Fractional Optimization Problem Research Articles

A Bregman proximal subgradient algorithm for nonconvex and nonsmooth fractional optimization problems

In this paper, we study a class of nonconvex and nonsmooth fractional optimization problem, where the numerator of which is the sum of a nonsmooth and nonconvex function and a relative smooth nonconvex function, while the denominator is relative weakly convex nonsmooth function. We propose a Bregman proximal subgradient algorithm for solving this type of fractional optimization problems. Under moderate conditions, we prove that the subsequence generated by the proposed algorithm converges to a critical point, and the generated sequence globally converges to a critical point when the objective function satisfies the Kurdyka-Łojasiewicz property. We also obtain the convergence rate of the proposed algorithm. Finally, two numerical experiments illustrate the effectiveness and superiority of the algorithm. Our results give a positive answer to an open problem proposed by Bot et al. [14].

Neutrosophic linear fractional optimization problem using fuzzy goal programming approach

This research paper develops an algorithm to solve a neutrosophic linear fractional optimization problem, where the cost of the objective functions, the technology coefficients, and the resources are neutrosophic numbers (triangular and trapezoidal numbers). Two defuzzification methods are used here to convert the neutrosophic linear fractional optimization problem into a multi-objective linear fractional optimization problem (MOLFOP). The fractional objective function is defuzzified using the component-wise optimization method, and the linear constraints are defuzzified using the aggregation ranking function strategy. Hence, the MOLFOP is then solved using the fuzzy goal programming method. In order to further explain our suggested algorithm, two numerical examples are solved at the end, and the outcomes obtained by our method are compared with the outcomes obtained using the other algorithm.

A note on “A new method for intuitionistic fuzzy multi-objective linear fractional optimization problem and its application in agricultural land allocation problem”

Moges et al. (Inf. Sci. 625 (2023) 457–475) proposed a method to find a Pareto-optimal solution (POS) of intuitionistic fuzzy multi-objective linear fractional optimization problem (MOLFOP). In this method, firstly, an intuitionistic fuzzy MOLFOP is transformed into a crisp MOLFOP with the help of a ranking function. Then, using the existing method (Fuzzy Sets Syst. 125 (2002) 335–342), the transformed crisp MOLFOP is transformed into a crisp multi-objective linear optimization problem (MOLOP). Finally, it is assumed that to find a POS of the considered intuitionistic fuzzy MOLFOP is equivalent to find a POS of the transformed crisp MOLOP. It is pertinent to mention that this assumption is valid only if the transformed crisp MOLFOP is equivalent to the considered intuitionistic fuzzy MOLFOP as well as the transformed crisp MOLOP is equivalent to the transformed crisp MOLFOP. In this note, a numerical example is considered to show that the transformed crisp MOLFOP is not equivalent to the considered intuitionistic fuzzy MOLFOP. Also, on the basis of the existing results (Fuzzy Sets Syst. 246 (2014) 156–159), it is pointed out that the transformed crisp MOLOP is not equivalent to the transformed crisp MOLFOP. Therefore, Moges et al.’s method is not valid and hence, the results of agricultural planning problem of Ethopia for cultivation of different crops, obtained by Moges et al. with their proposed method, are not correct.

On application of linear fractional programming problem using e-education system

This article focuses on an area of nonlinear programming problems known as linear fractional programming problems with multiple objectives. When tackling real-world linear fractional optimization problems, ambiguity and uncertainty in decision-making are inherent. This research aims to present a simple and computationally quick approach to solving multiple objective linear fractional programming problems with all decision variables and parameters described in terms of crisp. The proposed solution algorithm is based primarily on the fuzzy-based technique, and a membership function strategy. To resolve the multi-objective linear fractional programming problem, first consider the problem as a single objective function and along with the fuzzy programming model obtain the optimal solution using LINGO software. LINGO is a software application primarily used for solving linear, nonlinear, and integer optimization problems Moreover, an e-education setup problem demonstrates the steps of the proposed method.

Learnable bilevel optimization method for electrical capacitance tomography

The positive role of electrical capacitance tomography technology depends on high-precision tomographic images. Despite its success, one of the main barriers is the low-quality tomogram. A new learnable bilevel optimization imaging method is proposed to address this problem in this study, in which the image prior and model parameters can be learned from the collected datasets. The upper level optimization problem learns the regularization parameter under the constraint of the lower level optimization problem that implements image reconstruction. A new lower level optimization problem with the introduced machine learning prior is built, which leverages the prior knowledge from collected datasets, imaging targets and imaging mechanisms. The machine learning prior is learned through extreme learning machine, and the training is reformulated into a fractional optimization problem with the physical mechanisms of imaging as a constraint. A new optimizer is proposed to solve the learnable bilevel optimization imaging problem. The effectiveness has been demonstrated by the reconstruction of higher precision images and better noise immunity in comparison with advanced imaging techniques. The new imaging method unifies imaging mechanisms and machine learning, and promotes the complementarity of image priors, which offers new opportunities to unlock the potential of the measurement technique.

New Approach to Solving Fuzzy Multiobjective Linear Fractional Optimization Problems

In this paper, an iterative approach based on the use of fuzzy parametric functions is proposed to find the best preferred optimal solution to a fuzzy multiobjective linear fractional optimization problem. From this approach, the decision-maker imposes tolerance values or termination conditions for each parametric objective function. Indeed, the fuzzy parametric values are computed iteratively, and each fuzzy fractional objective is transformed into a fuzzy non-fractional parametric function using these values of parameters. The core value of fuzzy numbers is used to transform the fuzzy multiobjective non-fractional problem into a deterministic multiobjective non-fractional problem, and the ε-constraint approach is employed to obtain a linear single objective optimization problem. Finally, by setting the value of parameter ε, the Dangtzig simplex method is used to obtain an optimal solution. Therefore, the number of solutions is equal to the number of used values, and the optimal solution is chosen according to the preference of the decision-maker. We have provided a didactic example to highlight the step of our approach and its numerical performances.

Enhanced Solutions to Intuitionistic Fuzzy Multiobjective Linear Fractional Optimization Problems via Lexicographic Method

Optimization involving fuzzy numbers generally, and intuitionistic fuzzy numbers particularly is more and more an essential part of any computational intelligence method; and might be of interest in modeling fuzzy control systems, or carrying out a fuzzy sensitivity analysis. The recent literature includes many research papers related to both theoretical modelling, and practical implementation of fuzzy decision support systems. Fuzzy optimization is one of the main part in such decisionmaking tools. A realistic solution to a fuzzy optimization problem is always desired, and similarly, a Pareto optimal solution to a multiple objective optimization problem is always a need. In this paper we analyze the shortcomings; and eliminate the weaknesses of a solution approach from the literature proposed by Moges et al. in 2023. Firstly, Moges et al. used an accuracy function to de-fuzzify the original intuitionistic fuzzy optimization problem; then linearized the obtained crisp problem; and finally involved fuzzy goals to solve the derived multiple objective linear problem. We present two faulty key points of their approach, namely the de-fuzzification and linearization steps; prove the inappropriateness of their results; and propose some improvements. The final aim in addressing the first key point is to clarify the border between defuzzifications made in accordance to true theoretical statements, and those made for modeling reasons. The methodological improvement we propose is related to the second key point and it assures that Pareto optimal solutions - highly required in multiple objective optimization - are obtained. To illustrate our point of view, we use a numerical example from the literature, and report better numerical results derived by the improved methodology.

정수 선형 분수 계획법 문제에 대한 파라메트릭 접근법

In this paper, we focus on the integer linear fractional optimization problem, a special case of fractional programming where all functions in the objective function and constraints are linear and all variables are bounded and discrete. To derive the optimal solution, the parametric algorithms are considered, and the efficiency of the algorithms is investigated computationally. Unlike traditional parametric algorithms such as Newton's method, which create a unidirectional sequence approaching the optimal function value, our proposed algorithm generates both upper and lower bounds converging to this value. To demonstrate its effectiveness across various production and operations management problems, the suggested algorithms are used to solve the fractional knapsack problem by comparison to other algorithms (e.g., Newton’s method) under the randomized experimental conditions. The relative practical performance measured by the number of function calls demonstrates that the proposed algorithms are fast and robust for solving the linear fractional programs with discrete variables. Leveraging this algorithm holds the potential to overcome situations in traditional production and operations problems where non-fractional objective functions were previously unconsidered, thereby expecting to derive new outcomes and significance.

On Mond–Weir-Type Robust Duality for a Class of Uncertain Fractional Optimization Problems

This article is focused on the investigation of Mond–Weir-type robust duality for a class of semi-infinite multi-objective fractional optimization with uncertainty in the constraint functions. We first establish a Mond–Weir-type robust dual problem for this fractional optimization problem. Then, by combining a new robust-type subdifferential constraint qualification condition and a generalized convex-inclusion assumption, we present robust ε-quasi-weak and strong duality properties between this uncertain fractional optimization and its uncertain Mond–Weir-type robust dual problem. Moreover, we also investigate robust ε-quasi converse-like duality properties between them.

Componentwise Dinkelbach algorithm for nonlinear fractional optimization problems

The paper deals with fractional optimization problems where the objective function (ratio of two functions) is defined on a Cartesian product of two real normed spaces X and Y. Within this framework, we are interested to determine the so-called partial minimizers, i.e. points in with the property that any of its variables minimizes the objective function, restricted to this variable, with respect to the other one. While any global minimizer is obviously a partial minimizer, the reverse implication holds true only under additional assumptions (e.g. separability properties of the involved functions). By exploiting the particularities of the objective function, we deliver a Dinkelbach type algorithm for computing partial minimizers of fractional optimization problems. Further assumptions on the involved spaces and functions, such as Lipschitz-type continuity, partial Fréchet differentiability, and coercivity, enable us to establish the convergence of our algorithm to a partial minimizer.

Minimizing a complex quadratic fractional optimization problem with two second-order cone constraints

Minimizing a complex quadratic fractional optimization problem with two second-order cone constraints

Cross-Layer Design for Energy-Efficient Reliable Multi-Path Transmission in Event-Driven Wireless Sensor Networks.

In event-driven wireless sensor networks (WSNs), a reliable, efficient, and scalable routing solution is required for the reliable delivery of sensory data to the base station (BS). However, existing routing algorithms rarely address the issue of energy efficiency under multi-path conflicts for multi-event-driven scenarios. In order to maximize energy efficiency while maintaining a manageable conflict probability, this paper investigates a cross-layer design of routing and power control for multi-event-driven WSNs. We first develop a mathematical characterization of the conflict probability in multi-path routing, and we then formulate the energy efficiency maximization problem as a non-convex combinatorial fractional optimization problem subject to a maximum conflict probability constraint. By utilizing non-linear fractional programming and dual decomposition, an iterative search algorithm was used to obtain near-optimal power allocation and routing solutions. Extensive results demonstrate that our proposed algorithm achieved a gain of 9.09% to 35.05% in energy efficiency compared to other routing algorithms, thus indicating that our proposed algorithm can avoid unnecessary control overhead from multi-path conflicts with a lower conflict probability and can ensure maximum energy efficiency through routing and power control design.

Robust Optimality Conditions for a Class of Fractional Optimization Problems

In this paper, by considering the parametric technique, we study a class of fractional optimization problems involving data uncertainty in the objective functional. We formulate and prove the robust Karush-Kuhn-Tucker necessary optimality conditions and provide their sufficiency by considering the convexity and/or concavity assumptions of the involved functionals. In addition, to complete the study, an illustrative example is presented.

Dinkelbach Type Approximation Algorithms for Nonlinear Fractional Optimization Problems

In this paper we establish some approximation versions of the classical Dinkelbach algorithm for nonlinear fractional optimization problems in Banach spaces. We start by examining what occurs if at any step of the algorithm, the generated point desired to be a minimizer can only be determined with a given error. Next we assume that the step error tends to zero as the algorithm advances. The last version of the algorithm we present is making use of Ekeland’s variational principle for generating the sequence of minimizer-like points. In the final part of the article we deliver some results in order to achieve a Palais-Smale type compactness condition that guarantees the convergence of our Dinkelbach-Ekeland algorithm.

Inertial Proximal Block Coordinate Method for a Class of Nonsmooth Sum-of-Ratios Optimization Problems

In this paper, we consider a class of nonsmooth sum-of-ratios fractional optimization problems with block structure. This model class is ubiquitous and encompasses several important nonsmooth optimization problems in the literature. We first propose an inertial proximal block coordinate method for solving this class of problems by exploiting the underlying structure. The global convergence of our method is guaranteed under the Kurdyka--Lojasiewicz (KL) property and some mild assumptions. We then identify the explicit exponents of the KL property for three important structured fractional optimization problems. In particular, for the sparse generalized eigenvalue problem with either cardinality regularization or sparsity constraint, we show that the KL exponents are 1/2, and so, the proposed method exhibits linear convergence rate. Finally, we illustrate our theoretical results with both analytic and simulated numerical examples.

Parametric approach for multi-objective enhanced interval linear fractional programming problem

The design (decision) variables in the presented article of a multi-objective interval fractional optimization problem based on a linear function are assumed to take the form of a closed interval using the concept of the parametric form of an interval. The original problem is initially changed into equivalent multi-objective interval linear programming with the design variables as closed intervals. Further, it is made free from interval uncertainty by changing into a classical single-objective problem using the weighted-sum method. The solutions of the model are theoretically justified by its existence. Finally, a numerical example and a case study on the agricultural planting structure optimization problem with hypothetical data are presented to support the recommended technique for the model.

Optimality theorems for linear fractional optimization problems involving integral functions defined on Cn [0,1

Optimality theorems for linear fractional optimization problems involving integral functions defined on Cn [0,1

Resource Allocation for an IRS-Assisted Dual-Functional Radar and Communication System: Energy Efficiency Maximization

Dual-functional radar and communication (DFRC) techniques are promising in the next generation wireless communication network. An intelligent reflecting surface (IRS) can improve the energy efficiency (EE) of the DFRC system, which is of crucial importance in practical applications. However, the EE achieved by using the sum-rate can be limited by the channel conditions and the radar performance requirement. Thus, in this paper, the EE maximization problem is studied in an IRS-assisted DFRC system. The EE is maximized by jointly optimizing the transmit beamforming and the IRS phase shift matrix under both the perfect channel state information (CSI) and the imperfect CSI cases. To tackle the challenging nonconvex fractional optimization problem, an alternating optimization algorithm is proposed. The high-order challenging radar performance constraint is tackled by using properties of the convex function and Hermitian matrix involved in the analysis. In the imperfect CSI case, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal{S}}$ </tex-math></inline-formula> -Procedure is applied to tackle the semi-infinite inequality arising from the channel uncertainty. Simulation results demonstrate that our proposed design enjoys fast convergence and achieves a higher EE compared to the benchmark schemes, even under imperfect CSI case. Furthermore, simulation results demonstrate that the EE of the DFRC system can be significantly improved by applying IRS.

A Convex Approximation for a PDE Constrained Fractional Optimization Problem with an Application to Photonic Crystal Design

A Convex Approximation for a PDE Constrained Fractional Optimization Problem with an Application to Photonic Crystal Design

A new method for intuitionistic fuzzy multi-objective linear fractional optimization problem and its application in agricultural land allocation problem

This paper presents a new method for solving an intuitionistic fuzzy multi-objective linear fractional optimization (IFMOLFO) problem with crisp and intuitionistic fuzzy constraints. Here, all uncertain parameters are represented as triangular intuitionistic fuzzy numbers. We used an accuracy ranking function and variable transformation in the proposed method to convert an IFMOLFO problem into a crisp multi-objective linear optimization problem. Then, we formulated the first phase of the weighted intuitionistic fuzzy goal programming (WIFGP) model to obtain an intuitionistic fuzzy non-dominant (IFND) solution for the IFMOLFO problem. Several strategies for obtaining an IFND solution to the IFMOLFO problem have been proposed in the literature. However, in addition to constructing the phase-I WIFGP model, this study shows that the IFND solution may not be Pareto-optimal when some of the under-deviation variables are zero. As a result, the second phase of the WIFGP model is applied to address this issue. The benefits of both models are merged to provide a novel method, unlike any other method in the literature, for producing optimal solutions that satisfy both IFND and Pareto-optimal requirements. The suggested algorithm’s efficiency and reliability are demonstrated by addressing a real-life case study of an agricultural production planning problem and followed by solving a numerical example from literature.