The robust bilevel continuous knapsack problem with uncertain follower's objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack and the follower chooses an optimal packing according to his own profits, which may differ from those of the leader. To this bilevel problem, we add uncertainty in a natural way, assuming that the leader does not have full knowledge about the follower's problem. More precisely, adopting the robust optimization approach and assuming that the follower's profits belong to a given uncertainty set, our aim is to compute a solution that optimizes the worst-case follower's reaction from the leader's perspective. By investigating the complexity of this problem with respect to different types of uncertainty sets, we make first steps towards better understanding the combination of bilevel optimization and robust combinatorial optimization. We show that the problem can be solved in polynomial time for both discrete and interval uncertainty, but that the same problem becomes NP-hard when each coefficient can independently assume only a finite number of values. In particular, this demonstrates that replacing uncertainty sets by their convex hulls may change the problem significantly, in contrast to the situation in classical single-level robust optimization. For general polytopal uncertainty, the problem again turns out to be NP-hard, and the same is true for ellipsoidal uncertainty even in the uncorrelated case. All presented hardness results already apply to the evaluation of the leader's objective function

Fuzzy multilevel programming with a hybrid intelligent algorithm

AbstractIn order to model fuzzy decentralized decision-making problem, fuzzy expected value multilevel programming and chance-constrained multilevel programming are introduced. Furthermore, fuzzy simulation, neural network, and genetic algorithm are integrated to produce a hybrid intelligent algorithm for finding the Stackelberg-Nash equilibrium. Finally, two numerical examples are provided to illustrate the effectiveness of the hybrid intelligent algorithm

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

Modèles biniveaux pour la réponse de la demande dans les réseaux électriques intelligents

This thesis focuses on bilevel optimization, some variants, and an application to optimal price-setting in smart power grids.Bilevel optimization problems are a special subclass of constrained mathematical optimization problems where another problem, the lower level is embedded in the constraints.We consider their application to the optimal pricing of a Time-and-Level-of-Use Demand Response program, allowing an electricity supplier to leverage the flexibility of users through an economic incentive.A generalized form of bilevel optimization is also proposed where the lower level may pick a solution that is not optimal as typically assumed but near-optimal, that is feasible and within a fixed tolerance from an optimal solution.Solving this variant of bilevel optimization requires anticipation of the deviation from optimality and a guarantee that a solution remains feasible even with this deviation.Cette thèse étudie l'optimisation bi-niveau, certaines variantes et une application à la tarification dans les réseaux électriques intelligents.Les problèmes d'optimisation bi-niveaux sont une sous-catégorie de problèmes d'optimisation mathématique contrainte où un deuxième problème ou deuxième niveau estprésent dans les contraintes.Nous étudions leur application à un tariff variable en temps et en niveau de consommation, permettant à un fournisseur d'énergie d'exploiter la flexibilité de consommateurs par des incitations économiques.Une généralisation des problèmes bi-niveaux est également proposée, dans laquelle le deuxième niveau peut sélectionner une solution qui n'est pas optimale contrairement au modèle bi-niveau classique mais quasi-optimale.Résoudre cette variante de problèmes bi-niveaux demande l'anticipation de cette déviation de la solution de deuxième niveau de l'optimalité et garantit qu'une solution au problème bi-niveau sera réalisable malgré cette déviation

Robust optimization methods for chance constrained, simulation-based, and bilevel problems

The objective of robust optimization is to find solutions that are immune to the uncertainty of the parameters in a mathematical optimization problem. It requires that the constraints of a given problem should be satisfied for all realizations of the uncertain parameters in a so-called uncertainty set. The robust version of a mathematical optimization problem is generally referred to as the robust counterpart problem. Robust optimization is popular because of the computational tractability of the robust counterpart for many classes of uncertainty sets, and its applicability in wide range of topics in practice. In this thesis, we propose robust optimization methodologies for different classes of optimization problems. In Chapter 2, we give a practical guide on robust optimization. In Chapter 3, we propose a new way to construct uncertainty sets for robust optimization using the available historical data information. Chapter 4 proposes a robust optimization approach for simulation-based optimization problems. Finally, Chapter 5 proposes approximations of a specific class of robust and stochastic bilevel optimization problems by using modern robust optimization techniques

The Stochastic Bilevel Continuous Knapsack Problem with Uncertain Follower's Objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack, while the follower chooses a feasible packing maximizing his own profit. The leader's aim is to optimize a linear objective function in the capacity and in the follower's solution, but with respect to different item values. We address a stochastic version of this problem where the follower's profits are uncertain from the leader's perspective, and only a probability distribution is known. Assuming that the leader aims at optimizing the expected value of her objective function, we first observe that the stochastic problem is tractable as long as the possible scenarios are given explicitly as part of the input, which also allows to deal with general distributions using a sample average approximation. For the case of independently and uniformly distributed item values, we show that the problem is #P-hard in general, and the same is true even for evaluating the leader's objective function. Nevertheless, we present pseudo-polynomial time algorithms for this case, running in time linear in the total size of the items. Based on this, we derive an additive approximation scheme for the general case of independently distributed item values, which runs in pseudo-polynomial time.Comment: A preliminary version of parts of this article can be found in Section 8 of arXiv:1903.02810v