Facets of the axial three-index assignment polytope

We revisit the facial structure of the axial 3-index assignment polytope. After reviewing known classes of facet-defining inequalities, we present a new class of valid inequalities, and show that they define facets of this polytope. This answers a question posed by Qi and Sun (2000). Moreover, we show that we can separate these inequalities in polynomial time. Finally, we assess the computational relevance of the new inequalities by performing (limited) computational experiments

An Experimental Comparison of Uncertainty Sets for Robust Shortest Path Problems

Through the development of efficient algorithms, data structures and preprocessing techniques, real-world shortest path problems in street networks are now very fast to solve. But in reality, the exact travel times along each arc in the network may not be known. This led to the development of robust shortest path problems, where all possible arc travel times are contained in a so-called uncertainty set of possible outcomes. Research in robust shortest path problems typically assumes this set to be given, and provides complexity results as well as algorithms depending on its shape. However, what can actually be observed in real-world problems are only discrete raw data points. The shape of the uncertainty is already a modelling assumption. In this paper we test several of the most widely used assumptions on the uncertainty set using real-world traffic measurements provided by the City of Chicago. We calculate the resulting different robust solutions, and evaluate which uncertainty approach is actually reasonable for our data. This anchors theoretical research in a real-world application and gives an indicator which robust models should be the future focus of algorithmic development

Fast separation for the three-index assignment problem

A critical step in a cutting plane algorithm is separation, i.e., establishing whether a given vector x violates an inequality belonging to a specific class. It is customary to express the time complexity of a separation algorithm in the number of variables n. Here, we argue that a separation algorithm may instead process the vector containing the positive components of x, denoted as supp(x), which offers a more compact representation, especially if x is sparse; we also propose to express the time complexity in terms of |supp(x)|. Although several well-known separation algorithms exploit the sparsity of x, we revisit this idea in order to take sparsity explicitly into account in the time-complexity of separation and also design faster algorithms. We apply this approach to two classes of facet-defining inequalities for the three-index assignment problem, and obtain separation algorithms whose time complexity is linear in |supp(x)| instead of n. We indicate that this can be generalized to the axial k-index assignment problem and we show empirically how the separation algorithms exploiting sparsity improve on existing ones by running them on the largest instances reported in the literature

Solving Optimal Transmission Switching Problem via DC Power Flow Approximation

The objective of the Optimal Transmission Switching (OTS) problem is to identify a topology of the power grid that minimizes the total energy production costs, while satisfying the operational and physical constraints of the power system. The problem is formulated as a non-convex mixed-integer nonlinear program, which poses extraordinary computational challenges. A common approach to solve the OTS problem is to replace its non-convex non-linear constraints with some linear constraints that turn the original problem into a mixed-integer linear programming, named DC OTS. Although there is plenty of work studying solution methods for the DC OTS in the literature, whether and how solutions of the DC OTS are actually useful for the original OTS problem is often overlooked. In this work, we investigate to what extent DC OTS solutions can be used as a fast heuristic to compute feasible solutions for the original OTS problem. Computational experiments on a set of PGLib benchmark instances highlighted that the optimal solution of the DC OTS is rarely feasible for the original OTS problem, which is consistent with the literature. However, we also find that easy-to-implement modifications of the solution procedure help to address this issue. Therefore, we suggest using DC OTS solutions as a complementary option to state-of-the-art heuristics to compute feasible solutions of the original OTS problem

An Efficient Approach to Distributionally Robust Network Capacity Planning

In this paper, we consider a network capacity expansion problem in the context of telecommunication networks, where there is uncertainty associated with the expected traffic demand. We employ a distributionally robust stochastic optimization (DRSO) framework where the ambiguity set of the uncertain demand distribution is constructed using the moments information, the mean and variance. The resulting DRSO problem is formulated as a bilevel optimization problem. We develop an efficient solution algorithm for this problem by characterizing the resulting worst-case two-point distribution, which allows us to reformulate the original problem as a convex optimization problem. In computational experiments the performance of this approach is compared to that of the robust optimization approach with a discrete uncertainty set. The results show that solutions from the DRSO model outperform the robust optimization approach on highly risk-averse performance metrics, whereas the robust solution is better on the less risk-averse metric

Algorithms and uncertainty sets for data-driven robust shortest path problems

We consider robust shortest path problems, where the aim is to find a path that optimizes the worst-case performance over an uncertainty set containing all relevant scenarios for arc costs. The usual approach for such problems is to assume this uncertainty set given by an expert who can advise on the shape and size of the set. Following the idea of data-driven robust optimization, we instead construct a range of uncertainty sets from the current literature based on real-world traffic measurements provided by the City of Chicago. We then compare the performance of the resulting robust paths within and outside the sample, which allows us to draw conclusions on the suitability of uncertainty sets. Based on our experiments, we then focus on ellipsoidal uncertainty sets, and develop a new solution algorithm that significantly outperforms a state-of-the art solver

Equilibrium Design by Coarse Correlation in Quadratic Games

In a public good provision or a public bad abatement situation, the non-cooperative interplay of the participants typically results in low levels of provision or abatement. In the familiar class of n-person quadratic games, we show that Coarse Correlated equilibria (CCEs) - simple mediated communication devices that do not alter the strategic structure of the game - can significantly outperform the Nash equilibrium in terms of the policy objective above

Pricing toll roads under uncertainty

We study the toll pricing problem when the non-toll costs on the network are not fixed and can vary over time. We assume that users who take their decisions, after the tolls are fixed, have full information of all costs before making their decision. Toll-setter, on the other hand, do not have any information of the future costs on the network. The only information toll-setter have is historical information (sample) of the network costs. In this work we study this problem on parallel networks and networks with few number of paths in single origin-destination setting. We formulate toll-setting problem in this setting as a distributionally robust optimization problem and propose a method to solve to it. We illustrate the usefulness of our approach by doing numerical experiments using a parallel network

Improving Abatement Levels and Welfare by Coarse Correlation in an Environmental Game

Coarse correlated equilibria (CCE, Moulin and Vial, 1978) can be used to substantially improve upon the Nash equilibrium solution of the well-analysed abatement game (Barrett, 1994). We show this by computing successively the CCE with the largest total utility, the one with the highest possible abatement levels and finally, the one with maximal abatement level while maintaining at least the level of utility from the Nash outcome