### Seminario Jueves 10 Enero 2019 (CMM – **Beauchef 851 ****, piso -1, ****Sala ****B08**)

14:30 – 16:00: Terry Rockafellar (Resumen) – Roberto Lucchetti (Resumen)

16:00 – 16:30: Pausa café

16:30 – 18:00: Patrick Combettes (Resumen) – Teemu Pennanen (Resumen)

### Seminario Miércoles 16 Enero 2019 (UOH – **Campus Rancagua, Sala C312**)

15:00 – 16:30: Boris Mordukhovich (Resumen) – Michele Palladino (Resumen)

16:30 – 17:00: Pausa café

17:00 – 17:45: Francisco Silva (Resumen)

17:45 — Cóctel de camaradería

### Summer School: Strategies and Dynamics in Games

En Lima, Perú, 14 – 18 de Enero, 2019 – http://gamesdyn.imca.edu.pe/

### Seminario Miércoles 23 Enero 2019 (CMM – **Beauchef 851 ****, piso -1, ****Sala ****B08**)

16:00 – 17:30: Vera Roshchina (Resumen) – Antoine Hochart (Resumen)

### Pasantías en Chile de investigadores invitados por la red de Modelamiento Matemático

- Roberto Lucchetti, del 1 al 13 de Enero
- Terry Rockafellar, del 2 al 11 de Enero del 2019
- Teemu Pennanen, del 2 al 13 de Enero del 2019
- Michele Palladino, del 7 al 18 de Enero 2019
- Patrick Combettes, del 8 al 16 de Enero 2019
- Boris Mordukhovich, del 11 al 19 de Enero 2019
- Vera Roshchina, del 12 al 25 de Enero 2019
- Francisco Silva, del 14 de Enero al 14 de Febrero 2019

**Resúmenes**

### Terry Rockafellar

*Achieving economic equilibrium by a process of optimization*

*Abstract*

In the most basic problem of economic equilibrium, individual agents start with holdings of various “goods” but might prefer to adjust those holdings according to preferences based on utility functions. Adjustments

could be facilitated by a market in which goods can be bought and sold. An equilibrium is achieved when the market prices and holdings have reached a state in which adjustments are no longer of interest.

In the classical approach of Walras, the issue is mainly whether prices exist such that if agents, in buying and selling, maximize utility subject to the constraint of keeping within the budget provided by the value of

their initial holdings, will reach holdings that furnish an equilibrium configuration. This involves optimization in a decentralized manner in which agents act independently of each other, but it doesn’t provide an

economic process for arriving at the right prices.

A new and different approach is to let agents interact in pairs, one good at a time, deciding how much of it to pass from one to the other for some amount of money. This relies on bilateral optimization in which marginal

utility has a key role in providing personalized ranges of money prices at which agents are willing to buy or sell. The main result is that if this is kept going with no agents or goods being left out, the holdings and

personalized prices will converge to a particular equilibrium.

### Roberto Lucchetti

*Efficient Nash equilibria and mixed strategies*

*Abstract*

*As it is very well known, Nash equilibria usually provide a very bad outcome in social terms. The prisoner dilemma is the most famous example of such a situation. Concepts like price of anarchy and of stability were defined to detect, in specific classes of games, how far from being efficient are the outcomes provided by Nash equilibria. It can also be argued that generically Nash equilibria are inefficient. I present some result in this setting, focusing on equilibria in mixed strategies.*

### Patrick Combettes

*Perspective maximum likelihood-type estimation via proximal decomposition*

*Abstract*

We introduce an optimization model for maximum likelihood-type estimation (M-estimation) that generalizes a large class of existing statistical models, including Huber’s concomitant M-estimator, Owen’s Huber/Berhu concomitant estimator, the scaled lasso, support vector machine regression, and penalized estimation with structured sparsity. The model, termed perspective M-estimation, leverages the observation that convex M-estimators with concomitant scale as well as various regularizers are instances of perspective functions. Such functions are amenable to proximal analysis, which leads to principled and provably convergent optimization algorithms via proximal splitting. Using a geometrical approach based on duality, we derive novel proximity operators for several perspective functions of interest. Numerical experiments on synthetic and real-world data illustrate the broad applicability of the proposed framework.

(based on joint work with C. L. Müller)

### Teemu Pennanen

*Double auctions in welfare economics*

*Abstract*

Welfare economics argues that competitive markets lead to efficient allocation of resources. The classical theorems are based on the Walrasian market model which assumes the existence of market clearing prices. The emergence of such prices remains debatable. We replace the Walrasian market model by double auctions and show that the conclusions of welfare economics remain largely the same. Double auctions are not only a more realistic description of markets but they explain how equilibrium prices and efficient allocations may emerge in practice.

### Boris Mordukhovich

*Criticality of Lagrange multipliers in variational systems with numerical applications*

*Abstract*

This talk concerns the study of criticality of Lagrange multipliers in variational systems that has been recognized in both theoretical and numerical aspects of optimization and variational analysis. In contrast

to the previous developments dealing with polyhedral KKT systems and the like, we now focus on

general nonpolyhedral systems that are associated, in particular, with problems of conic programming.

Developing a novel approach, which is mainly based on advanced techniques and tools of second-order

variational analysis and generalized differentiation, allows us to overcome principal challenges of

nonpolyhedrality and to establish complete characterizations on noncritical multipliers in such settings.

The obtained results are applied to developing a conic programming version of the SQP method with

establishing its superlinear convergence.

Based on joint works with Ebrahim Sarabi (Miami University, OH, USA)

### Michele Palladino

*On the Pontryaguin Maximum Prnciple in Optimal Control*

*Abstract*

### Francisco Silva

*On the convergence problem for first order mean field games*

*Abstract*

In this talk, based on a joint work with M. Fischer (University of Padua), we provide a simple justification of the first order MFG system, first introduced by Lasry and Lions in 2007, as a PDE characterization of Nash equilibria for symmetric deterministic differential games with a continuum of players. Our main result shows that such equilibria can be found as the limit of Nash equilibria of suitable differential games with finite number of players.

### Vera Roshchina

*Higher order Voronoi cells*

*Abstract*

The classic Voronoi cells can be generalized to a higher-order version by considering the cells of points for which a given k-element subset of the set of sites consists of the k closest sites. We study the structure of the k-order Voronoi cells and illustrate our theoretical findings with a case study of two-dimensional higher-order Voronoi cells for four points.

The talk is based on joint work with Juan Enrique Martínez-Legaz and Maxim Todorov, arXiv:1811.10257.

### Antoine Hochart

*A game-theory approach to nonlinear Perron-Frobenius theory*

*Abstract*

Nonlinear Perron-Frobenius theory deals with self-maps f of a closed convex cone in a Banach space, that are positively homogeneous of degree one and preserve the partial order induced by the cone. Such maps appear in several fields including population dynamics, dynamical systems, optimal control or zero-sum repeated games. In all these fields, a basic question is to understand the asymptotic behavior of the iterates of f, behavior which is best understood when f has an eigenvector in the interior of the cone. In this talk, we present a generalized Perron-Frobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any map f acting on the open orthant of R^n. This criterion requires the construction of an “abstract” two-player stochastic game, which only depends on the behavior of f “at infinity”. It is fomulated in terms of dominions, i.e., subsets of states that can be made invariant by one player. In this way, we characterize the situation in which all the so-called slice spaces (subsets of points x such that f(x)-x is bounded below and above by constants) are bounded in Hilbert’s projective metric. This property is also equivalent to the boundedness, with respect to this metric, of all the orbits of all the uniform perturbations of f. Another manifestation of the combinatorial nature of the dominion conditions is an equivalent directed hypergraph formulation. We illustrate these results by considering specific classes of nonlinear maps: Shapley operators and nonnegative tensors.