Abstract
This paper presents computational experience with a rather straight forward implementation of an edge search algorithm for obtaining the globally optimal solution for linear programs with an additional reverse convex constraint. The paper's purpose is to provide a collection of problems, with known optimal solutions, and performance information for an edge search implementation so that researchers may have some benchmarks with which to compare new methods for reverse convex programs or concave minimization problems. There appears to be nothing in the literature that provides computational experience with a basic edge search procedure. The edge search implementation uses a depth first strategy. As such, this paper's implementation of the edge search algorithm is a modification of Hillestad's algorithm [11]. A variety of test problems is generated by using a modification of the method of Sung and Rosen [20], as well as a new method that is presented in this paper. Test problems presented may be obtained at ftp://newton.ee.ucla.edu/nonconvex/pub/.
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References
Avriel, M. and Williams, A.C. (1970), Complementary Geometric Programming, SIAM Journal of Applied Mathematics 19, 125–141.
Avriel, M. and Williams, A.C. (1971). An Extension of Geometric Programming with Applications in Engineering Optimization, Journal of Engineering Mathematics 5, 187–194.
Bansal, P.P. and Jacobsen, S.E. (1975), Characterization of Basic Solutions For a Class of Nonconvex Programs, Journal of Optimization Theory and Applications 15, 549–564.
Bansal, P.P. and Jacobsen, S.E. (1975), An Algorithm for Optimizing Network Flow Capacity under Economies-of-Scale, Journal of Optimization Theory and Applications 15, 565–586.
BenSaad, S. and Jacobsen, S.E. (1990), A Level Set Algorithm for a Class of Reverse Convex Programs, Annals of Operations Research 25, 19–42.
Benson, H.P. and Sayin, S. (1994), A Finite Concave Minimization Algorithm Using Branch and Bound and Neighbor Generation, Journal of Global Optimization 5(1), 1–14.
Floudas, C.A. and Pardalos, P.M. (1990), A Collection of Test Problems for Constrained Global Optimization Algorithms, Springer-Verlag, Lecture Notes in Computer Sciences 455.
Gurlitz, T.R. and Jacobsen, S.E. (1991), On the Use of Cuts in Reverse Convex Programs, Journal of Optimization Theory and Application 68 (2).
Hillestad, R.J. and Jacobsen, S.E. (1980), Linear Programs with an Additional Reverse-Convex Constraint, Journal of Applied Mathematics and Optimization 6, 257–269.
Hillestad, R.J. and Jacobsen, S.E. (1980), Reverse-Convex Programming, Journal of Applied Mathematics and Optimization 6, 63–78.
Hillestad, R.J. (1975), Optimization Problems Subject to a Budget Constraint with Economies of Scale, Operations Research, Journal of Applied Mathematics and Optimization 23 (6), 1091–1098.
Horst, R. and Tuy, H. (1990), Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin-New York.
Horst, R. and Thoai, N.V. (1989), Modification, Implementation and Comparison of Three Algorithms for Globally Solving Linearly Constrained Concave Minimization Problems, Computing 42, 271–289.
Kalantari, B. and Rosen, J.B. (1986), Construction of Large-Scale Global Minimum Concave Quadratic Test Problems, Journal of Optimization Theory and Applications 48, 303–313.
Moshirvaziri, K. (1994), A Generalization of the Construction of Test Problems for Nonconvex Optimization, Journal of Global Optimization 5(1), 21–34.
Moshirvaziri, K. (1994), Construction of Test Problems for a Class of Reverse Convex Programs, Journal of Optimization Theory and Applications 81(2), 343–354.
Pardalos, Panos M. (1987), Generation of Large-Scale Quadratic Programs for Use as Global Optimization Test Problems, ACM Transaction on Mathematical Software 13 (2), 143–147.
Rosen, J.B. (1966), Iterative Solution of Non-linear Optimal Control Problems, SIAM Journal of Control 4, 223–244.
Rosen, J.B. (1983), Global Minimization of a Linearly Constrained Concave Function by Partition of Feasible Domain, Mathematics of Operations Research 8, 215–230.
Sung, Y. and Rosen, J.B. (1982), Global Minimum Test Problem Construction, Mathematical Programming 24, 353–355.
Thuong, Nguyen Van and Tuy, Hoang (1985), A Finite Algorithm for Solving Linear Programs with an Additional Reverse Convex Constraint, Springer-Verlag, Lecture Notes in Economics and Mathematical Systems 225, 291–302.
Tuy, Hoang (1987), Convex Programs with an Additional Reverse Convex Constraint, Journal of Optimization Theory and Applications 52(3), 463–485.
Tuy, Hoang (1985), A General Deterministic Approach to Global Optimization via D.C. Programming, In: Hiriart-Urruty, J.B. (ed.), Fermat Days: Mathematics for Optimization, Elsevier, Amsterdam, 137–162.
Ueing, U. (1972), A Combinatorial Method to Compute a Global Solution of Certain Non-convex Optimization Problems, in Numerical Methods for Nonlinear Optimization, F.A. Lootsma (ed.), Academic Press, 223–230.
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Jacobsen, S.E., Moshirvaziri, K. Computational experience using an edge search algorithm for linear reverse convex programs. J Glob Optim 9, 153–167 (1996). https://doi.org/10.1007/BF00121661
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DOI: https://doi.org/10.1007/BF00121661