Prof.dr. Constantin Niculescu

1. Background. Elements of Functional Analysis. Differential Calculus in Banach Spaces.

2. Inequality Constraints. Optimality Conditions. Theorems of the Alternative.

3. Fenchel Duality. Subgradients and Convex Functions. The Fenchel Conjugate.

4. Convex Analysis. Polar Calculus. Convex sets and Extreme Points. Fenchel Biconjugation. Lagrangian Duality. Duality for Linear and Semidefinite Programming. Convex Process duality.

5. Nonsmooth Optimization. Generalized Derivatives. Regularity and Strict Differentiability. Tangent Cones. The limiting Subdifferential. Karush-Kuhn-Tucker Theory.

6. Fixed Points. Variational Inequalities. Applications to PDE.

**Bibliography**

** **

1. H. W. Alt, *Lineare Funktionalanalysis*, Springer-Lehrbuch, Berlin, 1992.

2. J. M. Borwein and A. S. Lewis, *Convex Analysis and Nonlinear Optimization.
Theory and Examples*, CMS Books in Mathematics, Springer-Verlag, New York, 2000.

3. F. H. Clarke, *Optimization and Nonsmooth Analysis*, Wiley, New York, 1983.

4. J.-B. Hiriart-Urruty and C. Lemaréchal, *Convex Analysis and Minimization Algorithms*,
Springer-Verlag, New York, 1975.

5. C.P. Niculescu and L.-E. Persson,* Convex Functions and their applications. A Contemporary Approach. *
CMS Books in Mathematics, Springer-Verlag, New York, 2006.

6. R.T. Rockafellar,* Convex Analysis*, Princeton Univ. Press, Princeton, N.J., 1988.