Extensions of Dinkelbach's algorithm for solving non-linear fractional programming problems

Ródenas, Ricardo García; López, M. Luz; Verastegui, Doroteo

doi:10.1007/bf02564711

Cited by 41 publications

(32 citation statements)

References 19 publications

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“…Readers may refer to [42][43] and references therein for details. Example 3.1 Consider the following linear fractional optimization problem subject to a system of sup-T P equations, which was originally presented by Wu et al [33] : …”

Section: Dinkelbach's Algorithmmentioning

confidence: 99%

Minimizing a linear fractional function subject to a system of sup-T equations with a continuous Archimedean triangular norm

Fang

2009

J Syst Sci Complex

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This paper shows that the problem of minimizing a linear fractional function subject to a system of sup-T equations with a continuous Archimedean triangular norm T can be reduced to a 0-1 linear fractional optimization problem in polynomial time. Consequently, parametrization techniques, e.g., Dinkelbach's algorithm, can be applied by solving a classical set covering problem in each iteration. Similar reduction can also be performed on the sup-T equation constrained optimization problems with an objective function being monotone in each variable separately. This method could be extended as well to the case in which the triangular norm is non-Archimedean.

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Section: Dinkelbach's Algorithmmentioning

confidence: 99%

Minimizing a linear fractional function subject to a system of sup-T equations with a continuous Archimedean triangular norm

Fang

2009

J Syst Sci Complex

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“…Moreover, constraint is concave, while all the other constraints are linear in a q , k , ς . Therefore, the solution to this problem can be obtained by solving a sequence of maximization problems that satisfy optimality conditions …”

Section: Max‐min Energy‐efficient Power Allocationmentioning

confidence: 99%

“…Therefore, the solution to this problem can be obtained by solving a sequence of maximization problems that satisfy optimality conditions. 24,25 Based on Remark 5, and to solve problem R-MMEE-PA ( q,k, )…”

Section: Max-min Energy-efficient Power Allocationmentioning

confidence: 99%

Joint relay selection and max‐min energy‐efficient power allocation in downlink multicell NOMA networks: A matching‐theoretic approach

Baidas

Bahbahani

El-Sharkawi

et al. 2019

Trans Emerging Tel Tech

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In this paper, the problem of joint relay selection and max‐min energy‐efficient power allocation (J‐RS‐MMEE‐PA) in downlink multicell nonorthogonal multiple access (NOMA) networks is studied. In particular, the goal is to perform relay selection for each cellular user within each cell, so as to achieve max‐min energy efficiency while satisfying quality‐of‐service (QoS) constraints. However, the formulated J‐RS‐MMEE‐PA problem happens to be nonconvex (ie, computationally intensive). In turn, a solution procedure for max‐min energy‐efficient power allocation is devised to determine the energy efficiency of each user per potential relay while meeting the target minimum rate per user. After that, the relay selection problem is modeled as a student‐project allocation with preferences over projects matching problem. A polynomial‐time complexity stable matching (SM) algorithm is proposed, which takes into account the maximum number of users that can select a relay as well as the number of users that can be associated with a base station. However, the proposed SM algorithm can only guarantee an SM size that is at least half the size of the maximum “optimal” cardinality SM. Thus, the improved SM (I‐SM) algorithm with polynomial‐time complexity is devised, so as to guarantee an SM size that is at least two‐thirds of that of the optimal matching. Lastly, simulation results are presented to compare the proposed matching algorithms to the J‐RS‐MMEE‐PA scheme, where it has been shown that the I‐SM algorithm is superior to its SM counterpart and yields comparable energy efficiency per cellular user to the J‐RS‐MMEE‐PA scheme while satisfying QoS constraints.

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“…In particular, Cour and Shi (2007) showed that it is NP-hard, in general, to solve for general eigenvectors under linear inequalities. So, we propose an iterative algorithm to find the global minimum of this optimization problem using Dinkelbach's method for fractional programming (FP) (Dinkelbach 1967;Rodenas et al 1999). To make the paper self-contained, we give a brief description of this algorithm next.…”

Section: Dncut Framework Under Hard and Convex Constraintsmentioning

confidence: 99%

“…The Dinkelbach algorithm was extended by Rodenas et al (1999) to provide a general framework for FP's, summarized below as Algorithm 1. λ * is the global minimum value of the objective function. Here, we emphasize that the superlinear convergence property is only regarding the iterations needed to achieve λ * and not the convergence of each iteration.…”

Section: Dinkelbach Algorithm For Fractional Programmingmentioning

confidence: 99%

Dinkelbach NCUT: An Efficient Framework for Solving Normalized Cuts Problems with Priors and Convex Constraints

Ghanem

Ahuja

2010

Int J Comput Vis

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In this paper, we propose a novel framework, called Dinkelbach NCUT (DNCUT), which efficiently solves the normalized graph cut (NCUT) problem under general, convex constraints, as well as, under given priors on the nodes of the graph. Current NCUT methods use generalized eigen-decomposition, which poses computational issues especially for large graphs, and can only handle linear equality constraints. By using an augmented graph and the iterative Dinkelbach method for fractional programming (FP), we formulate the DNCUT framework to efficiently solve the NCUT problem under general convex constraints and given data priors. In this framework, the initial problem is converted into a sequence of simpler sub-problems (i.e. convex, quadratic programs (QP's) subject to convex constraints). The complexity of finding a global solution for each sub-problem depends on the complexity of the constraints, the convexity of the cost function, and the chosen initialization. However, we derive an initialization, which guarantees that each sub-problem is a convex QP that can be solved by available convex programming techniques. We apply this framework to the special case of linear constraints, where the solution is obtained by solving a sequence of sparse linear systems using the conjugate gradient method. We validate DNCUT by performing binary segmentation on real images both with and without linear/nonlinear constraints, as well as, multi-class segmentation. When possi-B. Ghanem ( ) · N. Ahuja ble, we compare DNCUT to other NCUT methods, in terms of segmentation performance and computational efficiency. Even though the new formulation is applied to the problem of spectral graph-based, low-level image segmentation, it can be directly applied to other applications (e.g. clustering).

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Extensions of Dinkelbach's algorithm for solving non-linear fractional programming problems

Abstract: Fractional Programming, Parametric Optimization, Nonconvex Programming, Nonlinear Programming, AMS subject classification, 90C32, 90C31, 90C26, 90C30,

Cited by 41 publications

References 19 publications

Minimizing a linear fractional function subject to a system of sup-T equations with a continuous Archimedean triangular norm

Minimizing a linear fractional function subject to a system of sup-T equations with a continuous Archimedean triangular norm

Joint relay selection and max‐min energy‐efficient power allocation in downlink multicell NOMA networks: A matching‐theoretic approach

Dinkelbach NCUT: An Efficient Framework for Solving Normalized Cuts Problems with Priors and Convex Constraints

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