On the connection of facially exposed and nice cones

A closed convex cone K is called nice, if the set K^* + F^\perp is closed for all F faces of K, where K^* is the dual cone of K, and F^\perp is the orthogonal complement of the linear span of F. The niceness property is important for two reasons: it plays a role in the facial reduction algorithm of Borwein and Wolkowicz, and the question whether the linear image of a nice cone is closed also has a simple answer. We prove several characterizations of nice cones and show a strong connection with facial exposedness. We prove that a nice cone must be facially exposed; in reverse, facial exposedness with an added condition implies niceness. We conjecture that nice, and facially exposed cones are actually the same, and give supporting evidence

C++ Standard Template Library by template specialized containers

The C++ Standard Template Library is the flagship example for libraries based on the generic programming paradigm. The usage of this library is intended to minimize the number of classical C/C++ errors, but does not warrant bug-free programs. Furthermore, many new kinds of errors may arise from the inaccurate use of the generic programming paradigm, like dereferencing invalid iterators or misunderstanding remove-like algorithms. In this paper we present some typical scenarios that may cause runtime or portability problems. We emit warnings and errors while these risky constructs are used. We also present a general approach to emit "customized" warnings. We support the so-called "believe-me marks" to disable warnings. We present another typical usage of our technique, when classes become deprecated during the software lifecycle

Governor\u27s Remarks

Remarks by Gov. George Pataki on Law Day at Pace University School of Law, May 1, 1996

Exact duality in semidefinite programming based on elementary reformulations

In semidefinite programming (SDP), unlike in linear programming, Farkas' lemma may fail to prove infeasibility. Here we obtain an exact, short certificate of infeasibility in SDP by an elementary approach: we reformulate any semidefinite system of the form Ai*X = bi (i=1,...,m) (P) X >= 0 using only elementary row operations, and rotations. When (P) is infeasible, the reformulated system is trivially infeasible. When (P) is feasible, the reformulated system has strong duality with its Lagrange dual for all objective functions. As a corollary, we obtain algorithms to generate the constraints of {\em all} infeasible SDPs and the constraints of {\em all} feasible SDPs with a fixed rank maximal solution.Comment: To appear, SIAM Journal on Optimizatio

Bad semidefinite programs: they all look the same

Conic linear programs, among them semidefinite programs, often behave pathologically: the optimal values of the primal and dual programs may differ, and may not be attained. We present a novel analysis of these pathological behaviors. We call a conic linear system $Ax <= b$ {\em badly behaved} if the value of $\sup { | A x <= b }$ is finite but the dual program has no solution with the same value for {\em some} $c.$ We describe simple and intuitive geometric characterizations of badly behaved conic linear systems. Our main motivation is the striking similarity of badly behaved semidefinite systems in the literature; we characterize such systems by certain {\em excluded matrices}, which are easy to spot in all published examples. We show how to transform semidefinite systems into a canonical form, which allows us to easily verify whether they are badly behaved. We prove several other structural results about badly behaved semidefinite systems; for example, we show that they are in $NP \cap co-NP$ in the real number model of computing. As a byproduct, we prove that all linear maps that act on symmetric matrices can be brought into a canonical form; this canonical form allows us to easily check whether the image of the semidefinite cone under the given linear map is closed.Comment: For some reason, the intended changes between versions 4 and 5 did not take effect, so versions 4 and 5 are the same. So version 6 is the final version. The only difference between version 4 and version 6 is that 2 typos were fixed: in the last displayed formula on page 6, "7" was replaced by "1"; and in the 4th displayed formula on page 12 "A_1 - A_2 - A_3" was replaced by "A_3 - A_2 - A_1