On Quadratic Internal Model Principle in Mathematical Programming

Neculai ANDREI
I C I Bucharest
(National Institute for R & D in Informatics)
Center for Advanced Modeling and Optimization
8-10 Averescu Blvd.
011455 Bucharest 1, Romania
e-mail: nandrei@ici.ro

Abstract: We show that for continuous mathematical programming problems every optimization algorithm must encapsulate implicitly or explicitly a quadratic internal model of the problem to be solved which represents the essence of the problem from the view point of the algorithm. Optimization models are coming from the conservation laws. For systems which obey the principle of least action the Noether’s theorem expresses the equivalence between the conservation laws and symmetries. But mathematically, symmetries are expressed by quadratic forms. Therefore, at the heart of every real optimization model is a quadratic form. The quadratic internal model principle says that this quadratic form modified in order to imbed the main ingredients of the optimization algorithm represents the quadratic internal model of the optimization algorithm.

MSC: 49M07, 49M10, 90C06, 65K.

Keywords: Nonlinear optimization, line search methods, Newton system, quadratic programming, Noether theorem.

Neculai Andrei graduated the Faculty of Mathematics, “A.I. Cuza” University in Iasi, and the Faculty of Automatics and Computer Science at “Politehnica” University in Bucharest. He took his PhD degree for his contributions to Digraph Approach of Large-Scale Sparse Linear Dynamic Systems in 1984. He is a senior scientific researcher at National Institute for R&D in Informatics – ICI, since 1973. He is author of 15 books and text books, as well as over 100 published papers and 400 Technical Reports in area of mathematical modeling and optimization. He is the key architect of ALLO language and compiler for modeling in mathematical programming, as well as of some professional packages for linear programming and large-scale unconstrained optimization. His main current scientific interests centre on modeling for mathematical programming and on large-scale non-linear optimization, conjugate gradient, interior point methods, penalty and barrier approaches. Dr. Andrei is a founding member of the Operations Research Society of Romania; founding member of the Center for Advanced Modeling and Optimization; member of the Editorial Board of Computational Optimization and Applications – Springer Verlag (since 1992); member of Computational Optimization and Applications – Optimization Forum; member of the Editorial Board of Journal of Systems Science and Complexity – Science Press – China and Allerton Press – USA (since 2001). He is Editor-in-Chief to the Advanced Modeling and Optimization – An Electronic International Journal – ICI Press (since 1999) and Editor-in-Chief to the Romanian Journal of Informatics and Automatics. Dr. Andrei is an Alexander-von-Humboldt fellow, member ERCIM and member of Academy of Romanian Scientists (since 2001).

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CITE THIS PAPER AS:
Neculai ANDREI, On Quadratic Internal Model Principle in Mathematical Programming, Studies in Informatics and Control, ISSN 1220-1766, vol. 18 (4), pp. 337-348, 2009.