Interior Gradient and Epsilon-Subgradient Descent Methods for Constrained Convex Minimization

Published Online:https://doi.org/10.1287/moor.1030.0062

References

  • Auslender A. Numerical methods for nondifferentiable convex optimization. Math. Programming Stud. (1987) 30:102–126CrossrefGoogle Scholar
  • Auslender A., Haddou M. An interior proximal method for convex linearly constrained problems and its extension to variational inequalities. Math. Programming (1995) 71:77–100CrossrefGoogle Scholar
  • Auslender A., Teboulle M., Butanariu D., Censor Y., Reich S. A log-quadratic projection method for convex feasibility problems. Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Vol. 8. Studies in Computational Mathematics (2001a) (Elsevier, Amsterdam, The Netherlands) 1–10CrossrefGoogle Scholar
  • Auslender A., Teboulle M. Entropic proximal decomposition methods for convex programs and variational inequalities. Math. Programming (2001b) 91:33–47CrossrefGoogle Scholar
  • Auslender A., Teboulle M., Daniel P., Gianessi F., Maugeri A. The log-quadratic proximal methodology in convex optimization algorithms and variational inequalities. Equilibrium Problems and Variational Models, Vol. 68, Nonconvex Optimization and Its Applications (2003) (Kluwer Academic Press, Dordrecht, The Netherlands) CrossrefGoogle Scholar
  • Auslender A., Teboulle M., Ben-Tiba S. A logarithmic-quadratic proximal method for variational inequalities. Comput. Optim. Appl. (1999a) 12:31–40CrossrefGoogle Scholar
  • Auslender A., Teboulle M., Ben-Tiba S. Interior proximal and multiplier methods based on second order homogeneous kernels. Math. Oper. Res. (1999b) 24:645–668LinkGoogle Scholar
  • Ben-Tal A., Zibulevsky M. Penalty-barrier methods for convex programming problems. SIAM J. Optim. (1997) 7:347–366CrossrefGoogle Scholar
  • Bertsekas D. P. On the Goldstein-Levitin-Polyak gradient projection method. IEEE Trans. Automatic Control (1976) AC-21:174–183CrossrefGoogle Scholar
  • Bertsekas D. P.Nonlinear Programming (1999) 2nd ed.(Athena Scientific, Belmont, MA) Google Scholar
  • Correa R., Lemarechal C. Convergence of some algorithm for convex programming. Math. Programming (1993) 62:261–275CrossrefGoogle Scholar
  • Eckstein J. Nonlinear proximal point algorithms using Bregman functions,with applications to convex programming. Math. Oper. Res. (1993) 18:202–226LinkGoogle Scholar
  • Eggermont P. P. B. Multiplicatively iterative algorithms for convex programming. Linear Algebra Its Appl. (1990) 130:25–42CrossrefGoogle Scholar
  • Fukushima M. A descent algorithm for nonsmooth convex programming. Math. Programming (1984) 30:163–175CrossrefGoogle Scholar
  • Gaudioso M., Monaco M. F. A bundle type approach to unconstrained minimization of convex nonsmooth functions. Math. Programming (1982) 23:216–226CrossrefGoogle Scholar
  • Güler O. On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. (1991) 29:403–419CrossrefGoogle Scholar
  • Iusem A. N. Interior point multiplicative methods for optimization under positivity constraints. Acta Applicandae Mathematicae (1995) 38:163–184CrossrefGoogle Scholar
  • Iusem A. N., Teboulle M. Convergence rate analysis of nonquadratic proximal and augmented Lagrangian methods for convex and linear programming. Math. Oper. Res. (1995) 20:657–677LinkGoogle Scholar
  • Iusem A. N., Svaiter B., Teboulle M. Entropy-like proximal methods in convex programming. Math. Oper. Res. (1994) 19:790–814LinkGoogle Scholar
  • Iusem A. N., Svaiter B., Teboulle M. Multiplicative interior gradient methods for minimization over the nonnegative orthant. SIAM J. Control Optim. (1996) 34:389–406CrossrefGoogle Scholar
  • Kabbadj M. Methodes proximales entropiques. (1994) . Ph.D. thesis, Université de Montpellier II, FranceGoogle Scholar
  • Kiwiel K. C. An aggregate subgradient method for nonsmooth convex optimization. Math. Programming (1983) 27:320–341CrossrefGoogle Scholar
  • Kiwiel K. C. Proximal minimization methods with generalized Bregman functions. SIAM J. Control Optim. (1997) 35:1142–1168CrossrefGoogle Scholar
  • Kiwiel K. C. Subgradient method with entropic projections for convex nondifferentiable minimization. J. Optim. Theory Appl. (1998) 96:159–173CrossrefGoogle Scholar
  • Kiwiel K. C. A bundle Bregman proximal method for nondifferentiable convex minimization. Math. Programming (1999) 85:241–258CrossrefGoogle Scholar
  • Knopp K.Infinite Sequences and Series (1956) (Dover Publications, Inc., New York) Google Scholar
  • Lemaire B., Penot J. P. The proximal algorithm. International Series of Numerical Mathematics, Vol. 87 (1989) (Birkhäuser Verlag, Basel, Switzerland) 73–87Google Scholar
  • Lemarechal C., Lemarechal C., Mifflin R. Bundle methods in nonsmooth optimization. Nonsmooth Optimization (1978) (Pergamon Press, Oxford, U.K.) Google Scholar
  • Martinet B. Regularisation d'inéquations variationnelles par approximations successive. Revue Française d'Automatique et Informatique Recherche Opérationnelle (1970) 4:154–159Google Scholar
  • Nesterov Y., Nemirovski A.Interior Point Polynomial Algorithms in Convex Programming (1994) (SIAM Publications, Philadelphia, PA) CrossrefGoogle Scholar
  • Polyak B. T. Minimization of unsmooth functionals. USSR Comput. Math. Phys. (1969) 9:14–29CrossrefGoogle Scholar
  • Polyak B. T.Introduction to Optimization (1987) (Optimization Software Inc., New York) Google Scholar
  • Polyak R. A. Nonlinear rescaling vs. smoothing technique in constrained optimization. Math. Programming (2002) 92:197–235CrossrefGoogle Scholar
  • Polyak R. A., Teboulle M. Nonlinear rescaling and proximal-like methods in convex optimization. Math. Programming (1997) 76:265–284CrossrefGoogle Scholar
  • Robinson S. M. Linear convergence of epsilon subgradients methods for a class of convex functions. Math. Programming (1999) 86:41–50CrossrefGoogle Scholar
  • Rockafellar R. T.Convex Analysis (1970) (Princeton University Press, Princeton, NJ) CrossrefGoogle Scholar
  • Rockafellar R. T. Monotone operators and the proximal point algorithm. SIAM J. Control Optim. (1976) 14:877–898CrossrefGoogle Scholar
  • Silva P. J., Eckstein J., Humes C. Rescaling and stepsize selection in proximal methods using separable generalized distances. SIAM J. Optim. (2001) 12:238–261CrossrefGoogle Scholar
  • Teboulle M. Entropic proximal mappings with application to nonlinear programming. Math. Oper. Res. (1992) 17:670–690LinkGoogle Scholar
  • Teboulle M. Convergence of proximal-like algorithms. SIAM J. Optim. (1997) 7:1069–1083CrossrefGoogle Scholar
  • Tseng P., Bertsekas D. On the convergence of the exponential multiplier method for convex programming. Math. Programming (1993) 60:1–19CrossrefGoogle Scholar
  • Yamashuta N., Kanzow C., Morimoto T., Fukushima M. An infeasible interior proximal methods for convex programming problems with linear constraints. J. Nonlinear Convex Anal. (2001) 2:139–156Google Scholar
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