Some geometric results in semidefinite programming
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
The purpose of this paper is to develop certain geometric results concerning the feasible regions of Semidefinite Programs, called hereSpectrahedra.
We first develop a characterization for the faces of spectrahedra. More specifically, given a pointx in a spectrahedron, we derive an expression for the minimal face containingx. Among other things, this is shown to yield characterizations for extreme points and extreme rays of spectrahedra. We then introduce the notion of an algebraic polar of a spectrahedron, and present its relation to the usual geometric polar.
- F. Alizadeh (1991), Combinatorial Optimization with Interior Point Methods and Semi-Definite Matrices, Ph.D. Thesis, Computer Science Department, University of Minnesota, Minneapolis, Minnesota, 1991.
- F. Alizadeh (1995), Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization,SIAM J. Optimization 5, No. 1.
- G. P. Barker and D. Carlson (1975), Cones of Diagonally Dominant Matrices,Pac. J. Math. 57, 15–31.
- P. Binding (1990), Simultaneous Diagonalization of Several Hermitian Matrices,SIAM J. Matrix Anal Appl. 11, 531–536.
- P. Binding and C.-K. Li (1991), Joint Ranges of Hermitian Matrices and Simultaneous Diagonalization,Linear Algebra Appl. 151, 157–167.
- A. Ben-Israel, A. Charnes, and K. Kortanek, (1969) Duality and Asymptotic Solvability over Cones,Bull. of AMS 75, 318–324.
- A. Berman (1973),Cone, Matrices, and Mathematical Programming; Lecture Notes in Economics and Mathematical Systems, Springer.
- J. Borwein and H. Wolkowicz (1981), Characterization of Optimality for the Abstract Convex Program with Finite Dimensional Range,J. Austral. Math. Soc., Series A30, 390–411.
- J. Cullum, W. E. Donath, and P. Wolfe (1975), The Minimization of Certain Nondifferentiable Sums of Eigenvalue Problems,Math. Prog. Study 3, 35–55.
- C. Delorme and S. Poljak (1993), Combinatorial Properties and the Complexity of a Max-Cut Approximation,Europ. J. Combinatorics 14, 313–333.
- R. Fletcher (1985), Semi-Definite Matrix Constraints in Optimization,SIAM J. Control and Optimization 23, 493–513.
- M. X. Goemans and D. P. Williamson (1995), Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming,Submitted to J. ACM. (contact email@example.com for copies)
- R. Grone, S. Pierce, and W. Watkins (1990), Extremal Correlation Matrices,Linear Algebra and its Applications 134, pp. 63–70.
- M. Grötschel, L. Lovásza, and A. Schrijver (1984), Polynomial Algorithms for Perfect Graphs,Annals of Discrete Mathematics 21, C. Berge and V. Chvátal, eds., North Holland.
- M. Grötschel, L. Lovász, and A. Schrijver (1988),Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin.
- R. Horn and C.R. Johnson,Matrix Analysis, Cambridge University Press, Cambridge, 1985.
- M. Kojima, S. Kojima, and S. Hara, Linear Algebra for Semidefinite Programming, TR B-290, Research Reports on Information Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1994.
- M. Laurent and S. Poljak, On a Positive Semideflnite Relaxation of the Cut Polytope, Technical Report, LIENS-93-27, Ecole Normale Supérieure, France, 1993. (Contact firstname.lastname@example.org for copies)
- L. Lovász and A. Schrijver, Cones of Matrices and Setfunctions, and 0–1 Optimization,SIAM J. Optimization 1 (1991).
- Y. Nesterov and A. Nemirovskii,Interior Point Polynomial Methods for Convex Programming: Theory and Applications, SIAM, 1994.
- M. L. Overton, Large-Scale Optimization of Eigenvalues,SIAM J. Optimization 2 (1992), pp. 88–120.
- M. L. Overton and R. S. Womersley, Optimality Conditions and Duality Theory for Minimizing Sums of the Largest Eigenvalues of Symmetric Matrices,Math. Prog., Series B62 (1993), pp. 321–357.
- P. M. Pardalos and S. A. Vavasis (1992), Open Questions in Complexity Theory for Numerical Optimization,Math. Prog. 57(2), 337–339.
- G. Pataki, Algorithms for Linear Programs over Cones and Semidefinite Programming, Technical Report, GSIA, Carnegie-Mellon University, Pittsburgh, 1993. (contact email@example.com for copies)
- G. Pataki, On the Facial Structure of Cone-LP's and Semidefinite Programs, Management Science Research Report # MSRR-595, GSIA, Carnegie-Mellon University, Pittsburgh, 1994.
- M. Ramana (1995), An Exact Duality Theory for Semidefinite Programming and its Complexity Implications, DIMACS TR 95-02R (http://www.dimacs.edu), Rutgers University; Submitted toMath Programming.
- M. V. Ramana (1993), An algorithmic analysis of multiquadratic and Semidefinite programming problems, Ph.D. Thesis, The Johns Hopkins University, Baltimore, 1993.
- M. V. Ramana and A. J. Goldman, Cutting Plane Techniques for Multiquadratic Programming, Under Preparation.
- M. V. Ramana and A. J. Goldman, Quadratic Maps with Convex Images, Submitted to Math of OR.
- T. R. Rockafellar,Convex Analysis, Princeton University Press, Princeton, 1970.
- L. Vandenberghe and S. Boyd (1994), Positive-Definite Programming,Mathematical Programming: State of the Art 1994, J. R. Birge and K. G. Murty (eds.), U. of Michigan.
- H. Wolkowicz, Some Applications of Optimization in Matrix Theory,Linear Algebra and its Applications 40 (1981), 101–118.
- Some geometric results in semidefinite programming
Journal of Global Optimization
Volume 7, Issue 1 , pp 33-50
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- Semidefinite programming
- convex geometry
- Industry Sectors