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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Bibliographic Details
Published in:Journal of global optimization Vol. 77; no. 2; pp. 301 - 318
Main Authors: Xia, Yong, Wang, Longfei, Wang, Xiaohui
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2020
Springer
Springer Nature B.V
Subjects:
Algorithms
Computational geometry
Computer Science
Convexity
Lower bounds
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Real Functions
90C20
Lower bound
Branch and bound
Convex program
90C22
90C26
Fractional programming
ISSN:0925-5001, 1573-2916
Online Access:Get full text
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Summary: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.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-019-00870-2