Abstract We study the overflowing bin packing problem (OBPP), a recently proposed one-dimensional packing problem with notable conceptual ties to the field of (just-in-time) scheduling. In this scenario, we are required to pack items of known sizes into bins of known capacities so that, in broad terms, each bin’s total load comes as close as possible to its capacity. To establish its status as an independent research branch, the OBPP is first distinguished from related optimization problems. We then review an assignment model from the literature and analyze its theoretical properties, providing a closed-form expression for the associated LP bound and characterizing its worst-case performance ratio. In the second part, we introduce a new flow-based approach for the OBPP and examine both its theoretical and numerical advantages, the latter based on extensive computational experiments with diverse benchmark sets. Overall, the new flow formulation consistently produces high-quality—mostly optimal—solutions in short computation times, significantly outperforming previous state-of-the-art methods.

Abstract In this paper, we investigate solution approaches in set optimization that are based on so-called vectorization strategies. Thereby, the original set-valued problems are reformulated as multi-objective optimization problems, whose optimal solution sets approximate those of the original ones in a certain sense. We consider both infinite-dimensional and finite-dimensional vectorization approaches. In doing so, we collect and discuss the most important properties and relationships between the approaches.

In this paper, we provide a systematization of the equilibrium concepts for vector-valued games. In general abstract setting, we give the definitions of Nash equilibrium, weak Nash equilibrium, and proper Nash equilibrium, we investigate the relationships among them, and we provide existence results for proper Nash equilibria and weak Nash equilibria. The link between the introduced equilibrium concepts and Nash equilibria of a game with scalar payoffs obtained by means of linear scalarization is investigated. For a vector matrix game, we show that the notions of proper Nash equilibrium and Nash equilibrium are equivalent, we prove existence of both Nash equilibria and weak Nash equilibria, we give characterizations of proper and weak best correspondence sets respectively, by defining characterizing functions from the matrices and the preference cones, and finally we list all best correspondence sets. The given definitions are illustrated by means of examples.

Suppose N independent Bernoulli trials with success probabilities p1,p2,… are observed sequentially at times of a mixed binomial process. The task is to maximise, by using a nonanticipating stopping strategy, the probability of stopping at the last success. The case pk=1/k has been studied by many authors as a version of the familiar best choice problem, where both N and the observation times are random. We consider a more general profile pk=θ/(θ+k-1) and assume that the prior distribution of N is negative binomial with shape parameter ν, so the arrivals occur at times of a mixed Poisson process. The setting with two parameters offers a high flexibility in understanding the nature of the optimal strategy, which we show is intrinsically related to monotonicity properties of the Gaussian hypergeometric function. Using this connection, we find that the myopic stopping strategy is optimal if and only if ν≥θ. Furthermore, we derive formulas to assess the winning probability and discuss limit forms of the problem for large N.

Abstract This paper deals with the reconstruction of the desired demand in an optimal control problem, stated over a tree-shaped transportation network which is governed by a linear hyperbolic conservation law. As desired demands typically undergo fluctuations due to seasonality or unexpected events making short-term adjustments necessary, such an approach can exemplary be used for forecasting from past data. We suggest to model this problem as a so-called inverse optimal control problem, i.e., a hierarchical optimization problem whose inner problem is the optimal control problem and whose outer problem is the reconstruction problem. In order to guarantee the existence of solutions in the function space framework, the hyperbolic conservation law is interpreted in weak sense allowing for control functions in Lebesgue spaces. For the computational treatment of the model, we transfer the hierarchical problem into a nonsmooth single-level one by plugging the uniquely determined solution of the inner optimal control problem into the outer reconstruction problem before applying techniques from nonsmooth optimization. Some numerical experiments are presented to visualize various features of the model including different types of noise in the demand and strategies of how to observe the network in order to obtain good reconstructions of the desired demand.

Abstract We present a simple iterative method for solving quasimonotone as well as classical variational inequalities without monotonicity. Strong convergence analysis is given under mild conditions and thus generalize the few existing results that only present weak convergence methods under restrictive assumptions. We give finite and infinite dimensional numerical examples to compare and illustrate the simplicity and computational advantages of the proposed scheme.

A non-negative transferable utility (TU) game is average monotonic if there exists a non-negative vector according to which the relative worth is not decreasing when enlarging the coalition. We generalize this definition to the nontransferable utility (NTU) case. It is shown that an average monotonic NTU game shares several properties with an average monotonic TU game. In particular it has a special core element and there exists a population monotonic allocation scheme. We show that an NTU bankruptcy game is average monotonic with respect to the claims vector.