Multi-leader-follower potential games

In this paper, we discuss a particular class of Nash games, where the participants of the game (the players) are divided into two groups (leaders and followers) according to their position or influence on the other players. Moreover, we consider the case, when the leaders’ and/or the followers’ game can be described as a potential game. This is a subclass of Nash games that has been introduced by Monderer and Shapley in 1996 and has beneficial properties to reformulate the bilevel Nash game. We develope necessary and sufficient conditions for Nash equilibria and present existence and uniqueness results. Furthermore, we discuss some Examples to illustrate our results. In this paper, we discussed analytical properties for multi-leader follower potential games, that form a subclass of hierarchical Nash games. The application of these theoretical results to various fields of applications are a future research topic. Moreover, they are meant to serve as a starting point for the developement of efficient numerical solution methods for multi-leader-follower games.

Robust classification with feature selection using an application of the Douglas-Rachford splitting algorithm

This paper deals with supervised classification and feature selection with application in the context of high dimensional features. A classical approach leads to an optimization problem minimizing the within sum of squares in the clusters (I2 norm) with an I1 penalty in order to promote sparsity. It has been known for decades that I1 norm is more robust than I2 norm to outliers. In this paper, we deal with this issue using a new proximal splitting method for the minimization of a criterion using I2 norm both for the constraint and the loss function. Since the I1 criterion is only convex and not gradient Lipschitz, we advocate the use of a Douglas-Rachford minimization solution. We take advantage of the particular form of the cost and, using a change of variable, we provide a new efficient tailored primal Douglas-Rachford splitting algorithm which is very effective on high dimensional dataset. We also provide an efficient classifier in the projected space based on medoid modeling. Experiments on two biological datasets and a computer vision dataset show that our method significantly improves the results compared to those obtained using a quadratic loss function.

Direct and indirect methods to optimize the muscular force response to a pulse train of electrical stimulation

Recent force-fatigue mathematical models in biomechanics [7] allow to predict the muscular force response to functional electrical stimulation (FES) and leads to the optimal control problem of maximizing the force. The stimulations are Dirac pulses and the control parameters are the pulses amplitudes and times of application, the number of pulses is physically limited and the model leads to a sampled data control problem. The aim of this article is to present and compare two methods. The first method is a direct optimization scheme where a further refined numerical discretization is applied on the dynamics. The second method is an indirect scheme: first-order Pontryagin type necessary conditions are derived and used to compute the optimal sampling times.

Box-constrained optimization for minimax supervised learning

In this paper, we present the optimization procedure for computing the discrete boxconstrained minimax classifier introduced in [1, 2]. Our approach processes discrete or beforehand discretized features. A box-constrained region defines some bounds for each class proportion independently. The box-constrained minimax classifier is obtained from the computation of the least favorable prior which maximizes the minimum empirical risk of error over the box-constrained region. After studying the discrete empirical Bayes risk over the probabilistic simplex, we consider a projected subgradient algorithm which computes the prior maximizing this concave multivariate piecewise affine function over a polyhedral domain. The convergence of our algorithm is established.

A pseudospectral method for budget-constrained infinite horizon optimal control problems

In this paper a generalization of the indirect pseudo-spectral method, presented in [17], for the numerical solution of budget-constrained infinite horizon optimal control problems is presented. Consideration of the problem statement in the framework of weighted functional spaces allows to arrive at a good approximation for the initial value of the adjoint variable, which is inevitable for obtaining good numerical solutions. The presented method is illustrated by applying it to the budget-constrained linear-quadratic regulator model. The quality of approximate solutions is demonstrated by an example.

Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation

We study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem $ \frac{f(a)-f(b)}{b-a}={f}^{\prime}(c)$we provide the expression of its gradient ∇c(d,d), thus giving the asymptotic behavior of c(a, b) when both a and b tend to the same point d. Under appropriate mild conditions on f, this result is “universal” in the sense that it does not depend on the point d or the function f. The key tool to get at this result turns out to be the Legendre-Fenchel transformation for convex functions.

Goal-oriented adaptive sampling under random field modelling of response probability distributions

In the study of natural and artificial complex systems, responses that are not completely determined by the considered decision variables are commonly modelled probabilistically, resulting in response distributions varying across decision space. We consider cases where the spatial variation of these response distributions does not only concern their mean and/or variance but also other features including for instance shape or uni-modality versus multi-modality. Our contributions build upon a non-parametric Bayesian approach to modelling the thereby induced fields of probability distributions, and in particular to a spatial extension of the logistic Gaussian model. The considered models deliver probabilistic predictions of response distributions at candidate points, allowing for instance to perform (approximate) posterior simulations of probability density functions, to jointly predict multiple moments and other functionals of target distributions, as well as to quantify the impact of collecting new samples on the state of knowledge of the distribution field of interest. In particular, we introduce adaptive sampling strategies leveraging the potential of the considered random distribution field models to guide system evaluations in a goal-oriented way, with a view towards parsimoniously addressing calibration and related problems from non-linear (stochastic) inversion and global optimisation.

On the numerical solution of a free end-time homicidal chauffeur game

A functional formulation of the classical homicidal chauffeur Nash game is presented and a numerical framework for its solution is discussed. This methodology combines a Hamiltonian based scheme with proximal penalty to determine the time horizon where the game takes place with a Lagrangian optimal control approach and relaxation to solve the Nash game at a fixed end-time.

Multilevel augmented Lagrangian solvers for overconstrained contact formulations

Multigrid methods for two-body contact problems are mostly based on special mortar discretizations, nonlinear Gauss-Seidel solvers, and solution-adapted coarse grid spaces. Their high computational efficiency comes at the cost of a complex implementation and a nonsymmetric master-slave discretization of the nonpenetration condition. Here we investigate an alternative symmetric and overconstrained segment-to-segment contact formulation that allows for a simple implementation based on standard multigrid and a symmetric treatment of contact boundaries, but leads to nonunique multipliers. For the solution of the arising quadratic programs, we propose augmented Lagrangian multigrid with overlapping block Gauss-Seidel smoothers. Approximation and convergence properties are studied numerically at standard test problems.