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Hartley properly and super nondominated solutions in vector optimization with a variable ordering structure

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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.

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Notes

  1. 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).

  2. 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 \(x_1>-x_2.\)

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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).

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Authors and Affiliations

  1. School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Enghelab Avenue, Tehran, Iran

    Shokouh Shahbeyk & Majid Soleimani-damaneh

  2. Department of Industrial Engineering, Anadolu University, Iki Eylul Campus 26555, Eskisehir, Turkey

    Refail Kasimbeyli

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  1. Shokouh Shahbeyk
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  2. Majid Soleimani-damaneh
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  3. Refail Kasimbeyli
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Correspondence to Majid Soleimani-damaneh.

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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

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