NONLINEAR PROGRAMS WITH UNBOUNDED LAGRANGE MULTIPLIER SETS

We investigate nonlinear programs that have a nonempty but possibly unbounded Lagrange multiplier set and that satisfy the quadraticgrowth condition. We show that such programs can be transformed, by relaxing the constraints and adding a linear penalty term to the objective function, into equivalent nonlinear programs that have diierentiable data and a bounded Lagrange multiplier set and that satisfy the quadratic growth condition. As a result we can deene, for this type of problem, algorithms that are linearly convergent, using only rst-order information, and superlinearly convergent.

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

[2]  A. Ioffe.  Necessary and Sufficient Conditions for a Local Minimum. 3: Second Order Conditions and Augmented Duality , 1979 .

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

[4]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[5]  R. Fletcher.  Practical Methods of Optimization , 1988 .

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

[7]  P. Toint,et al.  Lancelot: A FORTRAN Package for Large-Scale Nonlinear Optimization (Release A) , 1992 .

[8]  Alexander D. Ioffe,et al.  On Sensitivity Analysis of Nonlinear Programs in Banach Spaces: The Approach via Composite Unconstrained Optimization , 1994, SIAM J. Optim..

[9]  J. F. Bonnans,et al.  Local analysis of Newton-type methods for variational inequalities and nonlinear programming , 1994 .

[10]  J. Frédéric Bonnans,et al.  Second-order Sufficiency and Quadratic Growth for Nonisolated Minima , 1995, Math. Oper. Res..

[11]  Zhi-Quan Luo,et al.  Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints , 1996, Math. Program..

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

[13]  Stephen J. Wright.  Superlinear Convergence of a Stabilized SQP Method to a Degenerate Solution , 1998, Comput. Optim. Appl..

[14]  Michal Kočvara,et al.  Nonsmooth approach to optimization problems with equilibrium constraints : theory, applications, and numerical results , 1998 .

[15]  H. Wolkowicz,et al.  SQ2P, Sequential Quadratic Constrained Quadratic Programming , 1998 .

[16]  William W. Hager,et al.  Stabilized Sequential Quadratic Programming , 1999, Comput. Optim. Appl..

[17]  William W. Hager,et al.  Stability in the presence of degeneracy and error estimation , 1999, Math. Program..

[18]  Andreas Fischer,et al.  Modified Wilson's Method for Nonlinear Programs with Nonunique Multipliers , 1999, Math. Oper. Res..

[19]  L. Petzold,et al.  Model reduction for chemical kinetics: an optimization approach , 1999 .

[20]  R. Vanderbei.  LOQO:an interior point code for quadratic programming , 1999 .

[21]  Mihai Anitescu,et al.  Degenerate Nonlinear Programming with a Quadratic Growth Condition , 1999, SIAM J. Optim..

[22]  Stephen J. Wright,et al.  Superlinear Convergence of an Interior-Point Method Despite Dependent Constraints , 2000, Math. Oper. Res..

[23]  Stephen J. Wright.  Modifying SQP for Degenerate Problems , 2002, SIAM J. Optim..

[24]  Sven Leyffer,et al.  Nonlinear programming without a penalty function , 2002, Math. Program..

[25]  Mihai Anitescu,et al.  A Superlinearly Convergent Sequential Quadratically Constrained Quadratic Programming Algorithm for Degenerate Nonlinear Programming , 2002, SIAM J. Optim..

[26]  Michael A. Saunders,et al.  USER’S GUIDE FOR SNOPT 5.3: A FORTRAN PACKAGE FOR LARGE-SCALE NONLINEAR PROGRAMMING , 2002 .

[27]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .