A Cycle-Based Formulation and Valid Inequalities for DC Power Transmission Problems with Switching

It is well-known that optimizing network topology by switching on and off transmission lines improves the efficiency of power delivery in electrical networks. In fact, the USA Energy Policy Act of 2005 (Section 1223) states that the U.S. should "encourage, as appropriate, the deployment of advanced transmission technologies" including "optimized transmission line configurations". As such, many authors have studied the problem of determining an optimal set of transmission lines to switch off to minimize the cost of meeting a given power demand under the direct current (DC) model of power flow. This problem is known in the literature as the Direct-Current Optimal Transmission Switching Problem (DC-OTS). Most research on DC-OTS has focused on heuristic algorithms for generating quality solutions or on the application of DC-OTS to crucial operational and strategic problems such as contingency correction, real-time dispatch, and transmission expansion. The mathematical theory of the DC-OTS problem is less well-developed. In this work, we formally establish that DC-OTS is NP-Hard, even if the power network is a series-parallel graph with at most one load/demand pair. Inspired by Kirchoff's Voltage Law, we give a cycle-based formulation for DC-OTS, and we use the new formulation to build a cycle-induced relaxation. We characterize the convex hull of the cycle-induced relaxation, and the characterization provides strong valid inequalities that can be used in a cutting-plane approach to solve the DC-OTS. We give details of a practical implementation, and we show promising computational results on standard benchmark instances

Reformulated acyclic partitioning for rail-rail containers transshipment

Many rail terminals have loading areas that are properly equipped to move containers between trains. With the growing throughput of these terminals all the trains involved in a sequence of such movements may not ¿t in the loading area simultaneously, and storage areas are needed to place containers waiting for their destination train, although this storage increases the cost of the transshipment. This increases the complexity of the planning decisions concerning these activities, since now trains need to be packed in groups that ¿t in the loading area, in such a way that the number of containers moved to the storage area is minimized. Additionally, each train is only allowed to enter the loading area once. Similarly to previous authors, we model this situation as an acyclic graph partitioning problem for which we present a new formulation, and several valid inequalities based on its theoretical properties. Our computational experiments show that the new formulation outperforms the previously existing ones, providing results that improve even on the best exact algorithm designed so far for this problem.Peer ReviewedPostprint (author's final draft

Formulations and valid inequalities for the node capacitated graph partitioning problem

We investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition. The objective is to minimize the sum of the costs of the edges between the subsets of the partition. This problem has a variety of applications, for instance in the design of electronic circuits and devices. We present alternative integer programming formulations for this problem and discuss the links between these formulations. Having chosen to work in the space of edges of the multicut, we investigate the convex hull of incidence vectors of feasible multicuts. In particular, several classes of inequalities are introduced, and their strength and robustness are analyzed as various problem parameters change

Formulations and Valid Inequalities for the Node Capacitated Graph Partitioning Problem

We investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition. The objective is to minimize the sum of the costs of the edges between the subsets of the partition. This problem has a variety of applications, for instance in the design of electronic circuits and devices. We present alternative integer programming formulations for this problem and discuss the links between these formulations. Having chosen to work in the space of edges of the multicut, we investigate the convex hull of incidence vectors of feasible multicuts. In particular, several classes of inequalities are introduced, and their strength and robustness are analyzed as various problem parameters change.clustering, graph partitioning, equipartition, knapsack, integer programming, ear decomposition

Formulations and valid inequalities for the node capacitated graph partitioning problem

We investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition. The objective is to minimize the sum of the costs of the edges between the subsets of the partition. This problem has a variety of applications, for instance in the design of electronic circuits and devices. We present alternative integer programming formulations for this problem and discuss the links between these formulations. Having chosen to work in the space of edges of the multicut, we investigate the convex hull of incidence vectors of feasible multicuts. In particular, several classes of inequalities are introduced, and their strength and robustness are analyzed as various problem parameters change.74324726

Knapsack Problems with Side Constraints

The thesis considers a specific class of resource allocation problems in Combinatorial Optimization: the Knapsack Problems. These are paradigmatic NP-hard problems where a set of items with given profits and weights is available. The aim is to select a subset of the items in order to maximize the total profit without exceeding a known knapsack capacity. In the classical 0-1 Knapsack Problem (KP), each item can be picked at most once. The focus of the thesis is on four generalizations of KP involving side constraints beyond the capacity bound. More precisely, we provide solution approaches and insights for the following problems: The Knapsack Problem with Setups; the Collapsing Knapsack Problem; the Penalized Knapsack Problem; the Incremental Knapsack Problem. These problems reveal challenging research topics with many real-life applications. The scientific contributions we provide are both from a theoretical and a practical perspective. On the one hand, we give insights into structural elements and properties of the problems and derive a series of approximation results for some of them. On the other hand, we offer valuable solution approaches for direct applications of practical interest or when the problems considered arise as sub-problems in broader contexts

Solving Real-Life Hydroinformatics Problems with Operations Research and Artificial Intelligence

Many real life problems in the hydraulic engineering literature can be modelled as constrained optimisation problems. Often, they are addressed in the literature through genetic algorithms, although other techniques have been proposed. In this thesis, we address two of these real life problems through a variety of techniques taken from the Artificial Intelligence and Operations Research areas, such as mixed-integer linear programming, logic programming, genetic algorithms and path relinking, together with hybridization amongst these technologies and with hydraulic simulators. For the first time, an Answer Set Programming formulation of hydroinformatics problems is proposed. The two real life problems addressed hereby are the optimisation of the response in case of contamination events, and the optimisation of the positioning of the isolation valves. The constraints of the former describe the feasible region of the Multiple Travelling Salesman Problem, while the objective function is computed by a hydraulic simulator. A simulation–optimisation approach based on Genetic Algorithms, mathematical programming, and Path Relinking, and a thorough experimental analysis are discussed hereby. The constraints of the latter problem describe a graph partitioning enriched with a maximum flow, and it is a new variant of the common graph partitioning. A new mathematical model plus a new formalization in logic programming are discussed in this work. In particular, the technologies adopted are mixed-integer linear programming and Answer Set Programming. Addressing these two real applications in hydraulic engineering as constrained optimisation problems has allowed for i) computing applicable solutions to the real case, ii) computing better solutions than the ones proposed in the hydraulic literature, iii) exploiting graph theory for modellization and solving purposes, iv) solving the problems by well suited technologies in Operations Research and Artificial Intelligence, and v) designing new integrated and hybrid architectures for a more effective solving