Ratio-conjugation tool and its application in linear optimization models†

† This article has been published in Russian in Economics and Mathematical Methods, 2006, 42(2), pp. 60–78 by the publishing house, Nauka, who have given permission to publish it in this journal.

Abstract

The ratio-conjugation-tool (RC) is constructed systematically, and its applicability for analysis of economic models is demonstrated. One of the main results establishes a general formula for RC of aggregative function. This tool is applied to the analysis of linear economic dynamics models of Neumann–Gale type which can be solved by a dynamic programming method: for each direct problem we construct the corresponding dual RC-problem. Moreover, we consider three examples of the most important economic dynamics models and look at these now from a unified general position.

† This article has been published in Russian in Economics and Mathematical Methods, 2006, 42(2), pp. 60–78 by the publishing house, Nauka, who have given permission to publish it in this journal.

Keywords:

• Conjugate function
• Dynamic programming
• Bellman equation
• Monotonicity
• Homogeneity
• Superdifferential
• Conjugate operators
• Stationary optimization models
• 2000 Mathematics Subject Classifications: 49L20
• 90C39
• 90C90
• 91B50
• 91B52

Acknowledgement

The author is thankful to all the participants of theoretical seminar in CEMI RAS held by professors V.I. Danilov and V.M. Polterovich and wishes to specially thank professors A.M. Rubinov and A.A. Shananin for the fruitful discussions and proposals. The author is also greatful to B.R. Frenklin who helped in preparing this article successfully. The author thanks the anonymous referee who helped in including 31, Theorem 3.4 in this article, and finally thanks professor A.A. Shananin, for the direction to focus on the fact that work [11] contains a result close to the Proposition 1 of the present article. This work has the financial support of the Russian Fund for Humanitary Research (grant 03-02-00058) and of the Russian Fund for Basic Research (grant 05-01-00412).

Notes

† This article has been published in Russian in Economics and Mathematical Methods, 2006, 42(2), pp. 60–78 by the publishing house, Nauka, who have given permission to publish it in this journal.

¹ There are several works where some types of conjugate operations different from (2) have been studied, (see, par example, 12,18,22,32).

² Usually the superdifferential is designated by ∂ f (x), see 24, section 23.

³ More precisely, Sion's theorem deals with inf sup and sup inf, but in our case both extrema are attained.

⁴ Term “Bellman equation” was already used in section 3 (see Theorem 2), but there this is “small” (only technological) equation. Full “large” meaning of this term is gaining importance in optimization problems.

⁵ In strong words, the limit function V ⁰ may have a jump on the border of orthant but for functions of class this flaw may be eliminated by the procedure of closure described in detail in 2.