Investigating an overdetermined system of linear equations by using convex functions

Year 2017, Volume: 46 Issue: 5, 865 - 874, 01.10.2017

Zlatko Pavic Vedran Novoselac

Abstract

The paper studies the application of convex functions in order to prove the existence of optimal solutions of an overdetermined system of linear
equations. The study approaches the problem by using even convex functions instead of projections. The research also relies on some special
properties of unbounded convex sets, and the lower level sets of continuous functions.

Keywords

overdetermined system, convex function, global minimum

References

• P. Bloomeld, W. Steiger, Least Absolute Deviations: Theory, Applications and Algorithms, Birkhauser Basel, 1983.
• M. Fiedler, J. Nedoma, J. Ramik, J. Rohn, K. Zimmermann, Linear Optimization Problems with Inexact Data, Springer-Verlag US, 2006.
• B. Grünbaum, Convex Polytopes (Second Edition), Springer-Verlag, New York, 2003.
• C. P. Niculescu, L. E. Persson, Convex Functions and Their Applications, Springer Science+ Business Media New York, 2006.
• M. R. Osborne, Finite Algorithms in Optimization and Data Analysis, John Wiley & Sons New York, 1985.
• R. W. Owens, V. P. Sreedharan, Algorithms for solving overdetermined systems of linear equations in the lp-metric, 0 < p < 1, J. Approx. Theory, 24 (1978), 1-17.
• Z. Pavi¢, Extension of Jensen's inequality to ane combinations, J. Inequal. Appl., 2014 (2014), Article ID 298.
• Z. Pavi¢, Improvements of the Hermite-Hadamard inequality, J. Inequal. Appl., 2015 (2015), Article ID 222.
• A. W. Roberts, D. E. Varberg, Convex Functions, Academic Press New York and London, 1973.
• J. A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999.
• C. Udrişte, Convex Functions and Optimization Methods on Riemannian Manifolds, Kluwer Academic Publishers, Dordrecht, 1994.
• G. Williams, Overdetermined systems of linear equations, Amer. Math. Monthly, 97 (1990), 511-513.