Approximation of the Shannon Capacity Via Matrix Cone Programming

This paper proposes a novel formulation using the matrix cone programming to compute an upper bound of the Shannon capacity of graphs,which is theoretically superior to the Lovász number.To achieve this,a sequence of matrix cones is constructed by adding certain co-positive matrices to the positive semi-definite matrix cones during the matrix cone programming.We require the sequence of matrix cones to have the weak product property so that the improved result of the matrix cone programming remains an upper bound of the Shannon capacity.Our result shows that the existence of a sequence of suitable matrix cones with the weak product property is equivalent to the existence of a co-positive matrix with testable conditions.Finally,we give some concrete examples with special structures to verify the existence of the matrix cone sequence.

Joint Bandwidth Allocation and Path Selection in WANs with Path Cardinality Constraints

In this paper,we study the joint bandwidth allocation and path selection problem,which is an extension of the well-known network utility maximization(NUM)problem,via solving a multi-objective minimization problem under path cardinality constraints.Specifically,such a problem formulation captures various types of objectives including proportional fairness,average delay,as well as load balancing.In addition,in order to handle the"unsplittable flows",path cardinality constraints are added,making the resulting optimization problem quite challenging to solve due to intrinsic nonsmoothness and nonconvexity.Almost all existing works deal with such a problem using relaxation techniques to transform it into a convex optimization problem.However,we provide a novel solution framework based on the linearized alternating direction method of multipliers(LADMM)to split the original problem with coupling terms into several subproblems.We then derive that these subproblems,albeit nonconvex nonsmooth,are actually simple to solve and easy to implement,which can be of independent interest.Under some mild assumptions,we prove that any limiting point of the generated sequence of the proposed algorithm is a stationary point.Numerical simulations are performed to demonstrate the advantages of our proposed algorithm compared with various baselines.

Biquadratic tensors,biquadratic decompositions,and norms of biquadratic tensors

Biquadratic tensors play a central role in many areas of science.Examples include elastic tensor and Eshelby tensor in solid mechanics,and Riemannian curvature tensor in relativity theory.The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor,respectively.The tensor product operation is closed for biquadratic tensors.All of these motivate us to study biquadratic tensors,biquadratic decomposition,and norms of biquadratic tensors.We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure.Then,either the number of variables is reduced,or the feasible region can be reduced.We show constructively that for a biquadratic tensor,a biquadratic rank-one decomposition always exists,and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition.We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor.Finally,we define invertible biquadratic tensors,and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse,and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor,and the spectral norm of its inverse.