Globally minimizing the sum of a convex–concave fraction and a convex function based on wave-curve bounds
We consider the problem of minimizing the sum of a convex–concave function and a convex function over a convex set (SFC). It can be reformulated as a univariate minimization problem, where the objective function is evaluated by solving convex optimization. The optimal Lagrangian multipliers of the c...
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| Vydané v: | Journal of global optimization Ročník 77; číslo 2; s. 301 - 318 |
|---|---|
| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Springer US
01.06.2020
Springer Springer Nature B.V |
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| ISSN: | 0925-5001, 1573-2916 |
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| Abstract | We consider the problem of minimizing the sum of a convex–concave function and a convex function over a convex set (SFC). It can be reformulated as a univariate minimization problem, where the objective function is evaluated by solving convex optimization. The optimal Lagrangian multipliers of the convex subproblems are used to construct sawtooth curve lower bounds, which play a key role in developing the branch-and-bound algorithm for globally solving (SFC). In this paper, we improve the existing sawtooth-curve bounds to new wave-curve bounds, which are used to develop a more efficient branch-and-bound algorithm. Moreover, we can show that the new algorithm finds an
ϵ
-approximate optimal solution in at most
O
1
ϵ
iterations. Numerical results demonstrate the efficiency of our algorithm. |
|---|---|
| AbstractList | We consider the problem of minimizing the sum of a convex-concave function and a convex function over a convex set (SFC). It can be reformulated as a univariate minimization problem, where the objective function is evaluated by solving convex optimization. The optimal Lagrangian multipliers of the convex subproblems are used to construct sawtooth curve lower bounds, which play a key role in developing the branch-and-bound algorithm for globally solving (SFC). In this paper, we improve the existing sawtooth-curve bounds to new wave-curve bounds, which are used to develop a more efficient branch-and-bound algorithm. Moreover, we can show that the new algorithm finds an [Formula omitted]-approximate optimal solution in at most [Formula omitted] iterations. Numerical results demonstrate the efficiency of our algorithm. We consider the problem of minimizing the sum of a convex–concave function and a convex function over a convex set (SFC). It can be reformulated as a univariate minimization problem, where the objective function is evaluated by solving convex optimization. The optimal Lagrangian multipliers of the convex subproblems are used to construct sawtooth curve lower bounds, which play a key role in developing the branch-and-bound algorithm for globally solving (SFC). In this paper, we improve the existing sawtooth-curve bounds to new wave-curve bounds, which are used to develop a more efficient branch-and-bound algorithm. Moreover, we can show that the new algorithm finds an ϵ-approximate optimal solution in at most O1ϵ iterations. Numerical results demonstrate the efficiency of our algorithm. We consider the problem of minimizing the sum of a convex–concave function and a convex function over a convex set (SFC). It can be reformulated as a univariate minimization problem, where the objective function is evaluated by solving convex optimization. The optimal Lagrangian multipliers of the convex subproblems are used to construct sawtooth curve lower bounds, which play a key role in developing the branch-and-bound algorithm for globally solving (SFC). In this paper, we improve the existing sawtooth-curve bounds to new wave-curve bounds, which are used to develop a more efficient branch-and-bound algorithm. Moreover, we can show that the new algorithm finds an ϵ -approximate optimal solution in at most O 1 ϵ iterations. Numerical results demonstrate the efficiency of our algorithm. |
| Audience | Academic |
| Author | Wang, Longfei Xia, Yong Wang, Xiaohui |
| Author_xml | – sequence: 1 givenname: Yong orcidid: 0000-0002-3522-7446 surname: Xia fullname: Xia, Yong organization: LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University – sequence: 2 givenname: Longfei surname: Wang fullname: Wang, Longfei organization: School of Mathematics and Statistics, Henan University – sequence: 3 givenname: Xiaohui surname: Wang fullname: Wang, Xiaohui email: xhwang@buaa.edu.cn organization: School of Astronautics, Beihang University |
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| Cites_doi | 10.1007/978-3-662-06409-2 10.1002/zamm.19840640809 10.1016/j.orl.2017.11.010 10.1007/BF00121658 10.1287/mnsc.22.8.858 10.1007/BF00138693 10.1007/978-1-4615-2025-2_10 10.1137/18M1164639 10.1023/A:1008316327038 10.1007/s10898-008-9378-7 10.1137/050624418 10.1007/978-1-4757-9859-3 10.1002/nav.3800090303 10.1007/s10589-012-9479-6 10.1080/02331934.2015.1113532 10.1080/1055678031000105242 10.1007/s11590-010-0210-1 10.1016/j.cam.2013.08.005 10.1287/mnsc.13.7.492 |
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| References_xml | – reference: FangSCGaoDYSheuRLXingWXGlobal optimization for a class of fractional programming problemsJ. Glob. Optim.2009453337353255021410.1007/s10898-008-9378-7 – reference: XiaYWangLWangSMinimizing the sum of linear fractional functions over the cone of positive semidefinite matrices: approximation and applicationsOper. Res. Lett.20184617680375917910.1016/j.orl.2017.11.010 – reference: Grant, M., Boyd, S.: CVX: MATLAB Software for Disciplined Convex Programming, Version 2.1. March; 2014. Available from: http://cvxr.com/cvx (2014) – reference: FreundRWJarreFSolving the sum-of-ratios problem by an interior-point methodJ. Glob. Optim.20011983102181264910.1023/A:1008316327038 – reference: LbarakiTSchaibleSFractional programmingEur. J. Oper. Res.2004124325338703439 – reference: BorweinJMLewisASConvex Analysis and Nonlinear Optimization: Theory and Examples2000New YorkSpringer10.1007/978-1-4757-9859-3 – reference: SahinidisNVBARON: a general purpose global optimiztion sofgware packageJ. Glob. Optim.19968220120510.1007/BF00138693 – reference: LiWMLiangYCShenPPBranch-reduction-bound algorithm for linear sum-of-ratios fractional programsPac. J. Optim.2015111799933450341352.90077 – reference: CambiniAMarteinLSchaibleSOn maximizing a sum of ratiosJ. Inf. Optim. Sci.198910657910005790676.90081 – reference: BeckABen-TalAOn the solution of the Tikhonov regularization of the total least squares problemSIAM J. Optim.200617198118221914610.1137/050624418 – reference: SchaibleSHorstRPardalosPMFractional programmingHandbook of Global Optimization1995DordrechtKluwer Academic Publishers49560810.1007/978-1-4615-2025-2_10 – reference: CharnesACooperWWProgramming with linear fractional functionalsNaval Res. Logist. Q.1962918118615237010.1002/nav.3800090303 – reference: AvrielMDiewertWESchaibleSZangIGeneralized Concavity, Mathematical Concepts and Methods in Science and Engineering1988New YorkPlenum Press0679.90029 – reference: WangLXiaYA linear-time algorithm for globally maximizing the sum of a generalized rayleigh quotient and a quadratic form on the unit sphereSIAM J. Optim.201929318441869398026610.1137/18M1164639 – reference: Hiriart-UrrutyJBLemarechalCConvex Analysis and Minimization Algorithms, Grundlehren der Mathematischen Wissenschaften1993BerlinSpringer10.1007/978-3-662-06409-2 – reference: DinkelbachWOn nonlinear fractional programmingManag. Sci.196713749249824248810.1287/mnsc.13.7.492 – reference: FakhriAGhateeMMinimizing the sum of a linear and a linear fractional function applying conic quadratic representation: continuous and discrete problemsOptimization201665510231038347872110.1080/02331934.2015.1113532 – reference: MatsuiTNP-hardness of linear multiplicative programming and related problemsJ. Glob. Optim.19969113119141160310.1007/BF00121658 – reference: SchaibleSShiJMFractional programming: the sum-of-ratios caseOptim. Methods Softw.2003182219229199659010.1080/1055678031000105242 – reference: SchaibleSSimultaneous optimization of absolute and relative termsZ. Angew. Math. Mech.198464836336476568610.1002/zamm.19840640809 – reference: XuCXuXMWangHFThe fractional minimal cost flow problem on networkOptim. Lett.201152307317278247010.1007/s11590-010-0210-1 – reference: ZhangLHOn optimizing the sum of the Rayleigh quotient and the generalized Rayleigh quotient on the unit sphereComput. Optim. 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| SubjectTerms | Algorithms Computational geometry Computer Science Convexity Lower bounds Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Real Functions |
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| Title | Globally minimizing the sum of a convex–concave fraction and a convex function based on wave-curve bounds |
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