A modified Quasi-Newton method for vector optimization problem

In this paper, existence of critical point and weak efficient point of vector optimization problem is studied. A sequence of points in n-dimension is generated using positive definite matrices like Quasi-Newton method. It is proved that accumulation points of this sequence are critical points or weak efficient points under different conditions. An algorithm is provided in this context. This method is free from any kind of priori chosen weighting factors or any other form of a priori ranking or ordering information for objective functions. Also, this method does not depend upon initial point. The algorithm is verified in numerical examples.

[1]  Gabriele Eichfelder,et al.  Scalarizations for adaptively solving multi-objective optimization problems , 2009, Comput. Optim. Appl..

[2]  Ziga Povalej,et al.  Quasi-Newton's method for multiobjective optimization , 2014, J. Comput. Appl. Math..

[3]  Margaret M. Wiecek,et al.  Generating epsilon-efficient solutions in multiobjective programming , 2007, Eur. J. Oper. Res..

[4]  Bing Liang,et al.  Trust region methods for solving multiobjective optimisation , 2013, Optim. Methods Softw..

[5]  T. Q. Phong,et al.  Scalarizing Functions for Generating the Weakly Efficient Solution Set in Convex Multiobjective Problems , 2005, SIAM J. Optim..

[6]  Johannes Jahn,et al.  Vector optimization - theory, applications, and extensions , 2004 .

[7]  Jörg Fliege,et al.  Newton's Method for Multiobjective Optimization , 2009, SIAM J. Optim..

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

[9]  Alfredo N. Iusem,et al.  Proximal Methods in Vector Optimization , 2005, SIAM J. Optim..

[10]  Jen-Chih Yao,et al.  Steepest descent methods for critical points in vector optimization problems , 2012 .

[11]  Nelson Maculan,et al.  On the choice of parameters for the weighting method in vector optimization , 2008, Math. Program..

[12]  Mark Goh,et al.  Quasi-Newton methods for solving multiobjective optimization , 2011, Oper. Res. Lett..

[13]  John E. Dennis,et al.  Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems , 1998, SIAM J. Optim..

[14]  Joydeep Dutta,et al.  A new scalarization and numerical method for constructing the weak Pareto front of multi-objective optimization problems , 2011 .

[15]  A. M. Geoffrion.  Proper efficiency and the theory of vector maximization , 1968 .

[16]  R. K. Ursem.  Multi-objective Optimization using Evolutionary Algorithms , 2009 .

[17]  E. Polak,et al.  On Multicriteria Optimization , 1976 .

[18]  B. A. Ghaznavi-ghosoni,et al.  Scalarization for characterization of approximate strong/weak/proper efficiency in multi-objective optimization , 2013 .

[19]  A. Iusem,et al.  Full convergence of the steepest descent method with inexact line searches , 1995 .

[20]  G. W. Evans,et al.  An Overview of Techniques for Solving Multiobjective Mathematical Programs , 1984 .

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

[22]  P. Gordan.  Ueber die Auflösung linearer Gleichungen mit reellen Coefficienten , 1873 .

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

[24]  Shouyang Wang,et al.  ε-approximate solutions in multiobjective optimization , 1998 .

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

[26]  Enrico Miglierina,et al.  Box-constrained multi-objective optimization: A gradient-like method without "a priori" scalarization , 2008, Eur. J. Oper. Res..

[27]  P. Serafini,et al.  Scalarizing vector optimization problems , 1984 .

[28]  Jen-Chih Yao,et al.  Approximate proximal methods in vector optimization , 2007, Eur. J. Oper. Res..

[29]  Boris S. Mordukhovich,et al.  Hybrid approximate proximal algorithms for efficient solutions in vector optimization , 2011 .

[30]  Shapour Azarm,et al.  On improving normal boundary intersection method for generation of Pareto frontier , 2012 .