Computation of Minimal Projections and Extensions. (1st February 2016)

• Record Type: Journal Article
• Authors: Foucart, Simon
• Is Part Of: Numerical functional analysis and optimization. Volume 37:Number 2(2016)
• Page Start: 159
• Page End: 185
• Publication Date: 2016-02-01
• Subjects: Coordinate spaces -- matrix spaces -- minimal projections -- polynomial spaces -- projection constants -- robust optimization -- semidefinite programming; 46B20 -- 49M29 -- 65K05 -- 90C22 -- 90C47; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2015.1091014
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: The determination of minimal projections is examined from an optimization theory viewpoint. It is first shown how to transform the problem into a linear program for the coordinate spaces and . It is then shown how to transform the problem into a linear program for the matrix spaces and . The procedure is exploited to experimentally determine minimal projections onto various matrix subspaces. Moreover, a fully theoretical determination of minimal projections into zero-trace matrices is proposed when the matrix norm is unitarily invariant. Next, for polynomial spaces it is shown how to approximate the problem by a linear program or by a semidefinite program using techniques from robust optimization. It allows us to tabulate the relative projection constants of several polynomial subspaces. The article finishes by illustrating that the underlying method also applies to the determination of minimal extensions rather than merely minimal projections.