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Proceedings of the International Conference in Memoriam Gyula Farkas August 23–26, 2005, Cluj-Napoca

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Proceedings of the International Conference in Memoriam Gyula Farkas August 23–26, 2005, Cluj-Napoca Proceedings of the International Conference In Memoriam Gyula Farkas August 23–26, 2005, Cluj-Napoca Proceedings of the International Conference In Memoriam Gyula Farkas August 23–26, 2005 Cluj-Napoca Edited by Z. Kása, G. Kassay, J. Kolumbán Cluj University Press 2006 This conference was dedicated to the memory of Gyula Farkas (1847–1930), the famous professor in Mathematics and Physics of the University of Kolozsvár/Cluj between 1887–1915. ORGANIZERS Babe³-Bolyai University, Cluj-Napoca Hungarian Operational Research Society, Budapest Operations Research Committee of the Hungarian Academy of Sciences Sapientia University, Cluj-Napoca Gyula Farkas Association for Mathematics and Informatics, Cluj-Napoca Hungarian Technical Sciences Society of Transylvania, Cluj-Napoca Transylvanian Museum Association, Cluj-Napoca SCIENTIFIC ORGANIZING COMMITTEE Wolfgang BRECKNER (Cluj, Romania), Tibor CSENDES (Szeged, Hungary), Zoltán GÁBOS (Cluj, Romania), József KOLUMBÁN (Cluj, Romania), Sándor KOMLÓSI (Pécs, Hungary), Katalin MARTINÁS (Budapest, Hungary), Petru T. MOCANU (Cluj, Romania), Zsolt PÁLES (Debrecen, Hungary), Tamás RAPCSÁK (Budapest, Hungary), Paul SZILÁGYI (Cluj, Romania), Béla VIZVÁRI (Budapest, Hungary) LOCAL ORGANIZING COMMITTEE Gábor KASSAY (Cluj, Romania), Zoltán KÁSA (Cluj, Romania), Gábor KÖLLŐ (Cluj, Romania), Lehel KOVÁCS (Cluj, Romania), Marian MUREŞAN (Cluj, Romania), László NAGY (Cluj, Romania), Ferenc SZENKOVITS (Cluj, Romania) INVITED SPEAKERS Katalin MARTINÁS, Boris S. MORDUKHOVICH, András PRÉKOPA, Constantin ZĂLINESCU, Gert WANKA, László NAGY, Zoltán GÁBOS http://math.ubbcluj.ro/˜farkas Contents SCIENTIFIC PAPERS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 Linear Inequalities, Duality Theorems and their Financial Ap- plications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 András PRÉKOPA Composed convex programming: duality and Farkas-type results 22 Radu Ioan BOŢ, Ioan Bogdan HODREA, Gert WANKA Brézis–Haraux-type approximation of the range of a monotone operator composed with a linear mapping : : : : : : : : : : : : : : : : : : : : : : : 36 Radu Ioan BOŢ, Sorin-Mihai GRAD, Gert WANKA A Little Theory for the Control of an Assembly Robot Using Farkas Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 50 Zsolt ROBOTKA, Béla VIZVÁRI Invariant cones and polyhedra for dynamical systems : : : : : : : : : : : 65 Zoltán HORVÁTH Investigations of Gyula Farkas in the Fields of Electrodynamics and Relativity Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 75 Zoltán GÁBOS, László NAGY Proof of the existence of the integrating factor based on the Farkas Lemma : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 85 Katalin MARTINÁS, Fabian BODOKY, Ildikó BRODSZKY HISTORY AND EDUCATION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 95 Gyula Farkas and the Franz Joseph University from Cluj/Ko- lozsvár : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 96 György GAAL The Development of the Cluj/Kolozsvár School of Mathematics (A hundred years ago, even more. ) : : : : : : : : : : : : : : : : : : : : : : : : : : 111 József KOLUMBÁN Gyula Farkas as a Bolyai Researcher : : : : : : : : : : : : : : : : : : : : : : : : : : : : 132 Róbert OLÁH-GÁL Gyula Farkas and the Mechanical Principle of Fourier : : : : : : : : : : : 136 Ferenc SZENKOVITS 5 6 Contents Computer Assisted Teaching of Operations Research at the Uni- versity of Kaposvár : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 145 Eleonóra STETTNER, Gyöngyi BÁNKUTI List of participants : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 152 The talks of the conference : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 154 Scientific Papers Linear Inequalities, Duality Theorems and their Financial Applications András PRÉKOPA? RUTCOR, Rutgers Center for Operations Research 640 Bartholomew Road, Piscataway, NJ 08854-8003 and Department of Operations Research L. Eötvös University, Budapest, Hungary Abstract. The purpose of this paper is to commemorate the eminent Hungarian mathe- matician and physicist, Gyula Farkas (1847–1930) on the occasion of the 75th anniversary of his death. First we quote from letters of Farkas to show how he contributed to the develop- ment of the world famous Hungarian mathematical scholarship, not only by his own scientific achievements, but also by his organizational activities. Next we show how linear inequality theorems find one step application in financial theory and practice. Finally, we establish a du- ality relationship in connection with multiobjective optimization and probabilistic constrained stochastic programming. AMS 2000 subject classifications: 90-03, 90B99, 90C99 Key words and phrases: Gyula Farkas, linear inequality, duality theory 1 About Farkas as a Human Being It has been shown in a number of publications (see, e.g., Prékopa 1978, 1980, Gábor, 1990, Martinás and Brodszky, 2002) that Farkas contributed fundamen- tal results to both mathematics and physics. Less known are the facts, however, that he was a human being with noble personal qualities and a very efficient organizer. He was a somewhat reserved person but he thought highly of profes- sional and human qualities and helped a lot those whom he considered worthy of it. He did not seek cheap popularity and it was one of the reasons that he was highly respected, inside and outside the university, which he was able to use favorably in university administrative matters. His words had been decisive. The picture that we can form of his character becomes more complete if we quote from his obituary that appeared after his death in the newspaper of “Az Est”: “His colleagues, who new him more closely, enthusiastically admired him, his students adored him”. He succeeded to secure employments as professors at the University of Kolozs- vár, for Lajos Schlesinger in 1897, Lipót Fejér in 1905, Frigyes Riesz in 1911 and Alfréd Haar in 1912, who later on became of paramount importance in the development of the 20th century Hungarian mathematical school. ? email: [email protected] 8 Linear Inequalities, Duality Theorems and Applications 9 Below we quote from Farkas' letters to Lipót Fejér. After he succeeded to hire Fejér to the University of Kolozsvár, on July 9, 1905: “I learn from your letter, dear doctor, that you are spending your time in the family circle... ...please give my regards to your family members, too. Who thinks fondly about our past and future cooperation, Gyula Farkas” On October 3, 1911, while he prepares the position for Riesz, at the University of Kolozsvár: “The Committee accepted my proposal.. concerning Frigyes Riesz”. On November 4, 1911 he is already working on getting a position for Haar: “I am no longer afraid of loosing Haar to abroad, once he comes home”. 2 Financial Applications of Linear Inequality Theorems The Capital Asset Pricing Model (CAPM) establishes a regression relationship between the rate of return Ri on an asset i, and the market return RM which can be defined, e.g., by the use of the Dow Jones index. It is given by Ri = ai + biRM + i; i = 1; : : : ; m; where i, i = 1; : : : ; m are independent random variables with E(i) = 0, i = 1; : : : ; m. In this equation the values Ri; RM are also random variables, RM is independent of i, i = 1; : : : ; m. The model was formulated by Treynor (1961) and Sharpe (1963, 1964). Arbitrage pricing theory (APT), formulated by Ross in 1976, is more general than CAPM because it assumes that the rate of return on asset i is a linear function of more than one factors. The basic equation used by APT is the regression equation Ri = ai + bi1X1 + : : : + binXn + i; (1) where i, i = 1; : : : ; m are independent random variables; Xi, i = 1; : : : ; m are also random variables, independent of i, i = 1; : : : ; m, E(i) = E(Xi) = 0, i = 1; : : : ; m, and ai; bi1; : : : ; bin, i = 1; : : : ; m are constants. We would like to check, if it is possible to create a portfolio out of the assets i = 1; : : : ; m in such a way that we buy or sell an amount jwij of asset i = 1; : : : ; m and m m wi ≤ 0; Riwi > 0: (2) Xi=1 Xi=1 10 András Prékopa If such w1; : : : ; wm numbers exist, then arbitrage exists because without any positive investment we obtain positive return. The possibility of arbitrage is m excluded if for any wi; i = 1; : : : ; m for which we have (approximately) wii = i=1 0, the following condition holds: the relations P m wi ≤ 0 Xi=1 m wibij = 0; j = 1; : : : ; n (3) Xi=1 imply that m wiai ≤ 0: (4) Xi=1 Note that ai = E(Ri); i = 1; : : : ; m. By Farkas' theorem (1901) the above relationship guarantees the existence of real numbers λ0 ≥ 0, λ1; : : : ; λm such that E(Ri) = λ0 + λ1bi1 + : : : + λnbin; i = 1; : : : ; m: (5) The number λ0 belongs to the first relation in (3), and can be identified as the risk free return Rf . The λ1; : : : ; λn are pricing the sensitivity coefficients in (1). They are also called risk premiums. Note that the inference from (3) and (4) to (5) is mathematically exact but before we came to that we had assumed that the multipliers w1; : : : ; wm eliminate the randomness from the portfolio, i.e., m wii = 0. i=1 P What is today called the main arbitrage pricing theorem is formulated
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