In this paper, we introduce a new P-type condition for nonlinear functions defined over Euclidean Jordan algebras, and study a continuation method for nonlinear complementarity problems over symmetric cones. This new P-type condition represents a new class of nonmonotone nonlinear complementarity problems that can be solved numerically. Citation Research Report, Division of Mathematical Sciences, School … Read more
In \cite{aYOSHISE06}, the author proposed a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and show that the following properties hold: (a) There is a path that is bounded and has a trivial starting point without any regularity assumption concerning the existence of feasible or strictly feasible solutions. (b) Any accumulation point … Read more
We present a strong duality theory for optimization problems over symmetric cones without assuming any constraint qualification. We show important complexity implications of the result to semidefinite and second order conic optimization. The result is an application of Borwein and Wolkowicz’s facial reduction procedure to express the minimal cone. We use Pataki’s simplified analysis and … Read more
This short note extends the sparse SOS (sum of squares) and SDP (semidefinite programming) relaxation proposed by Waki, Kim, Kojima and Muramatsu for normal POPs (polynomial optimization problems) to POPs over symmetric cones, and establishes its theoretical convergence based on the recent convergence result by Lasserre on the sparse SOS and SDP relaxation for normal … Read more
In this paper, we propose a homogeneous model for solving monotone mixed complementarity problems over symmetric cones, by extending the results in \cite{YOSHISE04} for standard form of the problems. We show that the extended model inherits the following desirable features: (a) A path exists, is bounded and has a trivial starting point without any regularity … Read more
We illustrate the usefulness of Jordan-algebraic technique for nonconvex optimization by considering a potential-reduction algorithm for a nonconvex quadratic function over the domain obtained as the intersection of a symmetric cone with an affine subspace Citation Preprint, September,2004 Article Download View Jordan-algebraic aspects of nonconvex optimization over symmetric cones
We extend the Q method to the symmetric cone programming. An infeasible interior point algorithm and a Newton-type algorithm are given. We give convergence results of the interior point algorithm and prove that the Newton-type algorithm is good for Citation AdvOl-Report#2004/18 McMaster University, Advanced Optimization Laboratory Hamilton, Ontario, Canada October 2004 Article Download View The … Read more
We study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones. While the analysis in Faybusovich (1997) depends on the optimization theory of convex log-barrier functions, our approach is based on the paper of Monteiro and Pang (1998), where a vast set of conclusions concerning continuous trajectories is shown for monotone … Read more
This paper is based on a recent work by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. Let ${\cal E}$ and ${\cal E}_+$ be a finite dimensional real vector space and a symmetric cone embedded in ${\cal E}$; examples of $\calE$ and $\calE_+$ include a pair of the … Read more
We establish polynomial-time convergence of infeasible-interior-point methods for conic programs over symmetric cones using a wide neighborhood of the central path. The convergence is shown for a commutative family of search directions used in Schmieta and Alizadeh. These conic programs include linear and semidefinite programs. This extends the work of Rangarajan and Todd, which established … Read more