On complexity of unconstrained hyperbolic 0–1 programming problems

Prokopyev, Oleg A.; Huang, Hong-Xuan; Pardalos, Pãnos M.

doi:10.1016/j.orl.2004.05.011

Cited by 43 publications

(34 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where all coefficients are positive, is known to be NP-Hard (see Prokopyev et al [11]). However, our problem (6) is a particular case of (7), where the variables are linked with less degree of freedom in such a way that problem (7) cannot be reduced to it, even for values L = 2, n ≥ 2.…”

Section: Complexity Of the Column Generation Subproblemmentioning

confidence: 99%

A Column Generation Algorithm for Choice-Based Network Revenue Management

Miranda-Bront

Méndez-Díaz

Vulcano

2009

Operations Research

275

222

View full text Add to dashboard Cite

In the last few years, there has been a trend to enrich traditional revenue management models built upon the independent demand paradigm by accounting for customer choice behavior. This extension involves both modeling and computational challenges.One way to describe choice behavior is to assume that each customer belongs to a segment, which is characterized by a consideration set, i.e., a subset of the products provided by the firm that a customer views as options. Customers choose a particular product according to a multinomial-logit criterion, a model widely used in the marketing literature.In this paper, we consider the choice-based, deterministic, linear programming model (CDLP) of Gallego et al. [6], and the follow-up dynamic programming (DP) decomposition heuristic of van Ryzin and Liu [16], and focus on the more general version of these models, where customers belong to overlapping segments. To solve the CDLP for real-size networks, we need to develop a column generation algorithm. We prove that the associated column generation subproblem is indeed NP-Complete, and propose a simple, greedy heuristic to overcome the complexity of an exact algorithm. Our computational results show that the heuristic is quite effective, and that the overall approach has good practical potential and leads to high quality solutions.

show abstract

Section: Complexity Of the Column Generation Subproblemmentioning

confidence: 99%

A Column Generation Algorithm for Choice-Based Network Revenue Management

Miranda-Bront

Méndez-Díaz

Vulcano

2009

Operations Research

275

222

View full text Add to dashboard Cite

show abstract

“…Conversely, if ADz = 0, 0 = z ≥ 0, then for δ = 0 we have that (z, δ) = 0 and z + eδ ≥ 0. Now, by using Gordan's theorem of the alternative [10, Theorem 2.4.5], condition (18) is equivalent to DA T r > 0 having a solution r, which is the desired condition (15) Note that Proposition 2 can be easily implemented by solving the linear program (10) and checking that its minimum objective function value is δ = 1. Also, if there exists a dual optimal solution with d ′ u = δ = 1 which has the property that for all i either v i or w i is strictly positive, then it follows from the complementarity conditions: v T (−x + e) = 0 and w T (x + e) = 0 for the primal optimal solution x, that x ∈ {−1, 1} n .…”

Section: Proposition 2 Uniqueness Of Solution Of (1) That Is Integer mentioning

confidence: 99%

“…However our condition (15), which is equivalent to that of the columns of the matrix AD lying in the same hemisphere of R m , is key in deriving our probalistic results of Section 3.…”

Section: Proposition 2 Uniqueness Of Solution Of (1) That Is Integer mentioning

confidence: 99%

See 1 more Smart Citation

Probability of unique integer solution to a system of linear equations

Mangasarian

Recht

2011

European Journal of Operational Research

View full text Add to dashboard Cite

Abstract. We consider a system of m linear equations in n variables Ax = d and give necessary and sufficient conditions for the existence of a unique solution to the system that is integer: x ∈ {−1, 1} n . We achieve this by reformulating the problem as a linear program and deriving necessary and sufficient conditions for the integer solution to be the unique primal optimal solution. We show that as long as m is larger than n/2, then the linear programming reformulation succeeds for most instances, but if m is less than n/2, the reformulation fails on most instances. We also demonstrate that these predictions match the empirical performance of the linear programming formulation to very high accuracy.

show abstract

“…We note here that characterizing uniqueness of solution for integer problems such as (1) is essentially an NP-hard problem [10,9,8]. Our interest here is in providing sufficient conditions for uniqueness of a given integer solution to (1), and utilizing these conditions as a computational tool for a class of integer problems.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of integer solution of linear equations

Mangasarian

Ferris²

2010

Optim Lett

View full text Add to dashboard Cite

Abstract. We consider the system of m linear equations in n integer variables Ax = d and give sufficient conditions for the uniqueness of its integer solution x ∈ {−1, 1} n by reformulating the problem as a linear program. Uniqueness characterizations of ordinary linear programming solutions are utilized to obtain sufficient uniqueness conditions such as the intersection of the kernel of A and the dual cone of a diagonal matrix of ±1's is the origin in R n . This generalizes the well known condition that ker(A) = 0 for the uniqueness of a non-integer solution x of Ax = d. A zero maximum of a single linear program ensures the uniqueness of a given integer solution of a linear equation.

show abstract

On complexity of unconstrained hyperbolic 0–1 programming problems

Cited by 43 publications

References 12 publications

A Column Generation Algorithm for Choice-Based Network Revenue Management

A Column Generation Algorithm for Choice-Based Network Revenue Management

Probability of unique integer solution to a system of linear equations

Uniqueness of integer solution of linear equations

Contact Info

Product

Resources

About