Abstract

This paper deals with an infinite-dimensional optimization approach to the strong separation of two bounded sets in a normed space. We present an approximation procedure, called Algorithm (A), such that a semi-infinite optimization problem must be solved at each step. Its global convergence is established under certain natural assumptions, and a stopping criterion is also provided. The particular case of strong separation in the space Lp(X,A,μ)L_{p} (\mathbb{X}, \mathcal{A}, \mu ) is approached in detail. We also propose Algorithm (B), which is an implementable modification of Algorithm (A) for separating two bounded sets in Lp([a,b])L_{p}([a,b]), with [a,b][a,b] being an interval in R\mathbb{R}. Some illustative computational experience is reported, and a particular stopping criterion is provided for the case of functions of bounded variation in L2([a,b])L_{2}([a,b]).

Author information

Contact details are reproduced from the original publication and may be historical.

Marco A. López

Dept. of Statistics and Operations Research, Alicante University, Ap. de Correos 99, 03080 Alicante, Spain

marco.antonio@ua.es

Soon-Yi Wu

National Cheng Kung University, Tainan, Taiwan

Chen Ling

Zhejiang University of Finance and Economics, Hangzhou, P. R. China

Liqun Qi

Polytechnic University of Hong Kong, Hong Kong, P. R. China

Keywords

  • Strong separation
  • infinite dimensional optimization
  • semi-infinite programming

Mathematics Subject Classification

90C48, 46A22, 90C90

References

  1. K. P. Bennet: Decision tree construction via linear programming, in: Proc. 4th Midwest Artificial Intelligence Cognitive Sci. Soc. Conf. (1992) 97–101.Google Scholar
  2. D. L. Cohn: Measure Theory, Birhhäuser, Boston (1980).https://doi.org/10.1007/978-1-4899-0399-0Google Scholar
  3. B. D. Coleman: Finitely convergent learning programs for the separation of shaded figures, Biol. Cybern. 25 (1976) 57–60.https://doi.org/10.1007/BF00337049Google Scholar
  4. M. Á. Goberna, M. A. López: Linear Semi-Infinite Optimization, Wiley, Chichester (1998).Google Scholar
  5. M. Á. Goberna, M. A. López, S.-Y. Wu: Separation by hyperplanes: a linear semi-infinite programming approach, in: Semi-Infinite Programming. Recent Advances, M. A. Goberna, M. A. López (eds.), Nonconvex Optim. Appl. 57, Kluwer, Dordrecht (2001) 255–269.https://doi.org/10.1007/978-1-4757-3403-4_12Google Scholar
  6. O. L. Mangasarian: Linear and nonlinear separation of patterns by linear programming, Oper. Res. 13 (1965) 444–452.https://doi.org/10.1287/opre.13.3.444Google Scholar
  7. O. L. Mangasarian: Mathematical programming in neural networks, ORSA J. Comput. 5 (1993) 349–360.https://doi.org/10.1287/ijoc.5.4.349Google Scholar
  8. J. B. Rosen: Pattern separation by convex programming, J. Math. Anal. Appl. 10 (1965) 123–134.https://doi.org/10.1016/0022-247X(65)90150-2Google Scholar

Suggested citation

M. A. López, S.-Y. Wu, C. Ling, L. Qi. “A Mathematical Programming Approach to Strong Separation in Normed Spaces.” Journal of Convex Analysis 17 (2010), No. 1, 211–227. https://doi.org/10.68381/jca17016

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.