A characterization of ill-posed data instances for convex programming

Abstract.Given a data instance of a convex program, we provide a collection of conic linear systems such that the data instance is ill-posed if and only if at least one of those systems is satisfied. This collection of conic linear systems is derived from a characterization of the boundary of the set of primal and dual feasible data instances associated with the given convex program.

[1]  J. Renegar Some perturbation theory for linear programming , 1994, Math. Program..

[2]  L. G. H. Cijan A polynomial algorithm in linear programming , 1979 .

[3]  Bud Mishra,et al.  Algorithmic Algebra , 1993, Texts and Monographs in Computer Science.

[4]  S. M. Robinson Stability Theory for Systems of Inequalities. Part I: Linear Systems , 1975 .

[5]  S. A. Ashmanov Stability conditions for linear programming problems , 1981 .

[6]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[7]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[8]  James Renegar,et al.  Incorporating Condition Measures into the Complexity Theory of Linear Programming , 1995, SIAM J. Optim..

[9]  Gene H. Golub,et al.  Matrix Computations, Third Edition , 1996 .

[10]  Robert M. Freund,et al.  Some characterizations and properties of the “distance to ill-posedness” and the condition measure of a conic linear system , 1999, Math. Program..

[11]  Stephen M. Robinson,et al.  A Characterization of Stability in Linear Programming , 1977, Oper. Res..

[12]  Núñez Araya,et al.  Condition numbers and properties of central trajectories in convex programming , 1997 .

[13]  Robert M. Freund,et al.  Condition-Based Complexity of Convex Optimization in Conic Linear Form via the Ellipsoid Algorithm , 1999, SIAM J. Optim..

[14]  Robert M. Freund,et al.  On the Complexity of Computing Estimates of Condition Measures of a Conic Linear System , 2003, Math. Oper. Res..

[15]  James Renegar,et al.  Linear programming, complexity theory and elementary functional analysis , 1995, Math. Program..

[16]  C. M. Shetty,et al.  Nonlinear Programming - Theory and Algorithms, Second Edition , 1993 .

[17]  Jorge R. Vera,et al.  On the complexity of linear programming under finite precision arithmetic , 1998, Math. Program..

[18]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[19]  L. Khachiyan Polynomial algorithms in linear programming , 1980 .

[20]  Olvi L. Mangasarian,et al.  A Stable Theorem of the Alternative: An Extension of the Gordan Theorem. , 1981 .

[21]  James Renegar,et al.  Condition Numbers, the Barrier Method, and the Conjugate-Gradient Method , 1996, SIAM J. Optim..

[22]  Robert M. Freund,et al.  Condition measures and properties of the central trajectory of a linear program , 1998, Math. Program..

[23]  D. Luenberger Optimization by Vector Space Methods , 1968 .