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Sufficient efficiency conditions for vector ratio problem on the second-order jet bundle. (English) Zbl 1242.49022

Summary: Motivated by its possible applications in mechanics and mechanical engineering, in our previous published work A. Pitea and M. Postolache [“Minimization of vectors of curvilinear functionals on the second order jet bundle. Necessary conditions,” Optimization Letters, vol. 6, no. 3, p. 459–470, 2011], we initiated an optimization theory for second-order jet bundles. We considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problems, subject to PDE and/or PDI constraints. Within this framework, we introduced necessary optimality conditions. As natural continuation of these results, the present work introduces a study of sufficient efficiency conditions.

MSC:

49J40 Variational inequalities
49Q20 Variational problems in a geometric measure-theoretic setting
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[1] A. Chinchuluun and P. M. Pardalos, “A survey of recent developments in multiobjective optimization,” Annals of Operations Research, vol. 154, pp. 29-50, 2007. · Zbl 1146.90060 · doi:10.1007/s10479-007-0186-0
[2] A. Chinchuluun, P. M. Pardalos, A. Migdalas, and L. Pitsoulis, Pareto Optimality, Game Theory and Equilibria, vol. 17 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2008, Edited by Altannar Chinchuluun, Panos M. Pardalos, Athanasios Migdalas and Leonidas Pitsoulis. · Zbl 1143.91004 · doi:10.1007/978-0-387-77247-9
[3] Z. A. Khan and M. A. Hanson, “On ratio invexity in mathematical programming,” Journal of Mathematical Analysis and Applications, vol. 205, no. 2, pp. 330-336, 1997. · Zbl 0872.90094 · doi:10.1006/jmaa.1997.5180
[4] L. V. Reddy and R. N. Mukherjee, “Some results on mathematical programming with generalized ratio invexity,” Journal of Mathematical Analysis and Applications, vol. 240, no. 2, pp. 299-310, 1999. · Zbl 0946.90089 · doi:10.1006/jmaa.1999.6334
[5] M. A. Hanson, “A duality theorem in non-linear programming with non-linear constraints,” The Australian Journal of Statistics, vol. 3, pp. 64-72, 1961. · Zbl 0102.15601 · doi:10.1111/j.1467-842X.1961.tb00310.x
[6] C. Singh and M. A. Hanson, “Multiobjective fractional programming duality theory,” Naval Research Logistics, vol. 38, no. 6, pp. 925-933, 1991. · Zbl 0749.90068 · doi:10.1002/nav.3800380610
[7] V. Jeyakumar and B. Mond, “On generalised convex mathematical programming,” Australian Mathematical Society Journal Series B, vol. 34, no. 1, pp. 43-53, 1992. · Zbl 0773.90061 · doi:10.1017/S0334270000007372
[8] Z.-A. Liang, H.-X. Huang, and P. M. Pardalos, “Efficiency conditions and duality for a class of multiobjective fractional programming problems,” Journal of Global Optimization, vol. 27, no. 4, pp. 447-471, 2003. · Zbl 1106.90066 · doi:10.1023/A:1026041403408
[9] A. Pitea and M. Postolache, “Minimization of vectors of curvilinear functionals on the second order jet bundle. Necessary conditions,” Optimization Letters, vol. 6, no. 3, pp. 459-470, 2011. · Zbl 1280.90109
[10] A. Pitea, Integral geometry and PDE constrained optimization problems, Ph.D. thesis, University “Politehnica” of Bucharest, 2008.
[11] C. Udri\cste and M. Postolache, Atlas of Magnetic Geometric Dynamics, vol. 3, Geometry Balkan Press, Bucharest, Romania, 2001. · Zbl 1018.70001
[12] C. Udri\cste, O. Dogaru, and I. \cTevy, “Null Lagrangian forms and Euler-Lagrange PDEs,” Journal of Advanced Mathematical Studies, vol. 1, no. 1-2, pp. 143-156, 2008. · Zbl 1169.49022
[13] A. Pitea and M. Postolache, “Duality theorems for a new class of multitime multiobjective variational problems,” Journal of Global Optimization. In press. · Zbl 1250.65084 · doi:10.1007/s10898-011-9740-z
[14] A. Pitea and C. Udri\cste, “Sufficient efficiency conditions for a minimizing fractional program,” “Politehnica” University of Bucharest. Scientific Bulletin, vol. 72, no. 2, pp. 13-20, 2010. · Zbl 1299.49024
[15] \cS. Mititelu, “Extensions in invexity theory,” Journal of Advanced Mathematical Studies, vol. 1, no. 1-2, pp. 63-70, 2008. · Zbl 1179.26048
[16] \cS. Mititelu and M. Postolache, “Mond-Weir dualities with Lagrangians for multiobjective fractional and non-fractional variational problems,” Journal of Advanced Mathematical Studies, vol. 3, no. 1, pp. 41-58, 2010. · Zbl 1202.65083
[17] \cS. Mititelu, V. Preda, and M. Postolache, “Duality of multitime vector integral programming with quasiinvexity,” Journal of Advanced Mathematical Studies, vol. 4, no. 2, pp. 59-72, 2011. · Zbl 1239.65040
[18] C. Nahak and R. N. Mohapatra, “Nonsmooth \rho -(\eta ,\theta )-invexity in multiobjective programming problems,” Optimization Letters, vol. 6, no. 2, pp. 253-260, 2012. · Zbl 1280.90108 · doi:10.1007/s11590-010-0239-1
[19] A. Pitea, “Null Lagrangian forms on 2nd order jet bundles,” Journal of Advanced Mathematical Studies, vol. 3, no. 1, pp. 73-82, 2010. · Zbl 1206.49026
[20] M. Postolache, “Minimization of vectors of curvilinear functionals on second order jet bundle: dual program theory,” Abstract and Applied Analysis, vol. 2012, Article ID 535416, 12 pages, 2012. · Zbl 1237.49048 · doi:10.1155/2012/535416
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