Advanced Continuous Optimization I
(J.-C. Gilbert, INRIA)
Jean-Charles Gilbert <jean-charles.gilbert@inria.fr>
Tentative (and maybe too ambitious) program
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Abstract optimality conditions
First order optimality conditions
Second order optimality conditions
Linearization methods
Semismooth Newton algorithm for nonlinear equations and optimization
Josephy-Newton algorithm for nonlinear inclusion
SQP algorithm in nonlinear optimization
Proximal and augmented Lagrangian methods
Proximal algorithm for solving inclusion and optimization problems
Augmented Lagrangian algorithm in optimization
Application to convex quadratic optimization (error bound and global linear convergence)
Application to SDP optimization
Interior point methods (should be reorganized)
Interior point methods in convex optimization
Examples of use in linear and SDP optimization
Interior point methods in nonlinear optimization
Schedule : Monday, 14h-18h15, from Sept 21 to Nov 9. Examination on Nov 16.
Where ? ENSTA, room 2330
This course will have no computer sessions, but the course A.C.O. II will be mostly devoted to
computer sessions completing A.C.O. I
Introduction to Operational Research and Combinatorics
(X. Allamigeon, INRIA and Polytechnique, and C. Gicquel and D. Quadri, U. Paris-Sud)
Xavier Allamigeon <xavier.allamigeon@inria.fr>,
Celine Gicquel <Celine.Gicquel@lri.fr>,
Dominique Quadri <dominique.quadri@lri.fr>
Operations Research is an area of applied mathematics aiming at helping organizations to make
better decisions. It relies on a variety of mathematical techniques to model and solve decision and
optimization problems arising in many different contexts such as manufacturing, transportation,
health care, telecommunications or financial planning. The objective of the course is to provide an
introduction to Operations Research by focusing mainly on the use of mathematical programming
while keeping a balance between theoretical material and practice.
In the first part of the course, we will explore some of the mathematical foundations of operations
research, through an introduction to linear programming and the theory of convex polyhedra. We
will address the notions of duality, integer linear programming, network flows, and will cover
combinatorial algorithms like the simplex method, etc. A particular attention will be paid to the
exposition of the computational complexity of the techniques which we will deal with. We will also
give an idea of the resolution of non linear programs for which basic solver are not able to provide
the optimal solution. In particular, we will focus on reformulation techniques.
The second part of the course will consist of an introduction to mixed-integer linear programming.
We will first show how many optimization problems can be modeled as a mixed-integer linear
program involving binary or integer decision variables and study their resolution by the Branch &
Bound algorithm. We will then devote some time on modeling and solving practical optimization
problems using the mathematical programming solver CPLEX.
Schedule
First part (X.A.): Wednesday 14h-16h + Thursday 9h30-12h30 from Sept 23 to Oct 8
Second part (C.G. + D.Q.) : Thursday 14h-17h, from Oct 8 to Nov 12.
Examination on the week of Nov 16.
Where ?
The lessons of Oct 15, 22 and Nov 5 will be computer sessions. They take place at ENSTA, room
1210.
All the other lessons are held at Polytechnique, room according to the following schedule.
Mercredi 23 Septembre 2015 de 14h00 – 16h00 (PC n° 7)
Jeudi 24 Septembre 2015 de 09h30 – 12h30 (PC n° 4)
Mercredi 30 Septembre 2015 de 14h00 – 16h00 (PC n° 8)
Jeudi 01 Octobre 2015 de 09h30 – 12h30 (PC n° 4)
Mercredi 07 Octobre 2015 de 14h00 – 16h00 (PC n° 7)
Jeudi 08 Octobre 2015 de 09h30 – 12h30 (PC n° 5)
Game Theory
(T. Tomala, HEC)
Tristan Tomala <tomala@hec.fr>
This course will present the main models of strategic games, related mathematical tools, important
results and applications.
1- Decision problems (one-player games): Preference relations, utility and payoff functions,
decision under risk and VnM axioms, representation theorems.
2- Zero-sum games: MinMax Theorems, pure and mixed strategies.
3- n-player games: Dominance and Rationalizability. Nash equilibria and fixed points, correlated
equilibria.
4- Extensive form games: perfect information and backward induction, Zermelo and Kuhn's
Theorems. Imperfect information, mixed and behavioral strategies, perfect recall and equivalence
Theorems.
5- Incomplete information and Bayesian games.
6- Repeated games and Folk Theorems.
Bibliography:
In French:
-Bases Mathématiques de la Théorie des jeux, R. Laraki, J. Renault et S. Sorin. Ed. Polytechnique.
-Théorie des jeux: introduction à la théorie des jeux répétés, R. Laraki, J. Renault et T. Tomala, Ed.
Polytechnique. http://www.math.polytechnique.fr/xups/vol06.html
In English:
-A Course in Game Theory, M. Osborne and A. Rubinstein, MIT Press.
-An Introductory Course on Mathematical Game Theory, J. Gonzalez-Diaz, I.Garcia-Jurado and G.
Fiestras-Janeiro. AMS.
Advanced:
-Game Theory, M. Maschler, E. Solan and S. Zamir, Cambridge Press.
Very Advanced:
-Repeated Games, J-F Mertens, S. Sorin and S. Zamir. Cambridge Press.
Schedule: Wednesday 9h-12h + Friday14h-16h, from Sept 23 to Nov 6.
Examination two weeks later.
Where ? Polytechnique, room PC n° 4 except the following dates
Vendredi 09 Octobre 2015 de 14h00 – 16h00 (PC n° 9)
Vendredi 16 Octobre 2015 de 14h00 – 16h00 (PC n° 9)
Mercredi 21 Octobre 2015 de 09h00 – 12h00 (ENSTA-Salle n° 1428 – 1er
Vendredi 23 Octobre 2015 de 14h00 – 16h00 (PC n° 9)
Dynamical Programming
Markov decision processes: dynamic programming and applications
(Marianne Akian, INRIA, and Jean-Philippe Chancelier, Ecole des Ponts)
Marianne Akian <marianne.akian@inria.fr>
Jean-Philippe Chancelier <jpc@cermics.enpc.fr>
The aim of this course is to introduce different stochastic control models and to present dynamic
programming as a tool for solving them. Illustrations selected among stockmanagement, portfolio
selection, Yield management, transportation or Web PageRank optimisationwill be presented.
We shall consider essentially stochastic dynamical systems with discrete time and finite state
space, or finite Markov chains. This framework already contains the essential difficulties (for
instance for long term problems), and allows one to give at the same time an insight of algorithms,
mathematical techniques and qualitative properties. We may however consider some examples
with infinite state space or continuous time.
We shall present the different types of stochastic control problems: complete and incomplete
observation problems, criteria with finite horizon, discounted infinite horizon, stopping time, ergodic
criteria, risk-sensitive criteria, constrainted problems, armed bandit problems. For some of
these criteria, we shall state the corresponding dynamic programming equations, study their
qualitative properties, and the algorithms for solving them (value iterations, policy iterations, linear
programming), and deduce in some cases the structure of optimal strategies.
This is a joint course with ENSTA: lectures 1-6 and 8 are common to ENSTA course and M2
course.
Schedule: Tuesday Sept 22 and 29, Oct 6, 13, and 20, Nov 3, 9h30-12h (Marianne Akian),
14h30-17h (Jean-Philippe Chancelier). Lecture 8 (Oct 13, 14h30-17h) will be a training (computer)
session. Final exam: Tuesday Nov 17, 9h-12h.
Where? ENSTA. room 2413 except on Oct 13 (room 1148 and 1149)
Key Words: Markov Decision processes, Stochastic control, Ergodic control, Risk-sensitive control, Dynamic programming, Max-plus algebra, Value iteration, Policy iteration.
Some references:
• D. P. Bertsekas. Dynamic programming. Prentice Hall Inc., Englewood Cliffs, NJ, 1987. Deterministic and stochastic models.
• E.B. Dynkin and A.A. Yushkevich. Controlled Markov processes. Springer-Verlag, New York,
• M. L. Puterman. Markov decision processes: discrete stochastic dynamic programming. Wiley
Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley
& Sons Inc., New York, 1994. , A Wiley-Interscience Publication.
Optimal Control of ODE
(J.F. Bonnans, Inria-Cmap, and H. Zidani, Ensta)
J. Frédéric Bonnans <Frederic.Bonnans@inria.fr>
Hasnaa Zidani <housnaa.zidani@ensta-paristech.fr>
The course will provide an introduction to the optimal control theory, through the study of the
optimality conditions (Pontryagin’s principle), the shooting algorithm, and the dynamic
programming approach (HJB equation).
It will be in two parts, the first common with a third year Ensta course (corresponding lessons
denoted by P1 and P2 below).
Precise schedule
23 09 2015
16:00 18:00
25 09 2015
30 09 2015
02 10 2015
07 10 2015
09 10 2015
14 10 2015
16 10 2015
09:00 12:15
16:00 18:00
09:00 12:15
16:00 18:00
09:00 12:15
16:00 18:00
09:00 12:15
21 10 2015
23 10 2015
04 11 2015
06 11 2015
20 11 2015
16:00 18:00
09:00 12:15
16:00 18:00
09:00 12:15
09:00 12:00
P 1: Introduction- Examples, differential calculus in function
spaces
P1: Pontryagin's principle (PMP)
P 1: Applications of the PMP
P 1: Minimal time function, optimal synthesis
P 1: Shooting methods
P 1: Shooting method (numerical simulations)
P 2 (début): State constraints (PMP).
P 1 (END): Controlled trajectories: Compactness and other
regularity properties
P 2: State constraints and shooting.
P 2: HJB approach. Value function, dynamic programming
P 2: Singular arcs.
P 2: HJB approach: verification
EXAM
Where ? ENSTA (room 2413, except on Oct 9, room 1212).
Advanced Continuous Optimization II
(J.-C. Gilbert, INRIA)
Jean-Charles Gilbert <jean-charles.gilbert@inria.fr>
This course will consist in two parts
Part 1 :
18h of computer sessions, supervised by J.-C. Gilbert, around the topics introduced from the
theoretical point of view in A.C.O. II
Part 2 :
An invited course given by an international expert, also open to PhD students, researchers and
R&D engineers.
This year we will host a course by Yuri Nesterov (U. Cath. Louvain)
Yurii Nesterov <yurii.nesterov@uclouvain.be>
The precise topic of the course will be announced.
Schedule :
Part 1 Monday, 14h-18h, from Nov 23 to Jan 4.
Part 2 will be concentrated on the days Jan 20-22
Where ? ENSTA for Part 1 (room 1210), part 2 to be announced
To validate the course one must attend both parts. However, also students not attending Part 1 are
encouraged to come to Part 2, for their own culture.
Calculus of Variations
(F. Santambrogio, U. Paris-Sud)
Filippo Santambrogio <filippo.santambrogio@math.u-psud.fr>
We will study existence (sometimes uniqueness) and characterization of the minimizers in
optimization problem in suitable classes of functions, with special attention to those problems
arising from physics, economics, or to the most natural geometric problems. We will also study the
notion of convergence of a sequence of variational problems, called Γ-convergence, with its most
interesting examples.
Content:
• Calculus of variations in dimension 1
• Semi-continuity of integral functionals
• Connection with elliptic PDEs and harmonic functions: some regularity results using minimality
• ∆_p, ∆_∞ and absolute minimizers
• Vector minimization under divergence constraints and applications to traffic (optimal flows) and
mechanics (compliance)
• Convex duality in calculus of variations and applications
• Some notions on the BV, space, isoperimetric problem, existence and applications.
• General theory of Γ-convergence.
• Phase transition, Modica-Mortola approximation
• Some words on other variational problems (Newton problem for the body of minimal resistance,
Mumford-Shah model for image segmentation…)
Bibliography
— G. Buttazzo, M. Giaquinta, S. Hildebrandt One-dimensional variational problems
— E. Giusti, Direct Methods in the Calculus of Variations
— A. Braides, Γ-convergence for beginners.
— other books or lecture notes, that will be sugested along the course
Schedule: Thursday, 14h-17h + Friday 9h-12h, for five weeks from Nov 26 to Jan 15 (except
vacations and absence).
Where ? Orsay
This course is in common with the AMS master program on PDEs
A list of exercises will be proposed, as well as an example of possible examination test.
A tutor will be appointed to this course (Paul Peon <paul.pegon@math.u-psud.fr>) and will propose
exercise sessions along the duration of the course.
Stochastic Optimization
(F. Bonnans, INRIA and Polytechnique, and P. Carpentier, ENSTA)
Frédéric Bonnans <Frederic.Bonnans@inria.fr >
Pierre Carpentier <pierre.Carpentier@ensta-paristech.fr>
The course will present both theoretical and numerical aspects of decision problems with
uncertainty, where in the presence of partial information (often corresponding to an unknown
future), one sets a probabilistic framework in order, for instance, to minimize the expectation of a
cost.
Such problems are of course well-motivated by decision problems in the industry. They also have a
deep mathematical content, especially in the dynamic case when only the past information is
available. One has to take then conditional (to the past) decisions. In this setting the decision is a
function in a high dimensional space and therefore the numerical aspects also are challenging. The
course is organized as follows.
(Each lesson = 1 day = 5hrs)
Lesson 1 :
Examples (maximum likelihood, optimal storage).
Convex duality, subdifferential calculus
Lesson 2:
Stochastic gradient: Robbins-Monroe strategy, convergence speed
Subdifferential of an expectation. Normal integrands. Information
constraints
Lesson 3:
Stochastic gradient with constraints, duality.
Dynamic optimization in the convex, stochastic setting
Lesson 4:
Applications of stochastic gradient: variance minimization,
aerospace application
Dynamic programmin, stochastic dual dynamic programming
Lesson 5:
Discretization of stochastic problems
Optimal management of a GNL portfolio
Lesson 6:
Decomposition approach.
Management of water resources for electricity production
Schedule: Wednesday, 9h-12h + 14h-16h, for six weeks from Nov 25 on.
Where ? ENSTA, room 2330
Derivative-Free Optimization
(Anne Auger, INRIA Saclay, Laurent Dumas, Université de Versailles)
Anne Auger <anne.auger@inria.fr>,
Laurent Dumas <laurent.dumas@uvsq.fr>
Optimization problems are encountered in many fields of engineering for which the associated cost
function may be of various type : black box or explicit, with continuous or discrete variables, costly
or not to compute, etc... In many cases, the gradient of the cost function is not easy or even
impossible to compute or it can exhibit many local minima leading to consider Derivative Free
Optimization (DFO) methods.
This course deals with a large number of Derivative Free Optimization methods that have been
recently developed, either local or global, deterministic or stochastic. It will be illustrated by various
examples issued from industrial or medical fields.
Outline :
- Direct methods (Nelder Mead, MDS)
- Trust region methods (NEWUOA)
- Global methods (DIRECT)
- Response surface methods (RBF, Krigeage)
- Evolutionary algorithms (simulated annealing, GA, ES, PSO)
References :
A. Conn, K. Scheinberg and L. Vincente, Introduction to Derivative Free Optimization, SIAM, 2009.
D.E. Goldberg. Genetic Algorithms in Search, Optimization and Machine Learning, AddisonWesley Longman Publishing Co., Inc. Boston, USA, 1989.
Schedule: Friday 14h-17h30, for eight weeks from Nov 25 on.
Where ? ENSTA
Dynamic Games
(Rida Laraki, CNRS and Ecole Polytechnique, Vianney Perchet, Paris Diderot)
Rida Laraki <rida.laraki@dauphine.fr>
Vianney Perchet <vianney.perchet@normalesup.org>
Outline (12 × 2h30, December 2015 – January 2016):
1. Repeated games with incomplete information on one side [Aumann- Maschler] & Approachability
2. Introduction to Differential Games [Cardaliaguet]
3. Calibration and Learning in Games
4. Asymptotic & Uniform value in stochastic games [Bewley-Kohlberg, Mertens-Neymann].
5. Asymptotic values in repeated with incomplete information on both sides [Mertens-Zamir].
6. Extensions, counter examples and link with differential games.
References
[1] Aumann R.J. and M. Maschler (1995) Repeated Games with Incomplete Information, M.I.T.
Press (with the collaboration of R. Stearns).
[2] Bewley T. and E. Kohlberg (1976a) The asymptotic theory of stochastic games, Mathematics of
Operations Research, 1, 197-208.
[3] Cardaliaguet, P. (2010) Introduction to differential games, lecture notes.
[4] Laraki R. (2014) Advances in zero-sum dynamic games, Handbook of Game Theory IV, 27-93.
[5] Mertens J.-F. and A. Neyman (1981) Stochastic games, International Journal of Game Theory,
10, 53-66. 1
[6] Mertens J.-F., S. Sorin and S. Zamir (2015) Repeated Games, Cambridge University Press.
[7] Mertens J.-F. and S. Zamir (1971) The value of two-person zero-sum repeated games with lack
of information on both sides, International Journal of Game Theory, 1, 39-64. 1-29.
[8] Perchet, V. (2014) Approachability, regret and calibration: Implications and equivalences,
Journal of Dynamics and Games, 1, 181 – 254.
[9] Sorin S. (2002) A First Course on Zero-Sum Repeated Games, Springer
Where: Polytechnique
Schedule:
Dec 1, 09h30 – 12h00 + 13h30–16h00 (Salle n° 65)
Dec 3, 09h30 – 12h00 (PC n° 5)
Dec 8, 09h30 – 12h00 + 13h30-16h00 (PC 4)
Dec 10, 09h30 – 12h00 (PC 5)
Jan 12, 09h30 – 12h00 + 13h30-16h00 (PC 4)
Jan 19, 09h30 – 12h00 + 13h30-16h00 (PC 4)
Jan 26, 09h30 – 12h00 + 13h30-16h00 (PC 8
2n term courses : SPECIALIZED COURSES
Optimal Transport Methods in Analysis and Geometry
(Y. Brenier, CNRS and Polytechnique)
Yann Brenier <yann.brenier@math.polytechnique.fr>
The theory of optimal transport has undergone great developments in the past
20 years, both because its elementary – yet very general – features and because
of the connection it established between very different branches of mathematics.
We will cover some geometrical applications (convex surfaces with prescribed
Gaussian curvature, isoperimetric inequalities, geodesics in the group
of diffeomorphisms preserving the volume) and some analytical ones (Monge-Ampère,
Hamilton-Jacobi and Euler equations).
Bibliography
1 H. Brezis, Analyse fonctionnelle, Masson 1983,
Functional analysis, Springer 2011.
2 C. Villani, Topics in optimal transportation,
American Mathematical Society 2003.
3 C. Villani, Optimal transport. Old and new, Springer 2009.
4 L. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in
the space of probability measures, ETH, Birkhäuser 2008.
5 L. Ambrosio, A. Figalli, Geodesics in the space of measure-preserving maps
and plans, Arch. Ration. Mech. Anal. 194 (2009) 421-62.
6 F. Santambrogio, Optimal Transport for Applied Mathematicians, Birkhäuser, meant to appear in
Sept 2015
This course is in common with the AMS and the AAGA2 master programs.
Schedule: Monday, 10h30-12h30 + 13h30-15h30. Precise dates : Feb 8 and 29 (morning only),
Mar 7 (morning only), 14 and 21, April 4 and 11.
Where ? Orsay, room 113-115 (except March 7, room 117-119).
A tutor (Thomas Gallouët, a post-doc working in this field) will be appointed to this course and will
propose exercise sessions along the duration of the course.
Geometric Control
(U. Boscain, CNRS and Polytechnique, Y. Chitour, U. Paris-Sud, and F. Jean, ENSTA)
Ugo Boscain <ugo.boscain@polytechnique.edu>,
Chitour Yacine <chitour@lss.supelec.fr>,
Frédéric Jean <Frederic.Jean@ensta-paristech.fr>,
The aim of the course is to provide an introduction to control theory with a geometric viewpoint, i.e.
using tools from differential geometry. The couse is divided into two parts. In the first, we will study
the geometry of families of dynamical system, exhibing in particular the structure of the set of
trajectories. In the second, we wil deal with optimal control theory, stressing in particular the
Pontryagin Maximum Principle and its underlying geometry, in connection with properties of hypoelliptic operators.
Content:
• Controllability
• definition of control systems as families of dynamical systems
• structural properties
• Accessibility, Controllability and Stabilization
• Lie bracket, the Lie algebraic condition and the Brockett obstruction
• Krener theorem, Chow theorem, the orbit theorem
• Poisson stability and recurrence
• Controllability for systems under the strong Hörmander condition and unbounded controls
• the Frobenius Theorem
• non-holonomic mechanics
• examples : Linear systems, control of satellites, rolling bodies, quantum systems
• Optimal Control
• The classical approach of the calculus of variations (Euler-Lagrange Equations)
• Legendre transformation and the Hamiltonian formalism
• existence of solutions : Filippov Theorem
• the general Statement of the Pontryagin Maximum Principle
• Singular trajectories and abnormal extremals
• Proof of the PMP in the case of systems affine in the control with quadratic cost
• Minimum time with bounded controls
• Sub-Riemannian geometry and the hypoelliptic Laplacian
Bibliography:
— A. A. Agrachev and Y. L. Sachkov. Control Theory from the Geometric
Viewpoint. Springer-Verlag, 2004.
— V. Jurdjevic. Geometric Control Theory. Cambridge University Press, 1997.
— F. Jean. Control of Nonholonomic Systems : from Sub-Riemannian Geometry to Motion
Planning. Springer International Publishing, Springer-Briefs in Mathematics, 2014.
Schedule: Friday, 9h-12h, starting Feb 4
Where ? Orsay, room 121-123
A tutor (Dario Prandi <dario.prn@gmail.com>) will be appointed to this course and will propose
exercise sessions along the duration of the course.
This course is in common with the AMS master program.
Optimal control of partial differential equations
(A. Kroener, INRIA)
Axel Kroener <axel.kroener@inria.fr>
1. Motivation, Examples
1.1 Introduction
1.2 Cooling in metallurgy
1.3 Quantum control
2. Control of elliptic problems
2.1 Optimal control of linear elliptic equations
2.1.1 Existence
2.1.2 Neumann boundary control
2.1.3 Dirichlet boundary control
2.2 Optimal control of semi-linear equations
2.2.1 Existence
2.3 Differentiability and convexity
2.4 Optimality conditions for unconstrained problems (two-norm discrepancy)
2.5 Optimality conditions for control constrained problems
3. Control of the heat equation
3.1 Existence
3.2 Necessary optimality conditions
4. Control of the wave equation
4.1 Existence
4.2 Necessary optimality conditions
4.3 Neumann boundary control problems
4.4 Dirichlet boundary control problems
5. Numerical methods
5.1 Gradient methods for optimization problems with PDEs
5.2 (Semi-smooth) Newton methods for (control constrained) optimization problems with PDEs
5.3 Finite element discretization and a priori error estimates for an elliptic control problem
This course is in common with the AMS master program
Schedule Thursday, 3.45-6.15pm
Where Polytechnique, room PC 17
Sequential learning, sequential optimization
(G. Stoltz, HEC)
Gilles Stoltz <stoltz@hec.fr>
We will study two sequential settings, a stochastic one and a non-stochastic one, where in both
cases good actions / forecasts are to be picked.
1. Multi-armed bandit problems: We consider finitely many arms, each associated with an unknown
distribution, or even a continuum of arms, associated with a function f : [0,1] -> R and some noise
structure. When pulling an arm we get a reward drawn at random according to the associated
distribution. The aim is to maximize the obtained rewards but to do so a tradeoff between exploring
the arms (to have an idea of the best arms) and exploiting the obtained information (to pull those
best arms) must be performed. On the optimization side, this problem has strong ties with the
identification of the maximum of f.
2. Aggregation of expert advice: At each round, some base forecasters (experts) provide forecasts
for a sequential and quantitative phenomenon (height of an ozone peak, exchange rate, electricity
consumption). Instead of selecting one such forecast we will combine and aggregate their values,
based on past performance. The aim is to obtain meta-forecasts that are almost as good as the
best global convex or linear combination. The latter can only be determined in hindsight while we
are subject to a constraint of sequential prediction. This is where the optimization bottleneck lies:
performing online almost as well as we would have done in hindsight.
Schedule Wednesday, 9.30am-12noon, from Feb 10 to Apr 6.
Where Orsay, room 121-123
Advanced Game Theory
(T. Tomala, HEC)
Tristan Tomala <tomala@hec.fr>
This course will study various model of dynamic games, the qualitative nature of equilibria and how
communication is used strategically in dynamic environments.
The focus of this course will be on dynamic games with complete and incomplete information. As a
benchmark, we will first review the model of repeated games and its variety of equilibrium behavior,
including cooperative or collusive ones. Then we will see how equilibria are affected by
informational frictions such as: imperfect observation, incomplete knowledge of the preferences of
the opponents, the ability to communicate, the presence of reliable third parties. We shall pay
special attention to models of social networks where players interact with their direct neighbors,
and study the information flow in the network. We will also show how the techniques developed for
dynamic games are useful for mechanism design problems and games of pure information
transmission.
References
-Jackson (2010), Social and Economic Networks, Princeton.
-Mailath and Samuelson (2006), Repeated Games and Reputations, Oxford.
-Maschler, Solan and Zamir (2013) Game Theory, Cambridge.
Schedule: Friday, 13h30-17h45 (dates: 29/01, 5/02, 12/02, 19/02, 11/03, 18/03).
Where: ENSAE (room to be fixed)
Tropical algebra applied to optimization and games
(Stéphane Gaubert, INRIA and Polytechnique)
Stephane Gaubert <Stephane.Gaubert@inria.fr>
Tropical algebra deals with structures of “characteristic one”, in which the addition is idempotent.
The max-plus semifield, which consists of the real numbers equipped with the maximum, thought
of as an additive operation, and the addition, thought of as a multiplication operation, is the
prototypical example of tropical structure. The tropical approach provides tools, of a combinatorial
nature, to deal with deterministic and stochastic optimal control, and more generally, with zero-sum
two player problems. It applies to the ergodic or long horizon problems, in which one is interested
in a mean payoff per time unit, or in the growth of the income as the horizon tends to infinity.
The objective of this lecture is to present tools and recent results inspired by tropical geometry,
concerning control and games problems, paying a special attention to combinatorial and
algorithmic results. Some results will be illustrated by examples from applications, including growth
optimisation in population dynamics or web ranking.
This lecture makes also use of the operator approach of repeated games, in which the dynamic
programming operators are studied from the metric geometry point of view. It develops some tools
from non-linear Perron-Frobenius theory, dealing with order preserving operators on cones.
This lecture has relatively few prerequisites, except elements of optimisation (convexity, polyhedra)
and analysis (fixed point theory). The needed results are systematically recalled. This lecture is
primarily of a mathematical nature, being research oriented, however, it comprises a significant
part concerning algorithms and modelling, which may be of interest to students interested by the
applications of operations research and by games in computer science.
1 Introduction
The max-plus and other tropical semifields. Examples of problems to which tropical methods can
be applied: dynamic programming, optimal control, optimal assignment, mean payoff games (a
challenging open problem in computational complexity); log-convexity of Laplace transform;
tropicalisation of polynomials, generalizations of Cauchy bound on polynomial roots.
2 Algebraic methods in dynamic programming
The deterministic dynamic programming problem. Representation of the tropical eigenspace.
Asymptotics of the value function and “turnpike” theorem (properties of optimal strategies when the
horizon tends to infinity). How to price via tropical eigenvectors a long term contract, to avoid the
“après nous le déluge” behavior.
3 Non-linear Perron-Frobenius theory
Order preserving maps on cones. Hilbert’s and Thompson’s metric. Nonexpansiveness properties
in these metrics. The ergodic non-linear eigenproblem, and its applications: Markov decision
processes, combinatorial games, matrix scaling problems (entropy maximization), population
dynamics.
4 Generalized Perron-Frobenius theorem
Nonlinear généralisations of the Perron-Frobenius theorem. Existence and uniqueness of nonlinear
eigenvectors. Collatz-Wielandt certificates. Analysis of non-linear power algorithsms / value
iteration.
5 Policy iteration for games
The theorem of Ye: policy iteration with a fixed discount rate is strongly polynomial. Policy iteration
for mean payoff problems.
Schedule: Wednesday, 14h-17h15 (dates: 3/2, 10/2, 17/2, 2/3, 9/3, 16/3, 30/3).
Where: Polytechnique, PC 4 except on 3/2 PC1.
This course is in common with the OJD program of Paris 6 and Polytechnique.