Generalized Concavity

Avriel, Mordecai; Diewert, W. Erwin; Schaible, Siegfried; Zang, Israël

doi:10.1007/978-1-4684-7600-2

Cited by 330 publications

(154 citation statements)

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“…for every x ~ 2, and X := {y E JR": y >1 O, ~2~ ~ ~ Yi = 1}. This is a direct consequence of the property that a quasiconvex function attains its maximum in a vertex of a convex polyhedron [1]. Moreover, by the assumptions on the vector functions f and g we obtain that the function x~yTf(x)/yTg(x)…”

Section: The Dual Problem and How To Solve Itmentioning

confidence: 94%

“…Moreover, the function c is semistrictly quasiconcave [1], since it is the infimum of semistrictly quasiconcave functions h(-, y) indexed by x. Thus, by (3), we need to solve the quasiconcave optimization problem:…”

Section: X~ ~" Yt G( X )mentioning

confidence: 99%

“…where a local maximum is a global maximum [1]. Notice that (Q) corresponds to a new "dual" problem of (P).…”

Section: X~ ~" Yt G( X )mentioning

confidence: 99%

“…Fractional programming, i.e., the minimization of a ratio of two functions subject to constraints, has been studied extensively during the last several decades [1,8,27,28]. In the 1980s the focus has shifted towards multi-ratio optimization problems.…”

Section: Introductionmentioning

confidence: 99%

“…Various algorithms in generalized fractional programming are surveyed in [9] whereas computational experience with some of these algorithms is reported in [5,7,14]. Meanwhile also a duality theory for generalized fractional programs has been developed [1,8].…”

Section: Introductionmentioning

confidence: 99%

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A new algorithm for generalized fractional programs

Barros

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Schaible

et al. 1996

Mathematical Programming

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A new dual problem for convex generalized fractional programs with no duality gap is presented and it is shown how this dual problem can be efficiently solved using a parametric approach. The resulting algorithm can be seen as "'dual'" to the Dinkelbach-type algorithm for generalized fractional programs since it approximates the optimal objective value of the dual (primal) problem from below. Convergence results for this algorithm are derived and an easy condition to achieve superlinear convergence is also established. Moreover, under some additional assumptions the algorithm also recovers at the same time an optimal solution of the primal problem. We also consider a variant of this new algorithm, based on scaling the "'dual" parametric function, The numerical results, in case of quadratic-linear ratios and linear constraints, show that the performance of the new algorithm and its scaled version is superior to that of the Dinkelbach-type algorithms. From the computational results it also appears that contrary to the primal approach, the "'dual" approach is less influenced by scaling.

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Section: The Dual Problem and How To Solve Itmentioning

confidence: 94%

Section: X~ ~" Yt G( X )mentioning

confidence: 99%

“…where a local maximum is a global maximum [1]. Notice that (Q) corresponds to a new "dual" problem of (P).…”

Section: X~ ~" Yt G( X )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A new algorithm for generalized fractional programs

Barros

Frenk

Schaible

et al. 1996

Mathematical Programming

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Queueing-location problems on the plane

Drezner

Schaible

Simchi‐Levi

1990

Naval Research Logistics

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A global optimization approach for generating efficient points for multiobjective concave fractional programs

Benson

2005

J. Multi-Crit. Decis. Anal.

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In this article, we present a global optimization approach for generating efficient points for multiobjective concave fractional programming problems. The main work of the approach involves solving an instance of a concave multiplicative fractional program (W). Problem (W) is a global optimization problem for which no known algorithms are available. Therefore, to render the approach practical, we develop and validate a branch and bound algorithm for globally solving problem (W). To illustrate the performance of the global optimization approach, we use it to generate efficient points for a sample multiobjective concave fractional program.

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Generalized Concavity

Cited by 330 publications

References 0 publications

A new algorithm for generalized fractional programs

A new algorithm for generalized fractional programs

Queueing-location problems on the plane

A global optimization approach for generating efficient points for multiobjective concave fractional programs

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