Abstract
We develop convergent decomposition branch and bound algorithms for solving a class of bilinear programming problems. As an application of the proposed method, we apply it to quadratic programs with a few negative eigenvalues, and to a class of mixed integer programming problems.
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F.A. Al-Khayyal and J.E. Falk, “Jointly constrained biconvex programming,” Mathematics of Operations Research, Vol. 8 pp. 273–286, 1983.
R. Horst and H. Tuy, “Global Optimization: Deterministic Approaches,” 2 edition Berlin, New York: Springer-Verlag, 1993.
H. Konno, “A cutting plane algorithm for solving bilinear programs,” Mathematical Programming, Vol. 11, pp. 14–27, 1976.
L.D. Muu and W. Oettli, “A method for minimizing a convex—concave function over a convex set,” J. of Optimization Theory and Applications, Vol. 70 pp. 337–384, 1991.
L.D. Muu and W. Oettli. “Combined branch and bound and cutting plane methods for solving a class of nonlinear programming problems,” J. of Global Optimization, Vol. 3, pp. 377–391, 1993.
L.D. Muu and B.T. Tam, “Efficient methods for solving certain bilinear programming problem,” to appear in Acta Mathematica Vietnamica.
P.M. Pardalos and J.B. Rosen, “Constrained global optimization: Algorithms and applications,” in G. Goos and J. Hartmans, editors, Lecture Notes in Computer Science, number 268. Berlin: Springer-Verlag, 1987.
P.M. Pardalos and S.A. Vavasis, “Quadratic programming with one negative eigenvalue is NP-hard,” J. of Global Optimization, Vol. 21, pp. 843–855, 1991.
A.T. Phillips and J.B. Rosen, “A parallel algorithm for constrained concave quadratic global minimization,” Mathematical Programming, Vol. 42, pp. 412–448, 1988.
T.Q. Phong, L.T.H. An, and Pham D. Tao, “On globally solving linearly constrained indefinite quadratic minimization problems by decomposition branch and bound method,” (submitted).
H.D. Sherali and A. Alameddine, “A new reformulation—linearization technique for bilinear programming problems,” J. of Global Optimization, Vol. 2, pp. 379–410, 1992.
H.D. Sherali and C.M. Shetty, “A finitely convergent algorithm for bilinear programming problem using polar cuts and disjunctive face cuts,” Mathematical Programming, Vol. 19, pp. 379–410, 1980.
Pham D. Tao “Un algorithme pour la résolution du programme linéaire général,” RAIRO-Recherche opérationnelle, Vol. 25, pp. 183–201, 1991.
T.V. Thieu, “Relationship between bilinear programming and concave programming,” Acta Mathematica Vietnamica, Vol. 2, pp. 106–113 (1980).
N. V. Thoai and H. Tuy, “Convergent algorithms for minimizing a concave function,” Mathematics of Operations Research, Vol. 5 pp. 556–566 (1980).
H. Vaish and C.M. Shetty, “The bilinear programming problems,” Naval Res. Log. Quar., Vol. 23, pp. 303–319, 1976.
H. Vaish and C.M. Shetty, “A cutting plane algorithm for bilinear programming problems,” Naval Res. Log. Quar., Vol. 24, pp. 83–94, 1976.
Y. Yajima and H. Konno, “Efficient algorithm for solving rank two and rank three bilinear programming problems,” J. of Global Optimization, Vol. 1, pp. 155–173, 1991.
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This paper was completed during the stay of the first author at LMI-INSA Rouen, CNRS URA 1378, France.
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Muu, L.D., Phong, T.Q. & Tao, P.D. Decomposition methods for solving a class of nonconvex programming problems dealing with bilinear and quadratic functions. Comput Optim Applic 4, 203–216 (1995). https://doi.org/10.1007/BF01300871
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DOI: https://doi.org/10.1007/BF01300871