Abstract
In this paper, we introduce Hartley properly and super nondominated solutions in vector optimization with a variable ordering structure. We prove the connections between Benson properly nondominated, Hartley properly nondominated, and super nondominated solutions under appropriate assumptions. Moreover, we establish some necessary and sufficient conditions for newly-defined solutions invoking an augmented dual cone approach, the linear scalarization, and variational analysis tools. In addition to the theoretical results, various clarifying examples are given.
Similar content being viewed by others
Notes
To prove (7), we define an auxiliary function \(\varphi :\mathbb {R}\longrightarrow \mathbb {R}\) as \(\varphi (\beta )=-\beta k+\sqrt{1-\beta ^2}\Big (k^2-k^2\sqrt{1-\frac{1}{k^2}}\Big )-1.\) Taking \(a^2+b^2=1\) into account, it is not difficult to see that (7) holds if and only if \(\varphi (a)<0.\) On the other hand, we have \(\varphi '(\beta )\le 0\) for any \(\beta \in [0,1),\) which implies that \(\varphi \) attains its maximum on [0, 1) at \(\beta =0\). So, \(\varphi (a)\le \varphi (0)=k^2-k^2\sqrt{1-\frac{1}{k^2}}-1\) (notice that, due to (6), \(a\ne 1\)). As \(k>1\), we get \(k^2-k^2\sqrt{1-\frac{1}{k^2}}-1<0\), leading to \(\varphi (a)<0\). This proves (7).
As \(bx_1+ax_2>0\), the inequality in (18) holds if and only if \(-ax_1-bx_2\le bx_1+ax_2\) which is equivalent to \((a+b)(x_1+x_2)\ge 0\). The last inequality is valid because \(a,b\ge 0\) and \(x_1>-x_2.\)
References
Bao, T.Q., Mordukhovich, B.S., Soubeyran, A.: Variational analysis in psychological modeling. J. Optim. Theory Appl. 164, 290–315 (2015)
Bao, T.Q., Mordukhovich, B.S.: Necessary condition for super minimizers in constrained multiobjective optimization. J. Global Optim. 43, 533–552 (2009)
Bao, T.Q., Mordukhovich, B.S.: Necessary nondomination conditions in set and vector optimization with variable ordering structures. J. Optim. Theory Appl. 162, 350–370 (2014)
Beck, A.: Introduction to Nonlinear Optimization. SIAM, Philadelphia (2014)
Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979)
Bergstresser, K., Charnes, A., Yu, P.L.: Generalization of domination structures and nondominated solutions in multicriteria decision making. J. Optim. Theory Appl. 18, 3–13 (1976)
Borwein, J.M.: Proper efficient points for maximization with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977)
Borwein, J.M., Zhuang, D.: Superefficiency in vector optimization. Trans. Am. Math. Soc. 338, 105–122 (1993)
Chen, G.Y.: Existence of solutions for a vector variational inequality: an extension of the Hartmann-Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992)
Chen, G.Y., Yang, X.Q.: Characterizations of variable domination structures via nonlinear scalarization. J. Optim. Theory Appl. 112, 97–110 (2002)
Eichfelder, G.: Vector optimization in medical engineering, In: Pardalos, P.M., Rassias T.M. (eds.) Mathematics Without Boundaries, pp. 181–215. Springer, New York (2014)
Eichfelder, G.: Variable Ordering Structures in Vector Optimization. Springer, Berlin (2014)
Eichfelder, G.: Optimal elements in vector optimization with a variable ordering structure. J. Optim. Theory Appl. 151, 217–240 (2011)
Eichfelder, G., Kasimbeyli, R.: Properly optimal elements in vector optimization with variable ordering structure. J. Global Optim. 60, 689–712 (2014)
Eichfelder, G., Gerlach, T.: Characterization of properly optimal elements with variable ordering structures. Optimization 65, 571–588 (2016)
Geoffrion, A.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)
Guerraggio, A., Molho, E., Zaffaroni, A.: On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 82, 1–21 (1994)
Hartley, R.: On cone efficiency, cone convexity, and cone compactness. SIAM J. Appl. Math. 34, 211–222 (1978)
Henig, M.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982)
Kasimbeyli, R.: A nonlinear cone separation theorem and scalarization in nonconvex vector optimization. SIAM J. Optim. 20, 1591–1619 (2010)
Kasimbeyli, R.: A conic scalarization method in multi-objective optimization. J. Global Optim. 56(2), 279–297 (2013)
Kasimbeyli, N., Kasimbeyli, R.: A representation theorem for Bishop-Phelps cones. Pac. J. Optim. 13(1), 55–74 (2017)
Khaledian, K., Khorram, E., Soleimani-damaneh, M.: Strongly proper efficient solutions: efficient solutions with bounded trade-offs. J. Optim. Theory Appl. 168, 864–883 (2016)
Kobis, E.: Set optimization by means of variable order relations. Optimization 66(12), 1991–2005 (2017)
Kuhn, H., Tucker, A.: Nonlinear programming. In: Neyman, J. (ed.) Proceeding of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492. University of California Press, Berkeley, California (1951)
Luc, D.T., Soubeyran, A.: Variable preference relations: existence of maximal elements. J. Math. Econ. 49, 251–262 (2013)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)
Rockafellar, R.T., West, R.J.B.: Varititional Analysis. Springer, Berlin (1998)
Sayadi-bander, A., Kasimbeyli, R., Pourkarimi, L.: A coradiant based scalarization to characterize approximate solutions of vector optimization problems with variable ordering structures. Oper. Res. Lett. 45(1), 93–97 (2017)
Soleimani, B., Tammer, C.: Concepts for approximate solutions of vector optimization problems with variable order structures. Vietnam J. Math. 42, 543–566 (2014)
Soleimani, B.: Characterization of approximate solutions of vector optimization problems with a variable order structure. J. Optim. Theory Appl. 162, 605–632 (2014)
Soleimani, B., Tammer, C.: Optimality conditions for approximate solutions of vector optimization problems with variable ordering structures. Bull. Iran. Math. Soc. 42(7), 5–23 (2016)
Yu, P.L.: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14, 319–377 (1974)
Zamani, M., Soleimani-damaneh, M., Kabgani, A.: Robustness in nonsmooth nonlinear multi-objective programming. Eur. J. Oper. Res. 247, 370–378 (2015)
Acknowledgements
The authors would like to express their gratitude to two anonymous referees and the associate editor for their helpful comments on the earlier versions of the paper. The research of the second author was in part supported by a grant from the Iran National Science Foundation (INSF) (No. 95849588).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shahbeyk, S., Soleimani-damaneh, M. & Kasimbeyli, R. Hartley properly and super nondominated solutions in vector optimization with a variable ordering structure. J Glob Optim 71, 383–405 (2018). https://doi.org/10.1007/s10898-018-0614-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-018-0614-5