Characterizing Global Minimizers of the Difference of Two Positive Valued Affine Increasing and Co-radiant Functions

Document Type: Research Paper

Authors

1 Mathematics, Mathematics and Computer Science, Shahid Bahonar University of Kerman, Kerman, Iran

2 Shahid Bahonar University of Kerman

10.22072/wala.2019.94381.1198

Abstract

‎Many optimization problems can be reduced to a
problem with an increasing and co-radiant objective function by a suitable transformation of variables. Functions, which are increasing and co-radiant, have found many applications in microeconomic analysis. In this paper, the abstract convexity of positive valued affine increasing and co-radiant (ICR) functions are discussed. Moreover, the basic properties of this class of functions such as support set, subdifferential and maximal elements of support set are characterized. Finally, as an application, necessary and sufficient conditions for the global minimum of the difference of two strictly positive valued affine ICR functions are presented.

Keywords


[1] M.H.  Daryaei and  H. Mohebi, Global minimization of the difference of strictly non-positive valued affine 

     ICR functions, J. Glob Optim., 61 (2005), 311-323.

[2] A.R. Doagooei and H. Mohebi, Monotonic analysis over ordered topological vector spaces IV, J. Glob. 

     Optim., 45} (2009), 355-369.

[3] A.R. Doagooei and  H. Mohebi, Optimization of the difference of ICR functions, Nonlinear Anal. Theory 

     Methods Appl., 71 (2009), 4493-4499.

[4] J. Dutta, J.E. Martinez-Legaz and A.M. Rubinov, Monotonic analysis over ordered topological vector  

     spaces:  I, Optim., 53 (2006), 129-146.

[5] J. Dutta, J.E. Martinez-Legaz and  A.M. Rubinov, Monotonic analysis over ordered topological vector 

     spaces: III, J. Convex Anal., 15 (2006), 581-592.

[6] B.M. Glover  and   A.M. Rubinov, Toland-Singer formula cannot distinguish a global minimizer from a 

     choice of stationary points, Numer. Funct. Anal. Optim., 20 (1999), 99-119.

[7] A.M. Rubinov, Abstract Convexity and Global Optimization, Kluwer AcademicPublishers, Boston, Dordrecht,  

     London, 2000.

[8] A.M. Rubinov and B.M. Glover, Increasing convex-along-rays functions with applications to global 

     optimization, J. Optim. Theory Appl.,  102(3) (1999), 615-642.


[9] I. Singer, Abstract Convex Analysis, Wiley-Interscience, New York, 1997.


[10] H. Tuy, Monotonic optimization: Problems and solution approach,
  

       SIAM J. Optim., 11(2) (2000), 464-494.