## Abstract and Applied Analysis

### $\sigma$-Porosity in monotonic analysis with applications to optimization

A. M. Rubinov

#### Abstract

We introduce and study some metric spaces of increasing positively homogeneous (IPH) functions, decreasing functions, and conormal (upward) sets. We prove that the complements of the subset of strictly increasing IPH functions, of the subset of strictly decreasing functions, and of the subset of strictly conormal sets are $\sigma$-porous in corresponding spaces. Some applications to optimization are given.

#### Article information

Source
Abstr. Appl. Anal., Volume 2005, Number 3 (2005), 287-305.

Dates
First available in Project Euclid: 25 July 2005

https://projecteuclid.org/euclid.aaa/1122298430

Digital Object Identifier
doi:10.1155/AAA.2005.287

Mathematical Reviews number (MathSciNet)
MR2197121

Zentralblatt MATH identifier
1102.46016

#### Citation

Rubinov, A. M. $\sigma$-Porosity in monotonic analysis with applications to optimization. Abstr. Appl. Anal. 2005 (2005), no. 3, 287--305. doi:10.1155/AAA.2005.287. https://projecteuclid.org/euclid.aaa/1122298430

#### References

• F. S. De Blasi and J. Myjak, On a generalized best approximation problem, J. Approx. Theory 94 (1998), no. 1, 54--72.
• R. Deville and J. P. Revalski, Porosity of ill-posed problems, Proc. Amer. Math. Soc. 128 (2000), no. 4, 1117--1124.
• J. Dutta, J. E. Martínez-Legaz, and A. M. Rubinov, Monotonic analysis over cones. I, Optimization 53 (2004), no. 2, 129--146.
• M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, (L. F. Boron, ed.), P. Noordhoff, Groningen, 1964, translated from Russian by R. E. Flaherty.
• S. Reich and A. J. Zaslavski, Asymptotic behavior of dynamical systems with a convex Lyapunov function, J. Nonlinear Convex Anal. 1 (2000), no. 1, 107--113.
• --------, The set of divergent descent methods in a Banach space is $\sigma$-porous, SIAM J. Optim. 11 (2001), no. 4, 1003--1018.
• A. M. Rubinov, Abstract Convexity and Global Optimization, Nonconvex Optimization and Its Applications, vol. 44, Kluwer Academic Publishers, Dordrecht, 2000.
• --------, Monotonic analysis: convergence of sequences of monotone functions, Optimization 52 (2003), no. 6, 673--692.
• A. M. Rubinov and X. Yang, Lagrange-Type Functions in Constrained Non-Convex Optimization, Applied Optimization, vol. 85, Kluwer Academic Publishers, Massachusetts, 2003.
• A. M. Rubinov and A. J. Zaslavski, Two porosity results in monotonic analysis, Numer. Funct. Anal. Optim. 23 (2002), no. 5-6, 651--668.
• A. J. Zaslavski, On a generic existence result for a class of optimization problems, preprint, 2003.