×

Exactness property of the exact absolute value penalty function method for solving convex nondifferentiable interval-valued optimization problems. (English) Zbl 1386.49045

Summary: In the paper, the classical exact absolute value function method is used for solving a nondifferentiable constrained interval-valued optimization problem with both inequality and equality constraints. The property of exactness of the penalization for the exact absolute value penalty function method is analyzed under assumption that the functions constituting the considered nondifferentiable constrained optimization problem with the interval-valued objective function are convex. The conditions guaranteeing the equivalence of the sets of LU-optimal solutions for the original constrained interval-valued extremum problem and for its associated penalized optimization problem with the interval-valued exact absolute value penalty function are given.

MSC:

49M30 Other numerical methods in calculus of variations (MSC2010)
90C25 Convex programming
90C30 Nonlinear programming

Software:

PLCP
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Wu, H.C.: The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur. J. Oper. Res. 176, 46-59 (2007) · Zbl 1137.90712 · doi:10.1016/j.ejor.2005.09.007
[2] Wu, H.C.: Wolfe duality for interval-valued optimization. J. Optim. Theory Appl. 138, 497-509 (2008) · Zbl 1191.90073 · doi:10.1007/s10957-008-9396-0
[3] Ahmad, I., Jayswal, A., Banerjee, J.: On interval-valued optimization problems with generalized invex functions. J. Inequal. Appl. 2013, 313 (2013) · Zbl 1292.90312 · doi:10.1186/1029-242X-2013-313
[4] Jayswal, A., Stancu-Minasian, I., Ahmad, I.: On sufficiency and duality for a class of interval-valued programming problems. Appl. Math. Comput. 218, 4119-4127 (2011) · Zbl 1268.90087
[5] Jayswal, A., Ahmad, I., Banerjee, J.: Nonsmooth interval-valued optimization and saddle-point optimality criteria. Bull. Malays. Math. Sci. Soc. 39, 1391-1411 (2016) · Zbl 1385.90020 · doi:10.1007/s40840-015-0237-7
[6] Sun, Y., Wang, L.: Saddle-point type optimality for interval-valued programming. In: Proceedings of 2nd International Conference on Uncertainty Reasoning and Knowledge Engineering (URKE), Jakarta, Indonesia (2012)
[7] Sun, Y., Wang, L.: Duality theory for interval-valued programming. Adv. Sci. Lett. 7, 643-646 (2012) · doi:10.1166/asl.2012.2694
[8] Sun, Y., Wang, L.: Optimality conditions and duality in nondifferentiable interval-valued programming. J. Ind. Manag. Optim. 9, 131-142 (2013) · Zbl 1263.90122 · doi:10.3934/jimo.2013.9.131
[9] Zhang, J.: Optimality condition and Wolfe duality for invex interval-valued nonlinear programming problems. J. Appl. Math. 2013, Article ID 641345 (2013) · Zbl 1397.90367
[10] Zhou, H.C., Wang, Y.J.: Optimality condition and mixed duality for interval-valued optimization. Fuzzy Inf. Eng. AISC 62, 1315-1323 (2009) · Zbl 1190.90226
[11] Steuer, R.E.: Algorithms for linear programming problems with interval objective function coefficients. Math. Oper. Res. 6, 333-348 (1981) · Zbl 0491.90059 · doi:10.1287/moor.6.3.333
[12] Ishibuchi, H., Tanaka, H.: Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48, 219-225 (1990) · Zbl 0718.90079 · doi:10.1016/0377-2217(90)90375-L
[13] Chanas, S., Kuchta, D.: Multiobjective programming in optimization of interval objective functions-a generalized approach. Eur. J. Oper. Res. 94, 594-598 (1996) · Zbl 1006.90506 · doi:10.1016/0377-2217(95)00055-0
[14] Jiang, C., Han, X., Liu, G.R., Liu, G.P.: A nonlinear interval number programming method for uncertain optimization problems. Eur. J. Oper. Res. 188, 1-13 (2008) · Zbl 1135.90044 · doi:10.1016/j.ejor.2007.03.031
[15] Gabrel, V., Murat, C., Remli, N.: Linear programming with interval right hand sides. Int. Trans. Oper. Res. 17, 397-408 (2010) · Zbl 1187.90188 · doi:10.1111/j.1475-3995.2009.00737.x
[16] Hladík, M.: Optimal value bounds in nonlinear programming with interval data. TOP 19, 93-106 (2011) · Zbl 1225.90127 · doi:10.1007/s11750-009-0099-y
[17] Chalco-Cano, Y., Lodwick, W.A., Rufian-Lizana, A.: Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative. Fuzzy Optim. Decis. Mak. 12, 305-322 (2013) · Zbl 1428.90189 · doi:10.1007/s10700-013-9156-y
[18] Karmakar, S., Bhunia, A.K.: An alternative optimization technique for interval objective constrained optimization problems via multiobjective programming. J. Egypt. Math. Soc. 22, 292-303 (2014) · Zbl 1298.90094 · doi:10.1016/j.joems.2013.07.002
[19] Eremin, I.I.: The penalty method in convex programming. Cybern. Syst. Anal. 3, 53-56 (1967) · Zbl 0155.28405 · doi:10.1007/BF01071708
[20] Zangwill, W.I.: Nonlinear programming via penalty functions. Manag. Sci. 13, 344-358 (1967) · Zbl 0171.18202 · doi:10.1287/mnsc.13.5.344
[21] Antczak, T.: Exact penalty functions method for mathematical programming problems involving invex functions. Eur. J. Oper. Res. 198, 29-36 (2009) · Zbl 1163.90792 · doi:10.1016/j.ejor.2008.07.031
[22] Antczak T.: The exact \[l_1\] l1 penalty function method for constrained nonsmooth invex optimization problems. In: Hömberg, D. Tröltzsch, F. (eds.) System Modeling and Optimization Vol. 391 of the series IFIP Advances in Information and Communication Technology, pp. 461-470 (2013) · Zbl 1266.49024
[23] Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (1991) · Zbl 1140.90040
[24] Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, Inc., Cambridge (1982) · Zbl 0572.90067
[25] Bertsekas, D.P., Koksal, A.E.: Enhanced optimality conditions and exact penalty functions. In: Proceedings of Allerton Conference (2000) · Zbl 0491.90059
[26] Bonnans, J.F., Gilbert, JCh., Lemaréchal, C., Sagastizábal, C.A.: Numerical Optimization. Theoretical and Practical Aspects. Springer, Berlin (2003) · Zbl 1014.65045
[27] Charalambous, C.: A lower bound for the controlling parameters of the exact penalty functions. Math. Program. 15, 278-290 (1978) · Zbl 0395.90071 · doi:10.1007/BF01609033
[28] Di Pillo, G., Grippo, L.: Exact penalty functions in constrained optimization. SIAM J. Control Optim. 27, 1333-1360 (1989) · Zbl 0681.49035 · doi:10.1137/0327068
[29] Fletcher, R.: An exact penalty function for nonlinear programming with inequalities. Math. Program. 5, 129-150 (1973) · Zbl 0278.90063 · doi:10.1007/BF01580117
[30] Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, New York (2000) · Zbl 0905.65002 · doi:10.1002/9781118723203
[31] Janesch, S.M.H., Santos, L.T.: Exact penalty methods with constrained subproblems. Invest. Oper. 7, 55-65 (1997)
[32] Mangasarian, O.L.: Sufficiency of exact penalty minimization. SIAM J. Control Optim. 23, 30-37 (1985) · Zbl 0559.90072 · doi:10.1137/0323003
[33] Mongeau, M., Sartenaer, A.: Automatic decrease of the penalty parameter in exact penalty function methods. Eur. J. Oper. Res. 83, 686-699 (1995) · Zbl 0901.90165 · doi:10.1016/0377-2217(93)E0339-Y
[34] Nocedal, J., Wright, S.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, New York (2006) · Zbl 1104.65059
[35] Peressini, A.L., Sullivan, F.E., Uhl Jr., J.J.: The Mathematics of Nonlinear Programming. Springer, New York (1988) · Zbl 0663.90054 · doi:10.1007/978-1-4612-1025-2
[36] Wang, Z., Liu S.: A new smooth method for the \[l_1\] l1 exact penalty function for inequality constrained optimization. In: IEEE, Computational Science and Optimization (CSO), Third International Joint Conference on Computational Science and Optimization, pp. 110-113 (2010)
[37] Jayswal, A., Banerjee, J.: An exact \[l_1\] l1 penalty approach for interval-valued programming problem. J. Oper. Res. Soc. China 4, 461-481 (2016) · Zbl 1365.90216 · doi:10.1007/s40305-016-0120-8
[38] Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983) · Zbl 0552.65041
[39] Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) · Zbl 0193.18401 · doi:10.1515/9781400873173
[40] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) · Zbl 0582.49001
[41] Chankong, V., Haimes, Y.: Multiobjective Decision Making; Theory and Methodology. North-Holland, New York (1983) · Zbl 0622.90002
[42] Miettinen, K.M.: Nonlinear Multiobjective Optimization. International Series in Operations Research & Management Science, vol. 12. Kluwer Academic Publishers, Boston (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.