Approximate Solutions of Analytic Inequality Systems

We consider the stability problem for a system of equalities and inequalities. We prove that when such a system is satisfied approximately at a certain point, it is also satisfied exactly on a nearby point.

[1]  S. Smale.  Newton’s Method Estimates from Data at One Point , 1986 .

[2]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

[3]  Felipe Cucker,et al.  Complexity estimates depending on condition and round-off error , 1998, JACM.

[4]  J. Dedieu.  Penalty functions in subanalytic optimization , 1992 .

[5]  Uriel G. Rothblum,et al.  Approximations to Solutions to Systems of Linear Inequalities , 1995, SIAM J. Matrix Anal. Appl..

[6]  Adi Ben-Israel.  A Newton-Raphson method for the solution of systems of equations , 1966 .

[7]  S. Smale,et al.  Complexity of Bezout's theorem IV: probability of success; extensions , 1996 .

[8]  S. Łojasiewicz.  Ensembles semi-analytiques , 1965 .

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

[10]  Zhi-Quan Luo,et al.  Error bounds for analytic systems and their applications , 1994, Math. Program..

[11]  Olvi L. Mangasarian,et al.  A Condition Number for Differentiable Convex Inequalities , 1985, Math. Oper. Res..

[12]  S. M. Robinson.  An Application of Error Bounds for Convex Programming in a Linear Space , 1975 .

[13]  A. Lewis,et al.  Error Bounds for Convex Inequality Systems , 1998 .

[14]  D. Azé,et al.  Variational pairs and applications to stability in nonsmooth analysis , 2002 .

[15]  E. Bierstone,et al.  Semianalytic and subanalytic sets , 1988 .

[16]  F. Cucker,et al.  Decision Problems and Round-Off Machines , 2001, Theory of Computing Systems.

[17]  広中 平祐.  Introduction to real-analytic sets and real-analytic maps , 1973 .

[18]  Myong-Hi Kim,et al.  Newton's Method for Analytic Systems of Equations with Constant Rank Derivatives , 2002, J. Complex..

[19]  Zhi-Quan Luo,et al.  Extension of Hoffman's Error Bound to Polynomial Systems , 1994, SIAM J. Optim..

[20]  Alfred Auslender,et al.  Global Regularity Theorems , 1988, Math. Oper. Res..

[21]  James Renegar,et al.  Is It Possible to Know a Problem Instance Is Ill-Posed?: Some Foundations for a General Theory of Condition Numbers , 1994, J. Complex..

[22]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[23]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

[24]  A. Hoffman.  On approximate solutions of systems of linear inequalities , 1952 .

[25]  J. Warga.  A Necessary and Sufficient Condition for a Constrained Minimum , 1992, SIAM J. Optim..

[26]  A. Ioffe.  Regular points of Lipschitz functions , 1979 .