On some properties and an application of the logarithmic barrier method

We analyze the logarithmic barrier method for nonsmooth convex optimization in the setting of point-to-set theory. This general framework allows us to both extend and include classical results. We also propose an application for finding efficient points of nonsmooth constrained convex vector-valued problems.

[1]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[2]  G. Sonnevend.  An "analytical centre" for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming , 1986 .

[3]  R. Cominetti.  Nonlinear Averages and Convergence of Penalty Trajectories in Convex Programming , 1999 .

[4]  Nimrod Megiddo,et al.  Boundary Behavior of Interior Point Algorithms in Linear Programming , 1989, Math. Oper. Res..

[5]  Jörg Fliege,et al.  Steepest descent methods for multicriteria optimization , 2000, Math. Methods Oper. Res..

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

[7]  Renato D. C. Monteiro,et al.  Interior path following primal-dual algorithms. part I: Linear programming , 1989, Math. Program..

[8]  Alfred Auslender.  Penalty and Barrier Methods: A Unified Framework , 1999, SIAM J. Optim..

[9]  L. M. Graña Drummond,et al.  THE CENTRAL PATH IN SMOOTH CONVEX SEMIDEFINITE PROGRAMS , 2002 .

[10]  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..

[11]  R. Sundaram.  A First Course in Optimization Theory: Optimization in ℝ n , 1996 .

[12]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[13]  JAHN Johannes.  Scalarization in vector optimization , 1984 .

[14]  Roberto Cominetti,et al.  Asymptotic Analysis for Penalty and Barrier Methods in Convex and Linear Programming , 1997, Math. Oper. Res..

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

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

[17]  C. Tammer,et al.  Theory of Vector Optimization , 2003 .

[18]  R. C. Monteiro,et al.  Interior path following primal-dual algorithms , 1988 .

[19]  Etienne de Klerk,et al.  On the Convergence of the Central Path in Semidefinite Optimization , 2002, SIAM J. Optim..

[20]  S. Sinha.  A Duality Theorem for Nonlinear Programming , 1966 .

[21]  Nelson Maculan,et al.  On the choice of parameters for the weighting method in vector optimization , 2007, Math. Program..

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

[23]  Alfredo N. Iusem,et al.  A Projected Gradient Method for Vector Optimization Problems , 2004, Comput. Optim. Appl..

[24]  Clóvis C. Gonzaga,et al.  Examples of ill-behaved central paths in convex optimization , 2005, Math. Program..

[25]  B. Svaiter,et al.  A steepest descent method for vector optimization , 2005 .

[26]  Kaisa Miettinen,et al.  Nonlinear multiobjective optimization , 1998, International series in operations research and management science.

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

[28]  D. T. Luc.  Scalarization of vector optimization problems , 1987 .

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

[30]  P. Wolfe.  A duality theorem for non-linear programming , 1961 .