IJPAM: Volume 59, No. 2 (2010)

MIXED E-, E $^{\mathbf{\ast }}$-CONVEX FUNCTIONS

Emre Tokgöz
$^1$Department of Computer Science
University of Oklahoma
Norman, OK, 73019, USA
e-mail: Emre.Tokgoz-1@ou.edu

Abstract.Many minimization problems include functions with integer variables or a combination of integer and real variables. Tokgöz [#!2!#] defined mixed T and T$^{\ast }$ convex functions by using L and L$^{\natural }$ convex function definitions and introduced their corresponding mixed Hessian matrices for quadratic L and L$^{\natural }$ convex functions. In this paper, similar to the extension of L and L$^{\natural }$ convex functions to the mixed T and T$^{\ast }$-convex functions, the domain of M and M$^{\natural }$ convex functions are extended to include real variables in the domain. Mixed Hessian matrices are defined for quadratic mixed E and E$^{\ast }$ convex functions with properties similar to those of the Hessian matrix for real variables.

Received: December 17, 2009

AMS Subject Classification: 26B25

Key Words and Phrases: Hessian matrix, optimization, discrete convex function, real convex function

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 59
Issue: 2