Generalized Levitin--Polyak Well-Posedness in Constrained Optimization

In this paper, we consider Levitin--Polyak-type well-posedness for a general constrained optimization problem. We introduce generalized Levitin--Polyak well-posedness and strongly generalized Levitin--Polyak well-posedness. Necessary and sufficient conditions for these types of well-posedness are given. Relations among these types of well-posedness are investigated. Finally, we consider convergence of a class of penalty methods and a class of augmented Lagrangian methods under the assumption of strongly generalized Levitin--Polyak well-posedness.

[1]  M. Morari,et al.  Nonlinear parametric optimization using cylindrical algebraic decomposition , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[2]  Roberto Lucchetti,et al.  The EPI-Distance Topology: Continuity and Stability Results with Applications to Convex Optimization Problems , 1992, Math. Oper. Res..

[3]  T. Zolezzi,et al.  Well-posedness criteria in optimization with application to the calculus of variations , 1995 .

[4]  J. Revalski,et al.  Recent developments in well-posed variational problems , 1995 .

[5]  Jong-Shi Pang,et al.  Error bounds in mathematical programming , 1997, Math. Program..

[6]  T. Zolezzi.  Extended well-posedness of optimization problems , 1996 .

[7]  Asen L. Dontchev,et al.  Regularity and Conditioning of Solution Mappings in Variational Analysis , 2004 .

[8]  X. X. Huang,et al.  Extended and strongly extended well-posedness of set-valued optimization problems , 2001, Math. Methods Oper. Res..

[9]  Kok Lay Teo,et al.  Lower-Order Penalization Approach to Nonlinear Semidefinite Programming , 2007 .

[10]  Bethany L. Nicholson,et al.  Mathematical Programs with Equilibrium Constraints , 2021, Pyomo — Optimization Modeling in Python.

[11]  Roberto Lucchetti,et al.  Well Posedness, Towards Vector Optimization , 1987 .

[12]  Jean-Paul Penot,et al.  Metrically well-set minimization problems , 1992 .

[13]  P. Loridan.  Well-posedness in Vector Optimization , 1995 .

[14]  Tullio Zolezzi.  Well-Posedness and Optimization under Perturbations , 2001, Ann. Oper. Res..

[15]  X. X. Huang.  Extended Well-Posedness Properties of Vector Optimization Problems , 2000 .

[16]  Ewa M. Bednarczuk.  Well Posedness of Vector Optimization Problems , 1987 .

[17]  R. Tyrrell Rockafellar,et al.  Variational Analysis , 1998, Grundlehren der mathematischen Wissenschaften.

[18]  Julian P. Revalski.  Hadamard and Strong Well-Posedness for Convex Programs , 1997, SIAM J. Optim..

[19]  Sien Deng.  Coercivity properties and well-posedness in vector optimization , 2003, RAIRO Oper. Res..

[20]  T. Zolezzi,et al.  Well-Posed Optimization Problems , 1993 .

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

[22]  A. N. Tikhonov,et al.  On the stability of the functional optimization problem , 1966 .

[23]  J. Aubin,et al.  Applied Nonlinear Analysis , 1984 .

[24]  X. Q. Yang,et al.  Lower-order penalty methods for mathematical programs with complementarity constraints , 2004, Optim. Methods Softw..

[25]  Massimo Furi,et al.  About well-posed optimization problems for functionals in metric spaces , 1970 .

[26]  Diethard Klatte,et al.  On Procedures for Analysing Parametric Optimization Problems , 1982 .

[27]  J. P. Revalski,et al.  Constrained convex optimization problems-well-posedness and stability * , 1994 .

[28]  Xiaoqi Yang,et al.  Characterizing Nonemptiness and Compactness of the Solution Set of a Convex Vector Optimization Problem with Cone Constraints and Applications , 2004 .

[29]  Xiaoqi Yang,et al.  A Unified Augmented Lagrangian Approach to Duality and Exact Penalization , 2003, Math. Oper. Res..

[30]  René Henrion,et al.  Sufficient Conditions for Error Bounds and Applications , 2004 .