Quadratic Optimization in 0–1 Variables

Billionnet, Alain

doi:10.1002/9781118600245.ch8

Cited by 2 publications

(2 citation statements)

References 105 publications

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“…where Q is an n × n real matrix, c is a row vector in R n , and c 0 is a constant. The BQP is a well-studied problem in the operational research literature [6]. The focus of this paper is on a problem closely related to BQP, called the Bipartite (Unconstrained) Boolean Quadratic Programming Problem (BBQP) [23].…”

Section: Introductionmentioning

confidence: 99%

Markov Chain methods for the Bipartite Boolean Quadratic Programming Problem

Karapetyan

Punnen

Parkes

2017

European Journal of Operational Research

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We study the Bipartite Boolean Quadratic Programming Problem (BBQP) which is an extension of the well known Boolean Quadratic Programming Problem (BQP). Applications of the BBQP include mining discrete patterns from binary data, approximating matrices by rank-one binary matrices, computing the cut-norm of a matrix, and solving optimisation problems such as maximum weight biclique, bipartite maximum weight cut, maximum weight induced subgraph of a bipartite graph, etc. For the BBQP, we first present several algorithmic components, specifically, hill climbers and mutations, and then show how to combine them in a high-performance metaheuristic. Instead of hand-tuning a standard metaheuristic to test the efficiency of the hybrid of the components, we chose to use an automated generation of a multi-component metaheuristic to save human time, and also improve objectivity in the analysis and comparisons of components. For this we designed a new metaheuristic schema which we call Conditional Markov Chain Search (CMCS). We show that CMCS is flexible enough to model several standard metaheuristics; this flexibility is controlled by multiple numeric parameters, and so is convenient for automated generation. We study the configurations revealed by our approach and show that the best of them outperforms the previous state-of-the-art BBQP algorithm by several orders of magnitude. In our experiments we use benchmark instances introduced in the preliminary version of this paper and described here, which have already become the de facto standard in the BBQP literature. (Daniel Karapetyan), apunnen@sfu.ca (Abraham P. Punnen), andrew.parkes@nottingham.ac.uk (Andrew J. Parkes) i∈I c i + j∈J d j + (i,j)∈E q ij is maximised, where I ⊆ I, J ⊆ J and G is induced by I ∪ J . The Maximum Weight Induced Subgraph Problem on G is precisely the BBQP, where q ij = 0 if (i, j) / ∈ E. There are some other well known combinatorial optimisation problems that can be modelled as a BBQP. Consider the bipartite graph G = (I, J, E) with w ij being the weight of the edge (i, j) ∈ E. Then the Maximum Weight

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Section: Introductionmentioning

confidence: 99%

Markov Chain methods for the Bipartite Boolean Quadratic Programming Problem

Karapetyan

Punnen

Parkes

2017

European Journal of Operational Research

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“…where Q ′ is an n × n real matrix, c ′ is a row vector in R n , and c ′ 0 is a constant. QP is a well-studied problem in the operations research literature (Billionnet, 2004). The focus of this paper is on a problem closely related to the QP called the Bipartite Unconstrained 0-1 Quadratic Programming Problem (BQP) (Punnen et al, 2012).…”

Section: Introductionmentioning

confidence: 99%

Heuristic algorithms for the bipartite unconstrained 0-1 quadratic programming problem

Karapetyan,

Punnen

2012

Preprint

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We study the Bipartite Unconstrained 0-1 Quadratic Programming Problem (BQP) which is a relaxation of the Unconstrained 0-1 Quadratic Programming Problem (QP). Applications of the BQP include mining discrete patterns from binary data, approximating matrices by rank-one binary matrices, computing cut-norm of a matrix, and solving optimization problems such as maximum weight biclique, bipartite maximum weight cut, maximum weight induced subgraph of a bipartite graph, etc. We propose several classes of heuristic approaches to solve the BQP and discuss a number of construction algorithms, local search algorithms and their combinations. Results of extensive computational experiments are reported to establish the practical performance of our algorithms. For this purpose, we propose several sets of test instances based on various applications of the BQP. Our algorithms are compared with state-of-the-art heuristics for QP which can also be used to solve BQP with reformulation. We also study theoretical properties of the neighborhoods and algorithms. In particular, we establish complexity of all neighborhood search algorithms and establish tight worst-case performance ratio for the greedy algorithm.

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Quadratic Optimization in 0–1 Variables

Cited by 2 publications

References 105 publications

Markov Chain methods for the Bipartite Boolean Quadratic Programming Problem

Markov Chain methods for the Bipartite Boolean Quadratic Programming Problem

Heuristic algorithms for the bipartite unconstrained 0-1 quadratic programming problem

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