### 1 — 08:30 — The radius of statistical efficiency

Classical results in asymptotic statistics show that the Fisher information matrix controls the difficulty of estimating a statistical model from observed data. In this work, we introduce a companion measure of robustness of an estimation problem: the radius of statistical efficiency (RSE) is the size of the smallest perturbation to the problem data that renders the Fisher information matrix singular. This quantity arises naturally as a statistical analogue of the "distance to ill-posedness", commonly used in numerical analysis and optimization. We compute RSE up to numerical constants for a variety of test bed problems, including principal component analysis, generalized linear models, phase retrieval, blind deconvolution, and matrix completion. In all cases, the RSE quantifies the compatibility between the covariance of the population data and the underlying model parameter.

### 2 — 09:00 — The Role of Subgradients in Second-Order Variational Analysis

Any second-order variational construction is defined at a point in the graph of subgradient map- ings. The second element of such points is a subgradient. Does it make a difference which subgradients we choose in these constructions? This talk is going to answer this question and demonstrate that certain subgradients, indeed, allow us to obtain more information about func- tions under consideration. Focusing mainly on the behavior of the second subderivative and subgradient proto-derivative of certain composite functions, we demonstrate that choosing the underlying subgradient, utilized in the definitions of these concepts, from the relative interior of the subdifferential mapping ensures stronger second-order variational properties such as strict twice epi-differentiability and strict subgradient proto-differentiability. Using this observation, we provide a simple characterization of continuous differentiability of the proximal mapping of our composite functions. As another application, we discuss the equivalence of metric regularity and strong metric regularity of a class of generalized equations at their nondegenerate solutions. This talk is based on joint works with Nguyen T. V. Hang.

### 3 — 09:30 — Polyhedral Newton-min algorithms for complementarity problems

The semismooth Newton method is a very efficient approach for computing

a zero of a large class of nonsmooth equations. When the initial iterate

is sufficiently close to a regular zero and the function is strongly

semismooth, the generated sequence converges quadratically to that zero,

while the iteration only requires to solve a linear system.

If the first iterate is far away from a zero, however, it is difficult

to force its convergence using linesearch or trust regions because a

semismooth Newton direction may not be a descent direction of the

associated least-square merit function, unlike when the function is

differentiable. We explore this question in the particular case of a

nonsmooth equation reformulation of the nonlinear complementarity

problem, using the minimum function. We propose a globally convergent

algorithm using a modification of a semismooth Newton direction that

makes it a descent direction of the least-square function. Instead of

requiring that the direction satisfies a linear system, it must be a

feasible point of a convex polyhedron; hence, it can be computed in

polynomial time. This polyhedron is defined by the often very few

inequalities, obtained by linearizing pairs of functions that have close

negative values at the current iterate; hence, somehow, the algorithm

feels the proximity of a ``negative kink'' of the minimum function and acts

accordingly. This is the plain PNM (Polyhedral Newton-min) algorithm.

In order to avoid as often as possible the extra cost of having to find

a feasible point of a polyhedron, a hybrid algorithm (HNM) is also proposed,

in which the Newton-min direction is accepted if a

sufficient-descent-like criterion is satisfied, which is often the case

in practice. Global and fast convergence to regular points is proved.

PNM or HNM may be used for linear complementarity problems (LCPs) since

only linear systems and polyhedral feasibility computations are involved,

no LCP subproblem. Preliminary experiments on LCPs were encouraging, we

observed that in most instances, the Newton-min direction was always

accepted. Only in a few instances and in only one or two iterations,

the algorithm needed expensive feasibility steps and these steps were

helpful in avoiding failures.

### 4 — 10:00 — Levenberg-Marquardt globalization of the polyhedral Newton-min algorithm for complementarity problems

Complementarity problems, in their rather general formulation, consist in

finding some vector x such that 0 <= F(x) | G(x) => 0, where F and G are

smooth vector functions. Optimality conditions for problems with nonlinear

inequality contraints fall within this framework, but the problem has many

other important applications. Remarkably, it is possible to reformulate the

problem as a nonsmooth equation H(x) = 0. Then, a possible approach

consists in applying nonsmooth Newton-like iterations to this equation.

A method of choice is the semismooth Newton algorithm, which benefits

from fast local convergence but lacks some global convergence property.

When H = min(F,G), one can overcome this drawback by using the

polyhedral Newton-min (PNM) algorithm with linesearch. The price to

pay for this improvement is the need to find a point in a convex

polyhedron (instead of solving a linear system) at each iteration.

Furthermore, regularity conditions on the cluster points are required to

guarantee that these are solutions. One goal of the presented work is to

remedy to this latter restriction by using a Levenberg-Marquardt (LM)

globalization technique. This extension is made difficult in particular

because the LM method minimizes a sum of squared linearized functions,

which mixes the components, whereas the NM algorithm is

designed componentwise thus enjoys separability properties.

When at an iterate some components of F and G are equal, H is not

differentiable. A technique used in the PNM algorithm deals with

these components by combining F and G, while the NM algorithm only

uses one of them. The proposed method follows a similar approach,

using a componentwise convex combination of F and G. An appropriate

choice of these weights allows one to obtain a descent curved path for

the cost function, provided that the current iterate is not stationnary. A

tolerance is used to detect the proximity to the kinks of H, which

intervenes in the analysis and is also relevant for numerical reasons.