On Solving the Quadratic Shortest Path Problem

Hu, Hao; Sotirov, Renata

doi:10.1287/ijoc.2018.0861

Cited by 20 publications

(22 citation statements)

References 28 publications

(84 reference statements)

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“…Broumi et al [27] calculated MST in interval valued bipolar neutrosophic (IVBN) setting. Hu and Sotirov [28] proposed amenity of semi definite programming for the quadratic SPP and performed some arithmetic operations to solve the QSPP using branch and bound algorithm. Dragan and Leitert [29] solved SPP on minimal peculiarity.…”

Section: Literature Of Reviewmentioning

confidence: 99%

The shortest path problem in interval valued trapezoidal and triangular neutrosophic environment

Nagarajan

Bakali

Talea

et al. 2019

Complex Intell. Syst.

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Real-life decision-making problem has been demonstrated to cover the indeterminacy through single valued neutrosophic set. It is the extension of interval valued neutrosophic set. Most of the problems of real life involve some sort of uncertainty in it among which, one of the famous problem is finding a shortest path of the network. In this paper, a new score function is proposed for interval valued neutrosophic numbers and SPP is solved using interval valued neutrosophic numbers. Additionally, novel algorithms are proposed to find the neutrosophic shortest path by considering interval valued neutrosophic number, trapezoidal and triangular interval valued neutrosophic numbers for the length of the path in a network with illustrative example. Further, comparative analysis has been done for the proposed algorithm with the existing method with the shortcoming and advantage of the proposed method and it shows the effectiveness of the proposed algorithm.Keywords Interval valued triangular neutrosophic number · Interval valued trapezoidal neutrosophic number · Ranking methods · Deneutrosophication · Neutrosophic shortest path problem · Network Introduction and literature of reviewIn this part, introduction to the objective of the paper is given by presenting basic concepts and procedure of the shortest path problem (SPP) and the literature of review have been collected to know the recent work related to the presented concept which shows the novelty of the presented work * Said Broumi Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Section: Literature Of Reviewmentioning

confidence: 99%

The shortest path problem in interval valued trapezoidal and triangular neutrosophic environment

Nagarajan

Bakali

Talea

et al. 2019

Complex Intell. Syst.

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“…Recently, a promising alternative for solving largescale SDP relaxations based on alternating direction augmented Lagrangian methods has been investigated; see Burer and Vandenbussche (2006), Povh et al (2006), Wen et al (2010), Zhao et al (2010), andSun et al (2020). There exist several variants of alternating direction augmented Lagrangian methods for solving SDPs; see, for example, Povh et al (2006), Zhao et al (2010), He et al (2014He et al ( , 2016, Oliveira et al (2018), Hu et al (2019), and Hu and Sotirov (2020). A recent method for solving large-scale SDPs that is related to the augmented Lagrangian paradigm is the conditional gradient augmented Lagrangian method (Yurtsever et al 2019;Mai et al 2020a, b).…”

Section: A Cutting-plane Augmented Lagrangian Approachmentioning

confidence: 99%

SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning

Meijer

Sotirov

2021

INFORMS Journal on Computing

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We study the quadratic cycle cover problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the reduction and the structure of the graph, which is exploited in an efficient algorithm that constructs this matrix for any instance of the problem. To solve our relaxations, we propose an algorithm that incorporates an augmented Lagrangian method into a cutting-plane framework by utilizing Dykstra’s projection algorithm. Our algorithm is suitable for solving SDP relaxations with a large number of cutting-planes. Computational results show that our SDP bounds and efficient cutting-plane algorithm outperform other QCCP bounding approaches from the literature. Finally, we provide several SDP-based upper bounding techniques, among which is a sequential Q-learning method that exploits a solution of our SDP relaxation within a reinforcement learning environment. Summary of Contribution: The quadratic cycle cover problem (QCCP) is the problem of finding a set of node-disjoint cycles covering all the nodes in a graph such that the total interaction cost between successive arcs is minimized. The QCCP has applications in many fields, among which are robotics, transportation, energy distribution networks, and automatic inspection. Besides this, the problem has a high theoretical relevance because of its close connection to the quadratic traveling salesman problem (QTSP). The QTSP has several applications, for example, in bioinformatics, and is considered to be among the most difficult combinatorial optimization problems nowadays. After removing the subtour elimination constraints, the QTSP boils down to the QCCP. Hence, an in-depth study of the QCCP also contributes to the construction of strong bounds for the QTSP. In this paper, we study the application of semidefinite programming (SDP) to obtain strong bounds for the QCCP. Our strongest SDP relaxation is very hard to solve by any SDP solver because of the large number of involved cutting-planes. Because of that, we propose a new approach in which an augmented Lagrangian method is incorporated into a cutting-plane framework by utilizing Dykstra’s projection algorithm. We emphasize an efficient implementation of the method and perform an extensive computational study. This study shows that our method is able to handle a large number of cuts and that the resulting bounds are currently the best QCCP bounds in the literature. We also introduce several upper bounding techniques, among which is a distributed reinforcement learning algorithm that exploits our SDP relaxations.

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“…By using Proposition 5 we computed the spanning set of linearizable matrices for the QSPP on tournament graphs, GRID1, GRID2 and PAR-K graphs up to certain size. We refer to Hu and Sotirov (2020) for the definition of these acyclic graphs. We have also computed the spanning set of linearizable matrices for the QAP for n ≤ 9 by brute-force search.…”

Section: The Lbb * Boundmentioning

confidence: 99%

“…Table 1 we shows numerical results for the first level RLT bound v RLT 1 for seven of the mentioned instances. We also compute the strongest semidefinite programming relaxation S D P N L from Hu and Sotirov (2020) for the mentioned instances. Optimal values are provided in the last column of the table . Table 1 shows that v L B B * dominates both v RLT 1 and v S D P N L for all instances.…”

Section: Proofmentioning

confidence: 99%

The linearization problem of a binary quadratic problem and its applications

Sotirov

2021

Ann Oper Res

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We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore–Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. We also present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem.

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On Solving the Quadratic Shortest Path Problem

Cited by 20 publications

References 28 publications

The shortest path problem in interval valued trapezoidal and triangular neutrosophic environment

The shortest path problem in interval valued trapezoidal and triangular neutrosophic environment

SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning

The linearization problem of a binary quadratic problem and its applications

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