Properties of the Log-Barrier Function on Degenerate Nonlinear Programs

We examine the sequence of local minimizers of the log-barrier function for a nonlinear program near a solution at which second-order sufficient conditions and the Mangasarian-Fromovitz constraint qualification are satisfied, but the active constraint gradients are not necessarily linearly independent. When a strict complementarity condition is satisfied, we show uniqueness of the local minimizer of the barrier function in the vicinity of the nonlinear program solution, and we obtain a semiexplicit characterization of this point. When strict complementarity does not hold, we obtain several other interesting characterizations, in particular, an estimate of the distance between the minimizers of the barrier function and the nonlinear program in terms of the barrier parameter, and a result about the direction of approach of the sequence of minimizers of the barrier function to the nonlinear programming solution.

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

[2]  L. Kantorovich,et al.  Functional analysis in normed spaces , 1952 .

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

[4]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[5]  W. Murray,et al.  Analytical expressions for the eigenvalues and eigenvectors of the Hessian matrices of barrier and penalty functions , 1971 .

[6]  Robert Mifflin.  Convergence bounds for nonlinear programming algorithms , 1975, Math. Program..

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

[8]  L. McLinden.  An analogue of Moreau's proximation theorem, with application to the nonlinear complementarity problem. , 1980 .

[9]  Lamberto Cesari,et al.  Optimization-Theory And Applications , 1983 .

[10]  N. Gould.  On the Accurate Determination of Search Directions for Simple Differentiable Penalty Functions , 1986 .

[11]  N. Megiddo.  Pathways to the optimal set in linear programming , 1989 .

[12]  Shinji Mizuno,et al.  Limiting Behavior of Trajectories Generated by a Continuation Method for Monotone Complementarity Problems , 1990, Math. Oper. Res..

[13]  Renato D. C. Monteiro,et al.  Limiting behavior of the affine scaling continuous trajectories for linear programming problems , 1991, Math. Program..

[14]  Margaret H. Wright,et al.  Interior methods for constrained optimization , 1992, Acta Numerica.

[15]  P. Toint,et al.  A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization , 1994 .

[16]  Margaret H. Wright,et al.  Some properties of the Hessian of the logarithmic barrier function , 1994, Math. Program..

[17]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

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

[19]  Jean-Pierre Dussault,et al.  A two parameter mixed interior-exterior penalty algorithm , 1995, Math. Methods Oper. Res..

[20]  R. Monteiro,et al.  Limiting behavior of the derivatives of certain trajectories associated with a monotone horizontal linear complementarity problem , 1996 .

[21]  T. Tsuchiya,et al.  On the formulation and theory of the Newton interior-point method for nonlinear programming , 1996 .

[22]  Stephen J. Wright,et al.  Superlinear convergence of an interior-point method for monotone variational inequalities , 1996 .

[23]  Renato D. C. Monteiro,et al.  On the Existence and Convergence of the Central Path for Convex Programming and Some Duality Results , 1998, Comput. Optim. Appl..

[24]  Michael L. Overton,et al.  A Primal-dual Interior Method for Nonconvex Nonlinear Programming , 1998 .

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

[26]  Anders Forsgren,et al.  Primal-Dual Interior Methods for Nonconvex Nonlinear Programming , 1998, SIAM J. Optim..

[27]  Margaret H. Wright,et al.  Ill-Conditioning and Computational Error in Interior Methods for Nonlinear Programming , 1998, SIAM J. Optim..

[28]  Stephen J. Wright,et al.  The role of linear objective functions in barrier methods , 1999, Math. Program..

[29]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[30]  Jorge Nocedal,et al.  A trust region method based on interior point techniques for nonlinear programming , 2000, Math. Program..

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

[32]  P. Toint,et al.  A primal-dual algorithm for minimizing a non-convex function subject to bound and linear equality constraints , 2000 .

[33]  Nicholas I. M. Gould,et al.  A primal-dual trust-region algorithm for non-convex nonlinear programming , 2000, Math. Program..

[34]  Stephen J. Wright.  On the convergence of the Newton/log-barrier method , 2001, Math. Program..

[35]  Stephen J. Wright.  Effects of Finite-Precision Arithmetic on Interior-Point Methods for Nonlinear Programming , 2001, SIAM J. Optim..

[36]  Garth P. McCormick,et al.  Logarithmic SUMT limits in convex programming , 2001, Math. Program..

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

[38]  Mihai Anitescu,et al.  On the rate of convergence of sequential quadratic programming with nondifferentiable exact penalty function in the presence of constraint degeneracy , 2002, Math. Program..

[39]  R. Tapia,et al.  Sphere of Convergence of Newton's Method on Two Equivalent Systems from Nonlinear Programming , 2004 .