September  2021, 11(3): 431-441. doi: 10.3934/naco.2020035

Properties of higher order preinvex functions

Department of Mathematics, COMSATS University Islamabad, Pakistan

* Corresponding author: Muhammad Aslam Noor

Received  December 2019 Revised  July 2020 Published  August 2020

In this paper, we define and introduce some new concepts of the higher order strongly preinvex functions and higher order strongly monotone operators involving an arbitrary bifunction. Some new relationships among various concepts of higher order strongly preinvex functions have been established. We have shown that the optimality conditions for the preinvex functions can be characterized by class of higher order strongly variational-like inequalities, which appears to be new ones. As a novel applications of the higher order strongly preinvex functions, we have obtained the parallelogram-like laws for the uniformly Banach spaces. As special cases, one can obtain various new and known results from our results. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

Citation: Muhammad Aslam Noor, Khalida Inayat Noor. Properties of higher order preinvex functions. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 431-441. doi: 10.3934/naco.2020035
References:
[1]

O. AlabdaliA. Guessab and G. Schmeisser, Characterizations of uniform convexity for differentiable functions, Appl. Anal. Discrte Math., 13 (2019), 721-732.  doi: 10.2298/aadm190322029a.  Google Scholar

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A. Ben-Isreal and B. Mond, What is invexity?, J. Austral. Math. Soc. Ser. B, 28 (1986), 1-9.  doi: 10.1017/S0334270000005142.  Google Scholar

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R. Cheng and C. B. Harris, Duality of the weak parallelogram laws on Banach spaces, J. Math. Anal. Appl., 404 (2013), 64-70.  doi: 10.1016/j.jmaa.2013.02.064.  Google Scholar

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R. Cheng and W. T. Ross, Weak parallelogram laws on Banach spaces and applications to prediction, Period. Math. Hung., 71 (2015), 45-58.  doi: 10.1007/s10998-014-0078-4.  Google Scholar

[6]

R. ChengJ. Mashreghi and W. T. Ross, Optimal weak parallelogram constants for $L_p $ space, Math. Inequal. Appl., 21 (2018), 1047-1058.  doi: 10.7153/mia-2018-21-71.  Google Scholar

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M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981), 545-550.  doi: 10.1016/0022-247X(81)90123-2.  Google Scholar

[8]

B. C. Joshi, S. K. Mishra and M. A. Noor, Semi-infinite mathematical programming problems with equilibrium constraints, Preprint, 2020. Google Scholar

[9]

T. Lara, N. Merentes and K. Nikodem, Strongly $h$-convexity and separation theorems, Int. J. Anal., 2016 (2016), Article ID 7160348, 5 pages. doi: 10.1155/2016/7160348.  Google Scholar

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G. H. Lin and M. Fukushima, Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints, J. Optim. Theory Appl., 118 (2003), 67-80.  doi: 10.1023/A:1024787424532.  Google Scholar

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S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901-908.  doi: 10.1006/jmaa.1995.1057.  Google Scholar

[12]

B. B. Mohsen, M. A. Noor, K. I. Noor and M. Postolache, Strongly convex functions of higher order involving bifunction, Mathematics, Appl. Math. Inform. Sci., 7 (2019), 1028. doi: 10.18576/amis/140117.  Google Scholar

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C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, Springer-Verlag, New York, 2018. doi: 10.1007/978-3-319-78337-6.  Google Scholar

[14]

K. Nikodem and Z. S. Pales, Characterizations of inner product spaces by strongly convex functions, Banach J. Math. Anal., 1 (2011), 83-87.  doi: 10.15352/bjma/1313362982.  Google Scholar

[15]

M. A. Noor, Variational-like inequalities, Optimization, 30 (1994), 323-330.  doi: 10.1080/02331939408843995.  Google Scholar

[16]

M. A. Noor, Invex equilibrium problems, J. Math. Anal. Appl., 302 (2005), 463-475.  doi: 10.1016/j.jmaa.2004.08.014.  Google Scholar

[17]

M. A. Noor and K. I. Noor, Higher order strongly generalized convex functions, Appl. Math. Inf. Sci., 14 (2020), 133-139.  doi: 10.18576/amis/140117.  Google Scholar

[18]

M. A. Noor and K. I. Noor, Some characterization of strongly preinvex functions, J. Math. Anal. Appl., 316 (2006), 697-706.  doi: 10.1016/j.jmaa.2005.05.014.  Google Scholar

[19]

M. A. Noor and K. I. Noor, Higher order strongly general convex functions and variational inequalities, AIMS Mathematics, 5 (2020), 3646-3663.  doi: 10.18576/amis/140117.  Google Scholar

[20]

B. T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl., 7 (1966), 2-75.   Google Scholar

[21]

G. Qu and N. Li, On the exponentially stability of primal-dual gradeint dynamics, IEEE Control Syst. Letters, 3 (2019), 43-48.   Google Scholar

[22]

H. K. Xu, Inequalities in Banach spaces with applications, Nonl. Anal.Theory, Meth. Appl., 16 (1991), 1127-1138.  doi: 10.1016/0362-546X(91)90200-K.  Google Scholar

[23]

X. M. YangQ. Yang and K. L. Teo, Criteria for generalized invex monotonicities, European J. Oper. Research, 164 (2005), 115-119.  doi: 10.1016/j.ejor.2003.11.017.  Google Scholar

[24]

T. Weir and B. Mond, Preinvex functions in multiobjective optimization, J. Math. Anal. Appl., 136 (1988), 29-38.  doi: 10.1016/0022-247X(88)90113-8.  Google Scholar

[25]

D. L. Zu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM Journal on Optimization, 6 (1996), 714-726.  doi: 10.1137/S1052623494250415.  Google Scholar

show all references

References:
[1]

O. AlabdaliA. Guessab and G. Schmeisser, Characterizations of uniform convexity for differentiable functions, Appl. Anal. Discrte Math., 13 (2019), 721-732.  doi: 10.2298/aadm190322029a.  Google Scholar

[2]

A. Ben-Isreal and B. Mond, What is invexity?, J. Austral. Math. Soc. Ser. B, 28 (1986), 1-9.  doi: 10.1017/S0334270000005142.  Google Scholar

[3]

W. L. Bynum, W., Weak parallelogram laws for Banach spaces, Can. Math. Bull., 19 (1976), 269-275.  doi: 10.4153/CMB-1976-042-4.  Google Scholar

[4]

R. Cheng and C. B. Harris, Duality of the weak parallelogram laws on Banach spaces, J. Math. Anal. Appl., 404 (2013), 64-70.  doi: 10.1016/j.jmaa.2013.02.064.  Google Scholar

[5]

R. Cheng and W. T. Ross, Weak parallelogram laws on Banach spaces and applications to prediction, Period. Math. Hung., 71 (2015), 45-58.  doi: 10.1007/s10998-014-0078-4.  Google Scholar

[6]

R. ChengJ. Mashreghi and W. T. Ross, Optimal weak parallelogram constants for $L_p $ space, Math. Inequal. Appl., 21 (2018), 1047-1058.  doi: 10.7153/mia-2018-21-71.  Google Scholar

[7]

M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981), 545-550.  doi: 10.1016/0022-247X(81)90123-2.  Google Scholar

[8]

B. C. Joshi, S. K. Mishra and M. A. Noor, Semi-infinite mathematical programming problems with equilibrium constraints, Preprint, 2020. Google Scholar

[9]

T. Lara, N. Merentes and K. Nikodem, Strongly $h$-convexity and separation theorems, Int. J. Anal., 2016 (2016), Article ID 7160348, 5 pages. doi: 10.1155/2016/7160348.  Google Scholar

[10]

G. H. Lin and M. Fukushima, Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints, J. Optim. Theory Appl., 118 (2003), 67-80.  doi: 10.1023/A:1024787424532.  Google Scholar

[11]

S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901-908.  doi: 10.1006/jmaa.1995.1057.  Google Scholar

[12]

B. B. Mohsen, M. A. Noor, K. I. Noor and M. Postolache, Strongly convex functions of higher order involving bifunction, Mathematics, Appl. Math. Inform. Sci., 7 (2019), 1028. doi: 10.18576/amis/140117.  Google Scholar

[13]

C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, Springer-Verlag, New York, 2018. doi: 10.1007/978-3-319-78337-6.  Google Scholar

[14]

K. Nikodem and Z. S. Pales, Characterizations of inner product spaces by strongly convex functions, Banach J. Math. Anal., 1 (2011), 83-87.  doi: 10.15352/bjma/1313362982.  Google Scholar

[15]

M. A. Noor, Variational-like inequalities, Optimization, 30 (1994), 323-330.  doi: 10.1080/02331939408843995.  Google Scholar

[16]

M. A. Noor, Invex equilibrium problems, J. Math. Anal. Appl., 302 (2005), 463-475.  doi: 10.1016/j.jmaa.2004.08.014.  Google Scholar

[17]

M. A. Noor and K. I. Noor, Higher order strongly generalized convex functions, Appl. Math. Inf. Sci., 14 (2020), 133-139.  doi: 10.18576/amis/140117.  Google Scholar

[18]

M. A. Noor and K. I. Noor, Some characterization of strongly preinvex functions, J. Math. Anal. Appl., 316 (2006), 697-706.  doi: 10.1016/j.jmaa.2005.05.014.  Google Scholar

[19]

M. A. Noor and K. I. Noor, Higher order strongly general convex functions and variational inequalities, AIMS Mathematics, 5 (2020), 3646-3663.  doi: 10.18576/amis/140117.  Google Scholar

[20]

B. T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl., 7 (1966), 2-75.   Google Scholar

[21]

G. Qu and N. Li, On the exponentially stability of primal-dual gradeint dynamics, IEEE Control Syst. Letters, 3 (2019), 43-48.   Google Scholar

[22]

H. K. Xu, Inequalities in Banach spaces with applications, Nonl. Anal.Theory, Meth. Appl., 16 (1991), 1127-1138.  doi: 10.1016/0362-546X(91)90200-K.  Google Scholar

[23]

X. M. YangQ. Yang and K. L. Teo, Criteria for generalized invex monotonicities, European J. Oper. Research, 164 (2005), 115-119.  doi: 10.1016/j.ejor.2003.11.017.  Google Scholar

[24]

T. Weir and B. Mond, Preinvex functions in multiobjective optimization, J. Math. Anal. Appl., 136 (1988), 29-38.  doi: 10.1016/0022-247X(88)90113-8.  Google Scholar

[25]

D. L. Zu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM Journal on Optimization, 6 (1996), 714-726.  doi: 10.1137/S1052623494250415.  Google Scholar

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