9 Abstract 10 We formulate two pairs of second order symmetric duality for nonlinear multiobjective mixed integer programs for 11 arbitrary cones. By using the concept of efficiency and second order invexity, we establish the weak, strong, converse 12 and self-duality theorems for our dual models. Several known results are obtained as special cases. 17 Following the earlier works of Dorn [5], D...

The Generalized Lax Conjecture asks whether every hyperbolicity cone is a section of a semidefinite cone of sufficiently high dimension. We prove that the space of hyperbolicity cones of hyperbolic polynomials of degree d in n variables contains (n/d) pairwise distant cones in the Hausdorff metric, and therefore that any semidefinite representation of such polynomials must have dimension at lea...

Self{scaled barrier functions on self{scaled cones were introduced through a set of axioms in 1994 by Y. E. Nesterov and M. J. Todd as a tool for the construction of long{step interior point algorithms. This paper provides rm foundation for these objects by exhibiting their symmetry properties, their intimate ties with the symmetry groups of their domains of deenition, and subsequently their de...

Self–scaled barrier functions on self–scaled cones were introduced through a set of axioms in 1994 by Y. E. Nesterov and M. J. Todd as a tool for the construction of long–step interior point algorithms. This paper provides firm foundation for these objects by exhibiting their symmetry properties, their intimate ties with the symmetry groups of their domains of definition, and subsequently their...

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...

There are three related concepts that arise in connection with the angular analysis of a convex cone: antipodality, criticality, and Nash equilibria. These concepts are geometric in nature but they can also be approached from the perspective of optimization theory. A detailed angular analysis of polyhedral convex cones has been carried out in a recent work of ours. This note focus on two import...