Abstract
In this note we show that various branch and bound methods for solving continuous global optimization problems can be readily adapted to the discrete case. As an illustration, we present an algorithm for minimizing a concave function over the integers contained in a compact polyhedron. Computational experience with this algorithm is reported.
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References
H.P. Benson, Algorithms for parametric nonconvex programming, J. Optim. Theory Appl. 38 (1982) 319–340.
H.P. Benson, A finite algorithm for concave minimization over a polyhedron, Naval Res. Logistics Quarterly 32 (1985) 165–177.
H.P. Benson, Separable concave minimization via partial outer approximation and branch and bound, Oper. Res. Lett., to appear.
J.E. Falk and K.R. Hoffman, Concave minimization via collapsing polytopes, Oper. Res. 34 (1986) 919–929.
J.E. Falk and K.R. Hoffman, A successive underestimation method for concave minimization problems, Math. Oper. Res. 1 (1976) 251–259.
J.E. Falk and R.M. Soland, An algorithm for separable nonconvex programming problems. Management Sci. 15 (1969) 550–569.
K.R. Hoffman, A method for globally minimizing concave functions over convex sets, Math. Programming 20 (1981) 22–32.
R. Horst An algorithm for nonconvex programming problems, Math. Programming 10 (1976) 312–321.
R. Horst, Deterministic global optimization with partition sets whose feasibility is not known. Application to concave minimization, reverse convex constraints, d.c.-programming and Lipschitzian optimization, J. Optim. Theory Appl. 58 (1988) 11–37.
R. Horst, Deterministic global optimization: recent advances and new fields of application, Naval Res. Logistics 37 (1990) 433–471.
R. Horst, A general class of branch and bound methods in global optimization with some new approaches for concave minimization, J. Optim. Theory Appl. 51 (1986) 271–291.
R. Horst, N.V. Thoai and H.P. Benson, Concave minimization via conical partitions and polyhedral outer approximation, Math. Programming, to appear.
R. Horst and H. Tuy,Global Optimization: Deterministic Approaches (Springer, Berlin, 1990).
R. Horst and H. Tuy, On the convergence of global methods in multiextremal optimization, J. Optim. Theory Appl. 54 (1987) 253–271.
H. Konno, Maximization of convex quadratic function subject to linear constraints, Math. Programming, 11 (1976) 117–127.
P.M. Pardalos and J.B. Rosen,Constrained Global Optimization: Algorithms and Applications (Springer, Berlin, 1987).
C.D. Pegden and C.C. Petersen, An algorithm (GIPC2) for solving integer programming problems with separable nonlinear objective functions, Naval Res. Logistics Quarterly 26 (1979) 595–609.
H. Tuy and R. Horst, Convergence and restart in branch-and-bound algorithms for global optimization: application to concave minimization and d.c. optimization problems, Math. Programming 41 (1988) 161–183.
H. Tuy, T.V. Thieu and N.Q. Thai, A conical algorithm for globally minimizing a concave function over a closed convex set, Math. Oper. Res. 10 (1985) 498–514.
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Benson, H.P., Selcuk Erenguc, S. & Horst, R. A note on adapting methods for continuous global optimization to the discrete case. Ann Oper Res 25, 243–252 (1990). https://doi.org/10.1007/BF02283698
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DOI: https://doi.org/10.1007/BF02283698