Abstract
This paper describes a new technique for generating convex, strictly concave and indefinite (bilinear or not) quadratic programming problems. These problems have a number of properties that make them useful for test purposes. For example, strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an exponential number of local minima with distinct function values and indefinite and jointly constrained bilinear problems with nonextreme global minima, can be generated.
Unlike most existing methods our construction technique does not require the solution of any subproblems or systems of equations. In addition, the authors know of no other technique for generating jointly constrained bilinear programming problems.
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References
F.A. Al-Khayyal, “Jointly constrained bilinear programs and related problems: an overview,”Computers and Mathematics with Applications 19 (1990) 53–62.
R.H. Bartels and N. Mahdavi-Amiri, “On generating test problems for nonlinear programming algorithms,”SIAM Journal on Scientific and Statistical Computing 7 (1986) 769–798.
P.H. Calamai and L. Vicente, “Generating quadratic bilevel programming test problems,” to appear in:ACM Transaction on Mathematical Software.
H.P. Crowder, R.S. Dembo and J.M. Mulvey, “Reporting computational experiments in mathematical programming,”Mathematical Programming 15 (1978) 316–329.
H.P. Crowder, R.S. Dembo and J.M. Mulvey, “On reporting computational experiments with mathematical software,”ACM Transactions on Mathematical Software 5 (1979) 193–203.
C.A. Floudas and P.M. Pardalos, “A collection of test problems for constrained global optimization,”Lecture Notes in Computer Science No. 455 (Springer, Berlin, 1990).
W.W. Hager, P.M. Pardalos, I.M. Roussos and H.D. Sahinoglou, “Active constraints, indefinite quadratic test problems, and complexity,”Journal of Optimization Theory and Applications 68 (1991) 499–511.
J.J. Júdice and A.M. Faustino, “A computational analysis of LCP methods for bilinear and concave quadratic programming,”Computers and Operations Research 18 (1991) 645–654.
B. Kalantari, “Construction of difficulty linearly constrained concave minimization problems,”Operations Research 33 (1985) 222–227.
B. Kalantari, “Quadratic functions with exponential number of local maxima,”Operations Research Letters 5 (1986) 47–49.
B. Kalantari and J.B. Rosen, “Construction of large-scale global minimum concave quadratic test problems,”Journal of Optimization Theory and Applications 48 (1986) 303–313.
M.K. Kozlov, S.P. Tarasov and L.G. Hačijan, “Polynomial solvability of convex quadratic programming,”Soviet Mathematics Doklady 20 (1979) 1108–1111.
M. Lenard and M. Minkoff, “Randomly generated test problems for positive definite quadratic programming,”ACM Transactions on Mathematical Software 10 (1984) 86–96.
F.A. Lootsma, “Comparative performance evaluation, experimental design, and generation of test problems in nonlinear optimization,” in: K. Schittkowski, ed.,Computational Mathematical Programming (Springer, Berlin, 1985) pp. 249–260.
W. Michaels and R.P. O'Neill, “A mathematical program generator MPGENR,”ACM Transactions on Mathematical Software 6 (1980) 31–44.
K.G. Murty and S.N. Kabadi, “Some NP-complete problems in quadratic and linear programming,”Mathematical Programming 39 (1987) 117–129.
P.M. Pardalos, “Construction of test problems in quadratic bivalent programming,”ACM Transactions on Mathematical Software 17 (1991) 74–87.
P.M. Pardalos, “Generation of large-scale quadratic programs for use as global optimization test problems,”ACM Transactions on Mathematical Software 13 (1987) 133–137.
P.M. Pardalos and J.B. Rosen, “Constrained global optimization: algorithms and applications,”Lecture Notes in Computer Science No. 268 (Springer, Berlin, 1987).
P.M. Pardalos and G. Schnitger, “Checking local optimality in constrained quadratic programming is NP-hard,”Operations Research Letters 7 (1988) 33–35.
P.M. Pardalos and S. Vavasis, “Quadratic programming with one negative eigenvalue is NP-hard,”Journal of Global Optimization 1 (1991) 15–22.
A.T. Phillips and J.B. Rosen, “A parallel algorithm for constrained concave quadratic global minimiation,”Mathematical Programming 42 (1988) 421–448.
J.B. Rosen, “Global minimization of a linearly constrained concave function by partition of feasible domain,”Mathematics of Operations Research 8 (1983) 215–230.
J.B. Rosen and S. Suzuki, “Construction of nonlinear programming test problems,”Communications of the ACM 8 (1965) 113.
Y.Y. Sung and J.B. Rosen, “Global minimum test problem construction,”Mathematical Programming 24 (1982) 353–355.
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Support of this work has been provided by the Instituto Nacional de Investigação Científica de Portugal (INIC) under contract 89/EXA/5 and by the Natural Sciences and Engineering Research Council of Canada operating grant 5671.
Much of this paper was completed while this author was on a research sabbatical at the Universidade de Coimbra, Portugal.
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Calamai, P.H., Vicente, L.N. & Júdice, J.J. A new technique for generating quadratic programming test problems. Mathematical Programming 61, 215–231 (1993). https://doi.org/10.1007/BF01582148
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DOI: https://doi.org/10.1007/BF01582148