Generalized linear multiplicative and fractional programming

Konno, Hiroshi; Kuno, Takahito

doi:10.1007/bf02283691

Cited by 62 publications

(25 citation statements)

References 3 publications

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“…The linear ratios can be replaced by a convex over a concave function, and the method in [36] is still valid. Such a parametric approach for solving a multiplicative program reduces to the one in [35] where p = 1 and both f i (x) and g i (x) are linear in (4.5). Konno and Fukaishi in [33] give a convex relaxation for problem (1.2).…”

Section: Algorithmsmentioning

confidence: 99%

Fractional programming: The sum-of-ratios case

Schaible

Shi²

2003

Optimization Methods and Software

143

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One of the most difficult fractional programs encountered so far is the sum-of-ratios problem. Contrary to earlier expectations it is much more removed from convex programming than other multi-ratio problems analyzed before. It really should be viewed in the context of global optimization. It proves to be essentially NP-hard in spite of its special structure under the usual assumptions on numerators and denominators. The article provides a recent survey of applications, theoretical results and various algorithmic approaches for this challenging problem.

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Section: Algorithmsmentioning

confidence: 99%

Fractional programming: The sum-of-ratios case

Schaible

Shi²

2003

Optimization Methods and Software

143

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“…81000650) and China Postdoctoral Science Foundation (No. 20100470206) bond portfolio optimization [2], VLSI chip design [3], multiple objective optimization [4] and so on. Problem (P x ) is known to be NP-hard, even in special cases such as when p = 2, X is a polyhedron, and f i is linear for each i = 1, 2 [5].…”

Section: Definition 2 the Multiplicative Programming Problem (P X )mentioning

confidence: 99%

An approximation algorithm for convex multiplicative programming problems

Shao¹,

Ehrgott²

2011

2011 IEEE Symposium on Computational Intelligence in Multicriteria Decision-Making (MDCM)

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Multiplicative programming problems are difficult global optimization problems known to be NP-hard. In this paper we propose a method for approximately solving convex multiplicative programming problems. This work is based on our previous work "An approximation algorithm for convex multiobjective programming problems". We show, by slightly changing the algorithm, that our method can be used to solve convex multiplicative programming problems. We provide an example that shows that our method has computational advantage compared with Benson's outcome space branch and bound outer approximation algorithm.

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“…Such problems arise in a variety of applications, including economic analysis [12], bond portfolio optimization [16], optimization of geometrical objects [19], and VLSI chip design [20]. In recent years, a number of algorithms have been proposed for globally solving various types of multiplicative and generalized multiplicative programming problems (see, e.g., [7,14,17,22,27,32], the survey in [18], and references therein).…”

Section: Applicationsmentioning

confidence: 99%

Concave envelopes of monomial functions over rectangles

Benson

2004

Naval Research Logistics

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Abstract:The construction of convex and concave envelopes of real-valued functions has been of interest in mathematical programming for over 3 decades. Much of this interest stems from the fact that convex and concave envelopes can play important roles in algorithms for solving various discrete and continuous global optimization problems. In this article, we use a simplicial subdivision tool to present and validate the formula for the concave envelope of a monomial function over a rectangle. Potential algorithmic applications of this formula are briefly indicated.

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Generalized linear multiplicative and fractional programming

Cited by 62 publications

References 3 publications

Fractional programming: The sum-of-ratios case

Fractional programming: The sum-of-ratios case

An approximation algorithm for convex multiplicative programming problems

Concave envelopes of monomial functions over rectangles

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