Degeneracy in NLP and the development of results motivated by its presence

We study notions of nondegeneracy and several levels of increasing degeneracy from the perspective of the local behavior of a local solution of a nonlinear program when problem parameters are slightly perturbed. Ideal nondegeneracy at a local minimizer is taken to mean satisfaction of second order sufficient conditions, linear independence and strict complimentary slackness. Following a brief exploration of the relationship of these conditions with the classical definition of nondegeneracy in linear programming, we recall a number of optimality and regularity conditions used to attempt to resolve degeneracy and survey results of Fiacco, McCormick, Robinson, Kojima, Gauvin and Janin, Shapiro, Kyparisis and Liu. This overview may be viewed as a structured survey of sensitivity and stability results: the focus is on progressive levels of degeneracy. We note connections of nondegeneracy with the convergence of algorithms and observe the striking parallel between the effects of nondegeneracy and degeneracy on optimality conditions, stability analysis and algorithmic convergence behavior. Although our orientation here is primarily interpretive and noncritical, we conclude that more effort is needed to unify optimality, stability and convergence theory and more results are needed in all three areas for radically degenerate problems.

[1]  G. McCormick Nonlinear Programming: Theory, Algorithms and Applications , 1983 .

[2]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[3]  Jerzy Kyparisis Sensitivity Analysis for Nonlinear Programs and Variational Inequalities with Nonunique Multipliers , 1990, Math. Oper. Res..

[4]  Anthony V. Fiacco,et al.  Sensitivity analysis for nonlinear programming using penalty methods , 1976, Math. Program..

[5]  Jiming Liu Linear Stability of Generalized Equations, Part II: Applications to Nonlinear Programming , 1994, Math. Oper. Res..

[6]  Jerzy Kyparisis Sensitivity analysis framework for variational inequalities , 1987, Math. Program..

[7]  A. Auslender,et al.  First and second order sensitivity analysis of nonlinear programs under directional constraint qualification conditions , 1990 .

[8]  A. Shapiro Sensitivity analysis of nonlinear programs and differentiability properties of metric projections , 1988 .

[9]  Jiming Liu Linear Stability of Generalized Equations Part I: Basic Theory , 1994, Math. Oper. Res..

[10]  S. M. Robinson Generalized equations and their solutions, part II: Applications to nonlinear programming , 1982 .

[11]  A. Fiacco,et al.  Sensitivity and stability analysis for nonlinear programming , 1991 .

[12]  N. Josephy Quasi-Newton methods for generalized equations , 1979 .

[13]  Jacques Gauvin,et al.  Directional Behaviour of Optimal Solutions in Nonlinear Mathematical Programming , 1988, Math. Oper. Res..

[14]  M. Kojima Strongly Stable Stationary Solutions in Nonlinear Programs. , 1980 .

[15]  Ekkehard W. Sachs Rates of convergence for adaptive Newton methods , 1986 .

[16]  N. Josephy Newton's Method for Generalized Equations. , 1979 .

[17]  Jerzy Kyparisis,et al.  On uniqueness of Kuhn-Tucker multipliers in nonlinear programming , 1985, Math. Program..

[18]  Stephen M. Robinson,et al.  Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms , 1974, Math. Program..

[19]  Alexander Shapiro Second order sensitivity analysis and asymptotic theory of parametrized nonlinear programs , 1985, Math. Program..

[20]  R. Rockafellar Directional differentiability of the optimal value function in a nonlinear programming problem , 1984 .

[21]  Stephen M. Robinson,et al.  Strongly Regular Generalized Equations , 1980, Math. Oper. Res..

[22]  M. J. D. Powell,et al.  Variable Metric Methods for Constrained Optimization , 1982, ISMP.