Inexact steepest descent algorithm for obtaining t-best approximation in a fuzzy normed space

Abstract

The main aim of this paper is to find t-best approximation in a fuzzy normed space via the optimization formulation. It is formulated as a constrained minimization problem which is solved by using penalty method. Each penalty is solved by using inexact steepest descent algorithm.

Keywords

t-best approximation fuzzy normed space steepest descent

1 Introduction

L. A. Zadeh [43] introduced the fuzzy sets theory in 1965. Later many researchers have applied this theory to the well-known results in the classical set theory. Therefore it has become an area of active research for nearly forty years. The fuzzy logic has been used not only in many engineering applications, for example, in the population dynamics [4], in the quantum physics [10], in the control of chaos [16], in decision making [38, 39], and in bifurcation of nonlinear dynamical systems [20], but also in various branches of mathematics, such as, in metric and topological spaces [1 , 22], in the approximation theory [2], in functional equations [15, 42], in the theory of functions [7, 21], in stability problem [26, 31], in fuzzy random variables [37], convergence of sequences of fuzzy real numbers [19], in statistical convergence [28, 30], and in nonlinear operators [29 , 40]. There are many studies in the field of fuzzy topology. Since, fuzzy topology has very important applications in the quantum particle physics [11, 12]. One of the most important problems in fuzzy topology is to obtain an appropriate concept of fuzzy metric space. This problem has been investigated by many authors [3 , 35] from the different points of view. Beg et al. used altering distance function to extend the notion of (φ, ψ)-weak contraction to intuitionistic fuzzy metric spaces [6]. In particular, George and Veeramani [17] have introduced and studied a notion of fuzzy metric space with the help of continuous t-norms, which constitutes a slight but appealing modification of the one due to Kramosil and Michalek [23]. Veeramani [36] in 2001 introduced the concept of t-best approximation in fuzzy metric spaces and Vaezpour and Karimi [24] introduced the concept of t-best approximation in fuzzy normed spaces. Recently Yilmaz introduced and studied the notion of weak and strong Schauder bases in fuzzy normed spaces [41]. In [18] Golet introduced a generalization of the concept of fuzzy normed spaces which includes some earlier defined fuzzy normed spaces as special cases. Eshaghi et al. established some stability results concerning the Cauchy-Jensen functional equation in generalized fuzzy normed spaces [15]. Also Eshaghi and Abbaszadeh surveyed Hyers-Ulam stability of Cauchy-Jensen functional inequality in fuzzy Banach spaces [14]. Mohiuddine in [27] defined a fuzzy 2-normed space and studied the concept of best approximation in fuzzy 2-normed spaces

On the other hand the problems of best approximation were initiated first by P. L. Chyebyshev in 1853. Problems of this kind deal with the search for a function in a prescribed class which has the least deviation from a given function, as measured in a prescribed metric. The theory of best approximation is mainly concerned with the questions of existence, uniqueness, characterizations and qualitative properties of the minimizing functions. It also deals with the continuity properties of the metric projection and the selection for a metric projection. There is a computational side of the theory which seeks algorithms that grind-out a sequence of functions which converge to a minimizing function, estimates rapidity of convergence, etc.

However there is no systematic way to obtain the t-best approximation in a given fuzzy normed space. In this paper the problem of finding t-best approximation will be considered as a minimization problem which is solved by using inexact steepest descent algorithm.

Definition 1.1. [24] A binary operation * : [0, 1] × [0, 1] → [0, 1] is said to be a continuous t-norm if ([0, 1] , *) is a topological monoid with unit 1 such that a * b ≤ c * d whenever a ≤ c and b ≤ d (a, b, c, d ∈ [0, 1]).

Definition 1.2. [35] The 3-tuple (X, N, *) is said to be a fuzzy normed space if X is a vector space, * is a continuous t-norm and N is a fuzzy set on X × (0, ∞) satisfying the following conditions for every x, y ∈ X and t, s > 0:

N (x, t) >0,

N (x, t) =1 ⇔ x = 0,

N (αx, t) = N (x, t/|α|), for all α ≠ 0,

N (x, t) * N (y, s) ≤ N (x + y, t + s),

N (x, .) : (0, ∞) ⟶ [0, 1] is continuous,

Lim_t→∞N (x, t) =1.

Definition 1.3. [36] Let A be a nonempty subset of a fuzzy normed space (X, N, *), for x ∈ X, t > 0 and let $d (A, x, t) = Sup {N (y - x, t) : y \in A} .$ (1)

An element y₀ is said to be a t-best approximation of x from A if $N (y_{0} - x, t) = d (A, x, t) .$ (2)

The paper is organized as follows: Section 2 introduces the proposed problem that must be solved. Section 3 presents the algorithm for solving the proposed problem and proves the algorithm finds the optimal solution. Finally, in Section 4 numerical examples are given to demonstrate the convergence and accuracy of the algorithm.

2 Problem statement

The constrained optimization formulation of the problem (1) is

$\begin{matrix} Max & f (x; t) & = N (x - x_{0}, t), \\ S . t . \\ x \in A & , \end{matrix}$ (3) where N (x - x₀, t) is the objective function and the set A is said to be the constraint set. Some important methods for constrained optimization replace the original problem by a sequence of unconstrained optimization problems in which the constraints are represented by terms added to the objective functions of the form $f (x) + \frac{C}{2} P (x)$ where P (x) named penalty function [25]. Then this unconstrained optimization problem is solved by the steepest descent method [5]. Now the above problem must be reformulated as an unconstrained optimization problem. By applying the penalty method [33], one can get,

$\begin{matrix} Max & F (x; t) & = N (x - x_{0}, t) - \frac{C}{2} P (x), \\ S . t . \\ x \in X & . \end{matrix}$ (4) where P and C are called the penalty function and parameter, respectively. $C = {c_{k}}_{k = 1}^{\infty}$ is chosen to be an increasing sequence of positive numbers [25].

3 Inexact steepest descent algorithm and its convergence

The method of steepest descent, proposed by Cauchy in 1847, is one of the most fundamental procedure for minimizing function of severable variables. The convergence analysis for an inexact steepest descent algorithm applied to a function $h : ℝ^{n} \to ℝ$ will be presented as follows.

Theorem 1.Let $h : ℝ^{n} \to ℝ$ be such that its gradient ∇h (x) is Lipschitz continuous with constantG > 0, onS (x₀) = {x : h (x) ≤ h (x₀)} for some given $x_{0} \in ℝ^{n}$ . Pick some fixed-step length parameter $\bar{λ}$ , and let 0 < ɛ < 1. Given any iteratex_k, define the search directiond_k = - ∇ h (x_k), and consider Armijou’s function $\hat{θ} (λ) = θ (0) + λ ɛ θ^{'} (0)$ , λ ≥ 0, whereθ (λ) = h (x_k + λd_k) is the line search function. ifd_k = 0, then stop. Otherwise find the smallest integerγ ≥ 0 for which $θ (\bar{λ} / 2^{γ}) \leq \hat{θ} (\bar{λ} / 2^{γ})$ and define the next iterate asx_k+1 = x_k + λ_kd_k, whereλ_k = λ/2^γ. Now suppose that starting with some iteratex₀, this procedure produces a sequence of iterationsx₀, x₁, x₂, ⋯. Then either the procedure terminates finitely with ∇h (x_k) =0 for somek, or else an infinite sequence {x_k} is generated such that the corresponding sequence ∇h (x_k) →0, ask→ ∞.

Proof: Refer to [5].

Let $X = ℝ^{n}$ . For a, b ∈ [0, 1], let a * b = ab. Suppose that $N : ℝ^{n} \times (0, \infty) \to [0, 1]$ , let (X, N, *) be a fuzzy normed space. Also let A ⊆ X be the constraint set. Then one can write $f (x, t) = N (x - x_{0}, t), x \in X = ℝ^{n} .$ (5)

In general the set A has the following form [5],

$\begin{matrix} g_{i} (x) \leq 0, i = 1, 2, \dots, m, \\ h_{i} (x) = 0, i = 1, 2, \dots, l . \end{matrix}$ (6)

For these constraints, a suitable penalty function P (x) is given by $P (x) = \sum_{i = 1}^{m} φ (g_{i} (x)) + \sum_{i = 1}^{l} ψ (h_{i} (x)),$ (7) where $φ (x) = max (0, x)^{4}, ψ (x) = x^{2} .$ (8)

Now, one can formulate the problem of obtaining t-best approximation in a fuzzy normed space as an optimization problem as follows:

$\begin{matrix} Max & f (x; t) \\ S . t . \\ g_{i} (x) \leq 0, i = 1, 2, \dots, m, ∥ \\ h_{i} (x) = 0, i = 1, 2, \dots, l . \end{matrix}$ (9)

By the penalty function (7) and changing maximization to minimization, one can get

$\begin{matrix} Min & F (x; t) = - f (x; t) + \frac{C}{2} P (x), \\ S . t . \\ x \in X = ℝ^{n} . \end{matrix}$ (10)

Now, suppose that f, g_i, i = 1, ⋯ , m and h_j, j = 1, 2, ⋯ , l, are 2-times continuously differentiable on any compact set in $ℝ^{n}$ . By the mean value theorem [34], ∇f, ∇g_i, i = 1, ⋯ , m and ∇h_j, j = 1, 2, ⋯ , l, are Lipschitz continuous. Therefore, for some x ∈ S (x₀), one can get $| \nabla f (x) - \nabla f (x_{0}) | \leq M_{0} ∥ x - x_{0} ∥,$ (11) $| \nabla g_{i} (x) - \nabla g_{i} (x_{0}) | \leq M_{i}^{'} ∥ x - x_{0} ∥,$ (12) $\begin{matrix} i = 1, 2, \dots, m \\ | \nabla h_{j} (x) - \nabla h_{j} (x_{0}) | \leq M_{j}^{″} ∥ x - x_{0} ∥, \\ j = 1, 2, \dots, l, \end{matrix}$ (13) then, one can obtain,

$\begin{matrix} | \nabla F (x) - \nabla F (x_{0}) | \leq | \nabla f (x) - \nabla f (x_{0}) | \\ + \frac{C}{2} {max | φ^{'} | \sum_{i = 1}^{m} | \nabla g_{i} (x) - \nabla g_{i} (x_{0}) | \\ + max | ψ^{'} | \sum_{J = 1}^{L} | \nabla h_{j} (x) - \nabla h_{j} (x_{0}) |} \\ \leq M ∥ x - x_{0} ∥, \end{matrix}$ (14) where $M = M_{0} + \frac{C}{2} {max | φ^{'} | \sum_{i = 1}^{m} M_{i}^{'} + max | ψ^{'} | \sum_{j = 1}^{L} M_{j}^{″}} .$ (15)

Then, ∇F (x) satisfies the Lipschitz condition on S (x₀) for some $x_{0} \in ℝ^{n}$ . Now by theorem mentioned above, one can conclude that the inexact steepest descent algorithm is convergent to the optimal solution for the t-best approximation problem.

4 Numerical results

In this section, to demonstrate the efficiency of the method for obtaining the t-best approximation of a point from a given set, the numerical examples will be solved. The results are presented in some tables.

Example 1. [24] Let $X = ℝ^{2}$ . For a, b ∈ [0, 1], let a * b = ab. Define $N : ℝ^{2} \times (0, \infty) \to [0, 1]$ by $N ((x_{1}, x_{2}), t) = \frac{1}{\exp (\frac{\sqrt{(} x_{1}^{2} + x_{2}^{2})}{t})} .$ (16)

Then (X, N, *) is a fuzzy normed space. Let $A = {(x_{1}, x_{2}) \in ℝ^{2} | 0 \leq x_{1} \leq 1, 0 \leq x_{2} \leq x_{1}^{2}}$ , and x₀ = (0, 3). Then for every t > 0,

$\begin{matrix} d (A, (0, 3), t) = \\ Sup {N ((x_{1}, x_{2}) - (0, 3), t) \\ : 0 \leq x_{1} \leq 1, 0 \leq x_{2} \leq x_{1}^{2}} . \end{matrix}$ (17)

The exact solution of this problem is (1, 1) [24]. So, for every t > 0, y₀ = (1, 1) is t-best approximation of (0, 3) from A. By the process discussed in the preceding section, one can rewrite (17) as a minimization problem,

$\begin{matrix} Min & f ((x_{1}, x_{2}), t) = \\ - \exp (\frac{\sqrt{(} x_{1}^{2} + (x_{2} - 3)^{2})}{t})^{- 1}, \\ S . t . \\ g_{1} (x_{1}, x_{2}) = x_{2} - x_{1}^{2} \leq 0, \\ g_{2} (x_{1}, x_{2}) = x_{1} - 1 \leq 0, \\ g_{3} (x_{1}, x_{2}) = - x_{1} \leq 0, \\ g_{4} (x_{1}, x_{2}) = - x_{2} \leq 0 . \end{matrix}$ (18)

This is a constrained optimization problem. In order to use the steepest descent (SD) algorithm, it must be transformed to an unconstrained one. This transformation is done by using a penalty function. Precisely f and g_i, i = 1, 2, 3, 4, satisfy the assumptions of theorem (1) mentioned in the preceding section. Thus the convergence to the optimal solution is guaranteed. Hence, the above problem is written as:

$\begin{matrix} Min F ((x_{1}, x_{2}), t; C_{k}), \\ S . t . \\ (x_{1}, x_{2}) \in ℝ^{2} . \end{matrix}$ (19) where

$\begin{matrix} F ((x_{1}, x_{2}); C_{k}) = \\ - \exp (\frac{\sqrt{(} x_{1}^{2} + (x_{2} - 3)^{2})}{t})^{- 1} \\ + \frac{C_{k}}{2} \sum_{i = 1}^{i = 4} φ (g_{i} (x_{1}, x_{2})), \end{matrix}$ (20) in which φ is the same as in Equation. (8) and C_k is the penalty parameter. To simulate the penalty problem (19) using SD algorithm a sequence of subproblems is chosen based on the value of the penalty parameter, that is, {C₁, C₂, C₃, C₄, C₅, ⋯ , C₃₅} = {10³, 10⁴, 10⁵, 10⁶, 10⁷, ⋯ , 10⁷}. The output point of each subproblem is used as the initial point for the next subproblem.

By choosing some initial points and running the SD algorithm, one can obtain the corresponding optimal solutions and their elapsed time, these results are shown in Table 1.

The penalty problem (19) was also simulated with Nelder-Mid (NM) algorithm [5]. For this algorithm one problem is solved with C = 10⁷ and the results of the NM algorithm and the elapsed time for each of the initial points are shown in Table 2. According to the data in Tables 1, 2 one can simply state that the the SD algorithm is more accurate than the NM algorithm but in terms of computational time it is seen that the NM algorithm is faster than SD one. The reason is that NM is a search algorithm that only uses function evaluations but SD is a derivative based algorithm. On the other hand the data in the Table 3 shows that the conjugate gradient (CG) algorithm [25] fails to get the optimal solutions when the same sequence of subproblems is run as in the case of SD one. Taking small nearly orthogonal step the SD algorithm has poor convergence and suffers from the zigzagging phenomenon specially at later stages.

In spite of the above mentioned weaknesses, the SD algorithm is the first algorithm to be used for solving any new problem. Also it is a benchmark to test new algorithms and many algorithms have been introduced to overcome some of its weaknesses. To be convergent, the Newton’s algorithm needs the initial point be sufficiently close to the optimal solution but SD one lacks this shortcoming, that is, Newton’s algorithm is locally convergent but SD is globally convergent.

Example 2. Let $X = ℝ$ . For a, b ∈ [0, 1] and a * b = ab. Define: $N : ℝ \times (0, \infty) \to [0, 1],$ (21) by $N (x, t) = \frac{t}{t + | x |} .$ (22)

Then (X, N, *) is a fuzzy normed space. Let A = [0, 1] and x₀ ∈ X, t > 0. Then

$\begin{matrix} d (A, x_{0}, t) = SupN (x - x_{0}, t) \\ = {Sup}_{0 \leq x \leq 1} \frac{t}{t + | x - x_{0} |} . \end{matrix}$ (23)

The objective is to obtain a point or points that are the t-best approximation of x₀ via the optimization process. Reformulating the above problem as an optimization problem yields

$\begin{matrix} Max & f (x; t) = \frac{t}{t + | x - x_{0} |}, \\ S . t . \\ 0 \leq x \leq 1, x \in ℝ . \end{matrix}$ (24)

Or equivalently,

$\begin{matrix} Max & f (x; t) = \frac{t}{t + | x - x_{0} |}, \\ S . t . \\ g_{1} (x) = x - 1 \leq 0, \\ g_{2} (x) = - x \leq 0, x \in ℝ . \end{matrix}$ (25)

Precisely f and g_i, i = 1, 2, satisfy the assumptions of theorem 1 mentioned in the preceding section. Thus the convergence to the optimal solution is guaranteed. Interchanging maximization to minimization and using the penalty function, one can get:

$\begin{matrix} Min & F (x; t) = - \frac{t}{t + | x - x_{0} |} \\ + \frac{C}{2} \sum_{i = 1}^{2} Φ (g_{i} (x)), \\ S . t . \\ x \in ℝ . \end{matrix}$ (26)

To run this problem using the SD algorithm two subproblems are chosen based on the value of the penalty parameter C_k, that is, {C₁, C₂} = {10⁵, 10⁵}. But in case of NM and CG algorithms only one problem is solved with C₁ = 10⁵.

Case 1: Let x₀ > 1. Then, for every t > 0, y₀ = 1 is the t-best approximation of x₀ > 1 from A. One can find this point as an optimal solution of the following minimization problem. Now, let for example x₀ = 2

$\begin{matrix} Min & F (x; t) = - \frac{t}{t + | x - 2 |} \\ + \frac{C}{2} \sum_{i = 1}^{2} Φ (g_{i} (x)), \\ S . t . \\ x \in ℝ . \end{matrix}$ (27)

Choosing some initial points, one can compute the optimal solution of the minimizing problem or the t-best approximation of x₀ = 2. Table 4 shows the optimal solutions and their elapsed time which are obtained by SD algorithm. One can simply see that the theoretical predict is true. In Table 5 the optimal solution and related elapsed time of every initial point, which are computed by NM algorithm, are given.

The comparison of the optimal solutions and the elapsed time in Tables 4, 5 shows that both SD and NM algorithms have almost the same accuracy and computational time. However comparing these results with that of CG algorithm, shown in Table 6, reveals that the CG algorithm is both the fastest and the most accurate algorithm.

Case 2: Let x₀ < 0. Then for every t > 0, t-best approximation of x₀ < 0 from A is y₀ = 0. For example, by choosing x₀ = -3, one can find the t-best approximation of this point as the optimal solution of the following minimization problem:

$\begin{matrix} Min & F (x; t) = - \frac{t}{t + | x + 3 |} \\ + \frac{C}{2} \sum_{i = 1}^{2} Φ (g_{i} (x)), \\ S . t . \\ x \in ℝ . \end{matrix}$ (28)

The results in Tables 7 and 8 are obtained using the SD and NM algorithm respectively. The comparison of these data related to the two algorithms shows that the accuracy of the two methods is the same. Also they have almost the same computational time. Once again, by Table 9, it is seen that CG is the most accurate and the fastest algorithm for solving the problem arising in finding t-best approximation in a fuzzy normed space.

5 Conclusion

In this paper, a novel approach was introduced for obtaining the t-best approximation of a point from a given set in a fuzzy normed space. At first, this process was reformulated as a constrained maximization problem. Then, by using the penalty method and interchanging maximization to minimization, the t-best approximation problem had been written as an unconstrained minimization problem. At last, it was solved using the inexact steepest descent algorithm. Moreover, to discuss the efficiency of this approach two numerical examples were solved using the mentioned algorithm. As future researches, p-best approximation in fuzzy normed space and t-best approximation in intuitionistic fuzzy normed spaces can also be formulated as a constrained optimization problem. Since steepest descent algorithm has almost low convergence rate for penalty problem, other optimization algorithms rather than steepest descent can be used to solve t- best approximation in fuzzy normed spaces either in constrained case or in unconstrained one. For constrained optimization problems one can use primal methods such as the gradient projection method and the reduced gradient method or use dual methods such as the augmented Lagrangian methods.

Footnotes

Acknowledgments

The authors would like to thank the referees for their helpful comments and a careful reading of the article.

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