Lexicographic differentiation was introduced by Yurii Nesterov in 1987. A recent and more accessible overview is given in his Mathematical Programming paper. At the CIAO workshop on 27 April I talk about lexicographic differentiation and mention two applications: the construction of directed subdifferential and geometric conditions for facial dual completeness of closed convex cones.
The function shown in the Mathematica rendering is lexicographically smooth, but is neither quasidifferentiable nor tame. The idea of this example was suggested by Jeffrey Pang (NUS).
I am giving a talk at Monash ACM seminar series on Friday, 10th of March, 2-3pm in Room 340.
Title: Open problems in convex geometry
Abstract: A convex model is the second best thing after a closed-form solution. Convex optimisation problems are often highly tractable, with a variety of numerical methods producing reliable approximations or exact solutions. The choice of the algorithms is vast, and includes general techniques such as subgradient descent or alternating projections, as well as highly specialised simplex and interior point methods. The major factors in the choice of the particular technique are the structure of the problem and the trade-offs between the resources available, the desired accuracy and the reliability of solutions.
Convexity has seemingly little to do with fractals, however convex sets with fractal facial structure are not hard to imagine. Read More