We describe a graph coloring problem associated with the determination of mathematical derivatives. The coloring instances are obtained as intersection graphs of row partitioned sparse derivative matrices. The size of the graph is dependent on the partition and can be varied between the number of columns and the number of nonzero entries. If solved exactly … Read more
Considering that preprocessing is an important phase in linear programming, it should be systematically more incorporated in semidefinite programming solvers. The conversion method proposed by the authors (SIAM Journal on Optimization, vol.~11, pp.~647–674, 2000, and Mathematical Programming, Series B, vol.~95, pp.~303–327, 2003) is a preprocessing of sparse semidefinite programs based on matrix completion. This article … Read more
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” polynomial as a sum of … Read more
Sequences of generalized Lagrangian duals and their SOS (sums of squares of polynomials) 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 … Read more
We build upon the work of Fukuda et al.\ \cite{FuKoMuNa01} and Nakata et al.\ \cite{NaFuFuKoMu01}, in which the theory of partial positive semidefinite matrices has been applied to the semidefinite programming (SDP) problem as a technique for exploiting sparsity in the data. In contrast to their work, which improves an existing algorithm that is based … Read more
We show that SDP (semidefinite programming) and SOCP (second order cone programming) relaxations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a generalization of the results by S.~Zhang for a subclass of quadratic maximization problems that have nonnegative off-diagonal coefficient matrices of objective quadratic functions and diagonal coefficient matrices … Read more