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We demonstrate that if A 1 ; :::; A m are symmetric positive semideenite n n matrices with positive deenite sum and A is an arbitrary symmetric n n matrix, then the quality of the semideenite relaxation (P) is not worse than 1 2 ln(2m 2). It is shown that this bound is sharp in order, as far as the dependence on m is concerned, and that a feasible solution… (More)

Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linear and convex quadratic optimization is a special case.… (More)

In this survey we review the many faces of the S-lemma, a result about the cor-rectness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry and linear algebra as well. These were active research areas, but as there was little… (More)

Sensitivity analysis is one of the most interesting and preoccupying areas in optimization. Many attempts are made to investigate the problem's behavior when the input data changes. Usually variation occurs in the right hand side of the constraints and/or the objective function coefficients. Degeneracy of optimal solutions causes considerable difficulties… (More)

A standard quadratic problem consists of nding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semideenite programming relaxation is strengthened by replacing the cone of positive semideenite matrices by the cone of completely positive matrices (the positive semideenite matrices which allow a factorization F F T… (More)

No part of this Journal may be reproduced in any form, by print, photoprint, microolm or any other means without written permis-Abstract: This paper surveys the origins and implications of ((nite) criss-cross methods in linear programming and related problems. Most pivot algorithms, like Dantzig's celebrated Simplex method, need a feasible basis to start… (More)

We investigate families of quadrics that have fixed intersections with two given hyper-planes. The cases when the two hyperplanes are parallel and when they are nonparallel are discussed. We show that these families can be described with only one parameter. In particular we show how the quadrics are transformed as the parameter changes. This research was… (More)