy of LMI relaxations of… (More)

- Hayato Waki, Masakazu Muramatsu
- J. Optimization Theory and Applications
- 2013

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 can be regarded… (More)

- Sunyoung Kim, Masakazu Kojima, Hayato Waki
- SIAM Journal on Optimization
- 2005

Sequences of generalized Lagrangian duals and their SOS (sums of squares of polynomi-als) relaxations for a POP (polynomial optimization problem) are introduced. Sparsity of polynomials in the POP is used to reduce the sizes of the Lagrangian duals and their SOS relaxations. It is proved that the optimal values of the Lagrangian duals in the sequence… (More)

- Hayato Waki
- Optimization Letters
- 2012

Examples of weakly infeasible semidefinite programs are useful to test whether semidefinite solvers can detect infeasibility. However, finding non trivial such examples is notoriously difficult. This note shows how to use Lasserre's semidefinite programming relax-ations for polynomial optimization in order to generate examples of weakly infeasible… (More)

SFSDP is a Matlab package for solving sensor network localization problems. The package contains four functions, SFSDP.m, SFSDPplus.m, generateProblem.m, test SFSDP.m, and some numerical examples. The function SFSDP.m is an Matlab implementation of the semidefinite programming (SDP) relaxation proposed in the recent paper by Kim, Kojima and Waki for sensor… (More)

- Masakazu Kojima, Sunyoung Kim, Hayato Waki
- Math. Program.
- 2005

Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and SDP (semidefinite programming) relaxation of polynomial optimization problems. We disscuss effective methods to obtain a simpler representation of a " sparse "… (More)