Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 366 https://internationalpubls.com Comparison Between Cauchy's Steepest Descent and Fletcher-Reeves Methods in Solving Unconstrained Neutrosophic Nonlinear Programming Problems G. Vanaja1 and K. Ganesan2* 1,2*Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, SRM Nagar, Kattankulathur, Tamilnadu, India; vg5308@srmist.edu.in. *Correspondence: ganesank@srmist.edu.in Article History: Received: 17-09-2024 Revised: 25-10-2024 Accepted: 05-11-2024 Abstract: The main objective of this study is to solve unconstrained single-valued Neutrosophic Nonlinear Programming Problems using Cauchy’s Steepest Descent Method(CSDM) and the Fletcher-Reeves Method(FRM). Our approach is based on new arithmetic operations and ranking on the parametric representations of Triangular Neutrosophic Numbers(TNN). We prove some important theorems for Cauchy’s Steepest Descent Method and the Fletcher-Reeves Method. Numerical examples are presented to illustrate the theory developed in this article. The outcomes of the proposed methods are compared. Keywords: Cauchy’s Steepest Descent Method; Fletcher Reeves method; Unconstrained optimization; Single-valued neutrosophic number; Arithmetic operations and Ranking 1. Introduction The notion of fuzzy sets, proposed by Zadeh [27], has played a vital role in addressing real-world challenges. Smarandache [21] introduced neutrosophic sets to shade the problem of fuzzy sets that handles less and inconsistent information. It was invented as an protracted system of fuzzy, intuitionistic and traditional sets. Here indeterminacy is included together with membership and non- membership. Hence each element is associated with three parameters i.e., truth, indeterminacy, and falsity. The neutrosophic sets and logic are the utmost way of handling real-life problems, which obliges a generalization of fuzzy sets. Al-Naemi [1] introduced a novel parameter 𝛜𝑘 𝐺ℎ derived from the memoryless self-scaling DFP quasi-Newton method. The author demonstrated that any line search technique ensuring sufficient descent property can be employed, and further established the validity of the Zoutendijk condition, proving the method’s global convergence through a specific step-length approach. Using imprecise parameters to manage independent parameters. According to Andrei [2], an iterative process comprises two distinct components: determining a descent search direction 𝑑𝑘followed by a line search to identify an appropriate step size 𝛌𝑘. The search direction in the CSDM is precisely the negative gradient. Antczak [3] discussed an optimization problem involving a fuzzy objective function with inequality and equality constraints, all of which had locally Lipschitz functions. They successfully demonstrated that for this particular non-differentiable fuzzy optimization problem, they could create an associated bi-objective optimization problem. They also established a relationship between the vector optimization problem of Pareto solutions and the non- dominated (weakly) solutions of the original non-differentiable fuzzy optimization problem. Basim Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 367 https://internationalpubls.com [4] the aim of this work has been to develop new, modified conjugate gradient formulas that perform better than the conventional FR-CG method for picture restoration. The outcomes validate the efficacy of the approach employed in this study to generate variations of the traditional CG procedure. Bellman et al. [5] delved into the field of decision-making processes under conditions of fuzziness. Chakraborty [6] discussed a trapezoidal neutrosophic number is a more useful concept than a triangular one. The removal of area method (i) and mean of interval method (ii) are two important de-neutrosophication methods that are effective in de-neutrosophicating the corresponding number. Hanachi et al. [7] introduced a novel hybrid approach in which the parameter is determined as a convex combination of three parameters 𝛜𝑘 𝐷𝑌, 𝛜𝑘 𝑃𝑅𝑃, 𝛜𝑘 𝑃𝑅 the adequacy of the descent and the global convergence has demonstrated. Practical results indicate that the chosen approach is better and more efficient when compared to the utilized methods. Hassan et al. [8] discussed that the research focus has centered on developing novel, modified conjugate gradient formulations that outperform the traditional FR CG approach in image restoration applications. The findings demonstrate the efficacy of the strategy employed in this research to derive variations of the traditional CG technique. The proposed methods have demonstrated global convergence under the rigorous Wolfe line search conditions. Husin et al. [9] presented a new steepest descent (SD) method, with the primary goal being the modification of the direction that has sufficient descent requirements and global convergence properties. The second strategy involves presenting regression analysis for real-world issues using the suggested modification of the SD method. Hepzibah et al. [10, 11] discussed how unconstrained optimization problems with fuzzy valued functions are taken into consideration as well as how to solve unconstrained optimization problems using Newton’s method with SVNTN coefficients. Moreover, single variable and multivariable fuzzy unconstrained optimization problems using Interval Newton’s Method are discussed with illustrations. Kaliyaperumal and Das [12] introduced a fuzzy version of the problem, which they addressed using the necessary and sufficient conditions of Lagrangian multipliers with a focus on fuzziness. They illustrated this approach with a numerical example. By solving two numerical examples on using membership functions (MFs) and the other using robust rankings they clarified the model’s effectiveness. This model is designed to tackle uncertainties and subjective experiences of decision- makers and can assist in resolving challenges associated with decision-making. Kanaya [13] discussed a method for solving multi-objective nonlinear programming problems involving fuzzy parameters in the objective functions. Utilizing an interactive cutting-plane algorithm, this method relies on the stability set corresponding to α-Pareto optimal solutions obtained through the same method. Comprehensive study on NLNNs and NLN-LPPs was presented by Lachhwani [14]. Based on the proposed modified possibility score function, the author developed a novel solution technique for NLN-LPPs. Nearly the whole range of NN values is covered by the suggested modified possibility score function. Maissam Jdid [15] demonstrated how binary integers can be used to convert certain nonlinear models into linear ones, resulting in integer solutions that are consistent with the nature of the problems being studied. Ming ma et al. [16] defined a new fuzzy arithmetic and applied to fuzzy linear equations. Nagoorgani and Sudha [17] studied the optimality requirements of fuzzy nonlinear unconstrained minimization issues. The cost coefficients are denoted by TFNs and illustrated with several numerical examples. An alternative definition for NLNN was provided by Reig-Mullor [18], which works over some of the limitations mentioned in the literature. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 368 https://internationalpubls.com Also, they developed the notions of 𝛌, 𝛜, 𝛟-cuts, possibilistic variance, possibilistic mean, and possibilistic standard deviation, which are among the primary characteristics of NLNN. Seikh Mijanur Rahaman [19] proposed solution methodologies for solving a unique matrix game, whereby the payoffs are expressed using single-valued neutrosophic numbers (SVNNs). Two non-linear multi-objective programming tasks are defined to determine the ideal values and strategies for the participants. These multi-objective programming issues are converted into bi-objective programming problems by assigning equal significance to the objective functions. Shilpa Ivin Emimal and Irene Hepzibah [20] introduced the optimization of intuitionistic fuzzy valued functions in unconstrained problems. The Interval Newton's Method using an Intuitionistic approach addresses both single and multivariable optimization problems. Fuzzy sets (FS) and neutrosophic sets (NS) were combined by Sujit Das [22] to formulate neutrosophic fuzzy sets (NFS). The primary characteristic of NFS is its ability to handle inconsistent and imprecise data, which is highly useful when managing uncertain and inconsistent real-world applications. Uma Maheswari and Ganesan [23] introduced a fuzzy interpretation of the KKT condition, specifically for FFNLPP, and successfully identified their optimal fuzzy outcomes. Utilizing the Gradient technique, also recognized as CSDM, they transformed it into an unconstrained optimization problem involving multiple fuzzy variables. Vanaja and Ganesan [24,25] developed an interior fuzzy penalty function method to solve Fuzzy Nonlinear Programming Problems (FNLPP). They introduced innovative fuzzy arithmetic and ranking techniques based on the parametric representation of triangular fuzzy numbers. Additionally, they applied the exterior penalty method using fuzzy-valued functions to solve these FNLPPs. Ye [26] presented the notions of NN linear and nonlinear functions, as well as inequalities that include indeterminacy I, along with some basic operations of neutrosophic numbers (NNs). Afterwards, the authors presented general solution techniques for a variety of optimization models for NN nonlinear programming (NN-NP) issues with unconstrained and restricted optimizations. In this paper, we discuss Single-Valued Neutrosophic Nonlinear Programming Problem (SVNNLPP) involving triangular neutrosophic numbers. We present a neutrosophic version of the Cauchy’s Steepest Descent Method (CSDM) and Fletcher- Reeves Method (FRM) approaches based on a new method of neutrosophic arithmetic operations on the parametric representations of neutrosophic numbers and ranking on the neutrosophic numbers of the parametric forms. We prove theorems for the solution of SVNNLPP without converting the given problem to its equivalent crisp form. Numerical examples are presented to illustrate the efficacy of the proposed solution methods. The rest of this article is organized as follows. Section 2 provides some basic preliminaries and results on neutrosophic numbers. Section 3 provides NNLPP. Theorems on CSDM and FRM are provided in section 4. Section 5 presents the algorithms for CSDM and FRM. Section 6 deals with numerical examples. Result and discussion are given section 7. Finally, a brief conclusion is given in section 8. 2. Preliminaries We explored Neutrosophic definitions, which capture the core of Neutrosophy by presenting subtle viewpoints on concepts frequently experienced in situations of uncertainty. By employing these definitions, our objective is to shed light the offer valuable perspectives on its impact on theoretical frameworks. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 369 https://internationalpubls.com \textcolor{white}{``} Definition 2.1 Let Y be a universe of discourse. A neutrosophic set ￜ̃ￜ𝔫 in υ is characterized by a truth-membership function Tￜ̃ￜ𝔫 a indeterminacy-membership function Iￜ̃ￜ𝔫 and a falsity-membership function Fￜ̃ￜ𝔫. Tￜ̃ￜ𝔫(υ); Iￜ̃ￜ𝔫(υ) and Fￜ̃ￜ𝔫(υ) are are real standard elements of [0,1]. It can be written as ￜ̃ￜ𝔫 = {< υ, (Tￜ̃ￜ𝔫(υ), Iￜ̃ￜ𝔫(υ),Fￜ̃ￜ𝔫(υ)) > |υ ∈ Y}. The function Tￜ̃ￜ𝔫(υ); Iￜ̃ￜ𝔫(υ) and Fￜ̃ￜ𝔫(υ) are real standard or non standard subsets of ]0−, 1+[ \textcolor{white}{''} i.e.,Tￜ̃ￜ𝔫(υ), Iￜ̃ￜ𝔫(υ),Fￜ̃ￜ𝔫:Y →]0−, 1+[ . There is No restriction is applied on the sum of Tￜ̃ￜ𝔫(υ); Iￜ̃ￜ𝔫(υ) and Fￜ̃ￜ𝔫(υ), so 0 − ≀ supTￜ̃ￜ𝔫(υ),+supIￜ̃ￜ𝔫(υ),+supFￜ̃ￜ𝔫(υ) ≀ 3 + . For a fixed υ = υ ∈ Y,Tￜ̃ￜ𝔫(υ), Iￜ̃ￜ𝔫(υ),Fￜ̃ￜ𝔫(υ), i.e., in simply, Tￜ̃ￜ𝔫(υ), Iￜ̃ￜ𝔫(υ),Fￜ̃ￜ𝔫(υ) is called Neutrosophic Number (NN). \textcolor{white}{``} Definition 2.2 Let Y be a universe of discourse. A single valued neutrosophic set (SVNS) ￜ̃ￜ𝔫 over Y is an object having the form ￜ̃ￜ𝔫 = {< υ, (Tￜ̃ￜ𝔫(υ), Iￜ̃ￜ𝔫(υ),Fￜ̃ￜ𝔫(υ)) > |υ ∈ Y} where Tￜ̃ￜ𝔫(υ):Y → ]0−, 1+[, Iￜ̃ￜ𝔫(υ):Y →]0−, 1+[,Fￜ̃ￜn:Y →]0−, 1+[ are truth, indeterminacy and falsity membership with 0 ≀ Tￜ̃ￜ𝔫(υ),+Iￜ̃ￜ𝔫(υ),+Fￜ̃ￜ𝔫(υ) ≀ 3 for all υ ∈ Y. \textcolor{white}{''} Definition 2.3 A Single Valued Neutrosophic Number (SVNN) ￜ̃ￜ𝔫 = {< υ, (Tￜ̃ￜ𝔫(υ), Iￜ̃ￜ𝔫(υ),Fￜ̃ￜ𝔫(υ)) > |υ ∈ Y}, subset of a real line, is called generalised neutrosophic number if \textcolor{white}{``} 1. ￜ̃ￜ𝔫 is neut-normal. 2. ￜ̃ￜ𝔫 is neut-convex. 3. Tￜ̃ￜ𝔫(υ) is upper semi-continuous, Iￜ̃ￜ𝔫(υ) is lower semi continuous and Fￜ̃ￜ𝔫(υ) is lower semi continuous. 4. ￜ̃ￜ𝔫 is support, i.e. S(ￜ̃ￜ𝔫) = υ ∈ Y:Tￜ̃ￜ𝔫 > 0, Iￜ̃ￜ𝔫 < 1,Fￜ̃ￜ𝔫 < 1 is bounded. \textcolor{white}{''} Definition 2.4 A single valued neutrosophic number ￜ̃ￜ𝔫 is Triangular Neutrosophic Number (TNN) and is denoted by ￜ̃ￜ𝔫 = (〈𝔞T1 , 𝔞T2 , 𝔞T3 〉; 〈𝔞I1 , 𝔞I2 , 𝔞I3 〉; 〈𝔞F1 , 𝔞F2 , 𝔞F3 〉) having the membership function, indeterminacy function and non-membership function as follows.𝒶 ÎŒ T (υ) = { υ−𝔞T1 𝔞T2 −𝔞T1 , 𝔞T1 ≀ υ ≀ 𝔞T2 𝔞T3 −υ 𝔞T3 −𝔞T2 , 𝔞T2 ≀ υ ≀ 𝔞T3 0, elsewhere ÎŒ I (υ) = { υ−𝔞I1 𝔞I2 −𝔞I1 , 𝔞I1 ≀ υ ≀ 𝔞I2 𝔞I3 −υ 𝔞I3 −𝔞I2 , 𝔞I2 ≀ υ ≀ 𝔞I3 0, elsewhere Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 370 https://internationalpubls.com ÎŒ F (υ) = { υ−𝔞F1 𝔞F2 −𝔞F1 , 𝔞F1 ≀ υ ≀ 𝔞F2 𝔞F3 −υ 𝔞F3 −𝔞F2 , 𝔞F2 ≀ υ ≀ 𝔞F3 0, elsewhere We use ℱ(ℛ) to represent the set of all triangular neutrosophic numbers defined on ℛ. Definition 2.5 The (α, β, γ)-cut of neutrosophic set is denoted by ℱ(α, β, γ), where (α, β, γ) ∈ [0,1] and are fixed numbers, such that α+ β+ γ ≀ 3 and is defined as ℱ(α, β, γ) = (ÎŒ T (υ),ÎŒ I (υ),ÎŒ F (υ)), where υ ∈ Y, ÎŒ T (υ) ≥ α, ÎŒ I (υ) ≀ β, ÎŒ F (υ) ≀ γ. Definition 2.6 A triangular neutrosophic number ￜ̃ￜ𝔫 can also be represented as a pair ￜ̃ￜT 𝔫 = (𝔞T; 𝔞T), ￜ̃ￜI 𝔫 = (𝔞I; 𝔞I), ￜ̃ￜF 𝔫 = (𝔞F; 𝔞F) of functions 𝔞T(r), 𝔞T(r), 𝔞I(r), 𝔞I(r), 𝔞F(r), 𝔞F(r) 0 ≀ r ≀ 1 which satisfy the following requirements: • 𝔞T(r) is a bounded monotonic increasing left continuous function for membership function. • 𝔞T(r) is a bounded monotonic decreasing left continuous function for membership function. • 𝔞I(r) is a bounded monotonic increasing left continuous function for indeterminacy function. • 𝔞I(r) is a bounded monotonic decreasing left continuous function for indeterminacy function. • 𝔞F(r) is a bounded monotonic increasing left continuous function for non-membership function. • 𝔞F(r) is a bounded monotonic decreasing left continuous function for non-membership function. • 𝔞T(r) ≀ 𝔞T(r), 𝔞I(r) ≀ 𝔞I(r), 𝔞F(r) ≀ 𝔞F(r),0 ≀ r ≀ 1. Definition 2.7 ( Parametric Form) Let ￜ̃ￜ𝔫 = (𝔞T, 𝔞I, 𝔞F) be a triangular neutrosophic number and 𝔞T(r) = 𝔞T3 − (𝔞T3 − 𝔞T2 )r, 𝔞T(r) = 𝔞T1 + (𝔞T2 − 𝔞T1 )r, 𝔞I(r) = 𝔞I3 − (𝔞I3 − 𝔞I2 )r, 𝔞I(r) = 𝔞I1 + (𝔞I2 − 𝔞I1 )r, 𝔞F(r) = 𝔞F3 − (𝔞F3 − 𝔞F2 )r, 𝔞F(r) = 𝔞F1 + (𝔞F2 − 𝔞F1 )r, r ∈ [0,1]. The parametric form of the TNN is defined as ￜ̃ￜ = (〈𝔞T0 , 𝔞T∗ , 𝔞T∗〉; 〈𝔞I0 , 𝔞I∗ , 𝔞I∗〉; 〈𝔞F0 , 𝔞F∗ , 𝔞F∗〉), where 𝔞T∗ = 𝔞T0 − 𝔞T and 𝔞T∗ = 𝔞T − 𝔞T0 , 𝔞I∗ = 𝔞I0 − 𝔞I and 𝔞I∗ = 𝔞I − 𝔞I0 , 𝔞F∗ = 𝔞F0 − 𝔞F and 𝔞F∗ = 𝔞F − 𝔞F0 are the left and right fuzziness index functions respectively. The number 𝔞T0 = ( 𝔞T(1)+𝔞T(1) 2 ) , 𝔞I0 = ( 𝔞I(1)+𝔞I(1) 2 ) , 𝔞F0 = ( 𝔞F(1)+𝔞F(1) 2 ) is called the location index number. When r = 1, we get 𝔞T0 = 𝔞T2 , 𝔞I0 = 𝔞I2 , 𝔞F0 = 𝔞F2 . 2.1 Arithmetic Operations on Neutrosophic Numbers We develop a new fuzzy arithmetic using the parametric form of triangular neutrosophic numbers. This involves expressing the numbers in terms of location index and fuzziness index functions for membership, indeterminacy, and non-membership functions. We propose a new neutrosophic arithmetic operation, where the lattice rule, which involves the least upper bound and greatest lower bound in the lattice L, defines the fuzziness index functions and the location index number corresponds to standard arithmetic. That is for 𝔞, 𝔟 ∈ L, 𝔞 √ 𝔟 = max{𝔞, 𝔟} and 𝔞 ∧ 𝔟 = min{𝔞, 𝔟}. For Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 371 https://internationalpubls.com any two fuzzy numbers ￜ̃ￜ𝔫 = (〈𝔞T0 , 𝔞T∗ , 𝔞T∗〉; 〈𝔞I0 , 𝔞I∗ , 𝔞I∗〉; 〈𝔞F0 , 𝔞F∗ , 𝔞F∗〉), ℬ̃ 𝔫 = (〈𝔟T0 , 𝔟T∗ , 𝔟T∗〉; 〈𝔟I0 , 𝔟I∗ , 𝔟I∗〉; 〈𝔟F0 , 𝔟F∗ , 𝔟F∗〉) the arithmetic operations are defined as ￜ̃ￜ𝔫 ∗ ℬ̃ 𝔫 = (〈𝔞T0 , 𝔞T∗ , 𝔞T∗〉; 〈𝔞I0 , 𝔞I∗ , 𝔞I∗〉; 〈𝔞F0 , 𝔞F∗ , 𝔞F∗〉) ∗ (〈𝔟T0 , 𝔟T∗ , 𝔟T∗〉; 〈𝔟I0 , 𝔟I∗ , 𝔟I∗〉; 〈𝔟F0 , 𝔟F∗ , 𝔟F∗〉) = (〈𝔞T0 ∗ 𝔟T0 , 𝔞T∗ √ 𝔟T∗ , 𝔞T∗ √ 𝔟T∗〉, 〈𝔞I0 ∗ 𝔟I0 , 𝔞I∗ √ 𝔟I∗ , 𝔞I∗ √ 𝔟I∗〉, 〈𝔞F0 ∗ 𝔟F0 , 𝔞F∗ √ 𝔟F∗ , 𝔞F∗ √ 𝔟F∗〉) In particular for ￜ̃ￜ𝔫 = (〈𝔞T0 , 𝔞T∗ , 𝔞T∗〉; 〈𝔞I0 , 𝔞I∗ , 𝔞I∗〉; 〈𝔞F0 , 𝔞F∗ , 𝔞F∗〉), ℬ̃ 𝔫 = (〈𝔟T0 , 𝔟T∗ , 𝔟T∗〉; 〈𝔟I0 , 𝔟I∗ , 𝔟I∗〉; 〈𝔟F0 , 𝔟F∗ , 𝔟F∗〉) ∈ ℱ(ℛ), we have Addition: ￜ̃ￜ𝔫 ∗ ℬ̃ 𝔫 = (〈𝔞T0 , 𝔞T∗ , 𝔞T∗〉; 〈𝔞I0 , 𝔞I∗ , 𝔞I∗〉; 〈𝔞F0 , 𝔞F∗ , 𝔞F∗〉) + (〈𝔟T0 , 𝔟T∗ , 𝔟T∗〉; 〈𝔟I0 , 𝔟I∗ , 𝔟I∗〉; 〈𝔟F0 , 𝔟F∗ , 𝔟F∗〉) = (〈𝔞T0 + 𝔟T0 , 𝔞T∗ √ 𝔟T∗ , 𝔞T∗ √ 𝔟T∗〉, 〈𝔞I0 + 𝔟I0 , 𝔞I∗ √ 𝔟I∗ , 𝔞I∗ √ 𝔟I∗〉, 〈𝔞F0 + 𝔟F0 , 𝔞F∗ √ 𝔟F∗ , 𝔞F∗ √ 𝔟F∗〉) Subtraction: ￜ̃ￜ𝔫 ∗ ℬ̃ 𝔫 = (〈𝔞T0 , 𝔞T∗ , 𝔞T∗〉; 〈𝔞I0 , 𝔞I∗ , 𝔞I∗〉; 〈𝔞F0 , 𝔞F∗ , 𝔞F∗〉) − (〈𝔟T0 , 𝔟T∗ , 𝔟T∗〉; 〈𝔟I0 , 𝔟I∗ , 𝔟I∗〉; 〈𝔟F0 , 𝔟F∗ , 𝔟F∗〉) = (〈𝔞T0 − 𝔟T0 , 𝔞T∗ √ 𝔟T∗ , 𝔞T∗ √ 𝔟T∗〉, 〈𝔞I0 − 𝔟I0 , 𝔞I∗ √ 𝔟I∗ , 𝔞I∗ √ 𝔟I∗〉, 〈𝔞F0 − 𝔟F0 , 𝔞F∗ √ 𝔟F∗ , 𝔞F∗ √ 𝔟F∗〉) Multiplication: ￜ̃ￜ𝔫 ∗ ℬ̃ 𝔫 = (〈𝔞T0 , 𝔞T∗ , 𝔞T∗〉; 〈𝔞I0 , 𝔞I∗ , 𝔞I∗〉; 〈𝔞F0 , 𝔞F∗ , 𝔞F∗〉) × (〈𝔟T0 , 𝔟T∗ , 𝔟T∗〉; 〈𝔟I0 , 𝔟I∗ , 𝔟I∗〉; 〈𝔟F0 , 𝔟F∗ , 𝔟F∗〉) = (〈𝔞T0 × 𝔟T0 , 𝔞T∗ √ 𝔟T∗ , 𝔞T∗ √ 𝔟T∗〉, 〈𝔞I0 × 𝔟I0 , 𝔞I∗ √ 𝔟I∗ , 𝔞I∗ √ 𝔟I∗〉, 〈𝔞F0 × 𝔟F0 , 𝔞F∗ √ 𝔟F∗ , 𝔞F∗ √ 𝔟F∗〉) Division: ￜ̃ￜ𝔫 ∗ ℬ̃ 𝔫 = (〈𝔞T0 , 𝔞T∗ , 𝔞T∗〉; 〈𝔞I0 , 𝔞I∗ , 𝔞I∗〉; 〈𝔞F0 , 𝔞F∗ , 𝔞F∗〉) ÷ (〈𝔟T0 , 𝔟T∗ , 𝔟T∗〉; 〈𝔟I0 , 𝔟I∗ , 𝔟I∗〉; 〈𝔟F0 , 𝔟F∗ , 𝔟F∗〉) = (〈𝔞T0 ÷ 𝔟T0 , 𝔞T∗ √ 𝔟T∗ , 𝔞T∗ √ 𝔟T∗〉, 〈𝔞I0 ÷ 𝔟I0 , 𝔞I∗ √ 𝔟I∗ , 𝔞I∗ √ 𝔟I∗〉, 〈𝔞F0 ÷ 𝔟F0 , 𝔞F∗ √ 𝔟F∗ , 𝔞F∗ √ 𝔟F∗〉) provided 𝔟T0 , 𝔟I0 , 𝔟F0 ≠ 0. 2.2 Ranking of neutrosophic Numbers The ranking of neutrosophic numbers plays an essential role in the decision-making process within a fuzzy environment. Various authors in the literature have proposed different types of ranking methods. This article uses a highly effective ranking technique based on the graded mean. For ￜ̃ￜ𝔫 = Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 372 https://internationalpubls.com (〈𝔞T0 , 𝔞T∗ , 𝔞T∗〉; 〈𝔞I0 , 𝔞I∗ , 𝔞I∗〉; 〈𝔞F0 , 𝔞F∗ , 𝔞F∗〉) ∈ ℱ(ℛ), define ℜ:ℱ(ℛ) → ℛ by ℜ(ￜ̃ￜ𝔫) = ( 〈𝔞T∗+4𝔞T0 +𝔞 T∗〉,〈𝔞I∗+4𝔞I0 +𝔞 I∗〉,〈𝔞F∗+4𝔞F0 +𝔞 F∗〉 6 ). For any two triangular fuzzy numbers ￜ̃ￜ𝔫 = (〈𝔞T0 , 𝔞T∗ , 𝔞T∗〉; 〈𝔞I0 , 𝔞I∗ , 𝔞I∗〉; 〈𝔞F0 , 𝔞F∗ , 𝔞F∗〉) and ℬ̃ 𝔫 = (〈𝔟T0 , 𝔟T∗ , 𝔟T∗〉; 〈𝔟I0 , 𝔟I∗ , 𝔟I∗〉; 〈𝔟F0 , 𝔟F∗ , 𝔟F∗〉) ∈ ℱ(ℛ), we have the following comparison: • If ℜ(ￜ̃ￜ𝔫) < ℜ(ℬ̃ 𝔫 ), then ￜ̃ￜ𝔫 ≺ ℬ̃ 𝔫 • If ℜ(ￜ̃ￜ𝔫) > ℜ(ℬ̃ 𝔫 ), then ￜ̃ￜ𝔫 ≻ ℬ̃ 𝔫 • If ℜ(ￜ̃ￜ𝔫) = ℜ(ℬ̃ 𝔫 ), then ￜ̃ￜ𝔫 ≈ ℬ̃ 𝔫 . Note 1. ℜ(ￜ̃ￜ𝔫 + ℬ̃ 𝔫 ) = ℜ(ￜ̃ￜ𝔫) +ℜ(ℬ̃ 𝔫 ) Note 2. ℜ(ￜ̃ￜ𝔫. ℬ̃ 𝔫 ) = ℜ(ￜ̃ￜ𝔫).ℜ(ℬ̃ 𝔫 ) Note 3. ℜ(ￜ̃ￜ𝔫/ℬ̃ 𝔫 ) = ℜ(ￜ̃ￜ𝔫)/ℜ(ℬ̃ 𝔫 ), provided ℜ(ℬ̃ 𝔫 ) ≠ 0. 3. Neutrosophic Non Linear Programming Problems (NNLPP) Let general NNLPP " 𝓂𝒟𝓃 𝑓𝒩(ￜ̃ￜ𝑖) 𝑠𝑢𝑏. 𝑡𝑜 ℎ̃𝒟 𝒩 (ￜ̃ￜ𝑖) ≈ 0̃ 𝑓𝑜𝑟 𝒟 = 1,2, ⋯ , ℓ ￜ̃ￜ𝑗 𝒩(ￜ̃ￜ𝑖) ⪯ 0̃ 𝑓𝑜𝑟 𝒿 = 1,2, ⋯ ,𝓂 ￜ̃ￜ𝑖 ⪰ 0̃" (1) where 𝑓𝒩 , ℎ̃1 𝒩 , ⋯ , ℎ̃𝑙 𝒩 , ￜ̃ￜ1 𝒩 , ⋯ , ￜ̃ￜ𝓂 𝒩 are continuous neutrosophic valued functions defined on 𝑅𝓃. A vector ￜ̃ￜ𝑖 = (ￜ̃ￜ1, ￜ̃ￜ2, ￜ̃ￜ3,⋯ , ￜ̃ￜ𝑛) is considered a feasible solution to the NNLPP (1) if it meets all the constraints and adheres to the non-negativity condition of the NNLPP. The collection of all such feasible solutions is known as the feasible region, which is defined by these criteria. ￜ̃ￜ = {ￜ̃ￜ𝒊 ∈ R𝓃/ℎ̃𝒟 𝒩 (ￜ̃ￜ) ≈ 0̃ 𝑓𝑜𝑟 𝒟 = 1,2, ⋯ , ℓ, ￜ̃ￜ𝒿 𝒩(ￜ̃ￜ)𝑓𝑜𝑟 𝒿 = 1,2,⋯ ,𝓂 𝑎𝑛𝑑 ￜ̃ￜ𝑖 ⪰ 0̃}. 4. Solution Methods We investigate two Neutrosophic optimization methods namely Neutrosophic Cauchy’s Steepest Descent Method and the Fletcher-Reeves Method (also called the Conjugate Gradient Method). These techniques give strong frameworks for improving objective functions when facing uncertainty, incompleteness, and inconsistency. Our goal is to explain the theoretical basics and real-world impacts of these Neutrosophic optimization methods using a series of theorems and proofs. This will show how effective they are in handling complicated optimization problems. 4.1 Neutrosophic Cauchy’s Steepest Descent Method Consider an unconstrained minimization problem min𝑓𝒩(ￜ̃ￜ1, ￜ̃ￜ2, 
 , ￜ̃ￜ𝔫). The goal is to discover a local minimizer, also known as a minimizer. 𝑓𝒩(ￜ̃ￜ) represents the objective function for this minimization job. Gradient-based approaches are based on the observation that 𝑓𝒩 decreases the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 373 https://internationalpubls.com fastest at a point 𝑝 in ℛ while advancing in the direction of −∇𝑓𝒩(𝑃). As a result, the iterative approach consists of, ￜ̃ￜ𝜅+1 = ￜ̃ￜ𝜅 + (ￜ̃ￜ𝜅ￜ̃ￜ(ￜ̃ￜ𝜅))𝒩 (2) where as ￜ̃ￜ𝜅 is the current estimate of ￜ̃ￜ∗, (ￜ̃ￜκ)𝒩 = (ￜ̃ￜ(ￜ̃ￜ𝜅))𝒩 is the search direction and 𝛌(κ) is the step length parameter in the space ℛ𝒩 of design variables. If we take (ￜ̃ￜκ)𝒩 = −(ￜ̃ￜκ)𝒩 = −∇𝑓𝒩(ￜ̃ￜκ), we get the method of steepest descent. We have ￜ̃ￜ(κ+1) = ￜ̃ￜ𝜅 − (𝛌𝜅(ￜ̃ￜκ))𝒩 , (ￜ̃ￜκ)𝒩 = ∇𝑓𝒩(ￜ̃ￜκ), where ￜ̃ￜ(κ) is the minimizer of the function 𝜙(ￜ̃ￜ) = 𝑓𝒩(ￜ̃ￜκ − (ￜ̃ￜￜ̃ￜκ))𝒩. To get the value of 𝛌(κ), one can use any of the one-dimensional search techniques. To start the iterative process effectively, the first approximation ￜ̃ￜ0 must be chosen carefully and according to the particular problem at hand. 4.1.1 Analyzing the Convergence of Quadratic Functions using the Neutrosophic steepest Descent Technique To demonstrate the convergence properties of gradient-based techniques, we will use a quadratic function represented as 𝑓𝒩(ￜ̃ￜ) ≈ 1 2 ￜ̃ￜ𝑇ￜ̃ￜￜ̃ￜ ≈ 1 2 〈ￜ̃ￜￜ̃ￜ, ￜ̃ￜ〉 (3) where ￜ̃ￜ is positive definite. If f̃ 𝒩 has a minimum at υ̃∗ ≈ 0̃ with f̃ 𝒩 (υ̃∗) ≈ 0̃, ￜ̃ￜ𝒩 ≈ ∇f̃ 𝒩 (υ̃) ≈ ￜ̃ￜυ̃ and ∇2f̃ 𝒩 (υ̃) ≈ ￜ̃ￜ . Then ℛ(Ί(γ̃𝒩)) = ℛ (f̃ 𝒩 (υ̃− γ̃ￜ̃ￜ𝒩)) = 1 2 ℛ(〈[ￜ̃ￜ − 2γ̃ￜ̃ￜ2 + γ̃2ￜ̃ￜ3]υ̃, υ̃〉) This gives ℛ(γ̃𝒩) = ℛ(〈ￜ̃ￜ2υ̃,υ̃)〉) ℛ(〈ￜ̃ￜ3υ̃,υ̃)〉) = ℛ(⟚ￜ̃ￜυ̃,ￜ̃ￜυ̃⟩) ℛ(⟚ￜ̃ￜ(ￜ̃ￜυ̃),ￜ̃ￜυ̃⟩) = ℛ(⟚ￜ̃ￜ𝒩 ,ￜ̃ￜ𝒩⟩) ℛ(⟚ￜ̃ￜￜ̃ￜ𝒩 ,ￜ̃ￜ𝒩⟩) Therefore, the iteration process of steepest descent for the quadratic function is expressed as ℛ(υ̃κ+1) = ℛ (υ̃κ − ⟹(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜκ)𝒩⟩ ⟚ￜ̃ￜ(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜκ)𝒩⟩ (ￜ̃ￜκ)𝒩) (4) Using the above formulation and the fact that ℛ (f̃ 𝒩 (υ̃κ+1)) = ℛ (f̃ 𝒩 (υ̃κ) − 1 2 [〈(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜκ)𝒩〉]2 〈ￜ̃ￜ(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜκ)𝒩〉 ) The gives ℛ ( f̃ 𝒩 (υ̃κ)−f̃ 𝒩 (υ̃κ+1) f̃ 𝒩 (υ̃κ) ) = ℛ ( [〈(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜκ)𝒩〉]2 〈ￜ̃ￜ(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜκ)𝒩〉〈(ￜ̃ￜκ)𝒩 ,ￜ̃ￜ−1(ￜ̃ￜκ)𝒩〉 ) and hence Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 374 https://internationalpubls.com ℛ(f̃ 𝒩 (υ̃κ+1)) = ℛ((1− ÎŒ κ )f̃ 𝒩 (υ̃κ)) (4a) where ÎŒ κ = [〈(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜκ)𝒩〉]2 〈ￜ̃ￜ(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜκ)𝒩〉〈(ￜ̃ￜκ)𝒩 ,ￜ̃ￜ−1(ￜ̃ￜκ)𝒩〉 Theorem 4.1 Let the neutrosophic quadratic function denoted as f̃ 𝒩 , defined by the expressions in (3), f̃ 𝒩 (υ̃) ≈ 1 2 υ̃Tￜ̃ￜυ̃ ≈ 1 2 〈ￜ̃ￜυ̃, υ̃〉 and (υ̃κ) represent the iterates generated by the process υ̃κ+1 = υ̃κ − 〈(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜκ)𝒩〉 〈ￜ̃ￜ(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜκ)𝒩〉 (ￜ̃ￜκ)𝒩 The sequence υ̃κ converges linearly towards the minimizer υ̃∗, which is approximately identical to 0̃, starting from any initial approximation υ̃0. Proof: To prove υ̃κ+1 = υ̃κ − 〈(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜκ)𝒩〉 〈ￜ̃ￜ(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜκ)𝒩〉 (ￜ̃ￜκ)𝒩, it is enough to prove ℛ(υ̃κ+1) = ℛ (υ̃κ − ⟹(ￜ̃ￜκ)𝒩 , (ￜ̃ￜκ)𝒩⟩ ⟚ￜ̃ￜ(ￜ̃ￜκ)𝒩 , (ￜ̃ￜκ)𝒩⟩ (ￜ̃ￜκ)𝒩) outlined in Equation (4) . That is it is enough to prove ℛ (f̃ 𝒩 (υ̃κ+1)) = ℛ((1− Ό̃ κ )f̃ 𝒩 (υ̃κ)) (5) This gives ℛ (f̃ 𝒩 (υ̃κ)) = ∏κ−1 i=0 ℛ((1− Ό̃ κ )f̃ 𝒩 (υ̃(0))), (υ̃κ) converges to ℛ(υ̃∗) = 0 if and only if ℛ(f̃ 𝒩 (υ̃κ)) → 0 and in the view the above equation, this is possible if and only if ∏∞ i=0 ℛ(1− Ό̃ κ ) = 0, which is true, because ℛ(1− Ό̃ κ ) ≀ ℛ ( γ̃ max −γ̃ min γ̃ max ) ≀ 1. As ℛ(υ̃∗) = 0 and ℛ (f̃ 𝒩 (υ̃)) = 1 2 ℛ(ￜ̃ￜυ̃, υ̃), Rayleigh inequality gives ℛ ( γ̃min 2 ∥ υ̃κ+1 − υ̃κ ∥2) ≀ ℛ(f̃ 𝒩 (υ̃κ+1 − υ̃(∗))) = ℛ(f̃ 𝒩 (υ̃κ+1)). (6) Similarly, ℛ ( γ̃ max 2 ∥ υ̃κ − υ̃(∗) ∥2) ≥ ℛ(f̃ 𝒩 (υ̃κ) − υ̃(∗)) = ℛ(f̃ 𝒩 (υ̃κ)) (7) Consequently (6) and (7), we get ℛ ( γ̃ min 2 ∥ υ̃κ+1 − υ̃κ ∥2) ≀ ℛ(f̃ 𝒩 (υ̃κ+1)) = ℛ((1− Ό̃ κ )f̃ 𝒩 (υ̃κ)) ≀ ℛ ( γ̃ max −γ̃ min 2 ∥ υ̃κ − υ̃∗ ∥2) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 375 https://internationalpubls.com Which implies ℛ ( ∥υ̃κ+1−υ̃∗∥ ∥υ̃κ−υ̃∗∥ ) ≀ ℛ (√ γ̃ max −γ̃ min γ̃ max ) as ℛ (√ γ̃ max −γ̃ min γ̃ max ) > 0 Hence ∥υ̃κ+1−υ̃∗∥ ∥υ̃κ−υ̃∗∥ ⪯ √ γ̃ max −γ̃ min γ̃ max . It implies that the convergence of υ̃κ to υ̃∗ is linear. 4.2 Exploring the Neutrosophic Fletcher-Reeves Technique (Neutrosophic Conjugate Gradient Method) In the Cauchy’s Steepest Descent Method, the determination of (d̃ κ )𝒩 as −(ￜ̃ￜκ)𝒩 leads the iterative process of υ̃κ towards the minimizer υ̃∗ in a zigzagging manner. Therefore, there is a necessity to generate a new search direction function (d̃ κ )𝒩 that will accelerate the convergence of the iterates {υ̃κ} towards υ̃∗. Consider the quadratic function given in (3), f̃ 𝒩 (υ̃) ≈ 1 2 υ̃Tￜ̃ￜυ̃ ≈ 1 2 〈ￜ̃ￜυ̃, υ̃〉 with ￜ̃ￜ positive definite. We shall generate ℛ((d̃ κ )𝒩) as manually conjugate direction with respect to ℛ(ￜ̃ￜ),ℛ(〈ￜ̃ￜ (d̃ j ) 𝒩 , (d̃ j )𝒩〉) = 0, i ≠ j. The procedure for conjugate direction generation is ℛ ((d̃ 0 )𝒩) = −ℛ((ￜ̃ￜ0)𝒩) = −ℛ (∇f̃ 𝒩 (υ̃0)) = −ℛ(ￜ̃ￜ𝒩(υ̃0)) with ℛ(υ̃0) being initial guess. Then ℛ(υ̃κ+1) = ℛ(υ̃κ − α̃κ(d̃ κ )𝒩), we get ℛ(α̃κ) = −ℛ ( 〈(d̃ κ )𝒩 ,(d̃ κ )𝒩〉 〈(ￜ̃ￜ(d̃ κ ))𝒩 ,(d̃ κ )𝒩〉 ). This ℛ(α̃κ) minimize the function ℛ(Ί̃(α̃)) = ℛ (f̃(υ̃ − α̃d̃ κ )) and hence ℛ(ￜ̃ￜκ+1) is orthognal to ℛ(d̃ κ ), for ℛ(α̃) that minimizes the equation is given by ℛ(ϕ̃ ′ (α̃)) = ℛ (〈∇f̃ 𝒩 (υ̃− α̃(d̃ κ ))𝒩 , (d̃ κ )𝒩〉) = 0 which is same as ℛ(〈(ￜ̃ￜκ+1)𝒩 , (d̃ κ )𝒩〉) = 0. The nextconjugate direction ℛ(d̃ κ+1 )𝒩 is given by ℛ(d̃ κ+1 )𝒩 +ℛ(β̃ κ (d̃ κ )𝒩) where ℛ((d̃ κ )𝒩) is so chosen that ℛ((d̃ κ )𝒩) is conjugate to ℛ ((d̃ κ+1 )𝒩) with respect to ℛ(ￜ̃ￜ). This gives ℛ(β̃ κ ) = ℛ ( (ￜ̃ￜκ+1)𝒩 ,ￜ̃ￜ(d̃ κ )𝒩 (d̃ κ )𝒩 ,ￜ̃ￜ(d̃ κ )𝒩 ). To evaluate ℛ(〈(d̃ κ )𝒩 , (ￜ̃ￜκ)𝒩〉) ℛ(⟹(d̃ κ )𝒩 , (ￜ̃ￜκ)𝒩⟩) = −ℛ(⟹(ￜ̃ￜκ)𝒩 , (ￜ̃ￜκ)𝒩⟩) +ℛ (β̃ κ−1 ⟹(d̃ κ−1 )𝒩 , (ￜ̃ￜκ)𝒩⟩) = −ℛ(〈(ￜ̃ￜκ)𝒩 , (ￜ̃ￜκ)𝒩〉) as ℛ((ￜ̃ￜκ)𝒩) ⊥ ℛ((d̃ κ−1 )𝒩) and hence ℛ(α̃κ) = −ℛ ( 〈(d̃ κ )𝒩 ,(ￜ̃ￜκ)𝒩〉 〈ￜ̃ￜ(d̃ κ )𝒩 ,(d̃ κ )𝒩〉 ). Theorem 4.2 The neutrosophic gradient vectors (ￜ̃ￜκ)𝒩 are mutually orthogonal and the direction search neutrosophic vectors (d̃ κ )𝒩 are mutually 𝒬-Conjugate. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 376 https://internationalpubls.com Proof: To prove the neutrosophic gradient vectors (ￜ̃ￜκ)𝒩 are mutually orthogonal and the direction search neutrosophic vectors (d̃ κ )𝒩 are mutually 𝒬-Conjugate, it is enough to prove the neutrosophic gradient vectors ℛ((ￜ̃ￜκ)𝒩) are mutually orthogonal and the direction search neutrosophic vectors ℛ((d̃ κ )𝒩) are mutually 𝒬-Conjugate. Let as prove the theorem by mathematical induction. The result is true for κ = 1, since ℛ((d̃ 0 )𝒩) and ℛ((d̃ 1 )𝒩) are 𝒬-conjugate by the choice of ℛ(β̃ 0 ). Also ℛ((ￜ̃ￜ0)𝒩) = ℛ((−d̃ 0 )𝒩) is orthogonal to ℛ((ￜ̃ￜ1)𝒩). Assume that ℛ((d̃ 0 )𝒩),ℛ((d̃ 1 )𝒩), . . . . ,ℛ((d̃ m )𝒩) are mutually 𝒬- conjugate and ℛ((ￜ̃ￜ0)𝒩),ℛ((ￜ̃ￜ1)𝒩), . . . . ,ℛ((ￜ̃ￜm)𝒩) are mutually orthogonal. That is the result is true for κ = m. We shall prove the result for κ = m+ 1. Then ℛ((ￜ̃ￜm+1)𝒩) = ℛ((ￜ̃ￜm)𝒩) +ℛ(α̃mￜ̃ￜ(d̃ 𝑚 )𝒩), where ℛ(α̃m) = −ℛ ( 〈(d̃ m )𝒩 ,(ￜ̃ￜm)𝒩〉 〈(ￜ̃ￜ(d̃ m ))𝒩 ,(d̃ m )𝒩〉 ) And hence ℛ(〈(ￜ̃ￜ(m+1))𝒩 , (ￜ̃ￜi)𝒩〉) = ℛ(⟹(ￜ̃ￜm)𝒩 , (ￜ̃ￜi)𝒩⟩ + ⟚α̃m⟩⟚ￜ̃ￜ(ￜ̃ￜm)𝒩 , (ￜ̃ￜi)𝒩〉) = ℛ(⟹(ￜ̃ￜm)𝒩 , (ￜ̃ￜi)𝒩⟩) +ℛ(⟚α̃m⟩⟚ￜ̃ￜ(ￜ̃ￜm)𝒩 , (ￜ̃ￜi)𝒩〉) = ℛ(⟹(ￜ̃ￜm)𝒩 , (ￜ̃ￜi)𝒩⟩) −ℛ(⟹(ￜ̃ￜm)𝒩 , (ￜ̃ￜi)𝒩⟩) = 0 Also ℛ((ￜ̃ￜm+1)𝒩) = −ℛ((ￜ̃ￜm+1)𝒩) +ℛ(β̃ m (d̃ 𝑚 )𝒩), where ℛ(β̃ m ) = ℛ( 〈(g̃m+1)𝒩 , ￜ̃ￜ(d̃ m )𝒩〉 〈(d̃ m )𝒩 , ￜ̃ￜ(d̃ m )𝒩〉 ) As ℛ(d̃ 0 )𝒩 ,ℛ(d̃ 1 )𝒩 , . . . . ,ℛ(d̃ m )𝒩 are 𝒬-conjugate. Taking inner product with ℛ(ￜ̃ￜ(d̃ m )𝒩), we have 〈ℛ(d̃ 𝑚+1 )𝒩 ,ℛ(ￜ̃ￜ(d̃ m )𝒩)〉 = −ℛ(〈(ￜ̃ￜm+1)𝒩 , ￜ̃ￜ(d̃ m )𝒩〉) +ℛ(〈β̃ m 〉〈(d̃ m )𝒩 , ￜ̃ￜ(d̃ m )𝒩〉) = −ℛ(〈(ￜ̃ￜm+1)𝒩 , ￜ̃ￜ(d̃ m )𝒩〉) +ℛ(〈(ￜ̃ￜm+1)𝒩 , ￜ̃ￜ(d̃ m )𝒩〉) = 0 Then, ℛ(ￜ̃ￜ(d̃ 𝑚 )𝒩) = ℛ ( (ￜ̃ￜm+1)𝒩−(ￜ̃ￜm)𝒩 α̃m ). and hence we have ℛ (〈(d̃ κ )𝒩 , ￜ̃ￜ(d̃ 𝑖 )𝒩〉) = ℛ ( 〈(ￜ̃ￜκ)𝒩 ,(ￜ̃ￜi+1)𝒩−(ￜ̃ￜi)𝒩〉 α̃i ) (8) Combining the result we have ℛ(〈(ￜ̃ￜκ)𝒩 , (ￜ̃ￜi)𝒩〉) = 0 = ℛ(〈(d̃ κ )𝒩 , ￜ̃ￜ(d̃ κ )𝒩〉), for all κ. 5. Algorithms This section provides the step-by-step procedures for the Neutrosophic steepest descent and the Fletcher-Reeves methods. These algorithms are developed to iteratively improve solutions by considering uncertainties and inconsistencies during the process. They represent the principles of Neutrosophy and provide valuable insights into managing uncertainty during optimization. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 377 https://internationalpubls.com 5.1 Algorithm for neutrosophic Cauchy’s Steepest Descent Method Step 1. Let us take the SVNNLPP with unconstrained optimization problem incorporate the neutrosophic triangular coefficient ￜ̃ￜ𝒩(υ̃(κ) 𝒩 ). Step 2. Convert the triangular neutrosophic number coeffecients into arithmetic neutrosophic number and then covert to parametric form. Step 3. Compute first and second derivative for the given function, and choose any initial points. Step 4. Input υ̃(0) 𝒩 , let κ ← 0 Step 5. Calculate S̃i 𝒩 , S̃i 𝒩 = −∇ￜ̃ￜi 𝒩, where ￜ̃ￜi 𝒩 = f̃ 𝒩 (υ̃), ∇ￜ̃ￜi 𝒩 = ∇f̃ 𝒩 (υ̃κ). Step 6. Find Hessian for ￜ̃ￜi 𝒩(x) Step 7. Find the (υ̃κ+1)𝒩 = (υ̃κ)𝒩 − (γ̃ i ￜ̃ￜκ)𝒩 then κ ← κ + 1. where γ̃ = −(∇ￜ̃ￜi 𝒩(x))TS̃i 𝒩 (S̃i 𝒩 )THi 𝒩 S̃i 𝒩 Step 8. Repeat the process until ∥ ∇ￜ̃ￜi 𝒩(υ̃κ) ∥< ε (or) ∥ (υ̃κ)𝒩 − (υ̃κ+1)𝒩 ∥< ε. Then halt the process if we get an optimal solution; otherwise, go to step 5. Step 9. Verify the optimum. 5.2 Algorithm for neutrosophic fletcher reeves method Step 1. Let us take the SVNNLPP with an unconstrained optimization problem and incorporate the neutrosophic triangular coefficient. ￜ̃ￜ𝒩(υ̃(κ) 𝒩 ). Step 2. Convert the triangular neutrosophic number coeffecients into arithmetic neutrosophic number and then covert to parametric form. Step 3. Compute first and second derivative for the given function, and choose any initial points. Step 4. Input υ̃(0) 𝒩 , let κ ← 0 Step 5. Calculate S̃i 𝒩 , S̃i 𝒩 = −∇ￜ̃ￜi 𝒩, where ￜ̃ￜi 𝒩 = f̃ 𝒩 (υ̃), ∇ￜ̃ￜi 𝒩 = ∇f̃ 𝒩 (υ̃κ). and S̃i+1 𝒩 = −∇ￜ̃ￜi 𝒩 + |∇ￜ̃ￜi 𝒩|2 |∇ￜ̃ￜi−1 𝒩 |2 S̃i 𝒩 Step 6. Find Hessian for ￜ̃ￜi 𝒩(υ̃) Step 7. Find the (υ̃κ+1)𝒩 = (υ̃κ)𝒩 − (γ̃ i ￜ̃ￜκ)𝒩 then κ ← κ + 1. where γ̃ = −(∇ￜ̃ￜi 𝒩(υ̃))TS̃i 𝒩 (S̃i 𝒩 )THi 𝒩 S̃i 𝒩 Step 8. Repeat the process until ∥ ∇ￜ̃ￜi 𝒩(υ̃κ) ∥< ε (or) ∥ (υ̃κ)𝒩 − (υ̃κ+1)𝒩 ∥< ε. Then halt the process if we get an optimal solution; otherwise, go to step 5. Step 9. Verify the optimum. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 378 https://internationalpubls.com 6. Numerical Examples This section presents numerical examples of two significant methods utilized in Neutrosophic optimization: the Fletcher-Reeves method (also known as the conjugate gradient method) and the Neutrosophic Steepest Descent. These examples provide an in-depth demonstration of how these algorithms are applied, discussing their convergence properties and their ability to traverse uncertain solution domains. Example 6.1 Consider the problem of unconstrained optimization with single-valued neutrosophic triangular coefficients. 𝑚𝑖𝑛 f̃(υ̃1, υ̃2) = (19,20,21); (0.69,0.51,0.52)υ̃1 + (25,26,27); (0.67,0.54,0.5)υ̃2 +(3,4,5); (0.74,0.75,0.68)υ̃1υ̃2 + (−5, −4, −3); (0.8,0.75,0.6)υ̃1 2 +(−4, −3, −2); (0.7,0.65,0.75)υ̃2 2 (9) Starting with initial point Ỹ 0 = ( (0,1,2); (0.65,0.6,0.5) (0,1,2); (0.65,0.6,0.5) ) Solution: The parametric form of the given SVNNLPP (only triplet)(9) is given by 𝑚𝑖𝑛 f̃(υ̃1, υ̃2) = (20,1− r, 1 − r); (0.69,0.51,0.52)υ̃1 + (26,1− r, 1− r); (0.67,0.54,0.5)υ̃2 + (4,1− r, 1− r); (0.74,0.75,0.68)υ̃1υ̃2 +(−4,1− r, 1 − r); (0.8,0.75,0.6)υ̃1 2 + (−3,1− r, 1 − r); (0.7,0.65,0.75)υ̃2 2 (10) Ỹ 0 = ( (1,1− r, 1− r); (0.65,0.6,0.5) (1,1− r, 1− r); (0.65,0.6,0.5) ) The condition required for υ̃ to be the optimum solution for (10) ∇f̃(υ̃1, υ̃2) ≈ 0̃. Hence we have ∂f̃ ∂υ̃1 = (20,1− r, 1 − r); (0.69,0.51,0.52) + (4,1− r, 1− r); (0.74,0.75,0.68)υ̃2 +(−8,1− r, 1− r); (0.8,0.75,0.6)υ̃1 ∂f̃ ∂υ̃2 = (26,1− r, 1 − r); (0.67,0.54,0.5) + (4,1− r, 1− r); (0.74,0.75,0.68)υ̃1 (−6,1− r, 1− r); (0.7,0.65,0.75)υ̃2 The condition required for υ̃ to be the optimum solution for (10) ∇2f̃(υ̃1, υ̃2) ≈ 0̃. Hence we have ∂2f̃ ∂υ̃1 2 = (−8,1− r, 1− r); (0.8,0.75,0.6), ∂2f̃ ∂υ̃2 2 = (−6,1− r, 1− r); (0.7,0.65,0.75) ∂2f̃ ∂υ̃1 ∂υ̃2 = (4,1− r, 1− r); (0.74,0.75,0.68), ∂2f̃ ∂υ̃2 ∂υ̃1 = (4,1− r, 1− r); (0.74,0.75,0.68) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 379 https://internationalpubls.com Table 1: Cauchy’s Steepest Descent Method Iteration 𝐘𝐢 = (𝛖𝟏, 𝛖𝟐) ￜ̃ￜ𝐢+𝟏 = (𝛖𝟏, 𝛖𝟐) 𝛁𝐟𝐢 1 ( (1,1− r, 1 − r); (0.65,0.6,0.5) (1,1− r, 1 − r); (0.65,0.6,0.5) ) ( (6.4736,1− r, 1− r); (0.8,0.75,0.75) (9.2104,1− r, 1− r); (0.8,0.75,0.75) ) ( (5.0528,1− r, 1− r); (0.8,0.75,0.75) (−3.368,1− r, 1− r); (0.8,0.75,0.75) ) 2 ( (6.4736,1− r, 1− r); (0.8,0.75,0.75) (9.2104,1− r, 1− r); (0.8,0.75,0.75) ) ( (6.9299,1− r, 1− r); (0.8,0.75,0.75) (8.9063,1− r, 1− r); (0.8,0.75,0.75) ) ( (0.186,1− r, 1− r); (0.8,0.75,0.75) (0.2818,1− r, 1− r); (0.8,0.75,0.75) ) 3 ( (6.9299,1− r, 1− r); (0.8,0.75,0.75) (8.9063,1− r, 1− r); (0.8,0.75,0.75) ) ( (6.993,1− r, 1− r); (0.8,0.75,0.75) (9.002,1− r, 1− r); (0.8,0.75,0.75) ) ( (0.064,1− r, 1− r); (0.8,0.75,0.75) (0.2818,1− r, 1− r); (0.8,0.75,0.75) ) 4 ( (6.993,1− r, 1− r); (0.8,0.75,0.75) (9.002,1− r, 1− r); (0.8,0.75,0.75) ) ( (6.9993,1− r, 1− r); (0.8,0.75,0.75) (9.002,1− r, 1− r); (0.8,0.75,0.75) ) ( (0.064,1− r, 1− r); (0.8,0.75,0.75) (−0.04,1− r, 1− r); (0.8,0.75,0.75) ) Table 2: Fletcher Reeves Method Iteration ￜ̃ￜ𝐢 = (𝛖𝟏, 𝛖𝟐) ￜ̃ￜ𝐢+𝟏 = (𝛖𝟏, 𝛖𝟐) 𝛁𝐟𝐢 1 ( (1,1− r, 1− r); (0.65,0.6,0.5) (1,1− r, 1− r); (0.65,0.6,0.5) ) ( (6.4736,1− r, 1− r); (0.8,0.75,0.75) (9.2104,1− r, 1− r); (0.8,0.75,0.75) ) ( (5.0528,1− r, 1− r); (0.8,0.75,0.75) (−3.368,1− r, 1− r); (0.8,0.75,0.75) ) 2 ( (6.4736,1− r, 1− r); (0.8,0.75,0.75) (9.2104,1− r, 1− r); (0.8,0.75,0.75) ) ( (6.9975,1− r, 1− r); (0.8,0.75,0.75) (9.000,1− r, 1− r); (0.8,0.75,0.75) ) ( (0.02,1− r, 1− r); (0.8,0.75,0.75) (−0.01,1− r, 1− r); (0.8,0.75,0.75) ) 3 ( (6.9975,1− r, 1− r); (0.8,0.75,0.75) (9.000,1− r, 1− r); (0.8,0.75,0.75) ) ( (6.999,1− r, 1− r); (0.8,0.75,0.75) (8.999,1− r, 1− r); (0.8,0.75,0.75) ) ( (0.001,1− r, 1− r); (0.8,0.75,0.75) (0.003,1− r, 1− r); (0.8,0.75,0.75) ) From the above tables(1),(2), we observed that the SDM converges at iteration 4 while the FRM at iteration 3. Hence the optimal solution of the given SVNNLPP (9) is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 380 https://internationalpubls.com υ̃1 = (6.999,1− r, 1 − r); (0.8,0.75,0.75), υ̃2 = (8.999,1− r, 1− r); (0.8,0.75,0.75) with f̃(υ̃1, υ̃2) = (186.999,1− r, 1− r); (0.8,0.75,0.75) That is the optimal solution of the SVNNLPP (9) is υ̃1 = (5.999+ r, 6.999,7.999− r); (0.8,0.75,0.75), υ̃2 = (7.999+ r, 8.999,9.999− r); (0.8,0.75,0.75) with 𝑚𝑖𝑛 f̃(υ̃1, υ̃2) = (185.999+ r, 186.999,187.999− r); (0.8,0.75,0.75). Example 6.2 Consider the problem of unconstrained optimization with single-valued neutrosophic triangular fuzzy coefficients, 𝑚𝑖𝑛 f̃(υ̃1, υ̃2) = (0,1,2); (0.68,0.51,0.55)υ̃1 − (0,1,2)(0.67,0.53,0.54)υ̃2 + (1,2,3); (0.7,0.65,0.72)υ̃1 2 + (1,2,3); (0.69,0.51,0.52)υ̃1υ̃2 +(0,1,2); (0.67,0.5,0.51)υ̃2 2 (11) Starting with initial point Ỹ 0 = ( (0,0,0); (0.66,0.52,0.53) (0,0,0); (0.66,0.52,0.53) ) Solution: The parametric form of the given SVNNLPP (only triplet)(11) is given by 𝑚𝑖𝑛 f̃(υ̃1, υ̃2) = (1,1− r, 1 − r); (0.68,0.51,0.55)υ̃1 − (1,1− r, 1− r); (0.67,0.53,0.54)υ̃2 + (2,1− r, 1− r); (0.7,0.65,0.72)υ̃1 2 +(2,1− r, 1− r); (0.69,0.51,0.52)υ̃1υ̃2 + (1,1− r, 1− r); (0.67,0.5,0.51)υ̃2 2 (12) Ỹ 0 = ( (0,1− r, 1− r); (0.66,0.52,0.53) (0,1− r, 1− r); (0.66,0.52,0.53) ) The condition required for υ̃ to be the optimum solution for (12) ∇f̃(υ̃1, υ̃2) ≈ 0̃. Hence we have ∂f̃ ∂υ̃1 = (1,1− r, 1 − r); (0.68,0.51,0.55) + (4,1− r, 1− r); (0.7,0.65,0.72)υ̃1 +(2,1− r, 1− r); (0.69,0.51,0.52)υ̃2 ∂f̃ ∂υ̃2 = −(1,1− r, 1− r); (0.67,0.53,0.54) + (2,1− r, 1 − r); (0.69,0.51,0.52)υ̃1 (2,1− r, 1 − r); (0.67,0.5,0.51)υ̃2 The condition required for υ̃ to be the optimum solution for (12) ∇f̃(υ̃1, υ̃2) ≈ 0̃. Hence we have ∂2f ∂υ̃1 2 = (4,1− r, 1− r); (0.7,0.65,0.72), ∂2f ∂υ̃2 2 = (2,1− r, 1− r); (0.67,0.5,0.51) ∂2f ∂υ̃1 ∂υ̃2 = (2,1− r, 1− r); (0.69,0.51,0.52), ∂2f ∂υ̃2 ∂υ̃1 = (2,1− r, 1− r); (0.69,0.51,0.52) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 381 https://internationalpubls.com Table 3: Cauchy’s Steepest Descent Method Iteration ￜ̃ￜ𝐢 = (𝛖𝟏, 𝛖𝟐) ￜ̃ￜ𝐢+𝟏 = (𝛖𝟏, 𝛖𝟐) 𝛁𝐟𝐢 1 ( (0,1− r, 1 − r); (0.66,0.52,0.53) (0,1− r, 1 − r); (0.66,0.52,0.53) ) ( (1,1− r, 1 − r); (0.7,0.65,0.72) (1,1− r, 1 − r); (0.7,0.65,0.72) ) ( (−1,1− r, 1− r); (0.7,0.65,0.72) (−1,1− r, 1− r); (0.7,0.65,0.72) ) 2 ( (1,1− r, 1 − r); (0.7,0.65,0.72) (1,1− r, 1 − r); (0.7,0.65,0.72) ) ( (−0.8,1− r, 1− r); (0.7,0.65,0.72) (1.2,1− r, 1− r); (0.7,0.65,0.72) ) ( (0.2,1− r, 1− r); (0.7,0.65,0.72) (−0.2,1− r, 1− r); (0.7,0.65,0.72) ) 3 ( (−0.8,1− r, 1− r); (0.7,0.65,0.72) (1.2,1− r, 1− r); (0.7,0.65,0.72) ) ( (−1,1− r, 1− r); (0.7,0.65,0.72) (1.4,1− r, 1− r); (0.7,0.65,0.72) ) ( (−0.2,1− r, 1− r); (0.7,0.65,0.72) (−0.2,1− r, 1− r); (0.7,0.65,0.72) ) 4 ( (−1,1− r, 1− r); (0.7,0.65,0.72) (1.4,1− r, 1− r); (0.7,0.65,0.72) ) ( (−0.96,1− r, 1− r); (0.7,0.65,0.72) (1.44,1− r, 1− r); (0.7,0.65,0.72) ) ( (0.04,1− r, 1− r); (0.7,0.65,0.72) (−0.04,1− r, 1− r); (0.7,0.65,0.72) ) 5 ( (−0.96,1− r, 1− r); (0.7,0.65,0.72) (1.44,1− r, 1− r); (0.7,0.65,0.72) ) ( (−1,1− r, 1− r); (0.7,0.65,0.72) (1.48,1− r, 1− r); (0.7,0.65,0.72) ) ( (−0.04,1− r, 1− r); (0.7,0.65,0.72) (−0.04,1− r, 1− r); (0.7,0.65,0.72) ) 6 ( (−1,1− r, 1− r); (0.7,0.65,0.72) (1.48,1− r, 1− r); (0.7,0.65,0.72) ) ( (−0.992,1− r, 1− r); (0.7,0.65,0.72) (1.488,1− r, 1− r); (0.7,0.65,0.72) ) ( (0.008,1− r, 1− r); (0.7,0.65,0.72) (−0.008,1− r, 1− r); (0.7,0.65,0.72) ) Table 4: Fletcher-Reeves Method Iteration ￜ̃ￜ𝐢 = (𝛖𝟏, 𝛖𝟐) ￜ̃ￜ𝐢+𝟏 = (𝛖𝟏, 𝛖𝟐) 𝛁𝐟𝐢 1 ( (0,1− r, 1 − r); (0.66,0.52,0.53) (0,1− r, 1 − r); (0.66,0.52,0.53) ) ( (1,1− r, 1 − r); (0.7,0.65,0.72) (1,1− r, 1 − r); (0.7,0.65,0.72) ) ( (−1,1− r, 1− r); (0.7,0.65,0.72) (−1,1− r, 1− r); (0.7,0.65,0.72) ) 2 ( (1,1− r, 1 − r); (0.7,0.65,0.72) (1,1− r, 1 − r); (0.7,0.65,0.72) ) ( (−1,1− r, 1− r); (0.7,0.65,0.72) (1.5,1− r, 1− r); (0.7,0.65,0.72) ) ( (0,1− r, 1 − r); (0.7,0.65,0.72) (0,1− r, 1 − r); (0.7,0.65,0.72) ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 382 https://internationalpubls.com From the above tables(3),(4), we observed that the CSDM converges at iteration 6 and the FRM converges at iteration 2. Hence the optimal solution of the given SVNNLPP (11) is υ̃1 = (−1,1− r, 1− r); (0.7,0.65,0.72), υ̃2 = (1.5,1− r, 1− r); (0.7,0.65,0.72) with f̃(υ̃1, υ̃2) = (−1.25,1− r, 1− r); (0.7,0.65,0.72) That is the optimal solution of the SVNNLPP (11) is υ̃1 = (−2+ r, −1,0− r); (0.7,0.65,0.72), υ̃2 = (0.5+ r, 1.5,2.5− r); (0.7,0.65,0.72) with f̃(υ̃1, υ̃2) = (−2.25+ r, −1.25, −0.25− r); (0.7,0.65,0.72) 7. Result and Discussion we focused on using the Fletcher-Reeves method Table (2) Table (4) and the Cauchy's Steepest Descent Method Table (1), Table (3) to solve two different problems, each with a unique set of notations. Our study determined that both methods produced the same solution, but the Fletcher- Reeves method converged to the solution in fewer iterations than the Steepest Descent Method, making it the more efficient method for obtaining the single-valued neutrosophic optimal solution to the problems. Table (5),(6) represent the single-valued neutrosophic optimal solution of the SVNNLPP (9) for various values of r. Table 5: Neutrosophic the optimum technique to solve the provided neutrosophic a triangular coefficients for different values of r ∈ [0,1] 𝐫 𝛖𝟏 𝛖𝟐 0 (5.999,6.999,7.999); (0.8,0.75,0.75) (7.999,8.999,9.999); (0.8,0.75,0.75) 0.25 (6.249,6.999,7.749); (0.8,0.75,0.75) (8.249,8.999,9.749); (0.8,0.75,0.75) 0.5 (6.499,6.999,7.499); (0.8,0.75,0.75) (8.499,8.999,9.499); (0.8,0.75,0.75) 0.75 (6.749,6.999,7.249); (0.8,0.75,0.75) (8.749,8.999,9.249); (0.8,0.75,0.75) 1 (6.999,6.999,6.999); (0.8,0.75,0.75) (8.999,8.999,8.999); (0.8,0.75,0.75) Table 6: Continue on the table (5) 𝐫 𝐟(𝛖𝟏, 𝛖𝟐) 0 (185.999,186.999,187.999); (0.8,0.75,0.75) 0.25 (186.249,186.999,187.749); (0.8,0.75,0.75) 0.5 (186.499,186.999,187.499); (0.8,0.75,0.75) 0.75 (186.749,186.999,187.249); (0.8,0.75,0.75) 1 (186.999,186.999,186.999); (0.8,0.75,0.75) When r = 1, we see that 𝛖𝟏 = (6.999,6.999,6.999); (0.8,0.75,0.75), 𝛖𝟐 = (8.999,8.999,8.999); (0.8,0.75,0.75) and 𝑚𝑖𝑛 𝐟(ￜ̃ￜ𝟏, 𝛖𝟐) = (186.999,186.999,186.999); (0.8,0.75,0.75) Table (7), (8) represent the single-valued neutrosophic optimal solution of the SVNNLPP (11) for various values of r. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 383 https://internationalpubls.com Table 7: Neutrosophic the optimum technique to solve the provided neutrosophic a triangular coefficients for different values of r ∈ [0,1] 𝐫 𝛖𝟏 𝛖𝟐 0 (−2, −1,0); (0.7,0.65,0.72) (0.5,1.5,2.5); (0.7,0.65,0.72) 0.25 (−1.75, −1, −0.25); (0.7,0.65,0.72) (0.75,1.5,2.25); (0.7,0.65,0.72) 0.5 (−1.5, −1, −0.5); (0.7,0.65,0.72) (1,1.5,2); (0.7,0.65,0.72) 0.75 (−1.25, −1, −0.75); (0.7,0.65,0.72) (1.25,1.5,1.75); (0.7,0.65,0.72) 1 (−1, −1, −1); (0.7,0.65,0.72) (1.5,1.5,1.5); (0.7,0.65,0.72) Table 8: Continue on the table (7) 𝐫 𝐟(𝛖𝟏, 𝛖𝟐) 0 (−2.25, −1.25, −0.25); (0.7,0.65,0.72) 0.25 (−2, −1.25, −0.5); (0.7,0.65,0.72) 0.5 (−1.75, −1.25, −0.75); (0.7,0.65,0.72) 0.75 (−1.25, −1.25, −1); (0.7,0.65,0.72) 1 (−1.25, −1.25, −1.25); (0.7,0.65,0.72) When r = 1, we see that 𝛖𝟏 = (−1, −1, −1); (0.7,0.65,0.72), 𝛖𝟐 = (1.5,1.5,1.5); (0.7,0.65,0.72) and 𝑚𝑖𝑛 𝐟(ￜ̃ￜ𝟏, 𝛖𝟐) = (−1.25, −1.25, −1.25); (0.7,0.65,0.72) The above tables (5), (6), (7), and (8) illustrate that the proposed method provides decision makers with the flexibility to select their preferred solutions by making appropriate choices of r. 8. Conclusion In this paper, we discussed a solution concept for SVNNLPP involving neutrosophic triangular numbers. First, the given SVNNLPP is expressed in terms of its location index number , left and right fuzziness index functions. In the parametric forms of neutrosophic numbers, a new type of neutrosophic arithmetic and neutrosophic ranking are introduced and utilized. The neutrosophic quadratic and neutrosophic conjugate gradient theorems for SVNNLPP are established. The neutrosophic versions of the CSDM and the FRM are used, and the neutrosophic optimal solution of the SVNNLPP is obtained without having to convert the given problem. The neutrosophic fuzzy optimal solution of the given SVNNLPP is tabulated for different values of r ∈ [0,1]. It is important to note that by utilizing the suggested procedure and selecting an appropriate value for r ∈ [0,1], the decision maker has the flexibility to select his or her preferred optimal solution based on the situation. The numerical solutions have been presented and discussed. Both methods provide same solution but FRM requires less number itreration comparing with CSDM. Conflicts of Interest: The authors have no conflicting interests related to the content of this article to declare. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 384 https://internationalpubls.com References [1] Al-Naemi, G. M., The Convergent Properties of a New Parameter for Unconstrained Optimization, European Journal of Pure and Applied Mathematics, (2022), 15(4): 1683-1693. https://doi.org/10.29020/nybg.ejpam.v15i4.4538. [2] Andrei, N., Steepest Descent Methods. In: Modern Numerical Nonlinear Optimization Springer Optimization and Its Applications, (2022), 195, Springer, Cham, https://doi.org/10.1007/978-3-031-08720-2_3 [3] Antczak, T., Optimality conditions for invex nonsmooth optimization problems with fuzzy objective functions, Fuzzy Optim. Decis. Making, (2023), 22, 121, https://doi.org/10.1007/s10700-022-09381-4. [4] Basim, A.H, Hawraz, N.J, Yoksal, A.L, Issam Abdul Rahman Moghrabi & Ali Joma Alissa, An enhanced fletcher- reeves-like conjugate gradient methods for image restoration, International Journal of Electrical and Computer Engineering (IJECE), (2023), 13(6), 6268-6276, DOI: 10.11591/ijece.v13i6.pp6268-6276. [5] Bellman, R.E. & Zadeh, L.A., Decision making in a fuzzy environment, Management Sci., (1970), 17, 141-164, https://doi.org/10.1287/mnsc.17.4.B141. [6] Chakraborty, A., Mondal, S.P., Mahata, A.& Alam, S., Different Linear and Non-Linear Form of Trapezoidal Neutrosophic Numbers, De-Neutrosophication Techniques and Its Application in Time-Cost Optimization Technique, Sequencing Problem, RAIRO - Operations Research, (2021), 55, S97-S118, doi:10.1051/ro/2019090. [7] Hanachi, S.B., Sellami, B. & Belloufi, M., New iterative conjugate gradient method for nonlinear unconstrained optimization, RAIRO-Operations Research, (2022), 56(4), 2315-2327, https://doi.org/10.1051/ro/2022109. [8] Hassan, B.A., Jabbar, H.N., Laylani, Y.A., Moghrabi, I.A.R., Alissa, A.J.: An enhanced fletcher-reeves-like conjugate gradient methods for image restoration. International Journal of Electrical and Computer Engineering (IJECE), (2023), 13(6), 6268-6276. [9] Husin, S.F., Mamat, M., Rivaie, M., Mohamed, M.A. & Ibrahim, H., Modification of search direction in steepest descent method and its application in regression analysis, International Journal of Advanced Trends in Computer Science and Engineering, (2020), 9(1), 1, DOI: 10.30534/ijatcse/2020/1791.12020. [10] Hepzibah R.I, & Shilpa Ivin Emimal S., On Solving Neutrosophic Unconstrained Optimization Problems by Steepest Descent Method and Fletcher Reeves Method, Intern. J. Fuzzy Mathematical Archive, (2021), 19(2), 111- 122, DOI: http://dx.doi.org/10.22457/ijfma.v19n2a02230. [11] Hepzibah, R.I. & Emimal, S.S.I., Solving fuzzy unconstrained optimization problems using interval newton’s method, Advances and Applications in Mathematical Sciences, (2022), 21(3), 1463-1477. [12] Kaliyaperumal, P., & Das, A., A mathematical model for nonlinear optimization which attempts membership functions to address the uncertainties, Multidisciplinary Digit. Publishing Inst. (Math.), (2022), 10(10), 1743, https://doi.org/10.3390/math10101743. [13] Kanaya, Z. A., An interactive method for fuzzy multi objective nonlinear programming problems, JKAU Sci., (2010), 22(1), 103-112, DOI: 10.4197 / Sci. 22-1.8. [14] Lachhwani, K., Solving Non-linear Neutrosophic Linear Programming Problems, Neutrosophic Sets and Systems, (2023), 60(1), 2. [15] Maissam Jdid & Florentin Smarandache., The Use of Neutrosophic Methods of Operation Research in the Management of Corporate Work, Neutrosophic Systems with Applications, (2023), 3, 1-16, https://doi.org/10.61356/j.nswa.2023.11. [16] Ming Ma, Menahem Friedman, & Abraham kandel, A new fuzzy arithmetic, Fuzzy Sets and Systems, (1999), 108(1), 83-90. https://doi.org/10.1016/S0165-0114(97)00310-2. [17] Nagoorgani, A. & Sudha, K., Optimality conditions for fuzzy non-linear unconstrained minimization problems, Bull. Pure Appl. Sci., (2019), 38(1), 378-384, DOI : 10.5958/2320-3226.2019.00041.9. [18] Reig-Mullor, J. & Salas-Molina, F., Non-Linear Neutrosophic Numbers and Its Application to Multiple Criteria Performance Assessment, International Journal of Fuzzy Systems, (2022), 24, 2889-2904, doi:10.1007/s40815-022- 01295-y. [19] Seikh, Mijanur Rahaman & Shibaji Dutta, A Nonlinear Programming Model to Solve Matrix Games with Pay-offs of Single-valued Neutrosophic Numbers, Neutrosophic Sets and Systems, (2021), 47, 1, https://digitalrepository.unm.edu/nss-journal/vol47/iss1/24. https://doi.org/10.29020/nybg.ejpam.v15i4.4538 https://doi.org/10.1007/978-3-031-08720-2_3 https://doi.org/10.1007/s10700-022-09381-4 https://doi.org/10.1287/mnsc.17.4.B141 https://doi.org/10.1051/ro/2022109 https://doi.org/10.3390/math10101743 https://doi.org/10.61356/j.nswa.2023.11 https://doi.org/10.1016/S0165-0114(97)00310-2 https://digitalrepository.unm.edu/nss-journal/vol47/iss1/24 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 385 https://internationalpubls.com [20] S. Shilpa Ivin Emimal, R. Irene Hepzibah, Solving Intuitionistic Fuzzy Unconstrained Optimization Problems Using Interval Newton’s Method, Communications on Applied Nonlinear Analysis, (2024), 31( 4s), 366-383. https://doi.org/10.52783/cana.v31.835 [21] Smarandache, F., Neutrosophy: neutrosophic probability set and logic, Rehoboth: American Research Press, (1998). [22] Sujit, D., Bikash, K.R., Mohuya ,B.K., Samarjit, K. & Dragan, P., Neutrosophic fuzzy set and its application in decision making, Journal of Ambient Intelligence and Humanized Computing,(2020),11(3), 5017 – 5029. https://doi.org/10.1007/s12652-020-01808-3. [23] Umamaheswari P. & Ganesan K., A New Approach for the Solution of Unconstrained Fuzzy Optimization Problems, AIP Conference Proceedings,(2019), 2112, 020004, https://doi.org/10.1063/1.5112189. [24] Vanaja, G., Ganesan, K., & Rathour Laxmi., Solving Fuzzy Nonlinear Programming Problems With Exterior Penalty Fuzzy Valued Functions, Advanced Mathematical Models & Applications, (2024), 9(1), 130–146. https://doi.org/10.62476/amma9130. [25] Vanaja, G., Ganesan, K., An interior penalty function method for solving fuzzy nonlinear programming problems, Bulletin of Electrical Engineering \& Informatics, (2024), 13(3), 2320-2330, DOI: 10.11591/eei.v13i3.7047. [26] Ye, Jun, Wenhua Cui, & Zhikang Lu., Neutrosophic Number Nonlinear Programming Problems and Their General Solution Methods under Neutrosophic Number Environments, Axioms, (2018), 7(1), 13. https://doi.org/10.3390/axioms7010013. [27] Zadeh, L.A., Fuzzy sets, Information and Control, (1965) 8(3), 338-353, https://doi.org/10.1016/ S0019-958 (65)90241-X. https://doi.org/10.52783/cana.v31.835 https://doi.org/10.1007/s12652-020-01808-3 https://doi.org/10.1063/1.5112189 https://doi.org/10.62476/amma9130 https://doi.org/10.3390/axioms7010013