DC Programming: Overview

Horst, Reiner; Thoại, Nguyễn Văn

doi:10.1023/a:1021765131316

Cited by 590 publications

(351 citation statements)

References 45 publications

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“…In our setting, each player's payoffs are given as difference of piecewise linear convex (DPLC) functions. As an aside, optimization problems involving the difference of convex functions are a well-studied class of nonconvex programs (see for example Horst and Thoai (1999)). …”

Section: Introductionmentioning

confidence: 99%

Rational Generating Functions and Integer Programming Games

Köppe

Ryan

Queyranne

2011

Operations Research

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We explore the computational complexity of computing pure Nash equilibria for a new class of strategic games called integer programming games with difference of piecewise linear convex payoffs. Integer programming games are games where players' action sets are integer points inside of polytopes. Using recent results from the study of short rational generating functions for encoding sets of integer points pioneered by Alexander Barvinok, we present efficient algorithms for enumerating all pure Nash equilibria, and other computations of interest, such as the pure price of anarchy, and pure threat point, when the dimension and number of "convex" linear pieces in the payoff functions are fixed. Sequential games where a leader is followed by competing followers (a Stackelberg-Nash setting) are also considered.

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Section: Introductionmentioning

confidence: 99%

Rational Generating Functions and Integer Programming Games

Köppe

Ryan

Queyranne

2011

Operations Research

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“…The special structure of the objective function of Problem (3) will be exploited in order to design a deterministic global optimization algorithm that allows us to find an optimal solution to the problem. More precisely, we will show that H(x) in (3) belongs to the broad class of DC functions, [11,10,21]. This key property will allow us to solve Problem (3) by branch-andbound algorithms, as the one described in Section 4, since lower and upper bounds can easily be obtained for DC functions as soon as a DC decomposition is available.…”

Section: Example 1 Let Us Consider the Network N = (A E) Withmentioning

confidence: 95%

“…An interesting property of the class of DC functions is that it is closed under the most common operations in optimization, [3,10,11,21,22]. In particular, if h 1 , .…”

Section: Example 1 Let Us Consider the Network N = (A E) Withmentioning

confidence: 99%

“…Contrary to the planar 1-median problem, known to be convex, [7], the 1-median problem on networks with continuous demand poses nontrivial challenges: we have no longer a simple expression for the objective, and its optimization calls for the use of global-optimization techniques. In particular, it is shown that the objective function is DC, [10,11,21,22] i.e., it can be written as the difference of two convex functions, and thus its optimization can be addressed via branch-and-bound methods customized for DC functions, [1,4,5].…”

Section: Introductionmentioning

confidence: 99%

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Solving the median problem with continuous demand on a network

Blanquero

Carrizosa

2013

Comput Optim Appl

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Where to locate one or several facilities on a network so as to minimize the expected users-closest facility transportation cost is a problem well studied in the OR literature under the name of median problem.In the median problem users are usually identified with nodes of the network. In many situations, however, such assumption is unrealistic, since users should be better considered to be distributed also along the edges of the transportation network. In this paper we address the median problem with demand distributed along edges and nodes. This leads to a globaloptimization problem, which can be solved to optimality by means of a branch-and-bound with DC bounds. Our computational experience shows that the problem is solved in short time even for large instances.

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“…Although it can be used to "model" a very wide range of mathematical problems [24] and has been studied extensively during the last thirty years (cf. [25,34,39]), it comes at a price: it is impossible to have an elegant theory and powerful algorithms for solving this problem without detailed structures on these arbitrarily given functions. As the result, even some very simple d.c. programming problems are considered as NP-hard [37].…”

Section: Problems and Motivationmentioning

confidence: 99%

On modeling and global solutions for d.c. optimization problems by canonical duality theory

Gao

2017

Applied Mathematics and Computation

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This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open problems are presented.

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DC Programming: Overview

Cited by 590 publications

References 45 publications

Rational Generating Functions and Integer Programming Games

Rational Generating Functions and Integer Programming Games

Solving the median problem with continuous demand on a network

On modeling and global solutions for d.c. optimization problems by canonical duality theory

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