Some physicists suggest that to more adequately describe the causal structure of space-time, it is necessary to go beyond the usual pseudoRiemannian causality, to a more general Finsler causality. In this general case, the set of all the events which can be influenced by a given event is, locally, a generic convex cone, and not necessarily a pseudoReimannian-style quadratic cone. Since all curr...

We propose a class of quadratic optimization problems whose exact optimal objective values can be computed by their completely positive cone programming relaxations. The objective function can be any quadratic form. The constraints of each problem are described in terms of quadratic forms with no linear terms, and all constraints are homogeneous equalities, except one inhomogeneous equality whe...

We address the iterative solution of symmetric KKT systems arising in the solution of convex quadratic programming problems. Two strictly related and well established formulations for such systems are studied with particular emphasis on the effect of preconditioning strategies on their relation. Constraint and augmented preconditioners are considered, and the choice of the augmentation matrix i...

where C is a convex closed cone in the Euclidean space IR, f : IR → IR and G : IR → Y is a mapping from IR into the space Y := S of m × m symmetric matrices. We refer to the above problem as a nonlinear semidefinite programming problem. In particular, if C = IR, the objective function is linear, i.e. f(x) := ∑n i=1 bixi, and the constraint mapping is affine, i.e. G(x) := A0 + ∑n i=1 xiAi where ...

We propose a class of quadratic optimization problems whose exact optimal objective values can be computed by their completely positive cone programming relaxations. The objective function can be any quadratic form. The constraints of each problem are described in terms of quadratic forms with no linear terms, and all constraints are homogeneous equalities, except one inhomogeneous equality whe...

We consider the strictly convex quadratic programming problem with bounded variables. A dual problem is derived using Lagrange duality. The dual problem is the minimization of an unconstrained, piecewise quadratic function. It involves a lower bound of λ1, the smallest eigenvalue of a symmetric, positive definite matrix, and is solved by Newton iteration with line search. The paper describes th...

For a convex body B in a vector space V , we construct its approximation Pk, k = 1, 2, . . . using an intersection of a cone of positive semidefinite quadratic forms with an affine subspace. We show that Pk is contained in B for each k. When B is the Symmetric Traveling Salesman Polytope on n cities Tn, we show that the scaling of Pk by n k + O ` 1 n ́ contains Tn for k ≤ b 2 c. Membership for P...

The Q method of semidefinite programming, developed by Alizadeh, Haeberly and Overton, is extended to optimization problems over symmetric cones. At each iteration of the Q method, eigenvalues and Jordan frames of decision variables are updated using Newton’s method. We give an interior point and a pure Newton’s method based on the Q method. In another paper, the authors have shown that the Q m...

A large class of separable quadratic programming problems is presented The problems in the class can be solved in linear time The class in cludes the separable convex quadratic transportation problem with a xed number of sources and separable convex quadratic programming with nonnegativity con straints and a xed number of linear equality constraints

We describe strong convex valid inequalities for conic quadratic mixed 0-1 optimization. The inequalities exploit the submodularity of the binary restrictions and are based on the polymatroid inequalities over binaries for the diagonal case. We prove that the convex inequalities completely describe the convex hull of a single conic quadratic constraint as well as the rotated cone constraint ove...