Generalized proximal point algorithms for multiobjective optimization problems

References

• Matinet , B . 1970 . Regularisation d'inéquations variationelles par approximations successives, Rev. . Francaise Inf. Rech, Oper. , 2 : 154 – 159 .
   Google Scholar
• Rockafellar , RT . 1976 . Monotone operators and the proximal point algorithm . SIAM J. Contr. Optim. , 14 ( 5 ) : 877 – 898 .
   Web of Science ®Google Scholar
• Auslander , A and Haddou , M . 1995 . An interior proximal point method for convex linearly constrained problems and its extension to variational inequalities . Math. Prog. , 71 : 77 – 100 .
   Web of Science ®Google Scholar
• Burachik , RS , Butnariu , D and Iusem , AN . 2000 . Iterative methods for solving stochastic convex feasibility problems and applications . Comput. Optim. Appl. , 15 : 269 – 307 .
   Web of Science ®Google Scholar
• Kamimura , S and Takahashi , W . 2003 . Strong convergence of a proximal-type algorithm in a Banach space . SIAM J. Optim. , 13 ( 3 ) : 938 – 945 .
   Web of Science ®Google Scholar
• Güler , O . 1991 . On the convergence of the proximal point algorithm for convex minimization . SIAM J. Contr. Optim. , 29 : 403 – 419 .
   Web of Science ®Google Scholar
• Pennanen , T . 2002 . Local convergence of the proximal point algorithm and multiplier methods without monotonicity . Math. Oper. Res. , 27 ( 1 ) : 170 – 191 .
   Web of Science ®Google Scholar
• Butnariu , D and Iusem , AN . 1997 . On a proximal point method for convex optimization in Banach space . Numer. Funct. Anal. Optim. , 18 : 723 – 744 .
   Web of Science ®Google Scholar
• Bauschke , HH and Borwein , JM . 1997 . Lengendre function and the method of random Bregman functions . J. Convex Anal. , 4 : 27 – 67 .
   Google Scholar
• Teboulle , M . 1997 . Convergence of proximal-like algorithms . SIAM J. Optim. , 7 : 1069 – 1083 .
   Web of Science ®Google Scholar
• Bregman , LM . 1967 . The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming . USSR Comput. Math. Math. Phys. , 7 : 165 – 186 .
   Google Scholar
• Censor , Y and Lent , A . 1981 . An iterative row-action method for interval convex programming . J. Optim. Theory Appl. , 34 : 321 – 353 .
   Web of Science ®Google Scholar
• Bertsekas , DP . 1982 . Constrained Optimization and Lagrange Multiplier Methods , New York : Academic Press .
   Google Scholar
• Censor , Y and Zenios , SA . 1992 . The proximal minimization algorithm with D-function . J. Optim. Theory Appl. , 73 : 451 – 464 .
   Web of Science ®Google Scholar
• Chen , G and Teboulle , M . 1993 . Convergence analysis of a proximal-like minimization algorithm using Bregman function . SIAM J. Optim. , 3 ( 3 ) : 538 – 543 .
   Web of Science ®Google Scholar
• De Pierro , A and Iusem , AN . 1986 . A relaxed version of Bregman's method for convex programming . J. Optim. Theory Appl. , 51 : 421 – 440 .
   Web of Science ®Google Scholar
• Eckstein , J . 1993 . Nonlinear proximal point algorithms using Bregman function, with applications to convex programming . Math. Oper. Res. , 18 ( 1 ) : 202 – 226 .
   Web of Science ®Google Scholar
• Kiwiel , KC . 1997 . Proximal minimization methods with generalized Bregman functions . SIAM J. Contr. Optim. , 35 : 326 – 349 .
   Web of Science ®Google Scholar
• Solodov , MV and Svaiter , BF . 2000 . Forcing strong convergence of proximal point iterations in a Hilbert space . Math. Prog. , 87 : 189 – 202 .
   Web of Science ®Google Scholar
• Burachik , RS and Iusem , AN . 1998 . A generalized proximal point algorithm for the variational inequality problem in a Hilbert space . SIAM J. Optim. , 8 ( 1 ) : 197 – 216 .
   Web of Science ®Google Scholar
• Burachik , RS and Scheimberg , S . 2001 . A proximal point algorithm for the variational inequality problem in Banach space . SIAM J. Contr. Optim. , 39 ( 5 ) : 1633 – 1649 .
   Web of Science ®Google Scholar
• Chen , GY , Huang , XX and Yang , XQ . 2005 . Vector Optimization: Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems , Vol. 541 , Berlin : Springer-Verlag .
   Google Scholar
• Jahn , J . 2004 . Vector Optimization: Theory, Applications and Extensions , Berlin : Springer-Verlag .
   Google Scholar
• Luc , TD . 1989 . Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems , Vol. 319 , Berlin : Springer-Verlag .
   Google Scholar
• Sawaragi , Y , Nakayama , H and Tanino , T . 1985 . The Theory of Multiobjective Optimization , New York : Academic Press .
   Google Scholar
• Fliege , J . 2006 . An efficient interior-point method for convex multicriteria optimization problems . Math. Oper. Res. , 31 : 825 – 845 .
   Web of Science ®Google Scholar
• Fliege , J and Svaiter , BF . 2000 . Steepest descent methods for multicriteria optimization . Math. Methods Oper. Res. , 51 : 479 – 494 .
   Web of Science ®Google Scholar
• Graña Drummond , LM and Iusem , AN . 2004 . A projected gradient method for vector optimization problems . Comput. Optim. Appl. , 28 : 5 – 30 .
   Web of Science ®Google Scholar
• Bonnel , H , Iusem , AN and Svaiter , BF . 2005 . Proximal methods in vector optimization . SIAM J. Optim. , 15 ( 4 ) : 953 – 970 .
   Web of Science ®Google Scholar
• Ceng , LC and Yao , JC . 2007 . Approximate proximal methods in vector optimization . Eur. J. Oper. Res. , 183 ( 1 ) : 1 – 19 .
   Web of Science ®Google Scholar
• Chen , Z , Xiang , CH , Zhao , KQ and Liu , XW . 2009 . Convergence analysis of Tikhonov-type regularization algorithms for multiobjective optimization problems . Appl. Math. Comput. , 211 : 167 – 172 .
   Web of Science ®Google Scholar
• Chen , Z , Huang , HQ and Zhao , KQ . 2009 . Approximate generalized proximal-type method for convex vector optimization problem in Banach spaces . Comput. Math. Appl. , 57 : 1196 – 1203 .
   Web of Science ®Google Scholar
• Chen , Z and Zhao , KQ . 2009 . A proximal-type method for convex vector optimization problem in Banach spaces . Numer. Funct. Anal. Optim. , 30 : 70 – 81 .
   Web of Science ®Google Scholar
• Ceng , LC , Mordukhovich , BS and Yao , JC . Hybrid approximate proximal method with auxiliary variational inequality for vector optimization . J. Optim. Theory Appl. (accepted) ,
   Google Scholar
• Chuong , TD , Mordukhovich , BS and Yao , JC . Hybrid approximate proximal algorithms for efficient solutions in vector optimization (preprint) ,
   Google Scholar
• T.D. Chuong and J.C. Yao, Viscosity-type approximation method for efficient solutions in vector optimization, Taiwanese J. Math. (2010) (to appear).
   Google Scholar
• Chuong , TD and Yao , JC . 2010 . Generalized Clarke epiderivatives of parametric vector optimization problems . J. Optim. Theory Appl. , (to appear)
   Web of Science ®Google Scholar
• Chuong , TD and Yao , JC . 2009 . Coderivatives of efficient point multifunctions in parametric vector optimization . Taiwanese J. Math. , 13 : 1671 – 1693 .
   Web of Science ®Google Scholar
• Auslander , A and Teboulle , M . 2003 . Asymptotic Cones and Functions in Optimization and Variational Inequalities , Berlin : Springer .
   Google Scholar
• Rockafellar , RT and Wets , JB . 1998 . Variational Analysis , Berlin : Springer .
   Google Scholar
• Flores-Bazan , F . 2002 . Ideal, weakly efficient solutions for vector problems . Math. Prog. , 93 : 453 – 475 .
   Web of Science ®Google Scholar
• Deng , S . 1998 . Characterizations of the nonemptiness and compactness of solution sets in convex vector optimization . J. Optim. Theory Appl. , 96 : 123 – 131 .
   Web of Science ®Google Scholar
• Phelps , RR . 1988 . Convex Functions, Montone Operators and Differentiability, Lecture Notes in Mathematics , Vol. 1364 , Berlin : Springer .
   Google Scholar