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  September 2021 Early access  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 and 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[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.

[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.

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