A Condition Number for Multifold Conic Systems

Let $A:Y\to X$ be a linear map and $K\subseteq X$ be a regular closed convex cone. Consider the problem of finding a nontrivial solution to the conic feasibility problem $Ay\in K$. Condition numbers for this problem (as well as for related ones) are studied to quantify various issues concerning properties of the conic feasibility problem. Some issues especially relevant are the behavior of the problem under data perturbations, the geometry of the set of solutions, and the complexity analyses of algorithms that solve the problem. In this paper we define and characterize a condition number that exploits the possible factorization of $K$ as a product of simpler cones. This condition number extends both Renegar's condition number and the one we defined in [Math. Program., 91 (2001), pp. 163-174] for polyhedral conic systems. We see these results as a step in developing a theory of conditioning that takes into account the structure of the problem.

[1]  John C. Baez,et al.  The Octonions , 2001 .

[2]  James Renegar,et al.  A mathematical view of interior-point methods in convex optimization , 2001, MPS-SIAM series on optimization.

[3]  Hans Jarchow,et al.  Topological Vector Spaces , 1981 .

[4]  Javier F. Pena,et al.  A Condition Number for Second Order Conic Systems , 2006 .

[5]  Felipe Cucker,et al.  A Primal-Dual Algorithm for Solving Polyhedral Conic Systems with a Finite-Precision Machine , 2002, SIAM J. Optim..

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

[7]  Javier Peña,et al.  On the block-structured distance to non-surjectivity of sublinear mappings , 2005, Math. Program..

[8]  R. Tennant.  Algebra , 1941, Nature.

[9]  J. Borwein,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

[10]  Heinz H. Bauschke,et al.  Hyperbolic Polynomials and Convex Analysis , 2001, Canadian Journal of Mathematics.

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

[12]  Felipe Cucker,et al.  A new condition number for linear programming , 2001, Math. Program..

[13]  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..

[14]  Felipe Cucker,et al.  Probabilistic Analysis of Condition Numbers for Linear Programming , 2002 .

[15]  Javier Peña,et al.  A characterization of the distance to infeasibility under block-structured perturbations , 2003 .

[16]  W. Rudin.  Real and complex analysis , 1968 .

[17]  Hugh Williams,et al.  On quaternions and octonions: Their geometry, arithmetic, and symmetry, by John H. Conway and Derek A. Smith. Pp. 159. $29. 2003. ISBN 1 5688 134 9 (A. K. Peters). , 2004, The Mathematical Gazette.

[18]  Felipe Cucker,et al.  On the Expected Condition Number of Linear Programming Problems , 2003, Numerische Mathematik.

[19]  Tevian Dray,et al.  The Exceptional Jordan Eigenvalue Problem , 1999 .

[20]  Fernando Ordóñez,et al.  Computational Experience and the Explanatory Value of Condition Measures for Linear Optimization , 2003, SIAM J. Optim..

[21]  D. Spielman,et al.  Smoothed Analysis of Renegar’s Condition Number for Linear Programming , 2002 .

[22]  John H. Conway,et al.  On Quaternions and Octonions , 2003 .

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

[24]  Donald Goldfarb,et al.  Second-order cone programming , 2003, Math. Program..

[25]  Adrian S. Lewis.  The Structured Distance to Ill-Posedness for Conic Systems , 2004, Math. Oper. Res..

[26]  Robert M. Freund,et al.  On the behavior of the homogeneous self-dual model for conic convex optimization , 2004, Math. Program..

[27]  Michael J. Todd,et al.  Primal-Dual Interior-Point Methods for Self-Scaled Cones , 1998, SIAM J. Optim..

[28]  L. Faybusovich.  Linear systems in Jordan algebras and primal-dual interior-point algorithms , 1997 .

[29]  J. Faraut,et al.  Analysis on Symmetric Cones , 1995 .

[30]  Jean-Louis Goffin,et al.  The Relaxation Method for Solving Systems of Linear Inequalities , 1980, Math. Oper. Res..

[31]  Tobias Müller,et al.  Algebraic Tail Decay of Condition Numbers for Random Conic Systems under a General Family of Input Distributions , 2006 .

[32]  Michael J. Todd,et al.  Self-Scaled Barriers and Interior-Point Methods for Convex Programming , 1997, Math. Oper. Res..

[33]  Yuval Rabani,et al.  Linear Programming , 2007, Handbook of Approximation Algorithms and Metaheuristics.

[34]  G. Dixon,et al.  On quaternions and octonions: Their geometry, arithmetic, and symmetry , 2004 .

[35]  Felipe Cucker,et al.  Tail Decay and Moment Estimates of a Condition Number for Random Linear Conic Systems , 2005, SIAM J. Optim..

[36]  James Renegar,et al.  Hyperbolic Programs, and Their Derivative Relaxations , 2006, Found. Comput. Math..