Linear multiplicative programming

Konno, Hiroshi; Kuno, Takahito

doi:10.1007/bf01580893

Cited by 108 publications

(48 citation statements)

References 25 publications

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“…The problem of optimizing a product of two variables over a polyhedron is called a linear multiplicative program ( [10,11]). In these references, parametric primal-dual simplex-type algorithms were presented to find the optimal value.…”

Section: Non-degenerate Situationsmentioning

confidence: 99%

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Enumerating the Nash equilibria of rank-1 games

Theobald¹

2009

Polyhedral Computation

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Abstract. A bimatrix game (A, B) is called a game of rank k if the rank of the matrix A + B is at most k. We consider the problem of enumerating the Nash equilibria in (non-degenerate) games of rank 1. In particular, we show that even for games of rank 1 not all equilibria can be reached by a LemkeHowson path and present a parametric simplex-type algorithm for enumerating all Nash equilibria of a non-degenerate game of rank 1.

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Section: Non-degenerate Situationsmentioning

confidence: 99%

“…This algorithm is based on the techniques of Konno and Kuno who have investigated linear multiplicative programs ( [10,11]). Our problem can be seen as an enumeration problem of generalized linear multiplicative programming.…”

Section: Introductionmentioning

confidence: 99%

Enumerating the Nash equilibria of rank-1 games

Theobald¹

2009

Polyhedral Computation

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“…, y k ) = k i=1 y i . It is known that such a function g is quasi-concave (Konno and Kuno 1992), and therefore its minimum is attained at an extreme point of the polytope. Multiplicative objective functions also arise in combinatorial optimization problems.…”

Section: Introductionmentioning

confidence: 99%

“…Approximation algorithms for minimizing a non-linear function over a polytope without the quasi-concavity assumption have not been studied in the literature so far. Konno and Kuno (1992) propose a parametric simplex algorithm for minimizing the product of two linear functions over a polytope. Benson and Boger (1997) give a heuristic algorithm for solving the more general linear multiplicative programming problem, in which the objective function can be a product of more than two linear functions.…”

Section: Introductionmentioning

confidence: 99%

An FPTAS for optimizing a class of low-rank functions over a polytope

Mittal

Schulz

2012

Math. Program.

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We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of nonlinear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Pareto-optimal front of the linear functions which constitute the given low-rank function. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasi-concavity on the objective function. For the special case of quasi-concave function minimization, we give an alternative FPTAS, which always returns a solution which is an extreme point of the polytope. Our technique can also be used to obtain an FPTAS for combinatorial optimization problems with non-linear objective functions, for example when the objective is a product of a fixed number of linear functions. We also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor, unless P = NP. We prove this by showing a similar hardness of approximation result for supermodular function minimization, a result that may be of independent interest. † Last updated on September 7, 2011.

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“…[3,4,19,20,21,32,34,35]). The multiplicative programming problem may find applications in such areas as microeconomics, geometric design, finance, VLSI chip design, and system reliability [4,11,20].…”

Section: Introductionmentioning

confidence: 99%

Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms

Zhao

2010

Appl Math Optim

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Abstract. While the product of finitely many convex functions has been investigated in the field of global optimization, some fundamental issues such as the convexity condition and the LegendreFenchel transform for the product function remain unresolved. Focusing on quadratic forms, this paper is aimed at addressing the question: When is the product of finitely many positive definite quadratic forms convex, and what is the Legendre-Fenchel transform for it? First, we show that the convexity of the product is determined intrinsically by the condition number of so-called 'scaled matrices' associated with quadratic forms involved. The main result claims that if the condition number of these scaled matrices are bounded above by an explicit constant (which depends only on the number of quadratic forms involved), then the product function is convex. Second, we prove that the Legendre-Fenchel transform for the product of positive definite quadratic forms can be expressed, and the computation of the transform amounts to finding the solution to a system of equations (or equally, finding a Brouwer's fixed point of a mapping) with a special structure. Thus, a broader question than the open "Question 11" in [SIAM Review, 49 (2007), 225-273] is addressed in this paper.

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Linear multiplicative programming

Cited by 108 publications

References 25 publications

Enumerating the Nash equilibria of rank-1 games

Enumerating the Nash equilibria of rank-1 games

An FPTAS for optimizing a class of low-rank functions over a polytope

Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms

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