For a proper cone K ⊂ R and its dual cone K∗ the complementary slackness condition x s = 0 defines an n-dimensional manifold C(K) in the space { (x, s) | x ∈ K, s ∈ K∗ }. When K is a symmetric cone, this fact translates to a set of n bilinear optimality conditions satisfied by every (x, s) ∈ C(K). This proves to be very useful when optimizing over such cones, therefore it is natural to look for...

Euclidean Jordan-algebra is a commonly used tool in designing interiorpoint algorithms for symmetric cone programs. T -algebra, on the other hand, has rarely been used in symmetric cone programming. In this paper, we use both algebraic characterizations of symmetric cones to extend the target-following framework of linear programming to symmetric cone programming. Within this framework, we desi...

Abstract A new primal-dual interior-point algorithm applicable to nonsymmetric conic optimization is proposed. It a generalization of the famous suggested by Nesterov and Todd for symmetric case, uses scalings cones proposed Tunçel. We specialize Tunçel’s important case 3 dimensional exponential-cones, resulting in practical with good numerical performance, on level standard cone ( e.g. , quadr...

In this paper we extend three classical and fundamental results in polyhedral geometry, namely, Carath\'{e}odory's theorem, the Minkowski-Weyl Gordan's lemma to infinite dimensional spaces, which considered cones monoids are invariant under actions of symmetric groups.