For a closed cone C in R, the completely positive cone of C is the convex cone KC in S generated by {uu : u ∈ C}. Such a cone arises, for example, in the conic LP reformulation of a nonconvex quadratic minimization problem over an arbitrary set with linear and binary constraints. Motivated by the useful and desirable properties of the nonnegative orthant and the positive semidefinite cone (and ...

In this paper we consider l0 regularized convex cone programming problems. In particular, we first propose an iterative hard thresholding (IHT) method and its variant for solving l0 regularized box constrained convex programming. We show that the sequence generated by these methods converges to a local minimizer. Also, we establish the iteration complexity of the IHT method for finding an -loca...

This article studies a pair of higher order nondifferentiable symmetric fractional programming problem over cones. First, cone convex function is introduced. Then using the properties this function, duality results are set up, which give legitimacy primal dual model.

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to symmetric cone constraints. The algorithm is based on a distance-like function that is an extension of the Kullback-Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithms for convex programming with nonnegative orthant cone constraints, we sho...

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to the symmetric cone constraint. The algorithm is based on a distance-like function that is an extension of the Kullback-Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithm for convex programming with nonnegative orthant cone constraint, we sh...

A fully symmetric duality model is presented which subsumes the classical treatments given by Duff in (1956), Eisenberg (1961) and Cottle (1963) for linear, homogeneous and quadratic convex programming. Moreover, a wide variety of other special objective functional structures, including homogeneity of any nonzero degree, is handled with equal ease. The model is valid in spaces of arbitrary dime...

An interior-point trust-region algorithm is proposed for minimization of general (perhaps, non-convex) quadratic objective function over the domain obtained as the intersection of a symmetric cone with an affine subspace. The algorithm uses a trust-region model to ensure descent on a suitable merit function. Convergence to first-order and second-order optimality conditions is proved. Numerical ...

For a closed cone C in Rn, the completely positive cone of C is the convex cone K in Sn generated by {uuT : u ∈ C}. Completely positive cones arise, for example, in the conic LP reformulation of a nonconvex quadratic minimization problem over an arbitrary set with linear and binary constraints. Motivated by the useful and desirable properties of the nonnegative orthant and the positive semidefi...

We call a positive semidefinite matrix whose elements are nonnegative a doubly nonnegative matrix, and the set of those matrices the doubly nonnegative cone (DNN cone). The DNN cone is not symmetric but can be represented as the projection of a symmetric cone embedded in a higher dimension. In [16], the authors demonstrated the efficiency of the DNN relaxation using the symmetric cone represent...

Let L be a linear transformation on a finite dimensional real Hilbert space H and K be a closed convex cone with dual K in H . The cone spectrum of L relative to K is the set of all real λ for which the linear complementarity problem x ∈ K, y = L(x)− λx ∈ K, and 〈x, y〉 = 0 admits a nonzero solution x. In the setting of a Euclidean Jordan algebra H and the corresponding symmetric cone K, we disc...