On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues

References

• Alizadeh F. Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. (1995) 5:13–51Crossref, Google Scholar
• Alizadeh F., Haeberly J.-P., Overton M. Complementarity and nondegeneracy in semidefinite programming. Math. Programming (1997) 77:111–128Crossref, Google Scholar
• Aubin J.-P., Frankowska H.Set-Valued Analysis (1990) (Birkhäuser, Boston) Google Scholar
• Barker G. P., Carlson D. Cones of diagonally dominant matrices. Pacific J. Math. (1975) 57:15–32Crossref, Google Scholar
• Clarke F. H.Optimization and Nonsmooth Analysis (1990) (Wiley, New York) Crossref, Google Scholar
• Cullum J., Donath W. E., Wolfe P. The minimization of certain nondifferentiable sums of eigenvalue problems. Math. Programming Stud. (1975) 3:35–55Crossref, Google Scholar
• Delorme C., Poljak S. Laplacian eigenvalues and the maximum cut problem. Math. Programming (1993) 62:557–574Crossref, Google Scholar
• Hiriart-Urruty J.-B., Ye D. Sensitivity analysis of all eigenvalues of a symmetric matrix. Numer. Math. (1995) 70:45–72Crossref, Google Scholar
• Horn R. A., Johnson C. R.Matrix Analysis (1987) (Cambridge University Press)Google Scholar
• Karger, Motwani D. R., Sudan M. Approximate graph coloring by semidefinite programming. Proc. 35th IEEE Symp. Foundations of Comput. Sci. (1994) 2–13Google Scholar
• Ky Fan. On a theorem of Weyl concerning the eigenvalues of linear transformations. Proc. Nat. Acad. Sci. USA (1949) 35:652–655Crossref, Google Scholar
• Lewis A. S. Convex analysis on the Hermitian matrices. SIAM J. Optim. (1996) 6:164–177Crossref, Google Scholar
• Lewis A. S., Overton M. Eigenvalue optimization. Acta Numerica (1996) 5:149–190Crossref, Google Scholar
• Nesterov Yu., Nemirovskii A.Interior Point Polynomial Algorithms in Convex Programming (1994) (SIAM, Philadelphia) Crossref, Google Scholar
• Overton M. L. Large-scale optimization of eigenvalues. SIAM J. Optim. (1992) 2:88–120Crossref, Google Scholar
• Overton M. L., Womersley R. S. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. Matrix Anal. Appl. (1992) 13:41–45Crossref, Google Scholar
• Overton M. L., Womersley R. S. Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices. Math. Programming (1993) 62:321–357Crossref, Google Scholar
• Pataki G.On the facial structure of cone-LPs and semidefinite programs (1994) . Management Science Research report MSRR-#595, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PAGoogle Scholar
• Ramana M. V., Goldman A. Some geometric results in semidefinite programming. J. Global Opt. (1995) 7:33–50Crossref, Google Scholar
• Rendl F., Wolkowicz H. A semidefinite framework for trust region subproblems with applications to large scale minimization. Math. Programming (1997) 77:273–299Google Scholar
• Rockafellar R. T.Convex Analysis (1970) (Princeton University Press, Princeton, New Jersey) Crossref, Google Scholar
• Shapiro A. Extremal problems on the set of nonnegative definite matrices. Linear Algebra Appl. (1985) 67:7–18Crossref, Google Scholar
• Shapiro A., Fan M. K. H. On eigenvalue optimization. SIAM J. Optim. (1995) 5:552–569Crossref, Google Scholar
• Stoer J., Witzgall C.Convexity and Optimization in Finite Dimensions (1970) (Springer-Verlag, New York) Crossref, Google Scholar
• Vandenberghe L., Boyd S. Semidefinite programming. SIAM Rev. (1996) 38:49–95Crossref, Google Scholar
• Wolkowicz H. Some applications of optimization in matrix theory. Linear Algebra Appl. (1981) 40:101–118Crossref, Google Scholar