A proximal point algorithm with a ϕ-divergence for quasiconvex programming

References

• Arrow , KJ and Enthoven , AC . 1961 . Quasi-concave programming . Econometrica , 29 : 779 – 800 .
   Web of Science ®Google Scholar
• Ashton , DJ and Atkins , DR . 1979 . Multicriteria programming for financial planning . J. Oper. Res. Soc. , 30 : 259 – 270 .
   Web of Science ®Google Scholar
• Attouch , H and Teboulle , M . 2004 . Regularized Lotka–Volterra dynamical system as continuous proximal-like method in optimization . J. Optimiz. Theory Appl. , 121 : 541 – 580 .
   Web of Science ®Google Scholar
• Bajona-Xandri , C and Martinez-Legaz , JE . 1999 . Lower subdifferentiability in minimax fractional programming . Optimization , 45 : 1 – 12 .
   Google Scholar
• Barron , EN and Liu , W . 1997 . Calculus of variation in L ∞ . Appl. Math. Optimiz. , 35 : 237 – 263 .
   Web of Science ®Google Scholar
• Bazaraa , MS , Sherali , HD and Shetty , CM . 1993 . Nonlinear Programming: Theory and Algorithms, , 2nd Edition , New York : John Wiley and Sons Inc .
   Google Scholar
• Boncompte , M and Martinez-Legaz , JE . 1991 . Fractional programming by lower subdifferentiability techniques . J. Optimiz. Theory App. , 68 : 95 – 116 .
   Web of Science ®Google Scholar
• Censor , Y and Zenios , A . 1992 . Proximal minimization algorithms with D-functions . J. Optimiz. Theory Appl. , 73 ( 3 ) : 451 – 464 .
   Web of Science ®Google Scholar
• Chen , G and Teboulle , M . 1993 . Convergence analysis of the proximal-like minimization algorithm using Bregman functions . SIAM J. Optimiz. , 3 : 538 – 543 .
   Web of Science ®Google Scholar
• Cruz Neto , JX , Ferreira , OP , Iusem , AN and Monteiro , RDC . 2007 . Dual convergence of the proximal point method with Bregman distances for linear programming . Optim. Method. Softw. , 22 : 339 – 360 .
   Web of Science ®Google Scholar
• Csiszar , I . 1967 . Information-type measures of difference of probability distributions and indirect observations . Studia Scientiarum Mathematicarum Hungarica , 2 : 299 – 318 .
   Google Scholar
• Eckstein , J . 1993 . Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming . Math. Oper. Res. , 18 : 202 – 226 .
   Web of Science ®Google Scholar
• Eggermont , PPB . 1990 . Multiplicative iterative algorithms for convex programming . Lin. Alg. Appl. , 130 : 25 – 42 .
   Web of Science ®Google Scholar
• Ferreira , OP and Oliveira , PR . 2002 . Proximal point algorithm on Riemannian manifolds . Optimization , 51 : 257 – 270 .
   Web of Science ®Google Scholar
• Gromicho , J . 1998 . Quasiconvex Optimization and Location Theory , Dordrecht, , The Netherlands : Kluwer Academic Publishers .
   Google Scholar
• Güler , O . 1991 . On the convergence of the proximal point algorithm for convex minimization . SIAM J. Cont. Optimiz. , 29 : 403 – 419 .
   Web of Science ®Google Scholar
• Iusem , AN , Svaiter , BF and Cruz Neto , JX . 1999 . Generalized proximal point methods and Cauchy trajectories in Riemannian manifolds . SIAM J. Cont. Optimiz. , 37 : 566 – 588 .
   Web of Science ®Google Scholar
• Iusem , AN , Svaiter , BF and Teboulle , M . 1994 . Entropy like proximal methods in convex programming . Math. Oper. Res. , 19 : 790 – 814 .
   Web of Science ®Google Scholar
• Iusem , AN and Teboulle , M . 1993 . On the convergence rate of entropic proximal optimization algorithms . Comput. Appl. Math. , 12 : 153 – 168 .
   Google Scholar
• Iusem , AN and Teboulle , M . 1995 . Convergence rate analysis of nonquadratic proximal methods for convex and linear programming . Math. Oper. Res. , 20 : 657 – 677 .
   Web of Science ®Google Scholar
• Kiwiel , KC . 1997 . Proximal minimization methods with generalized Bregman functions . SIAM J. Cont. Optimiz. , 35 : 1142 – 1168 .
   Web of Science ®Google Scholar
• Lemaire , B . 1992 . “ Advances in optimization ” . In Proc. Lambrecht 1991, Lecture Notes in Economics and Mathematical Systems , Vol. 378 , 39 – 51 . New York : Springer-Verlag .
   Google Scholar
• Martinet , B . 1978 . Perturbation des méthodes d'opimisation . Application, R.A.I.R.O, Analyse Numérique , 12 : 153 – 171 .
   Google Scholar
• Mas-Colell , A , Whinston , MD and Green , JR . 1995 . Microeconomic Theory , New York, NY, , USA : Oxford University Press .
   Google Scholar
• Quiroz , EAP and Oliveira , PR . 2009 . Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds . J. Convex Anal. , 16 : 49 – 70 .
   Web of Science ®Google Scholar
• Rockafellar , RT . 1976 . Augmented Lagrangians and applications of proximal point algorithm in convex programming . Math. Oper. Res. , 1 : 97 – 116 .
   Google Scholar
• Rockafellar , RT . 1976 . Monotone operators and the proximal point algorithm . SIAM J. Control Optimiz. , 14 : 877 – 898 .
   Web of Science ®Google Scholar
• Teboulle , M . 1992 . Entropic proximal mappings with applications to nonlinear programming . Math. Oper. Res. , 17 : 670 – 690 .
   Web of Science ®Google Scholar
• Teboulle , M . 1997 . Convergence of proximal-like algorithms . SIAM J. Optimiz. , 7 : 1069 – 1083 .
   Web of Science ®Google Scholar
• Tseng , P and Bertsekas , D . 1993 . On the convergence of the exponential multiplier method for convex programming . Math. Program. , 60 : 1 – 19 .
   Web of Science ®Google Scholar