Properties of the Log-Barrier Function on Degenerate Nonlinear Programs

References

  • Adler I., Monteiro R. D. C. Limiting behavior of the affine scaling continuous trajectories for linear programming problems. Math. Programming (1991) 50:29–51CrossrefGoogle Scholar
  • Anitescu M. On the rate of convergence of sequential quadratic programming with non-differentiable exact penalty function in the presence of constraint degeneracy. (1999) . Preprint ANL/MCS-P760-0699, Argonne National Laboratory, Argonne, ILGoogle Scholar
  • El Bakry A. S., Tapia R. A., Tsuchiya T., Zhang Y. On the formulation and theory of Newton interior point methods for nonlinear programming. J. Optim. Theory Appl. (1996) 89(3):507–541CrossrefGoogle Scholar
  • Benchakroun A., Dussault J-P., Mansouri A. A two parameter mixed interior-exterior penalty algorithm. ZOR-Math. Methods Oper. Res. (1995) 41:25–55CrossrefGoogle Scholar
  • Bonnans J-F., Ioffe A. Second-order sufficiency and quadratic growth for nonisolated minima. Math. Oper. Res. (1995) 20(4):801–819LinkGoogle Scholar
  • Byrd R. H., Gilbert J-Ch., Nocedal J. A trust region method based on interior point techniques for nonlinear programming. Math. Programming Ser. A (2000) . 〈 http://dx.doi.org/10.1007/s101070000189Google Scholar
  • Conn A. R., Gould N. I. M., Toint Ph. L. A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization. Numer. Math. (1994) 68:17–33CrossrefGoogle Scholar
  • Conn A. R., Gould N. I. M., Orban D., Toint Ph. L. A primal-dual trust-region algorithm for non-convex nonlinear programming. Math. Programming B (2000) 87(2):215–249CrossrefGoogle Scholar
  • Fiacco A. V., McCormick G. P.Nonlinear Programming: Sequential Unconstrained Minimization Techniques (1968) (John Wiley and Sons, New York) . Reprinted by SIAM Publications, 1990Google Scholar
  • Forsgren A., Gill P. E. Primal-dual interior methods for nonconvex nonlinear programming. SIAM J. Optim. (1998) 8(4):1132–1152CrossrefGoogle Scholar
  • Gauvin J. A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming. Math. Programming (1977) 12:136–138CrossrefGoogle Scholar
  • Gay D. M., Overton M. L., Wright M. H. A primal-dual interior method for nonconvex nonlinear programming. (1997) . Technical Report 97-4-08, Computing Sciences Research, Bell Laboratories, Murray Hill, NJGoogle Scholar
  • Gould N. I. M. On the accurate determination of search directions for simple differentiable penalty functions. IMA J. Numer. Anal. (1986) 6:357–372CrossrefGoogle Scholar
  • Hoffman A. J. On approximate solutions of systems of linear inequalities. J. Res. National Bureau Standards (1952) 49:263–265CrossrefGoogle Scholar
  • Kantorovich L. V., Akilov G. P.Functional Analysis in Normed Spaces (1964) (Pergamon Press, Oxford, U.K.) Google Scholar
  • Kojima M., Mizuno S., Noma T. Limiting behavior of trajectories by a continuation method for monotone complementarity problems. Math. Oper. Res. (1990) 15(4):662–675LinkGoogle Scholar
  • Lootsma F. A. Hessian matrices of penalty functions for solving constrained optimization. Philips Res. Rep. (1969) 24:322–331Google Scholar
  • McCormick G. P., Witzgall C. Logarithmic SUMT limits in convex programming. Math. Programming, Ser. A (2001) 90:113–145CrossrefGoogle Scholar
  • McLinden L. An analogue of Moreau's proximation theorem, with application to the nonlinear complementarity problem. Pacific J. Math. (1980a) 88:101–161CrossrefGoogle Scholar
  • McLinden L., Cottle R. W., Gianessi F., Lions J-L. The complementarity problem for maximal monotone multifunctions. Variational Inequalities and Complementarity Problems (1980b) (John Wiley and Sons, New York) 251–270chap. 17Google Scholar
  • Megiddo N., Megiddo N. Pathways to the optimal set in linear programming. Progress in Mathematical Programming: Interior Point and Related Methods (1989) (Springer Verlag, New York) 131–158CrossrefGoogle Scholar
  • Mifflin R. Convergence bounds for nonlinear programming algorithms. Math. Programming (1975) 8:251–271CrossrefGoogle Scholar
  • Monteiro R. C., Tshuchiya T. Limiting behavior of the derivatives of certain trajectories associated with a monotone horizontal linear complementarity problem. Math. Oper. Res. (1996) 21:793–814LinkGoogle Scholar
  • Monteiro R. C., Zhou F. On the existence and convergence of the central path for convex programming and some duality results. Comput. Optim. Appl. (1998) 10:51–77CrossrefGoogle Scholar
  • Murray W. Analytical expressions for the eigenvalues and eigenvectors of the Hessian matrices of barrier and penalty functions. J. Optim. Theory Appl. (1971) 7:189–196CrossrefGoogle Scholar
  • Nesterov Y., Nemirovskii A.Interior-Point Polynomial Algorithms in Convex Programming (1994) (SIAM Publications, Philadelphia, PA) CrossrefGoogle Scholar
  • Ortega J. M., Rheinboldt W. C.Iterative Solution of Nonlinear Equations in Several Variables (1970) (Academic Press, New York and London) Google Scholar
  • Ralph D., Wright S. J., Ferris M. C., Pang J-S. Superlinear convergence of an interior-point method for monotone variational inequalities. Complementarity and Variational Problems: State of the Art (1997) (SIAM Publications, Philadelphia, PA) 345–385Google Scholar
  • Ralph D., Wright S. J. Superlinear convergence of an interior-point method despite dependent constraints. Math. Oper. Res. (2000) 25(2):179–194LinkGoogle Scholar
  • Villalobos M. C., Tapia R. A., Zhang Y. The sphere of convergence of Newton's method on two equivalent systems from nonlinear programming. (1999) . Technical Report CRPC-TR9915, Department of Computational and Applied Mathematics, Rice University, Houston, TXGoogle Scholar
  • Wright M. H. Interior methods for constrained optimization. Acta Numerica (1992) (Cambridge University Press, Cambridge, U.K.) 341–407CrossrefGoogle Scholar
  • Wright M. H. Some properties of the Hessian of the logarithmic barrier function. Math. Programming (1994) 67:265–295CrossrefGoogle Scholar
  • Wright M. H. Ill-conditioning and computational error in interior methods for nonlinear programming. SIAM J. Optim. (1998) 9:84–111CrossrefGoogle Scholar
  • Wright S. J. Modifying SQP for degenerate problems. (1997) . Preprint ANL/MCS-P699-1097, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL. Revised June 2000Google Scholar
  • Wright S. J. Superlinear convergence of a stabilized SQP method to a degenerate solution. Comput. Optim. Appl. (1998) 11:253–275CrossrefGoogle Scholar
  • Wright S. J. Effects of finite-precision arithmetic on interior-point methods for nonlinear programming. SIAM J. Optim. (2001a) 12(1):36–78CrossrefGoogle Scholar
  • Wright S. J. On the convergence of the Newton/log-barrier method. Math. Programming Ser. A (2001b) 90:71–100CrossrefGoogle Scholar
  • Wright S. J., Jarre F. The role of linear objective functions in barrier methods. Math. Programming, Ser. A (1998) 84:357–373CrossrefGoogle Scholar
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