Abstract
In this paper we analyze difference-of-convex (d.c.) decompositions for indefinite quadratic functions. Given a quadratic function, there are many possible ways to decompose it as a difference of two convex quadratic functions. Some decompositions are dominated, in the sense that other decompositions exist with a lower curvature. Obviously, undominated decompositions are of particular interest. We provide three different characterizations of such decompositions, and show that there is an infinity of undominated decompositions for indefinite quadratic functions. Moreover, two different procedures will be suggested to find an undominated decomposition starting from a generic one. Finally, we address applications where undominated d.c.d.s may be helpful: in particular, we show how to improve bounds in branch-and-bound procedures for quadratic optimization problems.
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References
I.M. Bomze, “On standard quadratic optimization problems,” J. Global Optimiz., vol. 13, pp. 369-387, 1998.
I.M. Bomze, “Branch-and-bound approaches to standard quadratic optimization problems,” J. Global Optimiz., vol. 22, pp. 17-37, 2002.
M. Dür, “A parametric characterization of local optimality,” Math. Methods of Oper. Research, vol. 57, pp. 101-109, 2003.
P. Hansen, B. Jaumard, M. Ruiz, and J. Xiong, “Global minimization of indefinite quadratic functions subject to box constraints,” Nav. Res. Logist., vol. 40, pp. 373-392, 1993.
R. Horst and V.N. Thoai, “Modification, implementation and comparison of three algorithms for globally solving linearly constrained concave minimization problems,” Computing, vol. 42, pp. 271-289, 1989.
R. Horst and V.N. Thoai, “A newalgorithm for solving the general quadratic programming problem,” Comput. Optim. Appl., vol. 5, pp. 39-48, 1996.
R. Horst and V.N. Thoai, “DC programming: Overview,” J. Optimization Theory Appl., vol. 103, pp. 1-43, 1999.
R. Horst, V.N. Thoai, and J. de Vries, “On geometry and convergence of a class of simplicial covers,” Optimization, vol. 25, pp. 53-64, 1992.
R. Horst and H. Tuy, Global Optimization. Springer: Heidelberg, 1993.
D.S. Johnson and M.A. Trick (Eds.), “Cliques, coloring, and satisfiability: Second DIMACS implementation challenge,” DIMACS Series in Discrete Mathematics and Theoretical Computer Science 26. American Mathematical Society, Providence, RI, 1996.
A. Kuznetsova and A. Strekalovsky, “On solving the maximum clique problem,” J. Global Optimiz., vol. 21, pp. 265-288, 2001.
H.A. Le Thi and T. Pham Dinh, “Solving a class of linearly constrained indefinite quadratic problems by DC algorithms,” J. Global Optimiz., vol. 11, pp. 253-285, 1997.
H.A. Le Thi and T. Pham Dinh, “A branch and bound method via d.c. optimization algorithms and ellipsoidal technique for box constrained nonconvex quadratic problems,” J. Global Optimiz., vol. 13, pp. 171-206, 1998.
H.A. Le Thi and T. Pham Dinh, “A continuous approach for large-scale linearly constrained qudratic zero-one programming,” Optimization, vol. 50, pp. 93-120, 2001.
I. Nowak, “A new semidefinite programming bound for indefinite quadratic forms over a simplex,” J. Global Optimiz., vol. 14, pp. 357-364, 1999.
P.M. Pardalos and G. Schnitger, “Checking local optimality in constrained quadratic programming is NP-hard,” Oper. Res. Lett., vol. 7, pp. 33-35, 1988.
P.M. Pardalos and S.A. Vavasis, “Quadratic programming with one negative eigenvalue is NP-hard,” J. Global Optimiz., vol. 1, pp. 15-22, 1991.
B.N. Parlett, The Symmetric Eigenvalue Problem, SIAM: Philadelphia, PA, 1998.
T.Q. Phong, H.A. Le Thi, and T. Pham Dinh, “On globally solving linearly constrained indefinite quadratic minimization problems by decomposition branch and bound method,” RAIRO, Rech. Oper., vol. 30, pp. 31-49, 1996.
U. Raber, “A simplicial branch-and-bound method for solving nonconvex all-quadratic programs,” J. Global Optimiz., vol. 13, pp. 417-432, 1998.
J. Renegar, A Mathematical View of Interior-Point Methods in Convex Optimization, SIAM: Philadelphia, PA, 2001.
V. Stix, “Global optimization of standard quadratic problems including parallel approaches,” Ph.D. thesis, Univ. Vienna, 2000.
V. Stix, “Target-oriented branch-and-bound method for global optimization,” J. Global Optimiz., vol. 26, pp. 261-277, 2003.
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Bomze, I.M., Locatelli, M. Undominated d.c. Decompositions of Quadratic Functions and Applications to Branch-and-Bound Approaches. Computational Optimization and Applications 28, 227–245 (2004). https://doi.org/10.1023/B:COAP.0000026886.61324.e4
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DOI: https://doi.org/10.1023/B:COAP.0000026886.61324.e4