SeMA Journal

, Volume 75, Issue 2, pp 285–303

# A quasi-Newton method with rank-two update to solve fuzzy optimization problems

• Debdas Ghosh
• Debdulal Ghosh
• Sushil Kumar Bhuiya
Article

## Abstract

In this article, we develop a quasi-Newton method to obtain nondominated solutions of fuzzy optimization problems. The objective function of the optimization problem under consideration is a fuzzy-number-valued function. The notion of generalized Hukuhara difference of fuzzy numbers, and hence generalized Hukuhara differentiability for multi-variable fuzzy-number-valued functions are used to develop the quasi-Newton method. The proposed technique produces a sequence of positive definite inverse Hessian approximations to generate the iterative points. A sequential algorithm and the convergence result of the proposed method are also given. It is obtained that the sequence in the proposed method has superlinear convergence rate. The method is also found to have quadratic termination property. Two numerical examples are provided to illustrate the developed technique.

## Keywords

Quasi-Newton method Generalized-Hukuhara differentiability Fuzzy optimization Nondominated solution

## List of symbols

$$\mathbb {R}$$

The set of all real numbers

$$\widetilde{A}$$, $$\widetilde{B}$$, ...and $$\widetilde{a}$$, $$\widetilde{b}$$, ...

Fuzzy sets on $$\mathbb {R}$$

$$\mathcal {F}(\mathbb {R})$$

The set of all fuzzy numbers on $$\mathbb {R}$$

$$\mu (.|\widetilde{A})$$

Membership function of the fuzzy set $$\widetilde{A}$$

$$\widetilde{A}(\alpha )$$

$$\alpha$$-cut of the fuzzy set $$\widetilde{A}$$

(l / a / r)

A triangular fuzzy number at a

$${[}{\widetilde{a}}_\alpha ^{L}, {\widetilde{a}}_\alpha ^{U}{]}$$

$$\alpha$$-cut of the triangular fuzzy number $${\widetilde{a}}$$

$$\oplus$$

$$\ominus$$

Extended substraction

$$\odot$$

Extended multiplication

$$\ominus _{gH}$$

Generalized Hukuhara difference

$${[}\widetilde{f}_\alpha ^L(x), \widetilde{f}_\alpha ^U(x){]}$$

$$\alpha$$-cut of the fuzzy function $$\widetilde{f}$$ at x

$$\widetilde{f}_\alpha ^L$$

Lower function of the $$\alpha$$-cut of the fuzzy function $$\widetilde{f}$$

$$\widetilde{f}_\alpha ^U$$

Upper function of the $$\alpha$$-cut of the fuzzy function $$\widetilde{f}$$

$$\tfrac{\partial \widetilde{f}}{\partial x_i}(x)$$

Partial gH-derivative of $$\widetilde{f}$$ with respect to $$x_i$$ at x

$$\nabla \widetilde{f}(x)$$

gH-gradient of $$\widetilde{f}$$ at x

$$\nabla ^2 \widetilde{f}(x)$$

gH-Hessian of $$\widetilde{f}$$ at x

$$\widetilde{a} \preceq \widetilde{b}$$

$$\widetilde{a}^L_\alpha \le \widetilde{b}^L_\alpha$$ and $$\widetilde{a}^U_\alpha \le \widetilde{b}^U_\alpha$$ for all $$\alpha \in [0, 1]$$

$$\widetilde{a} \prec \widetilde{b}$$

$$\widetilde{a} \preceq \widetilde{b}$$ and $$\exists$$ $$\beta \in [0, 1]$$ such that $$\widetilde{a}^L_\beta < \widetilde{b}^L_\beta$$ or $$\widetilde{a}^U_\beta < \widetilde{b}^U_\beta$$.

90C70 90C29

## Notes

### Acknowledgements

The first author gratefully acknowledges the financial support through Early Career Research Award (ECR/2015/000467), Science and Engineering Research Board, Government of India.

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