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
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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