We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lov{\’a}sz’s theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its subgraphs. We develop an efficient, polynomial-time algorithm to extract a maximum stable set in a … Read more
The problem of automatic scheduling hypermedia documents consists in finding the optimal starting times and durations of objects to be presented, to ensure spatial and temporal consistency of a presentation while respecting limits on shrinking and stretching the ideal duration of each object. The combinatorial nature of the minimization of the number of objects whose … 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
The low-rank semidefinite programming problem (LRSDP_r) is a restriction of the semidefinite programming problem (SDP) in which a bound r is imposed on the rank of X, and it is well known that LRSDP_r is equivalent to SDP if r is not too small. In this paper, we classify the local minima of LRSDP_r and … 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 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
In this paper we study the distribution tails and the moments of a condition number which arises in the study of homogeneous systems of linear inequalities. We consider the case where this system is defined by a Gaussian random matrix and characterise the exact decay rates of the distribution tails, improve the existing moment estimates, … Read more
We introduce a new masked spectral bound for the maximum-entropy sampling problem. This bound is a continuous generalization of the very effective spectral partition bound. Optimization of the masked spectral bound requires the minimization of a nonconvex, nondifferentiable function over a semidefiniteness constraint. We describe a nonlinear affine scaling algorithm to approximately minimize the bound. … Read more
The nearest \cm\ problem is to find a positive semidefinite matrix with unit diagonal that is nearest in the Frobenius norm to a given symmetric matrix $A$. This problem can be formulated as an optimization problem with a quadratic objective function and semidefinite programming constraints. Using such a formulation, we derive and test a primal-dual … Read more
Recently, so-called self-regular barrier functions for primal-dual interior-point methods (IPMs) for linear optimization were introduced. Each such barrier function is determined by its (univariate) self-regular kernel function. We introduce a new class of kernel functions. The class is defined by some simple conditions on the kernel function and its derivatives. These properties enable us to … Read more