Abstract
We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints. Under a suitable simultaneous diagonalization assumption (which trivially holds for trust region type problems), we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints, which is also shown to be equivalent to a minimax convex problem. In both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts. Special cases and applications are also discussed. We outline interior-point polynomial-time algorithms for the solution of the equivalent convex programs.
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References
W. Achtziger, Minimax compliance truss topology subject to multiple loading. in: M.P. Bendsoe and C.A. Mota Soares, eds.,Topology Design of Structures (Kluwer, Dordrecht, 1993) 43–54.
A. Ben-Tal and A. Nemirovskii, Interior point polynomial time method for truss topology design. Research Report 3/92, Optimization Laboratory, Technion—Israel Institute of Technology (1992).
E. Eisenberg, Duality in homegeneous programming,Proceedings of the American Mathematical Society 12 (1961) 783–787.
P. Finsler, Über das Vorkommen definiter und semi-definiter Formen in scharen quadratische Formen,Commemarii Matematici Helvetici 9 (1937) 188–192.
R. Fletcher,Practical Methods of Optimization (Wiley, Chichester, 2nd ed., 1987).
O.E. Flippo and B. Jansen, Duality and sensitivity in quadratic optimization over a sphere, Technical Report 92-65, Technical University of Delft (1992).
W. Gander, Least squares with a quadratic constraint,Numerische Mathematik 36 (1981) 291–307.
D.M. Gay, Computing optimal locally constrained steps,SIAM Journal on Scientific and Statistical Computing 2 (1981) 186–197.
G.H. Golub and U. Von Matt, Quadratically constrained least squares and quadratic problems,Numerische Mathematik 59 (1991) 561–580.
R.A. Horn and C.R. Johnson,Matrix Analysis (Cambridge University Press, New York, 1985).
F. Korner, A tight bound for the boolean quadratic optimization problem and its use in a branch and bound algorithm.Optimization 19 (1988) 711–721.
J.J. Moré, Recent developments in algorithms and software for trust region methods, in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming, the State of the Art (Springer, New York, 1983) 268–285.
Y. Nesterov and A. Nemirovskii,Self-Concordant Functions and Polynomial Time Algorithms (CEMI, Moscow, 1989).
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
R.J. Stern and H. Wolkowicz, Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations,SIAM Journal on Optimization 5 (1995) 286–313.
Y. Ye, On affine scaling algorithms for nonconvex quadratic programming,Mathematical Programming 56 (3) (1992) 285–300.
Y. Yuan, On a subproblem of trust region algorithms for constrained optimization,Mathematical Programming 47 (1) (1990) 53–63.
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This author's work was partially supported by GIF, the German-Israeli Foundation for Scientific Research and Development and by the Binational Science Foundation.
This author's work was partially supported by National Science Foundation Grants DMS-9201297 and DMS-9401871.
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Ben-Tal, A., Teboulle, M. Hidden convexity in some nonconvex quadratically constrained quadratic programming. Mathematical Programming 72, 51–63 (1996). https://doi.org/10.1007/BF02592331
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DOI: https://doi.org/10.1007/BF02592331