Abstract
In this paper, we consider the problem on minimizing sums of the largest eigenvalues of a symmetric matrix which depends on the decision variable affinely. An important application of this problem is the graph partitioning problem, which arises in layout of circuit boards, computer logic partitioning, and paging of computer programs. Given ∈≥0, we first derive an optimality condition which ensures that the objective function is within ∈ error bound of the solution. This condition may be used as a practical stopping criterion for any algorithm solving the underlying problem. We also show that, in a neighborhood of the minimizer, the optimization problem can be equivalently formulated as a smooth constrained problem. An existing algorithm on minimizing the largest eigenvalue of a symmetric matrix is shown to be applicable here. This algoritm enjoys the property that if started close enough to the minimizer, then it will converge quadratically. To implement a practical algorithm, one needs to incorporate some technique to generate a good starting point. Since the problem is convex, this can be done by using an algorithm for general convex optimization problems (e.g., Kelley's cutting plane method or ellipsoid methods), or an algorithm specific for the optimization problem under consideration (e.g., the algorithm developed by Cullum, Donath, and Wolfe). Such union ensures that the overall algorithm has global convergence with quadratic rate. Finally, the results presented in this paper are readily extended on minimizing sums of the largest eigenvalues of a Hermitian matrix.
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References
M. Akgul, “Topics in relaxation and ellipsoidal methods,” Research Notes in Mathematics, Pitman, 1984.
F. Alizadeh, “Combinatorial optimization with semidefinite matrices,” Proc. of Second Annual Integer Programming and Combinatorial Optimization Conf., Carnegie-Mellon University, 1991.
E.R. Barnes, “An algorithm for partitioning the nodes of a graph,”SIAM J. Alg. and Disc. Math. 3, (1982).
A.E. Barnes, and A.J. Hoffman, “Partitioning, spectra, and linear programming,”Progress in Combinatorial Optimization, Academic Press: New York, NY, 1984.
J.A. Bondy, and U.S.R. Murty,Graph Theory with Applications, American Elsevier, 1976.
F.H. Clarke,Optimization and Nonsmooth Analysis, J. Wiley & Sons: New York, NY, 1983.
J. Cullum, W.E. Donath, and P. Wolfe, “The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices,”Math. Prog. Study 3, (1975), 35–55.
W.E. Donath, and A.J. Hoffman, “Algorithms for partitioning graphs and computer logic based on eigenvectors of connection matrices,”IBM Tech. Disclosures Bul. 15, (1972).
W.E. Donath, and A.J. Hoffman, “Lower bounds for the partitioning of graphs,”IBM J. Res. Dev.,17, (1973).
J.C. Doyle, “Analysis of feedback systems with structured uncertainties,”Proc. IEE-D 129, 6, (1982), 242–250.
M.K.H. Fan, “An algorithm to compute the structured singular value,” Technical report, SRC TR-86-8, Systems Research Center, University of Maryland, 1986.
M.K.H. Fan, “A quadratically convergent local algorithm on minimizing the largest eigenvalue of a symmetric matrix,”Linear Algebra and Its Applications on Numerical Linear Algebra Methods in Control, Signals and Systems, to appear.
M.K.H. Fan, and B. Nekooie, “On minimizing the largest eigenvalue of a symmetric matrix,”Linear Algebra and Its Applications, to appear.
J.E. Hauser, “Proximity algorithms: Theory and implementation,” Electronics Research Laboratory, University of California, Memo No. UCB/ERL M86/53, Berkeley, California, May 1986.
R.A. Horn, and C.R. Johnson,Matrix Analysis, Cambridge University Press, 1985.
J.E. Kelley, “The cutting-plane method for solving convex programs,”J. Soc. Indust. Appl. Math. 8, 2, (1960), 703–712.
C. Moler, J. Little, and S. Bangert,PRO-MATLAB User's Guide, The Math Works, Inc., Sherborn, MA, 1987.
G. Narasimhan, and R. Manber, “A generalization of Lovass's sandwich theorem, W. Cook & P. Seymour,”Polyhedral Combinatorics: Proc. of a DIMACS Workshop, American Mathematical Society, 1990.
B. Nekooie, and M.K.H. Fan, “A quadratically convergent local algorithm on minimizing sums of the largest eigenvalue of a symmetric matrix,” Proc. of 31th IEEE Conf. on Decision and Control Tucson, Arizona, December 1992, 1915–1920.
M.L. Overton, “On minimizing the maximum eigenvalue of a symmetric matrix,”SIAM J. Matrix Anal. Appl. 9 (1988), 256–268.
M.L. Overton, and R.S. Womersley, “On the sum of the largest eigenvalues of a symmetric matrix,”SIAM J. Matrix Anal. Appl. (1992), 41–45.
M.L. Overton, and R.S. Womersley, “Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices,”Math. Prog. to appear.
A. Shapiro, and M.K.H. Fan, “Generic analysis of optimization problems involving eigenvalues of symmetric matrices,” in preparation.
N.-K. Tsing, M.K.H. Fan, and E.I. Verriest, “On analyticity of functions involving eigenvalues,”Linear Algebra and Its Applications, to appear.
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Some of results in this paper were given in [19] without proofs.
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Nekooie, B., Fan, M.K.H. A quadratically convergent local algorithm on minimizing sums of the largest eigenvalues of a symmetric matrix. Comput Optim Applic 2, 107–127 (1993). https://doi.org/10.1007/BF01299152
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DOI: https://doi.org/10.1007/BF01299152