A spatial branch and bound algorithm for solving the sum of linear ratios optimization problem

In this paper, we consider the sum of linear ratios problem (SLR) that is known to be NP-hard and often arises in various practical applications such as data envelopment analysis and financial investment. We first introduce an equivalent problem (EP) of SLR that involves differences of square terms...

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Published in:	Numerical algorithms Vol. 93; no. 3; pp. 1373 - 1400
Main Authors:	Peiping, Shen, Yafei, Wang, Dianxiao, Wu
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.07.2023 Springer Nature B.V
Subjects:	Algebra Algorithms Approximation Branch and bound methods Computer Science Data envelopment analysis Iterative methods Linear functions Linear programming Mathematical analysis Numeric Computing Numerical Analysis Optimization Original Paper Rectangles Theory of Computation Variables Global optimization 90C30 Branch and bound 90C33 Second-order cone approximations 90C15 Sum of linear ratios problem Convergence analysis
ISSN:	1017-1398, 1572-9265
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Abstract	In this paper, we consider the sum of linear ratios problem (SLR) that is known to be NP-hard and often arises in various practical applications such as data envelopment analysis and financial investment. We first introduce an equivalent problem (EP) of SLR that involves differences of square terms in inequality constraints. Subsequently, the concave parts of the non-convex constraints in problem (EP) are replaced with the piecewise linear functions. Using the resulting second-order cone program (SOCP), we design a spatial branch and bound algorithm, which iteratively refines the piecewise linear approximations by dividing rectangles and solving a series of problems (SOCP) to obtain the solution of the original problem. Also, a region compression technique is proposed to accelerate the convergence of the algorithm. Furthermore, we demonstrate that the bound on the optimality gap is a function of approximation errors at the iteration and estimate that the worst-case number of iterations is in the order of O ( ε ) to attain an ε -optimal solution. Numerical results illustrate that the proposed algorithm scales better than both the existing LP-based algorithms and the off-the-shelf solvers SCIP to solve the problem (SLR). It is worth mentioning that the proposed algorithm takes significantly less time to reach four-digit accuracy than the time required by the known algorithms on small to medium problem instances.
AbstractList	In this paper, we consider the sum of linear ratios problem (SLR) that is known to be NP-hard and often arises in various practical applications such as data envelopment analysis and financial investment. We first introduce an equivalent problem (EP) of SLR that involves differences of square terms in inequality constraints. Subsequently, the concave parts of the non-convex constraints in problem (EP) are replaced with the piecewise linear functions. Using the resulting second-order cone program (SOCP), we design a spatial branch and bound algorithm, which iteratively refines the piecewise linear approximations by dividing rectangles and solving a series of problems (SOCP) to obtain the solution of the original problem. Also, a region compression technique is proposed to accelerate the convergence of the algorithm. Furthermore, we demonstrate that the bound on the optimality gap is a function of approximation errors at the iteration and estimate that the worst-case number of iterations is in the order of O ( ε ) to attain an ε -optimal solution. Numerical results illustrate that the proposed algorithm scales better than both the existing LP-based algorithms and the off-the-shelf solvers SCIP to solve the problem (SLR). It is worth mentioning that the proposed algorithm takes significantly less time to reach four-digit accuracy than the time required by the known algorithms on small to medium problem instances. In this paper, we consider the sum of linear ratios problem (SLR) that is known to be NP-hard and often arises in various practical applications such as data envelopment analysis and financial investment. We first introduce an equivalent problem (EP) of SLR that involves differences of square terms in inequality constraints. Subsequently, the concave parts of the non-convex constraints in problem (EP) are replaced with the piecewise linear functions. Using the resulting second-order cone program (SOCP), we design a spatial branch and bound algorithm, which iteratively refines the piecewise linear approximations by dividing rectangles and solving a series of problems (SOCP) to obtain the solution of the original problem. Also, a region compression technique is proposed to accelerate the convergence of the algorithm. Furthermore, we demonstrate that the bound on the optimality gap is a function of approximation errors at the iteration and estimate that the worst-case number of iterations is in the order of O(ε) to attain an ε-optimal solution. Numerical results illustrate that the proposed algorithm scales better than both the existing LP-based algorithms and the off-the-shelf solvers SCIP to solve the problem (SLR). It is worth mentioning that the proposed algorithm takes significantly less time to reach four-digit accuracy than the time required by the known algorithms on small to medium problem instances.
Author	Yafei, Wang Dianxiao, Wu Peiping, Shen
Author_xml	– sequence: 1 givenname: Shen surname: Peiping fullname: Peiping, Shen email: shenpeiping@163.com organization: School of Mathematics and Statistics, North China University of Water Resources and Electric Power – sequence: 2 givenname: Wang surname: Yafei fullname: Yafei, Wang organization: School of Mathematics and Statistics, North China University of Water Resources and Electric Power – sequence: 3 givenname: Wu surname: Dianxiao fullname: Dianxiao, Wu organization: School of Mathematics and Statistics, North China University of Water Resources and Electric Power
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Cites_doi	10.1007/BF01096535 10.1002/(SICI)1520-6750(199908)46:5<583::AID-NAV8>3.0.CO;2-5 10.1016/j.ejor.2006.08.036 10.1287/opre.24.3.452 10.1016/j.ejor.2014.02.039 10.1016/j.cam.2018.10.038 10.1023/A:1008316327038 10.1007/s40314-021-01614-3 10.1007/BF01585557 10.1002/nav.3800090303 10.1007/BF00120666 10.1016/j.ejor.2015.01.039 10.1023/A:1008376731013 10.1016/j.ejor.2013.03.025 10.1186/s13660-018-1651-9 10.1016/j.ejor.2013.02.023 10.1186/s13660-017-1420-1
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StartPage	1373
SubjectTerms	Algebra Algorithms Approximation Branch and bound methods Computer Science Data envelopment analysis Iterative methods Linear functions Linear programming Mathematical analysis Numeric Computing Numerical Analysis Optimization Original Paper Rectangles Theory of Computation Variables
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Title	A spatial branch and bound algorithm for solving the sum of linear ratios optimization problem
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Volume	93
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