Quasi interiors, lagrange multipliers, andLp spectral estimation with lattice bounds

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 theLp spectral estimation problem with the added constraint that the feasible vectors lie in a measurable strip [α, β].

[1]  Victor Klee,et al.  Extremal structure of convex sets. II , 1958 .

[2]  C. C. Braunschweiger,et al.  Quasi-interior points and the extension of linear functionals , 1966 .

[3]  Jonathan M. Borwein,et al.  Partially finite convex programming, Part I: Quasi relative interiors and duality theory , 1992, Math. Program..

[4]  Jonathan M. Borwein,et al.  A simple constraint qualification in infinite dimensional programming , 1986, Math. Program..

[5]  R. Holmes.  Geometric Functional Analysis and Its Applications , 1975 .

[6]  Jr. V. L. Klee.  Convex sets in linear spaces. III. , 1951 .

[7]  R. Rockafellar.  Conjugate Duality and Optimization , 1987 .

[8]  R. Goodrich,et al.  $L_2 $ Spectral Estimation , 1986 .

[9]  A. Peressini.  Ordered topological vector spaces , 1967 .

[10]  C. Micchelli,et al.  Smoothing and Interpolation in a Convex Subset of a Hilbert Space , 1988 .

[11]  V. L. KleeJr..  Extremal structure of convex sets , 1957 .

[12]  Allan O. Steinhardt,et al.  Spectral estimation via minimum energy correlation extension , 1985, IEEE Trans. Acoust. Speech Signal Process..

[13]  Marc Teboulle,et al.  A comparison of constraint quali cations in in nite-dimensional convex programming , 1990 .

[14]  Douglass J. Wilde,et al.  Foundations of Optimization. , 1967 .

[15]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[16]  H. H. Schaefer.  Banach Lattices and Positive Operators , 1975 .

[17]  Robert K. Goodrich,et al.  Lp-spectral estimation with anL∞-upper bound , 1993 .

[18]  Jonathan M. Borwein,et al.  Partially finite convex programming, Part II: Explicit lattice models , 1992, Math. Program..

[19]  M. Limber.  Quasi interiors of convex sets and applications to optimization , 1991 .