On Pareto equilibria for bi-objective diffusive optimal control problems

We investigate Pareto equilibria for bi-objective optimal control problems. Our framework comprises the situation in which an agent acts with a distributed control in a portion of a given domain, and aims to achieve two distinct (possibly conflicting) targets. We analyze systems governed by linear and semilinear heat equations and also systems with multiplicative controls. We develop numerical methods relying on a combination of finite elements and finite differences. We illustrate the computational methods we develop via numerous experiments

Optimal control and partial differential equations

In this work, some type of optimal control problems with equality constraints given by Partial Differential Equations (PDE) and convex inequality constraints are considered, obtaining their corresponding first order necessary optimality conditions by means of Dubovitskii-Milyutin (DM) method. Firstly, we consider problems with one objective functional (or scalar problems) but non-well posed equality constraints, where existence and uniqueness of state in function on control is not true (either one has existence but not uniqueness of state, or one has not existence of state for any control). In both cases, the classical Lions argument (re-writing the problem as an optimal control problem for the control without equality constraints, see for instance Lions, J. L. – Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1970) can not be applied. Afterwards, we consider multiobjective problems (or vectorial problems), considering three different concepts of solution: Pareto, Nash and Stackelberg. In all cases, an adequate abstract DM method is developed followed by an example

Multiobjective Design Optimization using Nash Games

International audienceIn the area of pure numerical simulation of multidisciplinary coupled systems, the computational cost to evaluate a configuration may be very high. A fortiori, in multi- disciplinary optimization, one is led to evaluate a number of different configurations to iterate on the design parameters. This observation motivates the search for the most in- novative and computationally efficient approaches in all the sectors of the computational chain : at the level of the solvers (using a hierarchy of physical models), the meshes and geometrical parameterizations for shape, or shape deformation, the implementation (on a sequential or parallel architecture; grid computing), and the optimizers (deterministic or semi-stochastic, or hybrid; synchronous, or asynchronous). In the present approach, we concentrate on situations typically involving a small number of disciplines assumed to be strongly antagonistic, and a relatively moderate number of related objective functions. However, our objective functions are functionals, that is, PDE-constrained, and thus costly to evaluate. The aerodynamic and structural optimization of an aircraft configuration is a prototype of such a context, when these disciplines have been reduced to a few major objectives. This is the case when, implicitly, many subsystems are taken into account by local optimizations. Our developments are focused on the question of approximating the Pareto set in cases of strongly-conflicting disciplines. For this purpose, a general computational technique is proposed, guided by a form of sensitivity analysis, with the additional objective to be more economical than standard evolutionary approaches

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

NEUMANN-DIRICHLET NASH STRATEGIES FOR THE SOLUTION OF ELLIPTIC CAUCHY PROBLEMS

International audienceWe consider the Cauchy problem for an elliptic operator, formulated as a Nash game. The overspecified Cauchy data are split between two players: the first player solves the elliptic equation with the Dirichlet part of the Cauchy data prescribed over the accessible boundary and a variable Neumann condition (which we call first player's strategy) prescribed over the inaccessible part of the boundary. The second player makes use correspondingly of the Neumann part of the Cauchy data, with a variable Dirichlet condition prescribed over the inaccessible part of the boundary. The first player then minimizes the gap related to the nonused Neumann part of the Cauchy data, and so does the second player with a corresponding Dirichlet gap. The two costs are coupled through a difference term. We prove that there always exists a unique Nash equilibrium, which turns out to be the reconstructed data when the Cauchy problem has a solution. We also prove that the completion Nash game has a stable solution with respect to noisy data. Some numerical two- and three-dimensional experiments are provided to illustrate the efficiency and stability of our algorithm

PDE-Constrained Equilibrium Problems under Uncertainty: Existence, Optimality Conditions and Regularization

In dieser Arbeit werden PDE-beschränkte Gleichgewichtsprobleme unter Unsicherheiten analysiert. Im Detail diskutieren wir eine Klasse von risikoneutralen verallgemeinerten Nash-Gleichgewichtsproblemen sowie eine Klasse von risikoaversen Nash Gleichgewichtsproblemen. Sowohl für die risikoneutralen PDE-beschränkten Optimierungsprobleme mit punktweisen Zustandsschranken als auch für die risikoneutralen verallgemeinerten Nash Gleichgewichtsprobleme wird die Existenz von Lösungen beziehungsweise Nash Gleichgewichten bewiesen und Optimalitätsbedingungen hergeleitet. Die Betrachtung von Ungleichheitsbedingungen an den stochastischen Zustand führt in beiden Fällen zu Komplikationen bei der Herleitung der Lagrange-Multiplikatoren. Nur durch höhere Regularität des stochastischen Zustandes können wir auf die bestehende Optimalitätstheorie für konvexe Optimierungsprobleme zurückgreifen. Die niedrige Regularität des Lagrange-Multiplikators stellt auch für die numerische Lösbarkeit dieser Probleme ein große Herausforderung dar. Wir legen den Grundstein für eine erfolgreiche numerische Behandlung risikoneutraler Nash Gleichgewichtsproblem mittels Moreau-Yosida Regularisierung, indem wir zeigen, dass dieser Regularisierungsansatz konsistent ist. Die Moreau-Yosida Regularisierung liefert eine Folge von parameterabhängigen Nash Gleichgewichtsproblemen und der Grenzübergang im Glättungsparameter zeigt, dass die stationären Punkte des regularisierten Problems gegen ein verallgemeinertes Nash Gleichgewicht des ursprünglich Problems schwach konvergieren. Die Theorie legt also nahe, dass auf der Moreau-Yosida Regularisierung eine numerische Methode aufgebaut werden kann. Darauf aufbauend werden Algorithmen vorgeschlagen, die aufzeigen, wie risikoneutrale PDE-beschränkte Optimierungsprobleme mit punktweisen Zustandsschranken und risikoneutrale PDE-beschränkte verallgemeinerte Nash Gleichgewichtsprobleme gelöst werden können. Für die Modellierung der Risikopräferenz in der Klasse von risikoaversen Nash Gleichgewichtsprobleme verwenden wir kohärente Risikomaße. Da kohärente Risikomaße im Allgemeinen nicht glatt sind, ist das resultierende PDE-beschränkte Nash Gleichgewichtsproblem ebenfalls nicht glatt. Daher glätten wir die kohärenten Risikomaße mit Hilfe einer Epi-Regularisierungstechnik. Sowohl für das ursprüngliche Nash Gleichgewichtsproblem als auch für die geglätteten parameterabhängigen Nash Gleichgewichtsprobleme wird die Existenz von Nash Gleichgewichten gezeigt, sowie Optimalitätsbedingungen hergeleitet. Wir liefern wertvolle Resultate dafür, dass dieser Glättungsansatz sich für die Entwicklung eines numerischen Verfahren eignet, indem wir beweisen können, dass sowohl eine Folge von stationären Punkten als auch eine Folge von Nash Gleichgewichten des epi-regularisierten Problems eine schwach konvergente Teilfolge hat, deren Grenzwert ein Nash Gleichgewicht des ursprünglichen Problems ist.In this paper, we analyze PDE-constrained equilibrium problems under uncertainty. In detail, we discuss a class of risk-neutral generalized Nash equilibrium problems and a class of risk-averse Nash equilibrium problems. For both, the risk-neutral PDE-constrained optimization problems with pointwise state constraints and the risk-neutral generalized Nash equilibrium problems, the existence of solutions and Nash equilibria, respectively, is proved and optimality conditions are derived. The consideration of inequality conditions on the stochastic state leads in both cases to complications in the derivation of the Lagrange multipliers. Only by higher regularity of the stochastic state we can resort to the existing optimality theory for convex optimization problems. The low regularity of the Lagrange multiplier also poses a major challenge for the numerical solvability of these problems. We lay the foundation for a successful numerical treatment of risk-neutral Nash equilibrium problems using Moreau-Yosida regularization by showing that this regularization approach is consistent. The Moreau-Yosida regularization yields a sequence of parameter-dependent Nash equilibrium problems and the boundary transition in the smoothing parameter shows that the stationary points of the regularized problem converge weakly against a generalized Nash equilibrium of the original problem. Thus, the theory suggests that a numerical method can be built on the Moreau-Yosida regularization. Based on this, algorithms are proposed to show how to solve risk-neutral PDE-constrained optimization problems with pointwise state bounds and risk-neutral PDE-constrained generalized Nash equilibrium problems. I n order to model risk preference in the class of risk-averse Nash equilibrium problems, we use coherent risk measures. Since coherent risk measures are generally not smooth, the resulting PDE-constrained Nash equilibrium problem is also not smooth. Therefore, we smooth the coherent risk measures using an epi-regularization technique. For both the original Nash equilibrium problem and the smoothed parameter-dependent Nash equilibrium problems, we show the existence of Nash equilibria, and derive optimality conditions. We provide valuable results for making this smoothing approach suitable for the development of a numerical method by proving that both, a sequence of stationary points and a sequence of Nash equilibria of the epi-regularized problem, have a weakly convergent subsequence whose limit is a Nash equilibrium of the original problem

A Nash-game approach to joint image restoration and segmentation

International audienceWe propose a game theory approach to simultaneously restore and segment noisy images. We define two players: one is restoration, with the image intensity as strategy, and the other is segmentation with contours as strategy. Cost functions are the classical relevant ones for restoration and segmentation, respectively. The two players play a static game with complete information, and we consider as solution to the game the so-called Nash Equilibrium. For the computation of this equilibrium we present an iterative method with relaxation. The results of numerical experiments performed on some real images show the relevance and efficiency of the proposed algorithm