Smoothed Analysis of Renegar’s Condition Number for Linear Programming

For any linear program, we show that a slight random relative perturbation of that linear program has small condition number with high probability. Following [ST01], we call this smoothed analysis of the condition number. Part of our main result is that the expectation of the log of the condition number of any appropriately scaled linear program subject to a Gaussian perturbation of variance σ is at most O(lognd/σ) with high probability. Since the condition number bounds the running time of many algorithms for convex programming, this may explain their observed fast convergence.

[1]  V. Klee,et al.  Helly's theorem and its relatives , 1963 .

[2]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[3]  R. Rao,et al.  Normal Approximation and Asymptotic Expansions , 1976 .

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

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

[6]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, STOC '84.

[7]  C. C. Gonzaga,et al.  An algorithm for solving linear programming programs in O(n3L) operations , 1988 .

[8]  C. C. Gonzaga,et al.  An Algorithm for Solving Linear Programming Problems in O(n 3 L) Operations , 1989 .

[9]  Pravin M. Vaidya,et al.  An algorithm for linear programming which requires O(((m+n)n2+(m+n)1.5n)L) arithmetic operations , 1990, Math. Program..

[10]  Michael J. Todd,et al.  Probabilistic Models for Linear Programming , 1991, Math. Oper. Res..

[11]  Keith Ball.  The reverse isoperimetric problem for Gaussian measure , 1993, Discret. Comput. Geom..

[12]  Michael J. Todd,et al.  A Lower Bound on the Number of Iterations of Primal-Dual Interior-Point Methods for Linear Programming , 1993 .

[13]  Shinji Mizuno,et al.  On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming , 1993, Math. Oper. Res..

[14]  Florian A. Potra,et al.  Average Performance of a Self–Dual Interior Point Algorithm for Linear Programming , 1993 .

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

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

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

[18]  Jacek Gondzio,et al.  Implementation of Interior Point Methods for Large Scale Linear Programming , 1996 .

[19]  Robert M. Freund,et al.  Interior point methods : current status and future directions , 1996 .

[20]  Jorge R. Vera.  Ill-Posedness and the Complexity of Deciding Existence of Solutions to Linear Programs , 1996, SIAM J. Optim..

[21]  R. Freund,et al.  Condition number complexity of an elementary algorithm for resolving a conic linear system , 1997 .

[22]  Anupam Gupta,et al.  An elementary proof of the Johnson-Lindenstrauss Lemma , 1999 .

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

[24]  Yinyu Ye,et al.  Probabilistic Analysis of an Infeasible-Interior-Point Algorithm for Linear Programming , 1999, Math. Oper. Res..

[25]  Robert M. Freund,et al.  Condition number complexity of an elementary algorithm for computing a reliable solution of a conic linear system , 2000, Math. Program..

[26]  Javier Peña,et al.  Understanding the Geometry of Infeasible Perturbations of a Conic Linear System , 1999, SIAM J. Optim..

[27]  Robert M. Freund,et al.  Opera Tions Research Center Working Paper Condition-measure Bounds on the Behavior of the Central Trajectory of a Semi-definite Program , 2022 .

[28]  Shang-Hua Teng,et al.  Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time , 2001, STOC '01.

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

[30]  Petra Huhn,et al.  Interior-Point Methods: Worst Case and Average Case Analysis of a Phase-I Algorithm and a Termination Procedure , 2002, J. Complex..

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

[32]  Robert M. Freund,et al.  A New Condition Measure, Preconditioners, and Relations Between Different Measures of Conditioning for Conic Linear Systems , 2002, SIAM J. Optim..

[33]  John Dunagan,et al.  Smoothed analysis of the perceptron algorithm for linear programming , 2002, SODA '02.

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

[35]  Shang-Hua Teng,et al.  Smoothed Analysis (Motivation and Discrete Models) , 2003, WADS.

[36]  Shang-Hua Teng,et al.  Smoothed analysis of termination of linear programming algorithms , 2003, Math. Program..

[37]  Robert M. Freund,et al.  Complexity of convex optimization using geometry-based measures and a reference point , 2015, Math. Program..

[38]  D. Spielman,et al.  Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time , 2004 .