Invited Session Wed.2.H 2035

Wednesday, 13:15 - 14:45 h, Room: H 2035

Cluster 24: Variational analysis [...]

Structural properties in variational analysis


Chair: Stephen Michael Robinson



Wednesday, 13:15 - 13:40 h, Room: H 2035, Talk 1

Boris Mordukhovich
Second-Order variational analysis and stability in optimization

Coauthor: Terry Rockafellar


We present new results on the second-order generalized differentiation theory of variational analysis with new applications to tilt and full stability in parametric constrained optimization in finite-dimensional spaces. The calculus results concern second-order subdifferentials (or generalized Hessians) of extended-real-valued functions, which are dual-type constructions generated by coderivatives of first-order subdifferential mappings. We develop general second-order chain rules for amenable compositions and calculate second-order subdifferentials for some major classes of piecewise linear-quadratic functions. These results are applied to characterizing tilt and full stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms.



Wednesday, 13:45 - 14:10 h, Room: H 2035, Talk 2

Adrian Lewis
Active sets and nonsmooth geometry

Coauthors: J. Bolte, A. Daniilidis, D, Drusvyatskiy, S. Wright


The active constraints of a nonlinear program typically define a surface central to understanding both theory and algorithms. The standard optimality conditions rely on this surface; they hold generically, and then the surface consists locally of all solutions to nearby problems. Furthermore, standard algorithms "identify'' the surface: iterates eventually remain there. A blend of variational and semi-algebraic analysis gives a more intrinsic and geometric view of these phenomena, attractive for less classical optimization models. A recent proximal algorithm for composite optimization gives an illustration.
%Joint work with J. Bolte, A. Daniilidis, D. Drusvyatskiy, and S. Wright.



Wednesday, 14:15 - 14:40 h, Room: H 2035, Talk 3

Shu Lu
Confidence regions and confidence intervals for stochastic variational inequalities


The sample average approximation (SAA) method is a basic approach for solving stochastic variational inequalities (SVI). It is well known that under appropriate conditions the SAA solutions provide asymptotically consistent point estimators for the true solution to an SVI. We propose a method to build asymptotically exact confidence regions for the true solution that are computable from the SAA solutions, by exploiting the precise geometric structure of the variational inequalities and by appealing to certain large deviations probability estimates. We justify this method theoretically by establishing a precise limit theorem, and apply this method in statistical learning problems.


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