Hacettepe Journal of Mathematics and Statistics Volume 46 (5) (2017), 865 874 Investigating an overdetermined system of linear equations by using convex functions Zlatko Pavi¢ ∗ y and Vedran Novoselac z Abstract The paper studies the application of convex functions in order to prove the existence of optimal solutions of an overdetermined system of lin- ear equations. The study approaches the problem by using even convex functions instead of projections. The research also relies on some spe- cial properties of unbounded convex sets, and the lower level sets of continuous functions. Keywords: overdetermined system, convex function, global minimum. 2000 AMS Classication: 15A06, 26B25. Received : 30.08.2016 Accepted : 19.12.2016 Doi : 10.15672/ HJMS.2017.423 1. Introduction We consider a system of m linear equations with n unknowns over the eld of real numbers given by a11x1 + ::: + a1nxn = b1 . (1.1) . .. : am1x1+ ::: + amnxn = bm Including the matrices 2 a11 : : : a1n 3 2 x1 3 2 b1 3 . (1.2) 6 . .. 7 6 . 7 6 . 7 A = 4 . 5 ; x = 4 . 5 ; b = 4 . 5 ; am1 : : : amn xn bm the given system gets the matrix form (1.3) Ax = b: ∗Department of Mathematics, Mechanical Engineering Faculty in Slavonski Brod, University of Osijek, Croatia, Email:
[email protected] yCorresponding Author. zDepartment of Mathematics, Mechanical Engineering Faculty in Slavonski Brod, University of Osijek, Croatia, Email:
[email protected] 866 n m Identifying the matrix A with a linear operator from R to R , the column matrix x n m with a vector of R , and the column matrices Ax and b with vectors of R , the given system takes the operator form.