Acceleration procedure for special classes of multi-extremal problems. (English) Zbl 1432.90142
Summary: We suggest an approach to solve special classes of multi-extreme problems to optimize the combination (e.g., sum, product) of several functions, under the assumption that the effective algorithms to optimize each of this item are known. The algorithm proposed is iterative. It realizes one of the idea of the branch-and-bound method and consists in successive correcting of the low and the upper bounds of optimal value of objective functions. In each iteration, the total area of the considered region that may contain the image optimal point, decreases at least twice. Various techniques that accelerate the process of finding solutions are discussed.
References:
| [1] | Antoshchenkova, I.V., Bykadorov, I.A.: Monopolistic competition model: the impact of technological innovation on equilibrium and social optimality. Autom. Remote Control 78, 537-556 (2017) · Zbl 1309.91055 · doi:10.1134/S0005117917030134 |
| [2] | Avriel, M.; Dewert, WE; Schaible, S.; Ziemba, WT; Schaible, S. (ed.); Ziemba, WT (ed.), Introduction in concave and generalized concave functions, 21-50 (1981), New York · Zbl 0539.90087 |
| [3] | Bykadorov, I.: Monopolistic competition model with different technological innovation and consumer utility levels. In: CEUR Workshop Proceeding, vol. 1987, pp 108-114 (2017) |
| [4] | Bykadorov, I.: Solution of special classes of multi-extremal problems. In: CEUR Workshop Proceeding, vol. 1987, pp. 115-122 (2017) |
| [5] | Bykadorov, I., Ellero, A., Funari, S., Kokovin, S., Pudova, M.: Chain store against manufacturers: regulation can mitigate market distortion. Lect. Notes Comput. Sci. 9869, 480-493 (2016) · Zbl 1407.91112 · doi:10.1007/978-3-319-44914-2_38 |
| [6] | Bykadorov, I., Ellero, A., Funari, S., Moretti, E.: Dinkelbach approach to solving a class of fractional optimal control problems. J. Optim. Theory Appl. 142(1), 55-66 (2009) · Zbl 1175.90385 · doi:10.1007/s10957-009-9540-5 |
| [7] | Bykadorov, I., Ellero, A., Moretti, E.: Minimization of communication expenditure for seasonal products. RAIRO Oper. Res. 36(2), 109-127 (2002) · Zbl 1062.90022 · doi:10.1051/ro:2002012 |
| [8] | Bykadorov, I., Ellero, A., Moretti, E., Vianello, S.: The role of retailer’s performance in optimal wholesale price discount policies. Eur. J. Oper. Res. 194(2), 538-550 (2009) · Zbl 1154.90356 · doi:10.1016/j.ejor.2007.12.008 |
| [9] | Bykadorov, I., Gorn, A., Kokovin, S., Zhelobodko, E.: Why are losses from trade unlikely? Econ. Lett. 129, 35-38 (2015) · Zbl 1321.91044 · doi:10.1016/j.econlet.2015.02.003 |
| [10] | Bykadorov, I., Kokovin, S.: Can a larger market foster R&D under monopolistic competition with variable mark-ups? Res. Econ. 71(4), 663-674 (2017) · doi:10.1016/j.rie.2017.10.006 |
| [11] | Bykadorov, I.A., Kokovin, S.G., Zhelobodko, E.V.: Product diversity in a vertical distribution channel under monopolistic competition. Autom. Remote Control 75(8), 1503-1524 (2014) · Zbl 1297.91069 · doi:10.1134/S0005117914080141 |
| [12] | Censor, Y., Segal, A.: Algorithms for the quasiconvex feasibility problem. J. Comput. Appl. Math. 185(1), 34-50 (2006) · Zbl 1080.65047 · doi:10.1016/j.cam.2005.01.026 |
| [13] | Charnes, A., Cooper, W.W.: Programming with linear fractional functionals. Naval Res. Logist. Q. 9(3-4), 181-186 (1962) · Zbl 0127.36901 · doi:10.1002/nav.3800090303 |
| [14] | Choo, E.U., Atkins, D.R.: Bicriteria linear fractional programming. J. Optim. Theory Appl. 36(2), 203-220 (1982) · Zbl 0452.90076 · doi:10.1007/BF00933830 |
| [15] | Diewert, W.E., Avriel, M., Zang, I.: Nine kinds of quasiconcavity and concavity. J. Econ. Theory 25(3), 397-420 (1981) · Zbl 0483.26007 · doi:10.1016/0022-0531(81)90039-9 |
| [16] | Falk, J.E., Palocsay, S.M.: Optimizing the Sum of Linear Fractional Functions, Collection: Recent Advances in Global Optimization, pp. 221-258. Princeton University Press, Princeton (1992) |
| [17] | Flores-Bazán, F., Flores-Bazán, F., Vera, C.: Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivatives. J. Global Optim. 63(1), 99-123 (2015) · Zbl 1327.90306 · doi:10.1007/s10898-015-0267-6 |
| [18] | Gruzdeva, T., Strekalovsky, A.: An approach to fractional programming via D.C. constraints problem: local search. Lect. Notes Comput. Sci. 9869, 404-417 (2016) · Zbl 1392.90112 · doi:10.1007/978-3-319-44914-2_32 |
| [19] | Gruzdeva, T., Strekalovsky, A.: On a solution of fractional programs via D.C. optimization theory. In: CEUR Workshop Proceeding, vol. 1987, pp. 246-252 (2017) |
| [20] | Hirche, J.: Zur Extremwertannahme und Dualitat bei Optimierungsproblemen mit Linearem und Gebrochen-Linearem Ziel funktionsanteil. Zeitschrift fur angewandte Mathematik und Mechanik 55, 184-185 (1975) · Zbl 0299.90026 · doi:10.1002/zamm.19750550310 |
| [21] | Horst, R., Pardalos, P.M. (eds.): Handbook of Global Optimization, p. 880. Kluwer Academic Publishers, Dordrecht (1995) · Zbl 0805.00009 |
| [22] | Horst, R., Tuy, H.: Global Optimization. Deterministic Approach, p. 730. Springer, Berlin (1996) · Zbl 0867.90105 · doi:10.1007/978-3-662-03199-5 |
| [23] | Konnov, I.V.: On convergence properties of a subgradient method. Optim. Methods Softw. 18(1), 53-62 (2003) · Zbl 1062.90048 · doi:10.1080/1055678031000111236 |
| [24] | Kuno, T.: A branch-and-bound algorithm for maximizing the sum of several linear ratios. J. Global Optim. 22(1-4), 155-174 (2002) · Zbl 1045.90071 · doi:10.1023/A:1013807129844 |
| [25] | Polak, B.T.: Minimization of unsmooth functionals. USSR Comput. Math. Math. Phys. 9(3), 14-29 (1969) · Zbl 0229.65056 · doi:10.1016/0041-5553(69)90061-5 |
| [26] | Schaible, S.: A note on the sum of a linear and linear-fractional function. Naval Res. Log. 24, 691-693 (1977) · Zbl 0377.90086 · doi:10.1002/nav.3800240416 |
| [27] | Stancu-Minasian, I.M.: Fractional Programming: Theory, Methods and Applications, p. 432. Kluwer Academic Publisher, Dordrecht (1997) · Zbl 0899.90155 · doi:10.1007/978-94-009-0035-6 |
| [28] | Teterev, A.G.: On generalization of linear and piecewise-linear programming. Matekon 63(3), 246-259 (1970) |
| [29] | Warburton, A.R.: Parametric solution of Bicriterion linear fractional programming. Oper. Res. 33(1), 74-84 (1985) · Zbl 0569.90088 · doi:10.1287/opre.33.1.74 |
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