A new linearization technique for multi-quadratic 0–1 programming problems

Chaovalitwongse, Wanpracha Art; Pardalos, Pãnos M.; Prokopyev, Oleg A.

doi:10.1016/j.orl.2004.03.005

Cited by 70 publications

(57 citation statements)

References 11 publications

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“…Note that (21) is implied in R-SAHLP and can be derived from (2) and (6). From the structure of (20) and the basic theory on Lagrangian multiplier, we obtain the next result, which allows us to apply classical subgradient method to obtain the strongest lower bound.…”

Section: Lagrangian Lower Boundmentioning

confidence: 86%

“…Constraints (6) and (8) ensure that the regular hubs and the backup hubs of any flow can only be the nodes chosen to be hubs and the regular hubs and the backup hubs must be different. Constraints (7) and (9) are used to eliminate the cases where either the source or the destination node of a flow is a hub.…”

Section: Literature Reviewmentioning

confidence: 99%

“…For example, for a small size problem with 10 nodes, linearization leads to roughly 100,000 new variables and 300, 000 new constraints, which is very difficult for professional solvers to generate solutions in a reasonable time. To address this issue, we adopt a recent linearization strategy [6,36,22] and create the following compact reformulation CptLinear. We mention that this linear formula distinguishes itself by the fact that the number of variables does little change and the number of constraints just linearly increases.…”

Section: Linear Models From Linear Reformulationmentioning

confidence: 99%

“…Results in [6,36,22] show that applying this type of linearization techniques to a 0 − 1 quadratic program will just linearly increase the number of variables and constraints. As…”

Section: Linear Models From Linear Reformulationmentioning

confidence: 99%

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The reliable hub-and-spoke design problem: Models and algorithms

An¹,

Zhang²,

Zeng³

2015

Transportation Research Part B: Methodological

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This paper presents a study on reliable single and multiple allocation hub-and-spoke network design problems where disruptions at hubs and the resulting hub unavailability can be mitigated by backup hubs and alternative routes. It builds nonlinear mixed integer programming models and presents linearized formulas. To solve those difficult problems, Lagrangian relaxation and Branch-and-Bound methods are developed to efficiently obtain optimal solutions. Numerical studies of proposed solution methods on practical instances are reported, along with a few insights of system design.Key words: reliable network design; hub-and-spoke; Lagrangian Relaxation; Branch-and- Bound Background and MotivationThe hub-and-spoke system has been widely employed in various industrial applications.It is a fully connected network with material/information flow between any two nodes being processed at a small number of critical nodes (i.e. hubs) and moved through interhub links. Compared with the one built with the point-to-point structure, it has a much smaller number of links. Also, because traffic flows are consolidated by hubs and interhub links, significantly less operating cost can be achieved because of economies of scale.Given such advantages, industry companies including airlines, logistics companies, and telecommunications firms extensively utilize the hub-and-spoke architecture to reduce the construction and operating costs.

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Section: Lagrangian Lower Boundmentioning

confidence: 86%

Section: Literature Reviewmentioning

confidence: 99%

Section: Linear Models From Linear Reformulationmentioning

confidence: 99%

“…Results in [6,36,22] show that applying this type of linearization techniques to a 0 − 1 quadratic program will just linearly increase the number of variables and constraints. As…”

Section: Linear Models From Linear Reformulationmentioning

confidence: 99%

See 2 more Smart Citations

The reliable hub-and-spoke design problem: Models and algorithms

An¹,

Zhang²,

Zeng³

2015

Transportation Research Part B: Methodological

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show abstract

“…Chaovalitwongse et al [97] and Sherali and Smith [98] provide recent, conceptually different O(n) linearization approaches.…”

Section: Quadratic Optimization With Binary Variablesmentioning

confidence: 99%

Non-convex mixed-integer nonlinear programming: A survey

Burer

Letchford

2012

Surveys in Operations Research and Management Science

352

229

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Concise RLT forms of binary programs: A computational study of the quadratic knapsack problem

Forrester

Adams

Hadavas

2009

Naval Research Logistics

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Abstract:The reformulation-linearization technique (RLT) is a methodology for constructing tight linear programming relaxations of mixed discrete problems. A key construct is the multiplication of "product factors" of the discrete variables with problem constraints to form polynomial restrictions, which are subsequently linearized. For special problem forms, the structure of these linearized constraints tends to suggest that certain classes may be more beneficial than others. We examine the usefulness of subsets of constraints for a family of 0-1 quadratic multidimensional knapsack programs and perform extensive computational tests on a classical special case known as the 0-1 quadratic knapsack problem. We consider RLT forms both with and without these inequalities, and their comparisons with linearizations derived from published methods. Interestingly, the computational results depend in part upon the commercial software used.

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A new linearization technique for multi-quadratic 0–1 programming problems

Cited by 70 publications

References 11 publications

The reliable hub-and-spoke design problem: Models and algorithms

The reliable hub-and-spoke design problem: Models and algorithms

Non-convex mixed-integer nonlinear programming: A survey

Concise RLT forms of binary programs: A computational study of the quadratic knapsack problem

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