Abstract
Parametric linear systems with interval coefficients arise in many practical applications in engineering and other related areas. In this paper, we derive a connection between parametric systems and interval linear programming, where interval parametric linear systems can be used to describe the weak optimal solution set. Then, we discuss the branch-and-prune framework for generating an approximation of the weak feasible set of such systems via a union of interval boxes. We propose two new pruning conditions that can be used to test infeasibility of interval boxes within the framework, based on interval linear algebra and on the geometry of zonotopes. Furthermore, we also design a condition for pruning the interval boxes by feasibility, which allows us to generate an inner approximation of the weak feasible set. Computational experiments illustrate competitiveness of the proposed pruning tests in comparison with other available pruning conditions. The experiments indicate that the test based on interval linear algebra usually produces the least number of boxes, while the zonotope-based test is right behind and runs several times faster.
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References
Akhmerov R (2005) Interval-Affine Gaussian algorithm for constrained systems. Reliab Comput 11:323–341. https://doi.org/10.1007/s11155-005-0040-5
Alefeld G, Kreinovich V, Mayer G (1998) The shape of the solution set for systems of interval linear equations with dependent coefficients. Mathematische Nachrichten 192(1):23–36. https://doi.org/10.1002/mana.19981920103
Alefeld G, Kreinovich V, Mayer G (2003) On the solution sets of particular classes of linear interval systems. J Comput Appl Math 152(1):1–15. https://doi.org/10.1016/S0377-0427(02)00693-3
Beeck H (1978) Linear programming with inexact data. Tech. Rep. TUM-ISU-7830, Technische Universität München. Institut für Statistik und Unternehmensforschung
Černý M, Rada M (2012) Polynomial time construction of ellipsoidal approximations of zonotopes given by generator descriptions. LNCS. Springer, Berlin, pp 156–163. https://doi.org/10.1007/978-3-642-29952-0_19
Chaiyakan S, Thipwiwatpotjana P (2021) Bounds on mean absolute deviation portfolios under interval-valued expected future asset returns. Comput Manag Sci 18:195–212. https://doi.org/10.1007/s10287-021-00392-x
D’Ambrosio C, Gentili M, Cerulli R (2020) The optimal value range problem for the interval (immune) transportation problem. Omega 95(102):059. https://doi.org/10.1016/j.omega.2019.04.002
Garajová E, Hladík M (2019) On the optimal solution set in interval linear programming. Comput Optim Appl 72(1):269–292. https://doi.org/10.1007/s10589-018-0029-8
Garajová E, Hladík M, Rada M (2017). In: Sforza A, Sterle C (eds) On the properties of interval linear programs with a fixed coefficient matrix, vol 217. Springer, Cham, pp 393–401. https://doi.org/10.1007/978-3-319-67308-0_40
Hladík M (2007) Solution set characterization of linear interval systems with a specific dependence structure. Reliab Comput 13(4):361–374. https://doi.org/10.1007/s11155-007-9033-x
Hladík M (2012) Enclosures for the solution set of parametric interval linear systems. Int J Appl Math Comput Sci 22(3):561–574. https://doi.org/10.2478/v10006-012-0043-4
Hladík M (2020) P-completeness of testing solutions of parametric interval linear systems. In: Ceberio M, Kreinovich V (eds) Decision making under constraints, studies in systems, decision and control. Springer, Berlin, pp 117–123. https://doi.org/10.1007/978-3-030-40814-5_14
Hladík M, Skalna I (2019) Relations between various methods for solving linear interval and parametric equations. Linear Algebra Appl 574:1–21. https://doi.org/10.1016/j.laa.2019.03.019
Horáček J (2019) Interval toolbox - LIME. https://kam.mff.cuni.cz/~horacek/lime.html
Jansson C (1991) Interval linear systems with symmetric matrices, skew-symmetric matrices and dependencies in the right hand side. Computing 46(3):265–274. https://doi.org/10.1007/BF02238302
Jasiński M, Pownuk A (2000) Modelling of heat transfer in biological tissue by interval FEM. pp 551–558
Jaulin L, Kieffer M, Didrit O et al (2001) Applied interval analysis. Springer, London. https://doi.org/10.1007/978-1-4471-0249-6
Kolev LV (2006) Improvement of a direct method for outer solution of linear parametric systems. Reliab Comput 12(3):193–202. https://doi.org/10.1007/s11155-006-7218-3
Kolev LV (2014) Parameterized solution of linear interval parametric systems. Appl Math Comput 246:229–246. https://doi.org/10.1016/j.amc.2014.08.037
Kulpa Z, Pownuk A, Skalna I (1998) Analysis of linear mechanical structures with uncertainties by means of interval methods. Comput Assist Mech Eng Sci 5(4):443–477
Mayer G (2012) An oettli-prager-like theorem for the symmetric solution set and for related solution sets. SIAM J Matrix Anal Appl 33(3):979–999. https://doi.org/10.1137/120867895
Mohammadi M, Gentili M (2021) The outcome range problem in interval linear programming. Comput Op Res 129(105):160. https://doi.org/10.1016/j.cor.2020.105160
Moore RE, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. Soc Indus Appl Math. https://doi.org/10.1137/1:9780898717716
Muhanna R, Mullen R (2001) Uncertainty in mechanics problems-interval-based approach. J Eng Mech 127:557–566. https://doi.org/10.1061/(ASCE)0733-9399(2001)127:6(557)
Narkawicz A, Muñoz C (2014) A formally verified generic branching algorithm for global optimization. In: Cohen E, Rybalchenko A (eds) Verified software: theories, tools, experiments. Springer, Berlin, pp 326–343
Nazari V, Notash L (2015) Motion analysis of manipulators with uncertainty in kinematic parameters. J Mech Robot 8(2):021014. https://doi.org/10.1115/1.4031657
Oettli W, Prager W (1964) Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides. Numer Math 6(1):405–409. https://doi.org/10.1007/BF01386090
Okumura K (1993) An application of interval operation to electric network analysis. Bullet Japan Soc Indus Appl Math 3(2):115–127. https://doi.org/10.11540/bjsiam.3.2_115
Popova ED (2004) Parametric interval linear solver. Numer Algorithms 37(1–4):345–356. https://doi.org/10.1023/B:NUMA.0000049480.57066.fa
Popova ED (2009) Explicit characterization of a class of parametric solution sets. Comptes Rendus de L’Academie Bulgare des Sciences 62(10):1207–1216
Popova ED (2012) Explicit description of AE solution sets for parametric linear systems. SIAM J Matrix Anal Appl 33(4):1172–1189. https://doi.org/10.1137/120870359
Popova ED (2015) Solvability of parametric interval linear systems of equations and inequalities. SIAM J Matrix Anal Appl 36(2):615–633. https://doi.org/10.1137/140966459
Popova ED, Krämer W (2008) Visualizing parametric solution sets. Bit Numer Math 48(1):95–115. https://doi.org/10.1007/s10543-007-0159-3
Rakpenthai C, Uatrongjit S, Premrudeepreechacharn S (2012) State estimation of power system considering network parameter uncertainty based on parametric interval linear systems. IEEE Trans Power Syst 27(1):305–313. https://doi.org/10.1109/TPWRS.2011.2162859
Rohn J, Kreinovich V (1995) Computing exact componentwise bounds on solutions of linear systems with interval data is NP-hard. SIAM J Matrix Anal Appl 16(2):415–420. https://doi.org/10.1137/S0895479893251198
Rump SM (2010) Verification methods: rigorous results using floating-point arithmetic. Acta Numer 19:287–449. https://doi.org/10.1017/S096249291000005X
Sharaya IA, Shary SP (2011) Tolerable solution set for interval linear systems with constraints on coefficients. Reliab Comput 15:345–357
Skalna I (2007) Evolutionary optimization method for approximating the solution set hull of parametric linear systems. In: Boyanov T, Dimova S, Georgiev K et al (eds) Numerical methods and applications. Springer, Berlin, pp 361–368. https://doi.org/10.1007/978-3-540-70942-8_43
Skalna I (2018) Parametric interval algebraic systems, studies in computational intelligence, vol 766. Springer International Publishing, Cham. https://doi.org/10.1007/978-3-319-75187-0
Skalna I, Hladík M (2017) A new method for computing a p-solution to parametric interval linear systems with affine-linear and nonlinear dependencies. BIT Numer Math 57(4):1109–1136. https://doi.org/10.1007/s10543-017-0679-4
Skalna I, Hladík M (2019) Direct and iterative methods for interval parametric algebraic systems producing parametric solutions. Numer Linear Algebra Appl 26(3):1–24. https://doi.org/10.1002/nla.2229
Vavasis SA (1991) Nonlinear optimization: complexity issues. Oxford University Press, New York
Zimmer M, Krämer W, Popova ED (2012) Solvers for the verified solution of parametric linear systems. Computing 94(2–4):109–123. https://doi.org/10.1007/s00607-011-0170-z
Funding
M. Rada, E. Garajová and M. Hladík were supported by the Czech Science Foundation under Grant P403-20-17529S. E. Garajová and M. Hladík were also supported by the Charles University project GA UK No. 180420.
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Communicated by Dario Pacciarelli.
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Rada, M., Garajová, E., Horáček, J. et al. New pruning tests for the branch-and-prune framework for interval parametric linear systems. Soft Comput 27, 12897–12912 (2023). https://doi.org/10.1007/s00500-022-06971-7
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DOI: https://doi.org/10.1007/s00500-022-06971-7