Fractional Programming

Frenk, Hans; Schaible, Siegfried

doi:10.1007/978-0-387-74759-0_189

Cited by 21 publications

(19 citation statements)

References 72 publications

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“…Hence, theρ j and ρ j are optimal values of certain fractional linear programs (see e.g. Frenk and Schaible 2004). Clearly λ (1) > 0 iff 1 − 4 > 0, λ (2) > 0 iff 3 − 4 < 0, λ (3) > 0 iff 1 − 2 < 0 and λ (4) > 0 iff 2 − 3 > 0.…”

Section: 4)mentioning

confidence: 97%

Multistationarity in mass action networks with applications to ERK activation

Conradi

Flockerzi

2011

J. Math. Biol.

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Ordinary Differential Equations (ODEs) are an important tool in many areas of Quantitative Biology. For many ODE systems multistationarity (i.e. the existence of at least two positive steady states) is a desired feature. In general establishing multistationarity is a difficult task as realistic biological models are large in terms of states and (unknown) parameters and in most cases poorly parameterized (because of noisy measurement data of few components, a very small number of data points and only a limited number of repetitions). For mass action networks establishing multistationarity hence is equivalent to establishing the existence of at least two positive solutions of a large polynomial system with unknown coefficients. For mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix, we present necessary and sufficient conditions for multistationarity that take the form of linear inequality systems. Solutions of these inequality systems define pairs of steady states and parameter values. We also present a sufficient condition to identify networks where the aforementioned conditions hold. To show the applicability of our results we analyse an ODE system that is defined by the mass action network describing the extracellular signal-regulated kinase (ERK) cascade (i.e. ERK-activation).

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Section: 4)mentioning

confidence: 97%

Multistationarity in mass action networks with applications to ERK activation

Conradi

Flockerzi

2011

J. Math. Biol.

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“…where X is a nonempty subset of R n , f i (x) and g i (x) are continuous on X and g i (x) are positive on X [10]. Further, if we assume 1.…”

Section: Generalized Fractional Programmingmentioning

confidence: 98%

Fast algorithms for L<inf>∞</inf> problems in multiview geometry

Agarwal

Snavely

Seitz

2008

2008 IEEE Conference on Computer Vision and Pattern Recognition

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Many problems in multi-view geometry, when posed as minimization of the maximum reprojection error across observations, can be solved optimally in polynomial time. We show that these problems are instances of a convex-concave generalized fractional program. We survey the major solution methods for solving problems of this form and present them in a unified framework centered around a single parametric optimization problem. We propose two new algorithms and show that the algorithm proposed by Olsson et al. [21] is a special case of a classical algorithm for generalized fractional programming. The performance of all the algorithms is compared on a variety of datasets, and the algorithm proposed by Gugat [12] stands out as a clearwinner. An open source MATLAB toolbox thats implements all the algorithms presented here is made available.

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“…can be solved by various algorithms (Frenk and Schaible (2001)) in which most of them are based on the Dinkelbach method, we adopt the Dinkelbach-type-2 algorithm (Crouzeix, Ferland and Schaible (1986)) for its refined function in order to enhance the convergent performance. This algorithm is introduced as follows:…”

Section: Preference Approach To Fuzzy Linear Inequalitiesmentioning

confidence: 99%

“…Based on the principle of maximizing the minimal degree of all membership functions, a non-linear auxiliary crisp model is derived in this study for the considered fuzzy optimization problem. Since the auxiliary model is characterized as a generalized fractional program (GFP), it can be solved by various existing algorithms (Frenk and Schaible (2001)), such as the Dinkelbach method. Then, a refined version, namely, the Dinkelbach-type-2 algorithm (Crouzeix, Ferland and Schaible (1986)), is introduced in this study.…”

Section: Introductionmentioning

confidence: 99%

Preference Approach to Fuzzy Linear Inequalities and Optimizations

Wang

2005

Fuzzy Optim Decis Making

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In this study we first present the preference structures in decision making as a generalized non-linear function. Then, we incorporate it into a fuzzy linear inequality of which the progressive and conservative manners are described by the concept of target hyperplanes. Thus, one's preference domain is said to be bounded by such soft constraint. When facing different situations, a membership function is defined as an evaluation function for incorporating one's optimistic or pessimistic attitude into this soft constraint. Once a goal is pursued, the problem is transformed into a symmetric fuzzy linear program. Based on max-min principle, an auxiliary crisp model in the form of a generalized fractional program is derived. Then, Dinkelbach-type-2 and the bisection algorithms are adopted for solution. Finally their simulation results of an uncertain production scheduling problem are reported.

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Fractional Programming

Cited by 21 publications

References 72 publications

Multistationarity in mass action networks with applications to ERK activation

Multistationarity in mass action networks with applications to ERK activation

Fast algorithms for L<inf>∞</inf> problems in multiview geometry

Preference Approach to Fuzzy Linear Inequalities and Optimizations

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Fractional Programming

Cited by 21 publications

References 72 publications

Multistationarity in mass action networks with applications to ERK activation

Multistationarity in mass action networks with applications to ERK activation

Fast algorithms for L<inf>&#x221E;</inf> problems in multiview geometry

Preference Approach to Fuzzy Linear Inequalities and Optimizations

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Fast algorithms for L<inf>∞</inf> problems in multiview geometry