Abstract
Lagrange multipliers useful in characterizations of solutions to spectral estimation problems are proved to exist in the absence of Slater's condition provided a new constraint involving the quasi-relative interior holds. We also discuss the quasi interior and its relation to other generalizations of the interior of a convex set and relationships between various constraint qualifications. Finally, we characterize solutions to theL p spectral estimation problem with the added constraint that the feasible vectors lie in a measurable strip [α, β].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Goodrich, R. K., andSteinhardt, A. O.,L 2 Spectral Estimation, SIAM Journal on Applied Mathematics, Vol. 46, pp. 417–426, 1986.
Goodrich, R. K., Steinhardt, A. O., andRoberts, R. A.,Spectral Estimation via Minimum Energy Correlation Extension, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 33, pp. 1509–1515, 1985.
Cole, R., andGoodrich, R. K.,L p Spectral Analysis with an L∞ Upper Bound, Journal of Optimization Theory and Applications, Vol. 72, No. 2, pp. 321–355, 1993.
Borwein, J. M., andLewis, A. S.,Partially Finite Convex Programming, Parts 1 and 2, Mathematical Programming, Vol. 57, pp. 15–48, 1992 and Vol. 57, pp. 49–83, 1992.
Cole, R.,L p Spectral Analysis with an L ∞ Upper Bound, PhD Thesis, University of Colorado, Boulder, Colorado, 1990.
Klee, V. L.,Convex Sets in Linear Spaces, Duke Mathematics Journal, Vol. 16, pp. 443–466, 1948.
Klee, V. L.,Extremal Structure of Convex Sets, Mathematische Zeitschrift, Vol. 69, pp. 90–104, 1958.
Peressini, A. L.,Ordered Topological Vector Spaces, Harper, New York, New York, 1967.
Gowda, M. S., andTeboulle, M.,A Comparison of Constraint Qualifications in Infinite-Dimensional Convex Programming, SIAM Journal on Control and Optimization, Vol. 28, pp. 925–935, 1990.
Fullerton, R. E., andBraunschweiger, C. C.,Quasi-Interior Points and the Extension of Linear Functionals, Mathematics Annalen, Vol. 162, pp. 214–224, 1966.
Fullerton, R. E., andBraunschweiger, C. C.,Quasi-Interior Points of Cones, Technical Report 2, University of Delaware, Newark, Delaware, 1963.
Schaefer, H. H.,Banach Lattices and Positive Operators, Springer-Verlag, New York, New York, 1974.
Bazarra, M. S., andShetty, C. M.,Foundations of Optimization. Springer-Verlag, New York, New York, 1976.
Slater, M.,Lagrange Multipliers Revisited: A Contribution to Nonlinear Programming, Discussion Paper 403, Cowles Commission, 1950.
Borwein, J. M., andWolkowicz, H.,A Simple Constraint Qualification in Infinite-Dimensional Programming, Mathematical Programming, Vol. 35, pp. 83–96, 1986.
Micchelli, C. A., andUtreras, F. I.,Smoothing and Interpolation in a Convex Subset of a Hilbert Space, SIAM Journal on Scientific and Statistical Computation, Vol. 9, pp. 728–746, 1988.
Irvine, L. D., andSmith, P. W.,Constrained Minimization in a Dual Space, International Series of Numerical Mathematics, Birkhäuser Verlag, Basel, Switzerland, Vol. 76, pp. 205–219, 1986.
Luenberger, D. G.,Optimization by Vector Space Methods, John Wiley and Sons, New York, New York, 1969.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Holmes, R. B.,Geometric Functional Analysis and Its Applications, Springer-Verlag, New York, New York, 1975.
Rockafellar, R. T.,Conjugate Duality and Optimization, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1974.
Limber, M. A.,Quasi Interiors of Convex Sets and Applications to Optimization, PhD Thesis, University of Colorado, Boulder, Colorado, 1991.
Dontchev, A. L.,Duality Methods for Constrained Best Interpolation, Mathematica Balkanica, Vol. 1, pp. 96–105, 1987.
Borwein, J. M., Lewis, A. S., andLimber, M. A.,Entropy Minimization with Lattice Bounds (to appear).
Borwein, J. M., andLimber, M. A.,A Comparison of Entropies in the Underdetermined Moment Problem (to appear).
Borwein, J. M., andLimber, M. A.,On Entropy Maximization via Convex Programming, IEEE Transactions on Signal Processing (to appear).
Author information
Authors and Affiliations
Additional information
Communicated by D. G. Luenberger
The authors wish to thank Jonathan M. Borwein and Adrian S. Lewis for many enlightening discussions and useful suggestions. The duality approach to the general problem inL p was suggested by J. M. Borwein.
Rights and permissions
About this article
Cite this article
Limber, M.A., Goodrich, R.K. Quasi interiors, lagrange multipliers, andL p spectral estimation with lattice bounds. J Optim Theory Appl 78, 143–161 (1993). https://doi.org/10.1007/BF00940705
Issue Date:
DOI: https://doi.org/10.1007/BF00940705