For a conic linear system of the form Ax \in K, K a convex cone, several condition measures have been extensively studied in the last dozen years. Herein we show that Renegar’s condition number is bounded from above and below by certain purely geometric quantities associated with A and K, and highlights the role of … Read more
We consider semidefinite programming (SDP) formulations of certain truss topology optimization problems, where a lower bound is imposed on the fundamental frequency of vibration of the truss structure. These SDP formulations were introduced in: [M. Ohsaki, K. Fujisawa, N. Katoh and Y. Kanno, Semi-definite programming for topology optimization of trusses under multiple eigenvalue constraints, Comp. … Read more
This paper focuses on monotone L\”{o}wner operators in Euclidean Jordan algebras and their applications to the symmetric cone complementarity problem (SCCP). We prove necessary and sufficient conditions for locally Lipschitz L\”{o}wner operators to be monotone, strictly monotone and strongly monotone. We also study the relationship between monotonicity and operator-monotonicity of L\”{o}wner operators. As a by-product … Read more
In this paper we devise a new version of primal simplex algorithms in which the classical iteration is decomposed two basic operations: the move and the pivot. The move operation decreases the primal objective value and the pivot operation increases the dual objective. We define the condition number of the pivot operation and present a … Read more
Linear Complementarity Problems ($LCP$s) belong to the class of $\mathbb{NP}$-complete problems. Therefore we can not expect a polynomial time solution method for $LCP$s without requiring some special property of the matrix coefficient matrix. Our aim is to construct some interior point algorithms which, according to the duality theorem in EP form, gives a solution of … Read more
The linear complementarity problem (LCP) belongs to the class of NP-complete problems. Therefore we can not expect a polynomial time solution method for LCPs without requiring some special property of the matrix of the problem. We show that the dual LCP can be solved in polynomial time if the matrix is row sufficient, moreover in … Read more
The famous Frank–Wolfe theorem ensures attainability of the optimal value for quadratic objective functions over a (possibly unbounded) polyhedron if the feasible values are bounded. This theorem does not hold in general for conic programs where linear constraints are replaced by more general convex constraints like positive-semidefiniteness or copositivity conditions, despite the fact that the … Read more
We show how to approximate the feasible region of structured convex optimization problems by a family of convex sets with explicitly given and efficient (if the accuracy of the approximation is moderate) self-concordant barriers. This approach extends the reach of the modern theory of interior-point methods, and lays the foundation for new ways to treat … Read more
We investigate hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph $G$. We introduce an operator $\Psi$ mapping any graph parameter $\beta(G)$, nested between the stability number $\alpha(G)$ and $\chi\left( {\ol G} \right)$, to a new graph parameter $\Psi_\beta(G)$, nested between $\alpha (\ol G)$ and $\chi(G)$; $\Psi_\beta(G)$ is polynomial time computable if … Read more
Recently we investigated in “The operator $\Psi$ for the Chromatic Number of a Graph” hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph $G$. In particular, we introduced two hierarchies of lower bounds, the `$\psi$’-hierarchy converging to the fractional chromatic number, and the `$\Psi$’-hierarchy converging to the chromatic number of a graph. … Read more