3,142 research outputs found
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
Polyhedral techniques in combinatorial optimization II: computations
Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience
Polyhedral techniques in combinatorial optimization II: applications and computations
The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formulation by adding violated strong valid inequalities, i.e., inequalities that are violated by the current fractional solution but satisfied by all feasible solutions, and that define high-dimensional faces, preferably facets, of the convex hull of feasible solutions. If we have the complete description of the convex hull of feasible solutions at hand all extreme points of this formulation are integral, which means that we can solve the problem as a linear programming problem. Linear programming problems are known to be computationally easy. In Part 1 of this article we discuss theoretical aspects of polyhedral techniques. Here we will mainly concentrate on the computational aspects. In particular we discuss how polyhedral results are used in cutting plane algorithms. We also consider a few theoretical issues not treated in Part 1, such as techniques for proving that a certain inequality is facet defining, and that a certain linear formulation gives a complete description of the convex hull of feasible solutions. We conclude the article by briefly mentioning some alternative techniques for solving combinatorial optimization problems
Polyhredral techniques in combinatorial optimization I: theory
Combinatorial optimization problems appear in many disciplines ranging
from management and logistics to mathematics, physics, and chemistry. These
problems are usually relatively easy to formulate mathematically, but most
of them are computationally hard due to the restriction that a subset of the
variables have to take integral values. During the last two decades there has
been a remarkable progress in techniques based on the polyhedral description
of combinatorial problems, leading to a large increase in the size of several
problem types that can be solved. The basic idea behind polyhedral techniques
is to derive a good linear formulation of the set of solutions by identifying
linear inequalities that can be proved to be necessary in the description of the
convex hull of feasible solutions. Ideally we can then solve the problem as
a linear programming problem, which can be done eciently. The purpose of
this manuscript is to give an overview of the developments in polyhedral theory,
starting with the pioneering work by Dantzig, Fulkerson and Johnson on the
traveling salesman problem, and by Gomory on integer programming. We also
present some modern applications, and computational experience
Advanced Design for Additive Manufacturing: 3D Slicing and 2D Path Planning
Commercial 3D printers have been increasingly implemented in a variety of fields due to their quick production, simplicity of use, and cheap manufacturing. Software installed in these machines allows automatic production of components from computer-aided design (CAD) models with minimal human intervention. However, there are fewer options provided, with a limited range of materials, limited path patterns, and layer thicknesses. For fabricating metal functional parts, such as laser-based, electron beam-based, and arc-welding-based additive manufacturing (AM) machines, usually more careful process design requires in order to obtain components with the desired mechanical and material properties. Therefore, advanced design for additive manufacturing, particularly slicing and path planning, is necessary for AM experts. This chapter introduces recent achievements in slicing and path planning for AM process
Revisiting the Evolution and Application of Assignment Problem: A Brief Overview
The assignment problem (AP) is incredibly challenging that can model many real-life problems. This paper provides a limited review of the recent developments that have appeared in the literature, meaning of assignment problem as well as solving techniques and will provide a review on a lot of research studies on different types of assignment problem taking place in present day real life situation in order to capture the variations in different types of assignment techniques. Keywords: Assignment problem, Quadratic Assignment, Vehicle Routing, Exact Algorithm, Bound, Heuristic etc
- …