Rates of Stability in Nonlinear Programming

We give conditions on a nonlinear programming problem for the set of feasible solutions to have stability on the order of a Lipschitz condition. These results then imply conditions for the optimal value of the objective function to satisfy a Lipschitz condition with respect to the right-hand side vector as well as for the set of ϵ-optimal solutions to have stability on the order of a Lipschitz condition. Results are obtained both with and without convexity assumptions.

[2]  W. Zangwill Nonlinear programming : a unified approach , 1972 .

[3]  R. Rockafellar,et al.  Duality and stability in extremum problems involving convex functions. , 1967 .

[4]  J. Daniel On Perturbations in Systems of Linear Inequalities , 1973 .

[5]  Harvey J. Greenberg,et al.  Extensions of the Evans-Gould Stability Theorems for Mathematical Programs , 1972, Oper. Res..

[6]  J. Joly,et al.  Stability and Duality in Convex Minimization Problems. , 1971 .

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

[8]  C. M. Shetty,et al.  Constraint Qualifications Revisited , 1972 .

[9]  Olvi L. Mangasarian,et al.  Nonlinear Programming , 1969 .

[10]  R. Meyer The Validity of a Family of Optimization Methods , 1970 .

[11]  A. M. Geoffrion Duality in Nonlinear Programming: A Simplified Applications-Oriented Development , 1971 .

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

[13]  James W. Daniel,et al.  Stability of the solution of definite quadratic programs , 1973, Math. Program..

[14]  Floyd J. Gould,et al.  Stability in Nonlinear Programming , 1970, Oper. Res..

[15]  A. Fiacco Convergence properties of local solutions of sequences of mathematical programming problems in general spaces , 1974 .

[16]  Douglas Gale A Geometric Duality Theorem with Economic Applications , 1967 .

[17]  W. Hogan Point-to-Set Maps in Mathematical Programming , 1973 .

[18]  S. M. Robinson Normed convex processes , 1972 .

[19]  G. Dantzig,et al.  On the continuity of the minimum set of a continuous function , 1967 .