A Davidon-Fletcher-Powell Type Quasi-Newton Method to Solve Fuzzy Optimization Problems

Abstract

In this article, a Davidon-Fletcher-Powell type quasi-Newton method is proposed to capture nondominated solutions of fuzzy optimization problems. The functions that we attempt to optimize here are multivariable fuzzy-number-valued functions. The decision variables are considered to be crisp. Towards developing the quasi-Newton method, the notion of generalized Hukuhara difference between fuzzy numbers, and hence generalized Hukuhara differentiability for multi-variable fuzzy-number-valued functions are used. In order to generate the iterative points, the proposed technique produces a sequence of positive definite inverse Hessian approximations. The convergence result and an algorithm of the developed method are also included. It is found that the sequence in the proposed method has superlinear convergence rate. To illustrate the developed technique, a numerical example is exhibited.

1 Introduction

In order to deal with imprecise nature of the objective and constraint functions in a decision-making problem, fuzzy optimization problems are widely studied since the seminal work by Bellman and Zadeh [6], in 1970. The research article by Cadenas and Verdegay [7], the monograph by Słowínski [29], and the references therein are rich stream of this topic. Very recently, in a survey article, Luhandjula [26] reported the milestones and perspective of the theories and applications of fuzzy optimization. The survey books by Lai and Hwang [20, 21] and by Lodwick and Kacprzyk [22] explored a perceptive overview on the development of fuzzy optimization problems. Recently, Ghosh and Chakraborty published a fuzzy geometrical view [9, 14, 15] on fuzzy optimization problems [16,17,18,19].

In order to solve an unconstrained fuzzy optimization problem, recently, Pirzada and Pathak [23] and Chalco-Cano et al. [8] developed a Newton method. Much similar to fuzzy optimization, Ghosh [12] derived a Newton method [11] and a quasi-Newton method [13] for interval optimization problems.

The real life optimization models often need to optimize a fuzzy function over a real data set. Mathematically, the problem is the following:

$$\begin{aligned} \displaystyle \min _{x \in \mathbb {R}^n} \widetilde{f}(x), \end{aligned}$$

where $\widetilde{f}$ is a fuzzy-number-valued function. There is plethora of studies and numerical algorithms [1, 28] to effectively tackle unconstrained conventional optimization problems. However, the way to apply those methods to solve fuzzy optimization problems is not apparent. In this article, we develop a Davidon-Fletcher-Powell type quasi-Newton method for an unconstrained fuzzy optimization problem.

In order to capture the optimum candidates [4, 28] for a smooth optimization problem, it is natural to use the notion of differentiability. However, to find the optimal points of a fuzzy optimization problem, in addition to the idea on differentiability of the fuzzy function, identification of an appropriate ordering of fuzzy numbers is of utmost importance. Because, unlike the real number set, the set of all fuzzy numbers is not linearly ordered [16].

Towards developing the notion of differentiability of fuzzy functions, Dubois and Prade [10] used the extension principle, and also exhibited that a fuzzy extension of the conventional definition of differentiation requires further enquiry. Recently, the idea of Hukuhara differentiability (H-differentiability) [3, 24] received substantial attention in fuzzy optimization theory. The concept of H-fuzzy-differentiation is rigorously discussed in [2]. Stefanini [30] proposed generalizations of H-differentiability (gH-differentiability) and its application in fuzzy differential equations. Bede and Stefanini [5] gave a generalized H-differentiability of fuzzy-valued functions. Chalco-Cano et al. [8] reported that gH-derivative is the most general concept for differentiability of fuzzy functions. Thus, in this paper, we employ the gH-derivative and its calculus [8, 33] to derive the quasi-Newton method for fuzzy optimization problems.

There have been extensive literature on ordering of fuzzy numbers including the research articles [25, 27, 31]. References of [32] reports main stream on ordering of fuzzy numbers. In this paper, we use the fuzzy-max ordering of fuzzy numbers of Ramík and Rimanek [27]. There are two reasons behind this choice. First, it is a partial ordering in the space of fuzzy numbers [23]. Second, it has insightful association [33] with the optimality notion on fuzzy optimization.

The rest of the article is organized in the following sequence. In Sect. 2, the notations and terminologies are given which are used throughout the paper. In Sect. 3, we derive a quasi-Newton method. Convergence analysis of the proposed method is presented in Sect. 4. Section 5 includes an illustrative numerical example. Finally, we give a brief conclusions and scopes for future research in Sect. 6.

2 Preliminaries

We use upper and lower case letters with a tildebar ($\widetilde{A}, \widetilde{B}, \widetilde{C}, \dots$ and $\widetilde{a}, \widetilde{b}, \widetilde{c}, \dots$) to denote fuzzy subsets of $\mathbb {R}$. The membership function of a fuzzy set $\widetilde{A}$ of $\mathbb {R}$ is represented by $\mu (x|\widetilde{A})$, for x in $\mathbb {R}$, with $\mu (\mathbb {R}) \subseteq [0, 1]$.

2.1 Fuzzy Numbers

Definition 1

($\alpha$-cut of a fuzzy set [16]). The $\alpha$-cut of a fuzzy set $\widetilde{A}$ of $\mathbb {R}$ is denoted by $\widetilde{A}(\alpha )$ and is defined by:

$$\begin{aligned} \widetilde{A}(\alpha ) = {\left\{ \begin{array}{ll} \{x : \mu (x | \widetilde{A})\ge \alpha \} &{} \,\,\, if \,\, 0 < \alpha \le 1\\ closure\{x : \mu (x|\widetilde{A})> 0\} &{} \,\,\, if \,\, \alpha = 0. \end{array}\right. } \end{aligned}$$

Definition 2

(Fuzzy number [16]). A fuzzy set $\widetilde{N}$ of $\mathbb {R}$ is called a fuzzy number if its membership function $\mu$ has the following properties:

• (i) $\mu (x|\widetilde{N})$ is upper semi-continuous,

• (ii) $\mu (x|\widetilde{N}) = 0$ outside some interval [a, d], and

• (iii) there exist real numbers b and c satisfying $a\le b\le c\le d$ such that $\mu (x|\widetilde{N})$ is increasing on [a, b] and decreasing on [c, d], and $\mu (x|\widetilde{N}) = 1$ for each x in [b, c].

In particular, if $b = c$, and the parts of the membership functions $\mu (x|\widetilde{N})$ in [a, b] and [c, d] are linear, the fuzzy number is called a triangular fuzzy number, denoted by (a/c/d). We denote the set of all fuzzy numbers on $\mathbb {R}$ by $\mathcal {F}(\mathbb {R})$.

Since $\mu (x|\widetilde{a})$ is upper semi-continuous for a fuzzy number $\widetilde{a}$ the $\alpha$-cut of $\widetilde{a}$, $\widetilde{a}(\alpha )$ is a closed and bounded interval of $\mathbb {R}$ for all $\alpha$ in [0, 1]. We write

$$\begin{aligned} \widetilde{a}(\alpha ) = \left[ \widetilde{a}_\alpha ^L,~ \widetilde{a}_\alpha ^U\right] . \end{aligned}$$

Let $\oplus$ and $\odot$ denote the extended addition and multiplication. According to the well-known extension principle, the membership function of $\widetilde{a} \otimes \widetilde{b}$ ($\otimes = \oplus$ or $\odot$) is defined by

$$\begin{aligned} \mu (z|\widetilde{a} \otimes \widetilde{b}) = \displaystyle \sup _{x \times y = z} \min \left\{ \mu (x|\widetilde{a}), ~\mu (y|\widetilde{b})\right\} . \end{aligned}$$

For any $\widetilde{a}$ and $\widetilde{b}$ in $\mathcal {F}(\mathbb {R})$, the $\alpha$-cut (for any $\alpha$ in [0, 1]) of their addition and scalar multiplication can be obtained by:

$$\begin{aligned} \big (\widetilde{a} \oplus \widetilde{b}\big )(\alpha )&= \left[ \widetilde{a}^L_\alpha + \widetilde{b}^L_\alpha ,~ \widetilde{a}^U_\alpha + \widetilde{b}^U_\alpha \right] \text { and } \\ \big (\lambda \odot \widetilde{a}\big )(\alpha )&= {\left\{ \begin{array}{ll} \left[ \lambda \widetilde{a}^L_\alpha ,~ \lambda \widetilde{a}^U_\alpha \right] &{}\text { if } \lambda \ge 0, \\ \left[ \lambda \widetilde{a}^U_\alpha , ~\lambda \widetilde{a}^L_\alpha \right] &{} \text { if } \lambda < 0. \end{array}\right. } \end{aligned}$$

Definition 3

(Generalized Hukuhara difference [30]). Let $\widetilde{a}$ and $\widetilde{b}$ be two fuzzy numbers. If there exists a fuzzy number $\widetilde{c}$ such that $\widetilde{c}$ $\oplus$ $\widetilde{b} = \widetilde{a}$ or $\widetilde{b} = \widetilde{a} \ominus \widetilde{c}$, then $\widetilde{c}$ is said to be generalized Hukuhara difference (gH-difference) between $\widetilde{a}$ and $\widetilde{b}$. Hukuhara difference between $\widetilde{a}$ and $\widetilde{b}$ is denoted by $\widetilde{a} \ominus _{gH} \widetilde{b}$.

In terms of $\alpha$-cut, for all $\alpha \in [0, 1]$, we have

$$\begin{aligned} \left( \widetilde{a} \ominus _{gH} \widetilde{b}\right) (\alpha ) = \left[ \min \left\{ \widetilde{a}^L_\alpha - \widetilde{b}^L_\alpha ,~ \widetilde{a}^U_\alpha - \widetilde{b}^U_\alpha \right\} , ~\max \left\{ \widetilde{a}^L_\alpha - \widetilde{b}^L_\alpha ,~ \widetilde{a}^U_\alpha - \widetilde{b}^U_\alpha \right\} \right] . \end{aligned}$$

2.2 Fuzzy Functions

Let $\widetilde{f}: \mathbb {R}^n \rightarrow \mathcal {F}(\mathbb {R})$ be a fuzzy function. For each x in $\mathbb {R}^n$, we present the $\alpha$-cuts of $\widetilde{f}(x)$ by

$$\begin{aligned} \widetilde{f}(x)(\alpha ) = \left[ \widetilde{f}_\alpha ^L(x),~ \widetilde{f}_\alpha ^U(x)\right] \text { for all } \alpha \in [0, 1]. \end{aligned}$$

The functions $\widetilde{f}_\alpha ^L$ and $\widetilde{f}_\alpha ^U$ are, evidently, two real-valued functions on $\mathbb {R}^n$ and are called the lower and upper functions, respectively.

With the help of gH-difference between two fuzzy numbers, gH-differentiability of a fuzzy function is defined as follows.

Definition 4

(gH-differentiability of fuzzy functions [8]). Let $\widetilde{f}: \mathbb {R}^n \rightarrow \mathcal {F}(\mathbb {R})$ be a fuzzy function and $x_0 = \left( x_1^0, x_2^0, \cdots , x_n^0\right)$ be an element of $\mathbb {R}^n$. For each $i = 1, 2, \cdots , n$, we define a fuzzy function $\widetilde{h}_i: \mathbb {R} \rightarrow \mathcal {F}(\mathbb {R})$ as follows

$$\begin{aligned} \widetilde{h}_i(x_i) = \widetilde{f}\left( x_1^0, \cdots , x_{i - 1}^0, x_i, x_{i + 1}^0, \cdots , x_n^0\right) . \end{aligned}$$

We say $h_i$ is gH-differentiable if the following limit exists

$$\begin{aligned} \lim _{t_i \rightarrow 0} \frac{\widetilde{h}_i(x_i^0 + t_i) \ominus _{gH} \widetilde{h}_i(x_i^0)}{t_i}. \end{aligned}$$

If $\widetilde{h}_i$ is gH-differentiable, then we say that $\widetilde{f}$ has the i-th partial gH-derivative at $x_0$ and is denoted by $\frac{\partial \widetilde{f}}{\partial x_i}(x_0)$.

The function $\widetilde{f}$ is said to be gH-differentiable at $x_0 \in \mathbb {R}^n$ if all the partial gH-derivatives $\frac{\partial \widetilde{f}}{\partial x_1}(x_0)$, $\frac{\partial \widetilde{f}}{\partial x_2}(x_0)$, $\cdots$, $\frac{\partial \widetilde{f}}{\partial x_n}(x_0)$ exist on some neighborhood of $x_0$ and are continuous at $x_0$.

Proposition 1

(See [8]). If a fuzzy function $\widetilde{f}: \mathbb {R}^n \rightarrow \mathcal {F}(\mathbb {R})$ is gH-differentiable at $x_0 \in \mathbb {R}^n$, then for each $\alpha \in [0, 1]$, the real-valued function $\widetilde{f}^L_\alpha + \widetilde{f}^U_\alpha$ is differentiable at $x_0$. Moreover,

$$\begin{aligned} \frac{\partial \widetilde{f}^L_\alpha }{\partial x_i}(x_0) + \frac{\partial \widetilde{f}^U_\alpha }{\partial x_i}(x_0) = \frac{\partial (\widetilde{f}^L_\alpha + \widetilde{f}^U_\alpha )}{\partial x_i}(x_0). \end{aligned}$$

Definition 5

(gH-gradient [8]). The gH-gradient of a fuzzy function $\widetilde{f}: \mathbb {R}^n \rightarrow \mathcal {F}(\mathbb {R})$ at a point $x_0 \in \mathbb {R}^n$ is defined by

$$\begin{aligned} \left( \frac{\partial \widetilde{f}(x_0)}{\partial x_1}, \frac{\partial \widetilde{f}(x_0)}{\partial x_2}, \cdots , \frac{\partial \widetilde{f}(x_0)}{\partial x_n} \right) ^t. \end{aligned}$$

We denote this gH-gradient by $\nabla \widetilde{f} (x_0)$.

We define an m-times continuously gH-differentiable fuzzy function $\widetilde{f}$ as a function whose all the partial gH-derivatives of order m exist and are continuous. Then, we have the following immediate result.

Proposition 2

(See [8]). Let $\widetilde{f}$ be a fuzzy function. Let at $x_0 \in \mathbb {R}^n$, the function $\widetilde{f}$ be m-times gH-differentiable. Then, the real-valued function $\widetilde{f}^L_\alpha + \widetilde{f}^U_\alpha$ is m-times differentiable at $x_0$.

Definition 6

(gH-Hessian [8]). Let the fuzzy function $\widetilde{f}$ be twice gH-differentiable at $x_0$. Then, for each i, the function $\frac{\partial \widetilde{f}}{\partial x_i}$ is gH-differentiable at $x^0$. The second order partial gH-derivative can be calculated through $\frac{\partial ^2 \widetilde{f}}{\partial x_i \partial x_j}$. The gH-Hessian of $\widetilde{f}$ at $x_0$ can be captured by the square matrix

$$\begin{aligned} \nabla ^2\widetilde{f}(x_0) = \left[ \frac{\partial ^2 \widetilde{f}}{\partial x_i \partial x_j}(x_0)\right] _{n\times n}. \end{aligned}$$

2.3 Optimality Concept

Definition 7

(Dominance relation between fuzzy numbers [23]). Let $\widetilde{a}$ and $\widetilde{b}$ be two fuzzy numbers. For any $\alpha \in [0, 1]$, let $\widetilde{a}_\alpha = \left[ \widetilde{a}^L_\alpha , \widetilde{a}^U_\alpha \right]$ and $\widetilde{b}_\alpha = \left[ \widetilde{b}^L_\alpha , \widetilde{b}^U_\alpha \right]$. We say $\widetilde{a}$ dominates $\widetilde{b}$ if $\widetilde{a}^L_\alpha \le \widetilde{b}^L_\alpha$ and $\widetilde{a}^U_\alpha \le \widetilde{b}^U_\alpha$ for all $\alpha \in [0, 1]$. If $\widetilde{a}$ dominates $\widetilde{b}$, then we write $\widetilde{a} \preceq \widetilde{b}$. The fuzzy number $\widetilde{b}$ is said to be strictly dominated by $\widetilde{a}$, if $\widetilde{a} \preceq \widetilde{b}$ and there exists $\beta \in [0, 1]$ such that $\widetilde{a}^L_\beta < \widetilde{b}^L_\beta$ or $\widetilde{a}^U_\beta < \widetilde{b}^U_\beta$. If $\widetilde{a}$ strictly dominates $\widetilde{b}$, then we write $\widetilde{a} \prec \widetilde{b}$.

Definition 8

(Non-dominated solution [23]). Let $\widetilde{f}: \mathbb {R}^n \rightarrow \mathcal {F}(\mathbb {R})$ be a fuzzy function and we intend to find a solution of ‘$\displaystyle \min _{x \in \mathbb {R}^n} \widetilde{f}(x)$’. A point $\bar{x} \in \mathbb {R}^n$ is said to be a locally non-dominated solution if for any $\epsilon >0$, there exists no $x \in N_{\epsilon }(\bar{x})$ such that $\widetilde{f}(x) \preceq \widetilde{f}(\bar{x})$, where $N_{\epsilon }(\bar{x})$ denotes $\epsilon$-neighborhood of $\bar{x}$. A local non-dominated solution is called a local solution of ‘$\displaystyle \min _{x \in \mathbb {R}^n} \widetilde{f}(x)$’.

For local non-dominated solution, the following result is proved in [8].

Proposition 3

(See [8]). Let $\widetilde{f}: \mathbb {R}^n \rightarrow \mathcal {F}(\mathbb {R})$ be a fuzzy function. If $x^*$ is a local minimizer of the real-valued function $\widetilde{f}^L_\alpha + \widetilde{f}^U_\alpha$ for all $\alpha \in [0, 1]$, then $x^*$ is a locally non-dominated solution of ‘$\displaystyle \min _{x \in \mathbb {R}^n} \widetilde{f}(x)$’.

3 Quasi-Newton Method

In this section, we consider to solve the following unconstrained Fuzzy Optimization Problem:

$$\begin{aligned} (FOP) \;\; \min _{x \in \mathbb {R}^n} \widetilde{f}(x), \end{aligned}$$

where $\widetilde{f}: \mathbb {R}^n \rightarrow \mathcal {F}(\mathbb {R})$ is a multi-variable fuzzy-number-valued function. On finding nondominated solution of the problem, we note that the existing Newton method (see [8, 23]) requires computation of the inverse of the concerned Hessian. However, computation of the inverse of Hessian is cost effective. Thus in this article, we intend to develop a quasi-Newton method to sidestep the computational cost of the existing Newton method. Towards this end, for the considered FOP we assume that at each of the following generated sequential points $x_k$, the function $\widetilde{f}$, its gH-gradient and gH-Hessian are well-defined. Therefore, according to Propositions 1 and 2, we can calculate $\widetilde{f}^L_\alpha (x_k)$, $\widetilde{f}^U_\alpha (x_k)$, $\nabla \widetilde{f}^L_\alpha (x_k)$, $\nabla \widetilde{f}^U_\alpha (x_k)$, $\nabla ^2 \widetilde{f}^L_\alpha (x_k)$ and $\nabla ^2 \widetilde{f}^U_\alpha (x_k)$ for all $\alpha \in [0, 1]$, for all $k = 0, 1, 2, \cdots$. Hence, we can have a quadratic approximations of the lower and the upper functions $\widetilde{f}^L_\alpha$ and $\widetilde{f}^U_\alpha$ at each $x_k$.

Let the quadratic approximation models of the functions $\widetilde{f}^L_\alpha$ and $\widetilde{f}^U_\alpha$ at $x_{k+1}$ be

$$\begin{aligned} h_\alpha ^L(x) = \widetilde{f}_\alpha ^L(x_{k+1}) + \nabla \widetilde{f}_\alpha ^L(x_{k+1})^t (x - x_{k+1}) + \tfrac{1}{2} (x - x_{k+1})^t~ \nabla ^2\widetilde{f}_\alpha ^L(x_{k+1})~ (x - x_{k+1}) \end{aligned}$$

and

$$\begin{aligned} h_\alpha ^U(x) = \widetilde{f}_\alpha ^U(x_{k+1}) + \nabla \widetilde{f}_\alpha ^U(x_{k+1})^t (x - x_{k+1}) + \tfrac{1}{2} (x - x_{k+1})^t~ \nabla ^2\widetilde{f}_\alpha ^U(x_{k+1})~ (x - x_{k+1}), \end{aligned}$$

which satisfy the interpolating conditions

$$\begin{aligned} h_\alpha ^L(x_{k+1}) = \widetilde{f}_\alpha ^L(x_{k+1}),~ h_\alpha ^U(x_{k+1}) = \widetilde{f}_\alpha ^U(x_{k+1}),~\nabla h^L_\alpha (x_{k+1}) = \nabla \widetilde{f}^L_\alpha (x_{k+1}), \text { and } \end{aligned}$$

$$\begin{aligned} \nabla h^U_\alpha (x_{k+1}) = \nabla \widetilde{f}^U_\alpha (x_{k+1}). \end{aligned}$$

The derivatives of $h_\alpha ^L$ and $h_\alpha ^U$ yield

$$\begin{aligned}&~\nabla h_\alpha ^L(x ) = \nabla \widetilde{f}_\alpha ^L(x_{k+1}) + \nabla ^2 \widetilde{f}_\alpha ^L(x_{k+1})~ (x - x_{k+1}) \nonumber \\ \text {and}&~\nabla h_\alpha ^U(x ) = \nabla \widetilde{f}_\alpha ^U(x_{k+1}) + \nabla ^2 \widetilde{f}_\alpha ^U(x_{k+1})~ (x - x_{k+1}). \end{aligned}$$

(1)

In the next, the Newton method (see [8, 23]) attempts to find x in terms of $x_{k+1}$ as follows

$$\begin{aligned} x = x_{k+1} - \left[ \nabla ^2 \phi (x_{k+1})\right] ^{-1} \nabla \phi (x_{k+1}), \end{aligned}$$

where $\phi (x) = \int _0^1 \left( \widetilde{f}_\alpha ^L(x) + \widetilde{f}_\alpha ^U(x)\right) d\alpha$. However, due to inherent computational difficulty to find $[\nabla ^2 \phi (x_{k+1})]^{-1}$, it is often suggested to consider an appropriate approximation. Let $A_{k+1}$ be an approximation of $[\nabla ^2 \phi (x_{k+1})]^{-1}$. Then from (1) setting $x = x_k$, $\delta _k = x_{k+1} - x_k$, $\beta _{\alpha k}^L = \nabla \widetilde{f}_\alpha ^L(x_{k+1}) - \nabla \widetilde{f}_\alpha ^L(x_{k})$ and $\beta _{\alpha k}^U = \nabla \widetilde{f}_\alpha ^U(x_{k+1}) - \nabla \widetilde{f}_\alpha ^U(x_{k})$, we obtain

$$\begin{aligned} \beta _{\alpha k}^L = \nabla ^2 \widetilde{f}_\alpha ^L(x_{k+1}) \delta _k \text { and } \beta _{\alpha k}^U = \nabla ^2 \widetilde{f}_\alpha ^U(x_{k+1}) \delta _k \end{aligned}$$

$$\begin{aligned} \Longrightarrow&~ \beta _{\alpha k}^L + \beta _{\alpha k}^U ~=~ \left( \nabla ^2 \widetilde{f}_\alpha ^L(x_{k+1}) + \nabla ^2 \widetilde{f}_\alpha ^U(x_{k+1})\right) \delta _k \text { for all } \alpha \in [0, 1]\\ \Longrightarrow&~ \int _0^1 \left( \beta _{\alpha k}^L + \beta _{\alpha k}^U\right) d \alpha ~=~ \left( \int _0^1 \nabla ^2\left( \widetilde{f}_\alpha ^L + \widetilde{f}_\alpha ^U\right) (x_{k+1}) d\alpha \right) \delta _k \\ \Longrightarrow&~ \int _0^1 \left( \nabla \phi _\alpha (x_{k+1}) - \nabla \phi _\alpha (x_k) \right) d \alpha ~=~ \left( \int _0^1 \nabla ^2 \phi _\alpha (x_{k+1}) d\alpha \right) \delta _k,\\&\text { where } \phi _\alpha (x) = \widetilde{f}_\alpha ^L(x) + \widetilde{f}_\alpha ^U(x)\\ \Longrightarrow&~ \nabla \phi (x_{k+1}) - \nabla \phi (x_k) = \nabla ^2 \phi (x_{k+1}) \delta _k, \text { where } \phi (x) = \int _0^1 \phi _\alpha (x) d\alpha \\ \Longrightarrow&~ A_{k+1} \varPhi _k = \delta _k, \text { where } \varPhi _k = \nabla \phi (x_{k+1}) - \nabla \phi (x_k). \end{aligned}$$

According to Proposition 3, to obtain nondominated solutions of (FOP), we need to have solutions of the equation $\nabla (\widetilde{f}_\alpha ^L + \widetilde{f}_\alpha ^U)(x) = 0$. In order to capture solutions of this equation, much similar to the Newton method [23], the above procedure clearly suggests to consider the following generating sequence

$$\begin{aligned}&~ \delta _k = x_{k+1} - x_k = A_{k+1} \varPhi _k \\ \Longrightarrow&~ x_{k+1} = x_k + A_{k+1} \varPhi _k. \end{aligned}$$

If $A_k$ is an appropriate approximation of the inverse Hessian matrix $\left[ \nabla ^2 \phi (x_{k+1})\right] ^{-1}$ and $\nabla (\widetilde{f}_\alpha ^L + \widetilde{f}_\alpha ^U)(x_{k+1}) \thickapprox 0$, then the equation $x_{k+1} = x_k + A_{k+1} \varPhi _k$ reduces to

$$\begin{aligned} x_{k+1} = x_k - \left[ \nabla ^2 \phi (x_{k+1})\right] ^{-1} \nabla \phi (x_k), \end{aligned}$$

which is the generating equation of the Newton method [23] and hence obviously will converge to the minimizer of $\widetilde{f}_\alpha ^L + \widetilde{f}_\alpha ^U$.

As we observe, the key point of the above method is to appropriately generate $A_{k}$’s. Due to the inherent computational difficulty to find the inverse of the Hessian $\nabla ^2 \phi (x_{k+1})$ we consider an approximation that should satisfy

$$\begin{aligned} A_{k+1} \varPhi _k = \delta _k. \end{aligned}$$

(2)

In this article we attempt to introduce a simple rank-two update of the sequence $\{A_k\}$ that satisfy the above quasi-Newton Eq. (2).

Let $A_k$ be the approximation of the k-th iteration. We attempt to update $A_k$ into $A_{k+1}$ by adding two symmetric matrices, each of rank one as follows:

$$\begin{aligned} A_{k+1} = A_k + p_k v_k v_k^t + q_k w_k w_k^t \end{aligned}$$

where $u_k$ and $v_k$ are two vectors in $\mathbb {R}^n$, and $p_k$ and $q_k$ are two scalars which are to be determined by the quasi-Newton Eq. (2). Therefore, we now have

$$\begin{aligned} A_k \varPhi _k + p_k v_k v_k^t \varPhi _k + q_k w_k w_k^t \varPhi _k = \delta _k. \end{aligned}$$

(3)

Evidently, $v_k$ and $w_k$ are not uniquely determined, but their obvious choices are $v_k = \delta _k$ and $w_k = A_k \varPhi _k$. Putting this values in (3), we obtain

$$\begin{aligned} p_k = \frac{1}{v_k^t \varPhi _k} = \frac{1}{\delta _k^t \varPhi _k} \text { and } q_k = - \frac{1}{w_k^t \varPhi _k} = - \frac{1}{\varPhi _k^t A_k \varPhi _k}. \end{aligned}$$

Therefore,

$$\begin{aligned} A_{k+1} = A_k + \frac{\delta _k \delta _k^t}{\delta _k^t \varPhi _k} - \frac{A_k \varPhi _k \varPhi _k^t A_k}{\varPhi _k^t A_k \varPhi _k}. \end{aligned}$$

(4)

We consider this equation to generate the sequence of above mentioned inverse Hessian approximation.

Therefore, accumulating all, we follow the following sequential way (Algorithm 1) to obtain an efficient solution of the considered (FOP) $\min _{x \in \mathbb {R}^n} \widetilde{f}(x)$.

4 Convergence Analysis

In this section, the convergence analysis of the proposed quasi-Newton method is performed. In the following theorem, it is found that the proposed method has superlinear convergence rate.

Theorem 1

(Superlinear convergence rate). Let $\widetilde{f}: \mathbb {R}^n \rightarrow \mathcal {F}(\mathbb {R})$ be thrice continuously gH-differentiable on $\mathbb {R}^n$ and $\bar{x}$ be a point such that

• (i) $\bar{x}$ is a local minimizer of $\widetilde{f}^L_\alpha$ and $\widetilde{f}^U_\alpha$,

• (ii) $\int _0^1 \nabla ^2 \widetilde{f}_\alpha ^L(x) d\alpha$ and $\int _0^1 \nabla ^2 \widetilde{f}_\alpha ^U(x) d\alpha$ are symmetric positive definite, and

• (iii) $\int _0^1 \nabla ^2 \widetilde{f}_\alpha ^L(x) d\alpha$ and $\int _0^1 \nabla ^2 \widetilde{f}_\alpha ^U(x) d\alpha$ are Lipschitzian with constants $\gamma ^L$ and $\gamma ^U$, respectively.

Then the iteration sequence $\{x_k\}$ in Algorithm 1 converges to $\bar{x}$ superlinearly if and only if

$$\begin{aligned} \lim _{k \rightarrow \infty } \frac{\left\| \left[ A_{k+1}^{-1} - \nabla ^2 \phi (\bar{x})\right] \delta _k\right\| }{\Vert \delta _k\Vert } = 0, \end{aligned}$$

where $\phi (x) = \int _0^1\left( \widetilde{f}_\alpha ^L(x) + \widetilde{f}_\alpha ^U(x)\right) d\alpha$.

Proof

According to the hypothesis (i), at $\bar{x}$ we have

$$\begin{aligned} \nabla \phi (\bar{x}) = \int _0^1 \left( \nabla \widetilde{f}_\alpha ^L(\bar{x}) + \nabla \widetilde{f}_\alpha ^U(\bar{x})\right) d \alpha = 0\end{aligned}$$

From hypothesis (ii), the Hessian matrix

$$\begin{aligned} \nabla ^2 \phi (\bar{x}) = \int _0^1 \nabla ^2 \widetilde{f}_\alpha ^L(\bar{x}) d \alpha + \int _0^1 \nabla ^2 \widetilde{f}_\alpha ^U(\bar{x}) d \alpha \end{aligned}$$

is positive definite and a symmetric matrix. With the help of hypothesis (iii), the function

$$\begin{aligned} \nabla ^2 \phi (x) = \int _0^1 \nabla ^2 \widetilde{f}_\alpha ^L(x) d \alpha + \int _0^1 \nabla ^2 \widetilde{f}_\alpha ^U(x) d \alpha \end{aligned}$$

is found to be Lipschitzian at $\bar{x}$ with constant $\gamma ^L + \gamma ^U$. Mathematically, there exists a neighborhood $N_\epsilon (\bar{x})$ where

$$\begin{aligned} \left\| \nabla ^2 \phi (x) - \nabla ^2 \phi (\bar{x})\right\| \le (\gamma ^L + \gamma ^U) \Vert x - \bar{x}\Vert \;\;\forall ~x \in N_\epsilon (\bar{x}). \end{aligned}$$

Towards proving the result, the following equivalence will be proved:

$$\begin{aligned}&~ \lim _{k \rightarrow \infty } \frac{\left\| \left[ A_{k+1}^{-1} - \nabla ^2 \phi (\bar{x})\right] \delta _k\right\| }{\Vert \delta _k\Vert } = 0\\ \Longleftrightarrow&~ \lim _{k \rightarrow \infty } \frac{\Vert \varPhi _{k+1}\Vert }{\Vert \delta _k\Vert } = 0 \\ \Longleftrightarrow&~ \lim _{k \rightarrow \infty } \frac{\Vert x_{k+1} - \bar{x}\Vert }{\Vert x_{k} - \bar{x}\Vert } = 0. \end{aligned}$$

With the help of quasi-Newton Eq. (1), we have

$$\begin{aligned}&~ \left[ A_{k+1}^{-1} - \nabla ^2 \phi (\bar{x})\right] \left[ x_{k+1} - x_k \right] \\ =&~ - \varPhi _k - \nabla ^2 \phi (\bar{x}) (x_{k+1} - x_k) \\ =&~ \left( \varPhi _{k+1} - \varPhi _k - \nabla ^2 \phi (\bar{x}) (x_{k+1} - x_k)\right) - \varPhi _{k+1}. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{\left\| \varPhi _{k+1}\right\| }{\Vert \delta _k\Vert } \le \frac{1}{\Vert \delta _k\Vert } \left[ \left\| \left( A_{k+1}^{-1} - \nabla ^2\phi (\bar{x}) \right) \delta _k \right\| +\left\| \varPhi _{k+1} - \varPhi _k - \nabla ^2 \phi (\bar{x}) \delta _k \right\| \right] . \end{aligned}$$

It is evident to note that

$$\begin{aligned}&~ \left\| \varPhi _{k+1} - \varPhi _k - \nabla ^2 \phi (\bar{x}) \delta _k\right\| \\ =&~ \left\| \left( \int _0^1 \nabla ^2 \left( \widetilde{f}_\alpha ^L + \widetilde{f}_\alpha ^U\right) (x_{k+1} + t (x_{k+1} - x_k)) dt\right) \delta _k - \nabla ^2 \phi (\bar{x}) \delta _k \right\| \\ =&~ \left\| \left( \int _0^1 \left( \nabla ^2 \phi (x_{k+1} + t \delta _k) - \nabla ^2 \phi (\bar{x})\right) dt\right) \delta _k \right\| \\ \le&~ (\gamma ^L + \gamma ^U)\left( \left\| x_{k+1} - \bar{x}\right\| + \Vert x_k - \bar{x}\Vert \right) . \end{aligned}$$

Since $\nabla ^2 \phi (\bar{x})$ is positive definite, we have $\xi > 0$ and $m \in \mathbb {N}$ such that

$$\begin{aligned} \Vert \varPhi _{k+1}\Vert = \Vert \varPhi _{k+1} - \nabla \phi (\bar{x})\Vert \ge \xi \Vert x_{k+1} - \bar{x}\Vert ~\text {for all}\,\,k\,\, \ge \,\,m. \end{aligned}$$

Hence, we now have

$$\begin{aligned} \frac{\Vert \varPhi _{k+1}\Vert }{\Vert \delta _k\Vert } \ge \frac{\xi \Vert x_{k+1} - \bar{x}\Vert }{\Vert x_{k+1} - \bar{x}\Vert + \Vert x_k - \bar{x}\Vert } = \xi \frac{c_k}{1 + c_k}, \end{aligned}$$

where $c_k = \frac{\Vert x_{k+1} - \bar{x}\Vert }{\Vert x_k - \bar{x}\Vert }$. This inequality gives

$$\begin{aligned}&~\lim _{k \rightarrow \infty } \frac{c_k}{1 + c_k} = 0 \\ \Longrightarrow&~\lim _{k \rightarrow \infty } c_k = 0 \\ \Longrightarrow&~\lim _{k \rightarrow \infty } \frac{\Vert x_{k+1} - \bar{x}\Vert }{\Vert x_k - \bar{x}\Vert } = 0. \end{aligned}$$

This completes the proof of superlinear convergence of the sequence $\{x_k\}$ in Algorithm 1.

Conversely, since $\{x_k\}$ converges superlinearly to $\bar{x}$ and $\nabla \phi (\bar{x}) = 0$, we must have $\beta > 0$ and $p \in \mathbb {N}$ such that

$$\begin{aligned} \Vert \varPhi _{k+1} \Vert \le \beta \Vert x_{k+1} - \bar{x}\Vert \text { for all } k \ge p. \end{aligned}$$

Again due to superlinear convergence of $\{x_k\}$, we have

$$\begin{aligned} 0 = \lim _{k \rightarrow \infty } \frac{\Vert x_{k+1} - \bar{x}\Vert }{\Vert x_k - \bar{x}\Vert } \ge \lim _{k \rightarrow \infty } \frac{\Vert \varPhi _{k+1}\Vert }{\beta \Vert x_k - \bar{x}\Vert } = \lim _{k \rightarrow \infty } \tfrac{1}{\beta } \frac{\Vert \varPhi _{k+1}\Vert }{\Vert x_{k+1} - x_k\Vert } \frac{\Vert x_{k+1} - x_k\Vert }{\Vert x_k - \bar{x}\Vert }. \end{aligned}$$

Since $\lim _{k \rightarrow \infty } \frac{\Vert x_{k+1} - x_k\Vert }{\Vert x_k - \bar{x}\Vert } = 1$, this inequality implies $\lim _{k \rightarrow \infty } \frac{\Vert \varPhi _k\Vert }{\Vert x_{k+1} - x_k\Vert } = 0$. Hence, the result follows.

5 Illustrative Example

In this section, an illustrative example is presented to explore the computational procedure of Algorithm 1.

Example 1

Consider the following quadratic fuzzy optimization problem:

Let us consider the initial approximation to the minimizer as $x_0 = \left( x_1^0, x_2^0\right) = (0, 0)$. With the help of fuzzy arithmetic, the lower and the upper function can be obtained as

$$\begin{aligned} \widetilde{f}_\alpha ^L(x_1, x_2) = {\left\{ \begin{array}{ll} \tfrac{1 + \alpha }{2} x_1^2 + \tfrac{\alpha }{2} x_2^2 + \tfrac{1+\alpha }{2} x_1 x_2 + \tfrac{\alpha }{2} x_1 - (1 - \tfrac{\alpha }{2}) x_2 &{} \text {if}\,\, x_1 \ge 0, x_2\ge 0\\ \tfrac{1 + \alpha }{2} x_1^2 + \tfrac{\alpha }{2} x_2^2 + \tfrac{3-\alpha }{2} x_1 x_2 + \tfrac{\alpha }{2} x_1 - \tfrac{\alpha }{2} x_2 &{} \text {if}\,\, x_1 \ge 0, x_2\le 0\\ \tfrac{1 + \alpha }{2} x_1^2 + \tfrac{\alpha }{2} x_2^2 + \tfrac{3-\alpha }{2} x_1 x_2 + (1-\tfrac{\alpha }{2}) x_1 - (1 - \tfrac{\alpha }{2}) x_2 &{} \text {if}\,\, x_1 \le 0, x_2\ge 0\\ \tfrac{1 + \alpha }{2} x_1^2 + \tfrac{\alpha }{2} x_2^2 + \tfrac{1+\alpha }{2} x_1 x_2 + (1-\tfrac{\alpha }{2}) x_1 - \tfrac{\alpha }{2} x_2 &{} \text {if}\,\, x_1 \le 0, x_2\ge 0\\ \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \widetilde{f}_\alpha ^U(x_1, x_2) = {\left\{ \begin{array}{ll} \tfrac{3 - \alpha }{2} x_1^2 + (1-\tfrac{\alpha }{2}) x_2^2 + \tfrac{3-\alpha }{2} x_1 x_2 + (1- \tfrac{\alpha }{2}) x_1 - \tfrac{\alpha }{2} x_2 &{} \text {if}\,\, x_1 \ge 0, x_2\ge 0\\ \tfrac{3- \alpha }{2} x_1^2 + (1-\tfrac{\alpha }{2}) x_2^2 + \tfrac{1+\alpha }{2} x_1 x_2 + (1-\tfrac{\alpha }{2}) x_1 - (1-\tfrac{\alpha }{2}) x_2 &{} \text {if}\,\, x_1 \ge 0, x_2\le 0\\ \tfrac{3 - \alpha }{2} x_1^2 + (1-\tfrac{\alpha }{2}) x_2^2 + \tfrac{1+\alpha }{2} x_1 x_2 + \tfrac{\alpha }{2} x_1 - \tfrac{\alpha }{2} x_2 &{} \text {if}\,\,x_1 \le 0, x_2\ge 0\\ \tfrac{3- \alpha }{2} x_1^2 + (1-\tfrac{\alpha }{2}) x_2^2 + \tfrac{3-\alpha }{2} x_1 x_2 + \tfrac{\alpha }{2} x_1 - (1-\tfrac{\alpha }{2}) x_2 &{} \text {if}\,\,x_1 \le 0, x_2\ge 0.\\ \end{array}\right. } \end{aligned}$$

Therefore,

$$\begin{aligned} \phi (x_1, x_2) = \int _0^1 \left( \widetilde{f}_\alpha ^L(x_1, x_2) + \widetilde{f}_\alpha ^U(x_1, x_2)\right) d\alpha = 2 x_1^2 + x_2^2 + 2x_1x_2 + x_1 - x_2. \end{aligned}$$

Here

$$\begin{aligned} \nabla \phi (x_1, x_2) = \begin{bmatrix} 4x_1 + 2x_2 + 1\\ 2x_1 + 2x_2 - 1 \end{bmatrix}. \end{aligned}$$

Considering the initial matrix $A_0 = I_2$, we calculate the sequence $\{x_k\}$, $x_k = \left( x_1^k, x_2^k\right)$, through the following equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} d_k &{} =~ -A_k \begin{bmatrix} 4x_1^k + 2x_2^k + 1\\ 2x_1^k + 2x_2^k - 1 \end{bmatrix}\\ \alpha _k &{} =~ \mathop {\text {argmin}}\nolimits _{\alpha \ge 0}~ \phi (x_k + \alpha d_k)\\ \delta _k &{} =~ \alpha _k d_k \\ x_{k+1} &{} =~ x_k + \delta _k \\ \varPhi _k &{} =~ \nabla \phi (x_{k+1}) - \nabla \phi (x_k) = \begin{bmatrix} 4(x_1^{k+1} - x_1^k) + 2(x_2^{k+1}-x_1^k)\\ 2(x_1^{k+1} - x_1^k) + 2(x_2^{k+1}-x_1^k \end{bmatrix}, \text { and } \\ A_{k+1} &{} =~ A_k + \frac{\delta _k \delta _k^t}{\delta _k^t \varPhi _k} - \frac{A_k \varPhi _k \varPhi _k^t A_k}{\varPhi _k^t A_k \varPhi _k}. \end{array}\right. } \end{aligned}$$

The initial iteration ($k = 0$)

$$\begin{aligned} {\left\{ \begin{array}{ll} x_0 &{}=~ (0, 0)\\ A_0 &{}=~ \begin{bmatrix} 1 &{} 0 \\ 0 &{} 1 \end{bmatrix} \\ \Vert \nabla \phi (x_0)\Vert &{}=~ \left\| (1, -1)^t \right\| = \sqrt{2} \ne 0 \\ d_0 &{}=~ - \begin{bmatrix} -1\\ 1\end{bmatrix} \\ \alpha _0 &{}=~ 1 \\ x_1 &{}=~ (-1, 1)\\ \delta _1 &{}=~ (-1, 1)\\ \varPhi _1 &{}=~ (-2, 0)\\ A_1 &{}=~ \begin{bmatrix} \tfrac{1}{2} &{} -\tfrac{1}{2} \\ -\tfrac{1}{2} &{} \tfrac{3}{2} \end{bmatrix}. \end{array}\right. } \end{aligned}$$

The second iteration ($k = 1$)

$$\begin{aligned} {\left\{ \begin{array}{ll} x_1 &{}=~ (-1, 1)\\ d_1 &{}=~ (0, 1)\\ \alpha _1 &{}=~ \tfrac{1}{2} \\ x_2 &{}=~ (-1, \tfrac{3}{2})\\ \nabla \phi (x_2) &{}=~ \begin{bmatrix} 0\\ 0 \end{bmatrix}. \end{array}\right. } \end{aligned}$$

Hence $\bar{x} = (-1, \tfrac{3}{2})$ is a nondominated solution of the considered problem. It is important to note that the method converged at the second iteration since the objective function is a quadratic fuzzy function.

6 Conclusion

In this paper, a quasi-Newton method with rank-two modification has been derived to find a non-dominated solution of an unconstrained fuzzy optimization problem. In the optimality concept, the fuzzy-max ordering of a pair of fuzzy numbers has been used. The gH-differentiability of fuzzy functions have been employed to find the non-dominated solution point. An algorithmic implementation and the convergence analysis of the proposed technique has also been presented. The technique is found to have superlinear convergence rate. A numerical example has been explored to illustrate the proposed technique.

It is to note that unlike Newton method for fuzzy optimization problems [8, 23], the proposed method is derived without using the inverse of the concerned Hessian matrix. Instead, a sequence of positive definite inverse Hessian approximation $\{A_k\}$ is used which satisfies the quasi-Newton equation $A_{k+1} \varPhi _k = \delta _k$. Thus the derived method sidestepped the inherent computational difficulty to compute inverse of the concerned Hessian matrix. In this way the proposed method is made more efficient than the existing Newton method [8, 23].

The method can be easily observed as much similar to the classical DFP method [28] for conventional optimization problem. In the analogous way of the presented technique, it can be further extended to a BFGS-like method [23] for fuzzy optimization. Instead of the rank-two modification in the approximation of the inverse of the Hessian matrix, a rank-one modification can also be done. It is also to be observed that in Algorithm 1, we make use the exact line search technique along the descent direction $d_k = - A_k \nabla \phi (x_k)$. However, an inexact line search technique [28] could have also been used. A future research on this rank-one modification and inexact line search technique for fuzzy optimization problem can be performed and implemented on Newton and quasi-Newton method.