Approximate affine linear relationship between L1 norm objective functional values and L2 norm constraint bounds

• Published in: IET signal processing Vol. 9; no. 9; pp. 670 - 680
• Main Authors: Yang, Zhijing, Ling, Bingo Wing-Kuen, Bingham, Chris
• Format: Journal Article
• Language: English
• Published: The Institution of Engineering and Technology 01.12.2015
• Subjects: approximate affine linear relationship; compressed sensing; L1 norm objective functional value; L2 norm constraint bound; linear programming; optimisation problem; signal representation; sparse signal representation; L2 norm constraint bound; compressed sensing; sparse signal representation; linear programming; approximate affine linear relationship; signal representation; L1 norm objective functional value; optimisation problem
• ISSN: 1751-9675; 1751-9683; 1751-9683
• DOI: 10.1049/iet-spr.2014.0402

For an optimisation problem with an L1 norm objective function subject to an L2 norm inequality constraint, this study shows that there is an approximately linear relationship between the L1 norm objective functional values and the L2 norm specifications. This relationship is verified through the use of random and real world industrial data. The obtained results can be employed for (i) estimating the L1 norm objective functional value without solving the optimisation problem numerically; (ii) providing an insight for defining the L2 norm specification in which a simple method is proposed in this study; and (iii) testing whether the obtained solutions are the globally optimal solutions or not. These advantages are demonstrated via the use of random data.