Pseudonormality and a Lagrange Multiplier Theory for Constrained Optimization

We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the Fritz–John theorem, and we introduce new and general conditions that extend and unify the major constraint qualifications. Among these conditions, two new properties, pseudonormality and quasinormality, emerge as central within the taxonomy of interesting constraint characteristics. In the case where there is no abstract set constraint, these properties provide the connecting link between the classical constraint qualifications and two distinct pathways to the existence of Lagrange multipliers: one involving the notion of quasiregularity and the Farkas lemma, and the other involving the use of exact penalty functions. The second pathway also applies in the general case where there is an abstract set constraint.

[1]  R. Srikant,et al.  The marginal user principle for resource allocation in wireless networks , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[2]  A. Ozdaglar,et al.  Min Common/Max Crossing Duality: A Simple Geometric Framework for Convex Optimization and Minimax Theory1 , 2003 .

[3]  Ian Gladwell.  Some Extensions of a Paper by Birkhoff and Priver , 1969 .

[4]  D. W. Peterson.  A REVIEW OF CONSTRAINT QUALIFICATIONS IN FINITE-DIMENSIONAL SPACES* , 1973 .

[5]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

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

[7]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[8]  Hector A. Rosales-Macedo.  Nonlinear Programming: Theory and Algorithms (2nd Edition) , 1993 .

[9]  Adrian S. Lewis,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

[10]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[11]  William M. Wells,et al.  Model-based object recognition using laser radar range imagery , 1999, Defense, Security, and Sensing.

[12]  M. Guignard.  Generalized Kuhn–Tucker Conditions for Mathematical Programming Problems in a Banach Space , 1969 .

[13]  T. Pietrzykowski.  An Exact Potential Method for Constrained Maxima , 1969 .

[14]  B. Mordukhovich.  Maximum principle in the problem of time optimal response with nonsmooth constraints PMM vol. 40, n≗ 6, 1976, pp. 1014-1023 , 1976 .

[15]  Asuman E. Ozdaglar,et al.  Routing and wavelength assignment in optical networks , 2003, TNET.

[16]  W. Zangwill.  Non-Linear Programming Via Penalty Functions , 1967 .

[17]  L. Hurwicz,et al.  Constraint Qualifications in Maximization Problems , 1961 .

[18]  F. J. Gould,et al.  Geometry of optimality conditions and constraint qualifications , 1972, Math. Program..

[19]  E. J. McShane.  The Lagrange Multiplier Rule , 1973 .

[20]  A. Ozdaglar,et al.  Optimal Solution of Integer Multicommodity Flow Problems With Application in Optical Networks , 2004 .

[21]  M. Bazaraa,et al.  Sufficient conditions for a globally exact penalty function without convexity , 1982 .

[22]  Frank H. Clarke,et al.  A New Approach to Lagrange Multipliers , 1976, Math. Oper. Res..

[23]  Asuman E. Ozdaglar,et al.  The relation between pseudonormality and quasiregularity in constrained optimization , 2004, Optim. Methods Softw..

[24]  Michael J. Grimble.  IEEE conference on decision and control , 1987 .

[25]  Michael J. Grimble.  Mathematical theory of networks and systems , 1987 .

[26]  M.I. Miller,et al.  Performance analysis for ground-based target orientation estimation: FLIR/LADAR sensor fusion , 1999, Conference Record of the Thirty-Third Asilomar Conference on Signals, Systems, and Computers (Cat. No.CH37020).

[27]  O. Mangasarian,et al.  The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints , 1967 .

[28]  J. Abadie.  ON THE KUHN-TUCKER THEOREM. , 1966 .

[29]  Jon W. Tolle,et al.  Exact penalty functions in nonlinear programming , 1973, Math. Program..

[30]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[31]  R. Tyrrell Rockafellar,et al.  Lagrange Multipliers and Optimality , 1993, SIAM Rev..

[32]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

[33]  Douglas Gale,et al.  Bayesian learning in social networks , 2003, Games Econ. Behav..

[34]  D. Bertsekas,et al.  Enhanced Optimality Conditions and Exact Penalty Functions , 2000 .

[35]  R. J. Paul,et al.  Optimization Theory: The Finite Dimensional Case , 1977 .

[36]  F. J. Gould,et al.  A NECESSARY AND SUFFICIENT QUALIFICATION FOR CONSTRAINED OPTIMIZATION , 1971 .

[37]  Jacques Gauvin,et al.  A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming , 1977, Math. Program..

[38]  R. Tyrrell Rockafellar,et al.  A dual approach to solving nonlinear programming problems by unconstrained optimization , 1973, Math. Program..