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Condition Measures and Properties of the Central Trajectory of a Linear Program
 Mathematical Programming
, 1997
"... Given a data instance d = (A; b; c) of a linear program, we show that certain properties of solutions along the central trajectory of the linear program are inherently related to the condition number C(d) of the data instance d = (A; b; c), where C(d) is a scaleinvariant reciprocal of a closelyrel ..."
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Cited by 34 (15 self)
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Given a data instance d = (A; b; c) of a linear program, we show that certain properties of solutions along the central trajectory of the linear program are inherently related to the condition number C(d) of the data instance d = (A; b; c), where C(d) is a scaleinvariant reciprocal of a closelyrelated measure ae(d) called the "distance to illposedness." (The distance to illposedness essentially measures how close the data instance d = (A; b; c) is to being primal or dual infeasible.) We present lower and upper bounds on sizes of optimal solutions along the central trajectory, and on rates of change of solutions along the central trajectory, as either the barrier parameter ¯ or the data d = (A; b; c) of the linear program is changed. These bounds are all linear or polynomial functions of certain natural parameters associated with the linear program, namely the condition number C(d), the distance to illposedness ae(d), the norm of the data kdk, and the dimensions m and n. 1 Introdu...
Interior Point Methods: Current Status And Future Directions
, 1997
"... This article provides a synopsis of the major developments in interior point methods for mathematical programming in the last thirteen years, and discusses current and future research directions in interior point methods, with a brief selective guide to the research literature. AMS Subject Classific ..."
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Cited by 13 (0 self)
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This article provides a synopsis of the major developments in interior point methods for mathematical programming in the last thirteen years, and discusses current and future research directions in interior point methods, with a brief selective guide to the research literature. AMS Subject Classification: 90C, 90C05, 90C60 Keywords: Linear Programming, Newton's Method, Interior Point Methods, Barrier Method, Semidefinite Programming, SelfConcordance, Convex Programming, Condition Numbers 1 An earlier version of this article has previously appeared in OPTIMA  Mathematical Programming Society Newsletter No. 51, 1996 2 M.I.T. Sloan School of Management, Building E40149A, Cambridge, MA 02139, USA. email: rfreund@mit.edu 3 The Institute of Statistical Mathematics, 467 MinamiAzabu, Minatoku, Tokyo 106 JAPAN. email: mizuno@ism.ac.jp INTERIOR POINT METHODS 1 1 Introduction and Synopsis The purpose of this article is twofold: to provide a synopsis of the major developments in ...