Abstract
We consider the problem of globally minimizing an abstract convex function called increasing positively homogeneous (IPH) function over a compact convex subset of an n −dimensional Euclidean space, for short, IPH optimization problem.
A method for solving IPH optimization problems called cutting angle algorithm was proposed by Rubinov and others in 1999. The principle of cutting angle algorithm is a generalization of the cutting plane method for convex programming problems, where the convex objective function is iteratively approximated by the maximum of a family of affine functions defined by its subgradients. In this article, we propose a method for solving IPH optimization problems which is a combination of the cutting angle algorithm with a branch and bound scheme successfully used in global optimization. The lower bounding procedure in the present algorithm is performed by solving ordinary convex (or even linear) programs. From preliminary computational results we hope that the proposed algorithm could work well for some problems with specific structures.
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References
Andramonov, M.Y., Rubinov, A.M., Glover, B.M.: Cutting angle method in global optimization. Applied Mathematics Letters 12, 95–100 (1999)
Bagirov, A.M., Rubinov, A.M.: Global Minimization of Increasing Positively Homogeneous Functions over the Unit Simplex. Annals of Operations Research 98, 171–187 (2000)
Bartels, S.G., Kuntz, L., Sholtes, S.: Continuous selections of linear functions and nonsmooth critical point theory. Nonlinear Analysis, TMA 24, 385–407 (1995)
Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis, Peter Lang, Frankfurt am Main (1995)
Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization, 2nd edn. Kluwer, Dordrecht (2000)
Horst, R., Thoai, N.V.: DC Programming: Overview. Journal of Optimization Theory and Applications 103, 1–43 (1999)
Horst, R., Thoai, N.V., Tuy, H.: Outer aproximation by polyhedral convex sets. O.R. Spektrum 9, 153–159 (1987)
Horst, R., Thoai, N.V., Tuy, H.: On an outer approximation concept in global optimization. Optimization 20, 255–264 (1989)
Horst, R., Thoai, N.V., Benson, H.P.: Concave minimization via conical partitions and polyhedral outer approximation. Mathematical Programming 50, 259–274 (1991)
Kelley, J.: The cutting plane method for solving convex programs. SlAM Journal 8, 703–712 (1960)
Ordin, B.: The Modified Cutting Angle Method for Global Minimization of Increasing Positively Homogeneous Functions over the Unit Simplex. Journal of Industrial and Management Optimization 5, 825–834 (2009)
Pallaschke, D., Rolewicz, S.: Foundations of Mathematical Optimization (Convex Analysis without Linearity). Kluwer Academic, Dordrecht (1997)
Rubinov, A.M.: Abstract Convexity and Global Optimization. Kluwer, Dordrecht (2000)
Rubinov, A.M., Andramonov, M.Y.: Minimizing increasing star-shaped functions based on abstract convexity. Journal of Global Optimization 15, 19–39 (1999)
Rubinov, A., Andramonov, M.Y.: Lipschitz programming via increasing convex-along-rays functions. Optimization Methods and Software 10, 763–781 (1999)
Singer, I.: Abstract Convex Analysis. Wilsey & Sons (1997)
Thoai, N.V.: Convergence and Application of a Decomposition Method Using Duality Bounds for Nonconvex Global Optimization. Journal of Optimization Theory and Applications 113, 165–193 (2002)
Thoai, N.V.: Decomposition Branch and Bound Algorithm for Optimization Problems over Efficient Sets. Journal of Industrial and Management Optimization 4, 647–660 (2008)
Thoai, N.V., Tuy, H.: Convergent algoritms for minimizing a concave function. Mathematics of Operations Research 5, 556–566 (1980)
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Van Thoai, N. (2013). Branch and Bound Algorithm Using Cutting Angle Method for Global Minimization of Increasing Positively Homogeneous Functions. In: Nguyen, N., van Do, T., le Thi, H. (eds) Advanced Computational Methods for Knowledge Engineering. Studies in Computational Intelligence, vol 479. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00293-4_2
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DOI: https://doi.org/10.1007/978-3-319-00293-4_2
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