|
Article information
2017 , Volume 22, ¹ 2, p.67-84
Vorontsova E.A.
Linear tolerance problem for input-output models with interval data
The paper considers economic input-output models proposed by W. Leontief. Inputoutput models are subject to uncertainty. The values of the technical coefficients of input-output models are usually evaluated with interval uncertainty. The final demand vector is also not precisely known and became an interval vector. Solution of linear tolerance problem (LTP) for input-output models is useful in predicting how various industrial sectors of the national (or regional) economy respond to changes in economic activity. For the interval linear system 𝐴𝑥 = 𝑏, the LTP requires inner evaluation of the tolerable solution set formed by all point vectors 𝑥 such that the product 𝐴𝑥 remains within interval vector 𝑏 for all possible point matrix 𝐴 within interval matrix 𝐴. A method based on the S.P. Shary’s recognizing functional (RF) of the tolerable solution set is applied to the problem of recognizing the solvability (emptyness or nonemptyness of the solution set) of the LTP for input-output models. A key step in the RF method is to solve the non-smooth concave maximization problem. The separating planes method (SPM) with additional clipping is proposed for the approximate numerical solution for different types of non-smooth optimization problem with convex structure. The latter problem can be reformulated as the computation of the convex conjugate functional value at the origin. SPM with clippings is a new effective black-box optimization method. The applications presented in this paper show how forecasting the development of regional economy on the basis of input-output tables can be done by means of RF method, SPM with clippings and N.Z. Shor’s r-algorithm. Results of numerical experiments are presented in the test case of economic modelling for Primorsky Krai region (Russian Federation). The results of computational experiments have shown the effectiveness of SMP with clippings and have confirmed the advantages of the RF method compared to other methods for solution of LTP problems.
[full text]
Keywords: interval system of linear algebraic equations, linear interval tolerance problem, input-output model, recognizing functional, separating planes methods, nonsmooth optimization, regional economy
Author(s):
Vorontsova Evgeniya Alexeevna
PhD.
Position: Senior Fellow
Office: Far Eastern Federal University
Address: 690950, Russia, Vladivostok, 8, Suhanova
E-mail: vorontsovaea@gmail.co
References:
[1] Voschinin, A.P. Data analysis under uncertainty - intervals and/or randomness? // Workshop on Interval Mathematics and Constraint Propagation Methods, Novosibirsk, 21-22 June 2004. Proceedings of the International Conference on Computational Mathematics ICCM- 2004. Workshops. Eds.: Yu.I. Shokin, A.M. Fedotov, S.P. Kovalyov, Yu.I. Molorodov, A.L. Semenov, S.P. Shary. Novosibirsk: ICM&MG Publisher; 2004: 147–158. (In Russ.) (In Russ.)
[2] Leontief, W. Studies in the structure of the american economy: Theoretical and empirical explorations in input-output analysis. New York: Oxford Univ. Press; 1953: 561.
[3] Jerrell, M.E. Applications of interval computations to regional economic input-output models. Applications of Interval Computations. Eds. R. B. Kearfott, V. Kreinovich. Kluver; 1996: 133–143.
[4] Rohn, J. Input-output model with interval data. Econometrica. 1980; (48):767–769.
[5] Kearfott, B., Nakao, M., Neumaier, A., Rump, S., Shary, S.P., Van Hentenryck, P. Standardized notation in interval analysis. Computational Technologies. 2010; 15(1):7–13.
[6] Shary, S.P. Solvability of interval linear equations and data analysis under uncertainty. Automation and Remote Control. 2012; 73(2):310–322.
[7] Shary, S.P., Sharaya, I.A. Recognizing solvability of interval equations and its application to data analysis. Computational Technologies. 2013; 18(3):80–109. (In Russ.)
[8] Vorontsova, E.A. A projective separating plane method with additional clipping for nonsmooth optimization. WSEAS Transactions on Mathematics. 2014; (13):115–121.
[9] Vorontsova, E.A., Nurminski, E.A. Synthesis of cutting and separating planes in a nonsmooth optimization method. Cybernetics and Systems Analysis. 2015; 51(4):619–631.
[10] Alefeld, G., Herzberger, J. Introduction to Interval Computation. New York: Academic Press; 1983: 333.
[11] Kalmykov, S.A., Shokin, Yu.I., Yuldashev, Z.Kh. Metody interval'nogo analiza [Methods of interval analysis]. Novosibirsk: Nauka; 1986: 224. (In Russ.)
[12] Shary, S.P. Konechnomernyy interval'nyy analiz [Finite-dimensional interval analysis]. Novosibirsk: XYZ Press; 2016: 617. (In Russ.) Available at: http://www.nsc.ru/interval/Library/InteBooks
[13] Neumaier, A. Interval methods for systems of equations. Cambridge: Cambridge Univ. Press; 1990: 255.
[14] Neumaier, A. Tolerance analysis with interval arithmetic. Freiburger Intervall-Berichte. 1986; (86/9):5–19.
[15] Rohn, J. Input-output planning with inexact data. Freiburger Intervall-Berichte. 1978; (9/78):1–16.
[16] Rohn, J. Inner solutions of linear interval systems. Series: Interval Mathematics. Lecture Notes in Computer Science. New York: Springer Verlag; 1986; (212): 157– 158.
[17] Khlebalin, N.A. Analiticheskiy metod sinteza regulyatorov v usloviyakh neopredelennosti parametrov ob"ekta [An analytical controller design under parameter uncertainty]. Saratov: Saratovskiy Politekhnicheskiy Institut; 1981:107–123. (In Russ.)
[18] Shary, S.P. Solving the linear interval tolerance problem. Mathematics and Computers in Simulation. 1995; (39):53–85.
[19] Nurminski, E.A. Separating plane algorithms for convex optimization. Mathematical Programming. 1997; (76):373–391.
[20] Rockafellar, R.T. Convex analysis. Princeton, New Jersey: Princeton Univ. Press; 1970: 451.
[21] Fourer, R., Gay, D.M., Kernighan, B.W. AMPL. A Modeling Language for Mathematical Programming. 2nd ed. Canada: Thomson Learning Academic Resource Center; 2003: 517.
[22] NEOS Server: State-of-the-Art Solvers for Numerical Optimization. Available at: http://neos-server.org/neos/
[23] Berghen, F.V. Optimization algorithm for non-linear, constrained, derivativefree optimization of continuous, high computing load, noisy objective functions: Technical report. IRIDIA, Univ. of Brussels, Belgium. May 2004. Available at: http://www.applied-mathematics.net
[24] Lancelot: a Fortran Package for Large-Scale Nonlinear Optimization (Release A). Available at: http://www.numerical.rl.ac.uk/lancelot/manual.html
[25] Shor, N.Z., Zhurbenko, N.G. A minimization method using the operation of space dilation in the direction of the difference of two successive gradients. Cybernetics. 1971; (7):450–459.
[26] Shor, N.Z., Zhurbenko, N.G., Likhovid, A.P., Stetsyuk, P.I. Algorithms of Nondifferentiable Optimization: Development and Application. Cybernetics and Systems Analysis. 2003; 39(4):537–548.
[27] Miller, R.E., Blair, P.D. Input—output analysis: Foundation and Extensions. Cambridge Univ. Press; 2009: 750.
[28] Mashunin, Yu.K., Mashunin, I.A. Forecasting the development of regional economy on the basis of input — output tables. Economy of Region. 2014; (2):276–289. (In Russ.)
[29] Primorskiy Kray. Sotsial'no-ekonomicheskie pokazateli: Statisticheskiy ezhegodnik [Socio-economic indicators: Statistical Yearbook]. Vladivostok: Primoskstat; 2014: 361. (In Russ.)
[30] Granberg, A.G. Osnovy regional'noy ekonomiki [The basics of regional economy]. Moscow: VshE; 2003: 495. (In Russ.)
[31] Sharaya, I.A. Tolerable solution set as a projection of the convex polyhedron. Computational Technologies. 2007; 12(6):124–137. (In Russ.)
[32] Dolan, E., More, J. Benchmarking optimization software with performance profiles. Mathematical Programming. 2002; (91):201–213.
[33] Conn, A.R., Gould, N.I.M., Toint, P.L. Numerical experiments with the LANCELOT package (Release A) for large-scale nonlinear optimization. Mathematical Programming. 1996; (73):73–110.
Bibliography link:
Vorontsova E.A. Linear tolerance problem for input-output models with interval data // Computational technologies. 2017. V. 22. ¹ 2. P. 67-84
|
