On qQuasiNewton’s Method for Unconstrained Multiobjective Optimization Problems
Abstract
:1. Introduction
2. Preliminaries
3. The $\mathit{q}$QuasiNewton Direction for Multiobjective
 1.
 $\psi \left(x\right)\le 0$ for all $x\in X$.
 2.
 The conditions below are equivalent:
 (a)
 The point x is non stationary.
 (b)
 ${d}_{q}\left(x\right)\ne 0$
 (c)
 $\psi \left(x\right)<0$.
 (d)
 ${d}_{q}\left(x\right)$ is a descent direction.
 3.
 The function ψ is continuous.
4. Algorithm and Convergence Analysis
Algorithm 1qGradient Algorithm 

Algorithm 2qQuasiNewton’s Algorithm for Unconstrained Multiobjective (qQNUM) 

 (a)
 $aI\le {W}_{j}\left(x\right)\le bI$ for all $x\in Y,$$j=1,\dots ,m,$
 (b)
 $\parallel {\nabla}_{q}^{2}{f}_{j}\left(y\right){\nabla}_{q}^{2}{f}_{j}\left(x\right),\parallel <\frac{\epsilon}{2}$ for all $x,y\in Y$ with $\parallel yx\parallel \in \delta ,$
 (c)
 $\parallel ({W}_{j}^{k}{\nabla}_{q}^{2}{f}_{j}\left({x}^{k}\right))(y{x}^{k})\parallel <\frac{\u03f5}{2}\parallel y{x}^{k}\parallel $ for all $k\ge {k}_{0}$, $y\in Y,$$j=1,\dots ,m,$
 (d)
 $\frac{\epsilon}{a}\le 1c,$
 (e)
 $B({x}^{0},r)\in Y,$
 (f)
 $\parallel {d}_{q}\left({x}^{0}\right)\parallel <min\{\delta ,r(1\frac{\epsilon}{a})\}.$
 1.
 $\parallel {x}^{k}{x}^{{k}_{0}}\parallel \le \parallel {d}_{q}\left({x}^{0}\right)\parallel \frac{1{\left(\frac{\epsilon}{a}\right)}^{k{k}_{0}}}{1\left(\frac{\epsilon}{a}\right)}$
 2.
 ${\alpha}_{k}=1$,
 3.
 $\parallel {d}_{q}\left({x}^{k}\right)\parallel \le \parallel {d}_{q}\left({x}^{{k}_{0}}\right)\parallel {\left(\frac{\epsilon}{a}\right)}^{k{k}_{0}},$
 4.
 $\parallel {d}_{q}\left({x}^{k+1}\right)\parallel \le \parallel {d}_{q}\left({x}^{k}\right)\parallel \frac{\epsilon}{a}.$
5. Numerical Results
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
 Eschenauer, H.; Koski, J.; Osyczka, A. Multicriteria Design Optimization: Procedures and Applications; Springer: Berlin, Germany, 1990. [Google Scholar]
 Haimes, Y.Y.; Hall, W.A.; Friedmann, H.T. Multiobjective Optimization in Water Resource Systems; Elsevier Scientific: Amsterdam, The Netherlands, 1975. [Google Scholar]
 Nwulu, N.I.; Xia, X. Multiobjective dynamic economic emission dispatch of electric power generation integrated with game theory based demand response programs. Energy Convers. Manag. 2015, 89, 963–974. [Google Scholar] [CrossRef]
 Badri, M.A.; Davis, D.L.; Davis, D.F.; Hollingsworth, J. A multiobjective course scheduling model: Combining faculty preferences for courses and times. Comput. Oper. Res. 1998, 25, 303–316. [Google Scholar] [CrossRef][Green Version]
 Ishibuchi, H.; Nakashima, Y.; Nojima, Y. Performance evaluation of evolutionary multiobjective optimization algorithms for multiobjective fuzzy geneticsbased machine learning. Soft Comput. 2011, 15, 2415–2434. [Google Scholar] [CrossRef]
 Liu, S.; Vicente, L.N. The stochastic multigradient algorithm for multiobjective optimization and its application to supervised machine learning. arXiv 2019, arXiv:1907.04472. [Google Scholar]
 Tavana, M. A subjective assessment of alternative mission architectures for the human exploration of mars at NASA using multicriteria decision making. Comput. Oper. Res. 2004, 31, 1147–1164. [Google Scholar] [CrossRef]
 Gass, S.; Saaty, T. The computational algorithm for the parametric objective function. Nav. Res. Logist. Q. 1955, 2, 39–45. [Google Scholar] [CrossRef]
 Miettinen, K. Nonlinear Multiobjective Optimization; Kluwer Academic: Boston, MA, USA, 1999. [Google Scholar]
 Fishbum, P.C. Lexicographic orders, utilities and decision rules: A survey. Manag. Sci. 1974, 20, 1442–1471. [Google Scholar] [CrossRef][Green Version]
 Coello, C.A. An updated survey of GAbased multiobjective optimization techniques. ACM Comput. Surv. (CSUR) 2000, 32, 109–143. [Google Scholar] [CrossRef]
 Fliege, J.; Svaiter, B.F. Steepest descent method for multicriteria optimization. Math. Method. Oper. Res. 2000, 51, 479–494. [Google Scholar] [CrossRef]
 Drummond, L.M.G.; Iusem, A.N. A projected gradient method for vector optimization problems. Comput. Optim. Appl. 2004, 28, 5–29. [Google Scholar] [CrossRef]
 Drummond, L.M.G.; Svaiter, B.F. A steepest descent method for vector optimization. J. Comput. Appl. Math. 2005, 175, 395–414. [Google Scholar] [CrossRef][Green Version]
 Branke, J.; Dev, K.; Miettinen, K.; Slowiński, R. (Eds.) Multiobjective Optimization: Interactive and Evolutionary Approaches; Springer: Berlin, Germany, 2008. [Google Scholar]
 Mishra, S.K.; Ram, B. Introduction to Unconstrained Optimization with R; Springer Nature: Singapore, 2019; pp. 175–209. [Google Scholar]
 Fliege, J.; Drummond, L.M.G.; Svaiter, B.F. Newton’s method for multiobjective optimization. SIAM J. Optim. 2009, 20, 602–626. [Google Scholar] [CrossRef][Green Version]
 Chuong, T.D. Newtonlike methods for efficient solutions in vector optimization. Comput. Optim. Appl. 2013, 54, 495–516. [Google Scholar] [CrossRef]
 Qu, S.; Liu, C.; Goh, M.; Li, Y.; Ji, Y. Nonsmooth Multiobjective Programming with QuasiNewton Methods. Eur. J. Oper. Res. 2014, 235, 503–510. [Google Scholar] [CrossRef]
 Jiménez, M.A.; Garzón, G.R.; Lizana, A.R. (Eds.) Optimality Conditions in Vector Optimization; Bentham Science Publishers: Sharjah, UAE, 2010. [Google Scholar]
 AlSaggaf, U.M.; Moinuddin, M.; Arif, M.; Zerguine, A. The qleast mean squares algorithm. Signal Process. 2015, 111, 50–60. [Google Scholar] [CrossRef]
 Aral, A.; Gupta, V.; Agarwal, R.P. Applications of qCalculus in Operator Theory; Springer: New York, NY, USA, 2013. [Google Scholar]
 Rajković, P.M.; Marinković, S.D.; Stanković, M.S. Fractional integrals and derivatives in qcalculus. Appl. Anal. Discret. Math. 2007, 1, 311–323. [Google Scholar]
 Gauchman, H. Integral inequalities in qcalculus. Comput. Math. Appl. 2004, 47, 281–300. [Google Scholar] [CrossRef][Green Version]
 Bangerezako, G. Variational qcalculus. J. Math. Anal. Appl. 2004, 289, 650–665. [Google Scholar] [CrossRef][Green Version]
 Abreu, L. A qsampling theorem related to the qHankel transform. Proc. Am. Math. Soc. 2005, 133, 1197–1203. [Google Scholar] [CrossRef]
 Koornwinder, T.H.; Swarttouw, R.F. On qanalogues of the Fourier and Hankel transforms. Trans. Am. Math. Soc. 1992, 333, 445–461. [Google Scholar]
 Ernst, T. A Comprehensive Treatment of qCalculus; Springer: Basel, Switzerland; Heidelberg, Germany; New York, NY, USA; Dordrecht, The Netherlands; London, UK, 2012. [Google Scholar]
 Noor, M.A.; Awan, M.U.; Noor, K.I. Some quantum estimates for HermiteHadamard inequalities. Appl. Math. Comput. 2015, 251, 675–679. [Google Scholar] [CrossRef]
 Pearce, C.E.M.; Pec̆arić, J. Inequalities for differentiable mappings with application to special means and quadrature formulae. Appl. Math. Lett. 2000, 13, 51–55. [Google Scholar] [CrossRef][Green Version]
 Ernst, T. A Method for qCalculus. J. Nonl. Math. Phys. 2003, 10, 487–525. [Google Scholar] [CrossRef][Green Version]
 Sterroni, A.C.; Galski, R.L.; Ramos, F.M. The qgradient vector for unconstrained continuous optimization problems. In Operations Research Proceedings; Hu, B., Morasch, K., Pickl, S., Siegle, M., Eds.; Springer: Heidelberg, Germany, 2010; pp. 365–370. [Google Scholar]
 Gouvêa, E.J.C.; Regis, R.G.; Soterroni, A.C.; Scarabello, M.C.; Ramos, F.M. Global optimization using qgradients. Eur. J. Oper. Res. 2016, 251, 727–738. [Google Scholar] [CrossRef]
 Chakraborty, S.K.; Panda, G. Newton like line search method using qcalculus. In International Conference on Mathematics and Computing. Communications in Computer and Information Science; Giri, D., Mohapatra, R.N., Begehr, H., Obaidat, M., Eds.; Springer: Singapore, 2017; Volume 655, pp. 196–208. [Google Scholar]
 Mishra, S.K.; Panda, G.; Ansary, M.A.T.; Ram, B. On qNewton’s method for unconstrained multiobjective optimization problems. J. Appl. Math. Comput. 2020. [Google Scholar] [CrossRef]
 Jackson, F.H. On qfunctions and a certain difference operator. Earth Environ. Sci. Trans. R. Soc. Edinb. 1908, 46, 253–281. [Google Scholar] [CrossRef]
 Bento, G.C.; Neto, J.C. A subgradient method for multiobjective optimization on Riemannian manifolds. J. Optimiz. Theory App. 2013, 159, 125–137. [Google Scholar] [CrossRef]
 Andrei, N. A diagonal quasiNewton updating method for unconstrained optimization. Numer. Algorithms 2019, 81, 575–590. [Google Scholar] [CrossRef]
 Nocedal, J.; Wright, S.J. Numerical Optimization, 2nd ed.; Springer Series in Operations Research and Financial Engineering; Springer: New York, NY, USA, 2006. [Google Scholar]
 Povalej, Z. QuasiNewton’s method for multiobjective optimization. J. Comput. Appl. Math. 2014, 255, 765–777. [Google Scholar] [CrossRef]
 Ye, Y.L. Dinvexity and optimality conditions. J. Math. Anal. Appl. 1991, 162, 242–249. [Google Scholar] [CrossRef][Green Version]
 Morovati, V.; Basirzadeh, H.; Pourkarimi, L. QuasiNewton methods for multiobjective optimization problems. 4ORQ. J. Oper. Res. 2018, 16, 261–294. [Google Scholar] [CrossRef]
 Samei, M.E.; Ranjbar, G.K.; Hedayati, V. Existence of solutions for equations and inclusions of multiterm fractional qintegrodifferential with nonseparated and initial boundary conditions. J. Inequal Appl. 2019, 273. [Google Scholar] [CrossRef][Green Version]
 Adams, C.R. The general theory of a class of linear partial difference equations. Trans. Am. Math.Soc. 1924, 26, 183–312. [Google Scholar]
 Sefrioui, M.; Perlaux, J. Nash genetic algorithms: Examples and applications. In Proceedings of the 2000 Congress on Evolutionary Computation, La Jolla, CA, USA, 16–19 July 2000; Volume 1, pp. 509–516. [Google Scholar]
 Huband, S.; Hingston, P.; Barone, L.; While, L. A review of multiobjective test problems and a scalable test problem toolkit. IEEE T. Evolut. Comput. 2006, 10, 477–506. [Google Scholar] [CrossRef][Green Version]
 Ikeda, K.; Kita, H.; Kobayashi, S. Failure of Paretobased MOEAs: Does nondominated really mean near to optimal? In Proceedings of the 2001 Congress on Evolutionary Computation, Seoul, Korea, 27–30 May 2001; Volume 2, pp. 957–962. [Google Scholar]
 Shim, M.B.; Suh, M.W.; Furukawa, T.; Yagawa, G.; Yoshimura, S. Paretobased continuous evolutionary algorithms for multiobjective optimization. Eng Comput. 2002, 19, 22–48. [Google Scholar] [CrossRef][Green Version]
 ValenzuelaRendón, M.; UrestiCharre, E.; Monterrey, I. A nongenerational genetic algorithm for multiobjective optimization. In Proceedings of the Seventh International Conference on Genetic Algorithms, East Lansing, MI, USA, 19–23 July 1997; pp. 658–665. [Google Scholar]
 Vlennet, R.; Fonteix, C.; Marc, I. Multicriteria optimization using a genetic algorithm for determining a Pareto set. Int. J. Syst. Sci. 1996, 27, 255–260. [Google Scholar] [CrossRef]
Problem  Source  [lb,ub]  (qQNUM)  (QNMO)  

$\mathit{iter}$  $\mathit{obj}$  $\mathit{grad}$  $\mathit{iter}$  $\mathit{obj}$  $\mathit{grad}$  
BK1  [46]  [−5, 10]  200  200  200  200  200  200 
MOP5  [46]  [−30, 30]  141  965  612  333  518  479 
MOP6  [46]  [0, 1]  250  2177  1712  181  2008  2001 
MOP7  [46]  [−400, 400]  200  200  200  751  1061  1060 
DG01  [47]  [−10, 13]  175  724  724  164  890  890 
IKK1  [47]  [−50, 50]  170  170  170  253  254  253 
SP1  [45]  [−3, 2]  200  200  200  525  706  706 
SSFYY1  [45]  [−2, 2]  200  200  200  200  300  300 
SSFYY2  [45]  [−100, 100]  263  277  277  263  413  413 
SK1  [48]  [−10, 10]  139  1152  1152  87  732  791 
SK2  [48]  [−3, 11]  154  1741  1320  804  1989  1829 
VU1  [49]  [−3, 3]  316  1108  1108  11,361  19,521  11,777 
VU2  [49]  [−3, 7]  99  1882  1882  100  1900  1900 
VFM1  [50]  [−2, 2]  195  195  195  195  290  290 
VFM2  [50]  [−4, 4]  200  200  200  524  693  678 
VFM3  [50]  [−3, 3]  161  1130  601  690  1002  981 
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Lai, K.K.; Mishra, S.K.; Ram, B. On qQuasiNewton’s Method for Unconstrained Multiobjective Optimization Problems. Mathematics 2020, 8, 616. https://doi.org/10.3390/math8040616
Lai KK, Mishra SK, Ram B. On qQuasiNewton’s Method for Unconstrained Multiobjective Optimization Problems. Mathematics. 2020; 8(4):616. https://doi.org/10.3390/math8040616
Chicago/Turabian StyleLai, Kin Keung, Shashi Kant Mishra, and Bhagwat Ram. 2020. "On qQuasiNewton’s Method for Unconstrained Multiobjective Optimization Problems" Mathematics 8, no. 4: 616. https://doi.org/10.3390/math8040616