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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Veröffentlicht in:Numerical algorithms Jg. 93; H. 3; S. 1373 - 1400
Hauptverfasser: Peiping, Shen, Yafei, Wang, Dianxiao, Wu
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.07.2023
Springer Nature B.V
Schlagworte:
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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Zusammenfassung: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.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-022-01471-z