On Convex Optimization in Hilbert Spaces

Year 2019, Volume: 1 Issue: 2, 89 - 95, 30.10.2019

Benard Okelo

Abstract

In this paper, convex optimization techniques are employed for convex optimization problems in infinite dimensional Hilbert spaces. A first order optimality condition is given. Let  $f : \mathbb{R}^{n}\rightarrow \mathbb{R}$ and let $x\in \mathbb{R}^{n}$ be a local solution to the problem $\min_{x\in \mathbb{R}^{n}} f(x).$ Then $f'(x,d)\geq 0$ for every direction $d\in \mathbb{R}^{n}$  for which $f'(x,d)$ exists. Moreover, Let  $f : \mathbb{R}^{n}\rightarrow \mathbb{R}$ be differentiable at  $x^{*}\in \mathbb{R}^{n}.$ If $x^{*}$ is a local minimum of $f$, then $\nabla f(x^{*}) = 0.$ A simple application involving the Dirichlet problem is also given. Lastly, we have given optimization conditions involving positive semi-definite matrices.

Keywords

Optimization, Convexity, Hilbert space

References

• [1]F. Albiac and N. J. Kalton, Topics in Banach space theory, Volume 233 of Graduate Texts in Mathematics. Springer, New York, 2006.
• [2] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, United Kingdom, 2004.
• [3] D. P. Bertsekas, Convex Analysis and Optimization, Athena Scienti.c, Belmont, MA, 2003.
• [4] N. Dunford and J. T. Schwartz, Linear operators, Part I. Wiley Classics Library. John Wiley and Sons Inc., New York, 1988.
• [5] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North Holland, Amsterdam, 1976.
• [6] I. Ekeland and T. Turnbull, Innite Dimensional Optimization and Convexity, The University of Chicago Press, Chicago, 1983.
• [7] R. Glowinski, J. L. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North Holland, Amsterdam, 1981.
• [8] M. Grasmair, Minimizers of optimization problems, To appear.
• [9] A.J. Kurdila and M. Zabarankin., Convex functional analysis, Systems and Control: Foundations and Applications. Birkhauser Verlag, Basel, 2005.
• [10] J.P. Vial, Strong convexity of set and functions, J. Math. Econom 9 (1982), 187-205.