EURO Journal on Computational Optimization最新文献

A conic optimization problem is a problem involving a constraint that the optimization variable be in some closed convex cone. Prominent examples are linear programs (LP), second order cone programs (SOCP), semidefinite problems (SDP), and copositive problems. We survey recent progress made in this area. In particular, we highlight the connections between nonconvex quadratic problems, binary quadratic problems, and copositive optimization. We review how tight bounds can be obtained by relaxing the copositivity constraint to semidefiniteness, and we discuss the effect that different modelling techniques have on the quality of the bounds. We also provide some new techniques for lifting linear constraints and show how these can be used for stable set and coloring relaxations.

In this manuscript, we consider smooth multi-objective optimization problems with convex constraints. We propose an extension of a multi-objective augmented Lagrangian Method from recent literature. The new algorithm is specifically designed to handle sets of points and produce good approximations of the whole Pareto front, as opposed to the original one which converges to a single solution. We prove properties of global convergence to Pareto stationarity for the sequences of points generated by our procedure. We then compare the performance of the proposed method with those of the main state-of-the-art algorithms available for the considered class of problems. The results of our experiments show the effectiveness and general superiority w.r.t. competitors of our proposed approach.

We consider an adjustable robust optimization problem arising in the area of supply chains: given sets of suppliers and demand nodes, we wish to find a flow that is robust with respect to failures of the suppliers. The objective is to determine a flow that minimizes the amount of shortage in the worst-case after an optimal mitigation has been performed. An optimal mitigation is an additional flow in the residual network that mitigates as much shortage at the demand sites as possible. For this problem we give a mathematical formulation, yielding a robust flow problem with three stages where the mitigation of the last stage can be chosen adaptively depending on the scenario. We show that already evaluating the robustness of a solution is $\mathrm{NP}$-hard. For optimizing with respect to this $\mathrm{NP}$-hard objective function, we compare three algorithms. Namely an algorithm based on iterative cut generation that solves medium-sized instances efficiently, a simple Outer Linearization Algorithm and a Scenario Enumeration algorithm. We illustrate the performance by numerical experiments. The results show that this instance of fully adjustable robust optimization problems can be solved exactly with a reasonable performance. We also describe possible extensions to the model and the algorithm.

Ailsa H. Land, who received the 2021 EURO Gold Medal, made some important contributions to the study of the Traveling Salesman Problem, which were published in a 1955 journal article and in a 1979 working paper. The purpose of this introductory note is to describe these contributions.

A simplex-based FORTRAN code, working entirely in integer arithmetic, has been developed for the exact solution of travelling-salesman problems. The code adds tour-barring constraints as they are found to be violated. It deals with fractional solutions by adding two-matching constraints and as a last resort by ‘Gomory’ cutting plane constraints of the Method of Integer Forms. Most of the calculations are carried out on only a subset of the variables, with only occasional passes through the whole set of possible variables. Computational experience on some 100-city problems is reported.

Adjustable robust optimization allows for some variables to depend upon the uncertain data after its realization. However, the uncertainty is often not revealed exactly. Incorporating inexactness of the revealed data in the construction of ellipsoidal uncertainty sets, we present an exact second-order cone program reformulation for robust linear optimization problems with inexact data and quadratically adjustable variables. This is achieved by establishing a generalization of the celebrated S-lemma for a separable quadratic inequality system with at most one non-homogeneous function. It allows us to reformulate the resulting separable quadratic constraints over an intersection of two ellipsoids in terms of second-order cone constraints. We illustrate our results via numerical experiments on adjustable robust lot-sizing problems with demand uncertainty, showing improvements over corresponding problems with affinely adjustable variables as well as with exactly revealed data.

In this paper, we introduce the Steiner Bi-objective Shortest Path Problem. This problem is defined on a directed graph $G=(V,A),$ with a subset $T\subset V$ of terminals. Arcs are labeled with travel time and cost. The goal is to find a complete set of efficient paths between every pair of nodes in $T$. The motivation behind this problem stems from data preprocessing for vehicle routing problems. We propose a solution method based on a labeling approach with a multi-objective A* search strategy guiding the search towards the terminals. Computational results based on instances generated from real road networks show the efficiency of the proposed algorithm compared to state-of-art approaches.

The goal of this literature review is to give an update on the recent developments for semi-infinite programs (SIPs), approximately over the last 20 years. An overview of the different solution approaches and the existing algorithms is given. We focus on deterministic algorithms for SIPs which do not make any convexity assumptions. In particular, we consider the case that the constraint function is non-concave with respect to parameters. Advantages and disadvantages of the different algorithms are discussed. We also highlight recent SIP applications. The article closes with a discussion on remaining challenges and future research directions.

Corporate mobility is often based on a fixed assignment of vehicles to employees. Relaxing this fixation and including alternatives such as public transportation or taxis for business and private trips could increase fleet utilization and foster the use of battery electric vehicles. We introduce the mobility offer allocation problemas the core concept of a flexible booking system for corporate mobility. The problem is equivalent to interval scheduling on dedicated unrelated parallel machines. We show that the problem is NP-hard to approximate within any factor. We describe problem specific conflict graphs for representing and exploring the structure of feasible solutions. A characterization of all maximum cliques in these conflict graphs reveals symmetries which allow to formulate stronger integer linear programming models. We also present an adaptive large neighborhood search based approach which makes use of conflict graphs as well. In a computational study, the approaches are evaluated. It was found that greedy heuristics perform best if very tight run-time requirements are given, a solver for the integer linear programming model performs best on small and medium instances, and the adaptive large neighborhood search performs best on large instances.

Linear bilevel optimization problems are known to be strongly NP-hard and the computational techniques to solve these problems are often motivated by techniques from single-level mixed-integer optimization. Thus, during the last years and decades many branch-and-bound methods, cutting planes, or heuristics have been proposed. On the other hand, there is almost no literature on presolving linear bilevel problems although presolve is a very important ingredient in state-of-the-art mixed-integer optimization solvers. In this paper, we carry over standard presolve techniques from single-level optimization to bilevel problems and show that this needs to be done with great caution since a naive application of well-known techniques does often not lead to correctly presolved bilevel models. Our numerical study shows that presolve can also be very beneficial for bilevel problems but also highlights that these methods have a more heterogeneous effect on the solution process compared to what is known from single-level optimization. As a side result, our numerical experiments reveal that there is an urgent need for better and more heterogeneous test instance libraries to further propel the field of computational bilevel optimization.