Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems

We present optimality conditions for bilevel optimal control problems where the upper level, to be solved by a leader, is a scalar optimal control problem and the lower level, to be solved by several followers, is a multiobjective convex optimal control problem. Multiobjective optimal control problems arise in many application areas where several conflicting objectives need to be considered. Minimize several objective functionals leads to solutions such that none of the objective functional values can be improved further without deteriorating another. The set of all such solutions is referred to as efficient (also called Pareto optimal, noninferior, or nondominated) set of solutions. The lower level of the semivectorial bilevel optimal control problems can be considered to be associated to a ”grande coalition” of a p-player cooperative differential game, every player having its own objective and control function. We consider situations in which these p-?players react as ”followers” to every decision imposed by a ”leader” (who acts at the so-called upper level). The best reply correspondence of the followers being in general non uniquely determined, the leader cannot predict the followers choice simply on the basis of his rational behavior. So, the choice of the best strategy from the leader point of view depends of how the followers choose a strategy among his best responses. In this paper, we will consider two (extreme) possibilities: (i) the optimistic situation, when for every decison of the leader, the followers will choose a strategy amongst the efficient controls which minimizes the (scalar) objective of the leader; in this case the leader will choose a strategy which minimizes the best he can obtain amongst all the best responses of the followers: (ii) the pessimistic situation, when the followers can choose amongst the efficient controls one which maximizes the (scalar) objective of the leader; in this case the leader will choose a strategy which minimizes the worst he could obtain amongst all the best responses of the followers. This paper continues the research initiated in [17] where existence results for these problems have been obtained.

Solving ill-posed bilevel programs

This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem an a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem

Notes on the value function approach to multiobjective bilevel optimization

This paper is concerned with the value function approach to multiobjective bilevel optimization which exploits a lower level frontier-type mapping in order to replace the hierarchical model of two interdependent multiobjective optimization problems by a single-level multiobjective optimization problem. As a starting point, different value-function-type reformulations are suggested and their relations are discussed. Here, we focus on the situations where the lower level problem is solved up to efficiency or weak efficiency, and an intermediate solution concept is suggested as well. We study the graph-closedness of the associated efficiency-type and frontier-type mappings. These findings are then used for two purposes. First, we investigate existence results in multiobjective bilevel optimization. Second, for the derivation of necessary optimality conditions via the value function approach, it is inherent to differentiate frontier-type mappings in a generalized way. Here, we are concerned with the computation of upper coderivative estimates for the frontier-type mapping associated with the setting where the lower level problem is solved up to weak efficiency. We proceed in two ways, relying, on the one hand, on a weak domination property and, on the other hand, on a scalarization approach. Throughout the paper, illustrative examples visualize our findings, the necessity of crucial assumptions, and some flaws in the related literature.Comment: 30 page

KKT reformulation and necessary conditions for optimality in nonsmooth bilevel optimization

For a long time, the bilevel programming problem has essentially been considered as a special case of mathematical programs with equilibrium constraints (MPECs), in particular when the so-called KKT reformulation is in question. Recently though, this widespread believe was shown to be false in general. In this paper, other aspects of the difference between both problems are revealed as we consider the KKT approach for the nonsmooth bilevel program. It turns out that the new inclusion (constraint) which appears as a consequence of the partial subdifferential of the lower-level Lagrangian (PSLLL) places the KKT reformulation of the nonsmooth bilevel program in a new class of mathematical program with both set-valued and complementarity constraints. While highlighting some new features of this problem, we attempt here to establish close links with the standard optimistic bilevel program. Moreover, we discuss possible natural extensions for C-, M-, and S-stationarity concepts. Most of the results rely on a coderivative estimate for the PSLLL that we also provide in this paper

STOCHASTIC OPTIMIZATION OVER A PARETO SET ASSOCIATED WITH A STOCHASTIC MULTI-OBJECTIVE OPTIMIZATION PROBLEM

ABSTRACT. We deal with the problem of minimizing the expectation of a real valued random function over the weakly Pareto or Pareto set associated with a Stochastic MultiObjective Optimization Problem (SMOP) whose objectives are expectations of random functions. Assuming that the closed form of these expectations is difficult to obtain, we apply the Sample Average Approximation method (SAA-N, where N is the sample size) in order to approach this problem. We prove that the Hausdorff-Pompeiu distance between the SAA-N weakly Pareto sets and the true weakly Pareto set converges to zero almost surely as N goes to infinity, assuming that all the objectives of our (SMOP) are strictly convex. Then we show that every cluster point of any sequence of SAA-N optimal solutions (N=1,2,. . . ) is almost surely a true optimal solution. To handle also the nonconvex case, we assume that the real objective to be minimized over the Pareto set depends on the expectations of the objectives of the (SMOP), i.e. we optimize over the outcome space of the (SMOP). Then, whithout any convexity hypothesis, we obtain the same type of results for the Pareto sets in the outcome spaces. Thus we show that the sequence of SAA-N optimal values (N=1,2 ...) converges almost surely to the true optimal value. Keywords: Optimization over a Pareto Set, Optimization over the Pareto Outcome Set, Multiobjective Stochastic Optimization, Multiobjective Convex Optimization, Sample Average Approximation Method AMS: 90C29, 90C25, 90C15, 90C26

On implicit variables in optimization theory

Implicit variables of a mathematical program are variables which do not need to be optimized but are used to model feasibility conditions. They frequently appear in several different problem classes of optimization theory comprising bilevel programming, evaluated multiobjective optimization, or nonlinear optimization problems with slack variables. In order to deal with implicit variables, they are often interpreted as explicit ones. Here, we first point out that this is a light-headed approach which induces artificial locally optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type necessary optimality conditions which correspond to treating the implicit variables as explicit ones on the one hand, or using them only implicitly to model the constraints on the other. A detailed comparison of the obtained stationarity conditions as well as the associated underlying constraint qualifications will be provided. Overall, we proceed in a fairly general setting relying on modern tools of variational analysis. Finally, we apply our findings to different well-known problem classes of mathematical optimization in order to visualize the obtained theory.Comment: 33 page

Analyse post-Pareto en optimisation vectorielle stochastique et déterministe : étude théorique et algorithmes.

This thesis explore related aspects to post-Pareto analysis arising from Stochastic Vector Optimization Problem. A Stochastic Vector Optimization Problem is to optimize a random vector objective function defined on an arbitrary set, and taking values in a partially ordered set. Its solution set (called Pareto set) consists of the feasible solutions which ensure some sort of equilibrium amongst the objectives. That is to say, Pareto solutions are such that noneof the objectives values can be improved further without deterioring another. Technically speaking, each Pareto solution is acceptable. The natural question that arises is : how to choose one solution ? One possible answer is to optimize an other objective over the Pareto set. Considering the existence of a decision-maker with its own criteria, we deal with the post-Pareto Stochastic Optimization Problem of minimizing its real-valued criteria over the Pareto set.Cette thèse relate certains aspects liés à l'analyse post-Pareto issue de Problèmes d'Optimisation Vectorielle Stochastique. Un problème d'optimisation Vectorielle Stochastique consiste à optimiser l'espérance d'une fonction vectorielle aléatoire définie sur un ensemble arbitraire et à valeurs dans un espace sectoriel ordonné. L'ensemble des solutions de ce problème (appelé ensemble de Pareto) est composé des solutions admissibles qui assurent un certain équilibre entre les objectifs : il est impossible d'améliorer la valeur d'un objectif sans détériorer celle d'un autre. D'un point de vue technique, chaque solution de Pareto est acceptable. Nous nous posons alors le problème de la sélection de l'une d'entre elles : en supposant l'existence d'un décideur qui aurait son propre critère de décision, nous considérons le problème post-Pareto Stochastique qui vise à minimiser cette fonctionnelle sur l'ensemble de Pareto associé à un Problème d'Optimisation Vectorielle Stochastique

On Optimistic and Pessimistic Bilevel Optimization Models for Demand Response Management

This paper investigates bilevel optimization models for demand response management, and highlights the often overlooked consequences of a common modeling assumption in the field. That is, the overwhelming majority of existing research deals with the so-called optimistic variant of the problem where, in case of multiple optimal consumption schedules for a consumer (follower), the consumer chooses an optimal schedule that is the most favorable for the electricity retailer (leader). However, this assumption is usually illegitimate in practice; as a result, consumers may easily deviate from their expected behavior during realization, and the retailer suffers significant losses. One way out is to solve the pessimistic variant instead, where the retailer prepares for the least favorable optimal responses from the consumers. The main contribution of the paper is an exact procedure for solving the pessimistic variant of the problem. First, key properties of optimal solutions are formally proven and efficiently solvable special cases are identified. Then, a detailed investigation of the optimistic and pessimistic variants of the problem is presented. It is demonstrated that the set of optimal consumption schedules typically contains various responses that are equal for the follower, but bring radically different profits for the leader. The main procedure for solving the pessimistic variant reduces the problem to solving the optimistic variant with slightly perturbed problem data. A numerical case study shows that the optimistic solution may perform poorly in practice, while the pessimistic solution gives very close to the highest profit that can be achieved theoretically. To the best of the authors’ knowledge, this paper is the first to propose an exact solution approach for the pessimistic variant of the problem