Error bounds for analytic systems and their applications

Using a 1958 result of Lojasiewicz, we establish an error bound for analytic systems consisting of equalities and inequalities defined by real analytic functions. In particular, we show that over any bounded region, the distance from any vectorx in the region to the solution set of an analytic system is bounded by a residual function, raised to a certain power, evaluated atx. For quadratic systems satisfying certain nonnegativity assumptions, we show that this exponent is equal to 1/2. We apply the error bounds to the Karush—Kuhn—Tucker system of a variational inequality, the affine variational inequality, the linear and nonlinear complementarity problem, and the 0–1 integer feasibility problem, and obtain new error bound results for these problems. The latter results extend previous work for polynomial systems and explain why a certain square-root term is needed in an error bound for the (monotone) linear complementarity problem.

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

[2]  L. Hörmander On the division of distributions by polynomials , 1958 .

[3]  S. Łojasiewicz Sur le problème de la division , 1959 .

[4]  S. M. Robinson Bounds for error in the solution set of a perturbed linear program , 1973 .

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

[6]  S. M. Robinson Some continuity properties of polyhedral multifunctions , 1981 .

[7]  O. Mangasarian Simple computable bounds for solutions of linear complementarity problems and linear programs , 1985 .

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

[9]  Olvi L. Mangasarian,et al.  Error bounds for monotone linear complementarity problems , 1986, Math. Program..

[10]  William J. Cook,et al.  Sensitivity theorems in integer linear programming , 1986, Math. Program..

[11]  Jong-Shi Pang,et al.  A Posteriori Error Bounds for the Linearly-Constrained Variational Inequality Problem , 1987, Math. Oper. Res..

[12]  O. Mangasarian,et al.  Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems , 1987 .

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

[14]  O. Mangasarian,et al.  Error bounds for strongly convex programs and (super)linearly convergent iterative schemes for the least 2-norm solution of linear programs , 1988 .

[15]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[16]  J. Pang,et al.  Error bounds for the linear complementarity problem with a P-matrix , 1990 .

[17]  I. Singer,et al.  The distance to a polyhedron , 1992 .

[18]  P. Tseng,et al.  On the linear convergence of descent methods for convex essentially smooth minimization , 1992 .

[19]  Paul Tseng,et al.  On a global error bound for a class of monotone affine variational inequality problems , 1992, Oper. Res. Lett..

[20]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[21]  O. Mangasarian Global error bounds for monotone affine variational inequality problems , 1992 .

[22]  Paul Tseng,et al.  Error Bound and Convergence Analysis of Matrix Splitting Algorithms for the Affine Variational Inequality Problem , 1992, SIAM J. Optim..

[23]  Wu Li The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program , 1993 .

[24]  Olvi L. Mangasarian,et al.  Nonlinear complementarity as unconstrained and constrained minimization , 1993, Math. Program..

[25]  Michael C. Ferris,et al.  Error bounds and strong upper semicontinuity for monotone affine variational inequalities , 1993, Ann. Oper. Res..

[26]  Olvi L. Mangasarian,et al.  New Error Bounds for the Linear Complementarity Problem , 1994, Math. Oper. Res..

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

[28]  J. Pang,et al.  On the Boundedness and Stability of Solutions to the Affine Variational Inequality Problem , 1994 .

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