Nonlinear optimization
Cetraro, July 1-7, 2007

Non-smooth optimization

Prof. Vladimir Demianov, St. Petersbourg State Univ. Russia


In the course, some tools and methods of nonsmooth analysis are presented which can be used for constructive solving problems of nondifferentiable optimization.

1. Smooth functions. Nonsmoothness. Optimality conditions (of the first and k-th order) of an arbitrary function on a metric space.

2. Some classes of Nonsmooth functions: convex and max-type functions. Properties. Subdifferentials of convex and max-type functions. Numerical methods for convex and max-type functions.

3. Quasidifferentiable functions. Calculus of quasidifferentials. Optimality conditions. Steepest descent and ascent directions. Numerical methods.

4. Codifferentiable functions. k-th order codifferentials and k-th order codifferentiable functions. Numerical aspects.

5. Constrained optimization problems. Exact penalties.

6. Directionally differentiable functions. Dini and Hadamard directional derivatives. Generalised directional derivatives. Clarke’s directional derivative.

7. Upper convex and lower concave approximations. Exhaustive families of approximations. Notions of upper and lower exhausters. Optimality conditions in terms of proper and adjoint exhausters. Descent and ascent directions.

8. Generalized Subdifferentials: Clarke’s subdifferential, Michel-Penot’s subdifferential, Frechet’s and Gateaux’ subdifferentials. Optimality conditions in terms of generalized derivatives and subdifferentials. Calculus of generalized subdifferentials via exhausters.

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