In this article we present methods based on Lagrangian duality and decomposition techniques for t... more In this article we present methods based on Lagrangian duality and decomposition techniques for the generalized knapsack problem with variable coefficients. The Lagrangian dual is solved with subgradient optimization or interval bisection. We also describe a heuristic that yields primal feasible solutions. Combining the Lagrangian relaxation with a primal (Benders) subproblem yields the subproblem phase in cross decomposition. By using averages in this procedure, we get the new mean-value cross-decomposition method. Finally, we describe how to insert this into a globally convergent generalized Benders decomposition framework, in the case that there is a duality gap. Encouraging computational results for the optimal generating unit commitment problem are presented.
Page 1. Oprimization, 1996, Vol. 36. pp. 139-152 1996 OPA (Overseas Publishers Association) Repri... more Page 1. Oprimization, 1996, Vol. 36. pp. 139-152 1996 OPA (Overseas Publishers Association) Reprints available directly from the publisher Amsterdam BV Published in The Netherlands under Photocopying permitted by license ...
When designing a telecommunication network, one often wish to include some kind of survivability ... more When designing a telecommunication network, one often wish to include some kind of survivability requirement, for example that the network should be two-connected. A two-connected network fulfills the requirement that there should be at least two paths with no links in common between all pairs of nodes. One form of design model is to prescribe that the network should be composed of connected rings of links. The network design problem is then to choose links from a given network, and compose them into a number of rings. A ring is reliable in the sense that there always exist two ways of sending traffic, clockwise or counter-clockwise, which means that a ring fulfills the two-connectivity requirement. There is often a number of requirements on a ring, such as a limited length and limited number of nodes connected to the ring. This means that a ring network will include a number of rings, and traffic between rings must be possible. The traffic between rings is usually made at certain nodes, called transit nodes. Therefore all rings should be connected to at least one of the transit nodes. We focus on the case where we have two transit nodes in the network.Each possible ring is associated with a certain fixed cost, and all links in a certain ring are given the same capacity. Reserve capacity is allocated according to certain principles. The number of possible rings in a network is an exponential function of the number of nodes in the network, so for larger networks is it impossible to a priori generate all possible rings.We describe the problem, and model it as a linear integer programming problem, where a set of rings are assumed to be known. The usage of rings, i.e., the allocation of demand to rings, is determined. In practice, too many rings can not be included in the model. Instead we must be able to generate useful rings. A Lagrangean relaxation of the model is formulated, and the dual solution is used in order to derive reduced costs which can be used to generate new better rings. The information generated describes only the physical structure of the ring, not the usage of it. The ring generation problem is a modified traveling salesman subtour problem, which is known to be difficult to solve. Therefore, we focus on heuristic solution methods for this problem.We also presents a column generation approach where the problem is modeled as a set covering problem. Here, a column describes both the topology of the ring and the exact usage of it. A similar ring generation problem appears as a subproblem, in order to generate new rings.All methods are computationally tested on both real life data and randomly generated data, similar to real life problems.
When designing a telecommunication network, one often wish to include some kind of survivability ... more When designing a telecommunication network, one often wish to include some kind of survivability requirement, for example that there should be at least two paths between every pair of nodes in the ...
We discuss the problem of projecting points on their convex hull. Points in the interior of the c... more We discuss the problem of projecting points on their convex hull. Points in the interior of the convex hull are moved outwards to the boundary of the convex hull. While finding the convex hull is a ...
Data from OpenStreetMap can be a valuable tool in route optimization of many kinds. With GPS data... more Data from OpenStreetMap can be a valuable tool in route optimization of many kinds. With GPS data, analyses of trips can be made. In this paper, we discuss how to extract such data and transform it ...
We present an exact formulation of the simple plant location problem with spatial interaction. Th... more We present an exact formulation of the simple plant location problem with spatial interaction. The formulation is based on the derivation of the spatially interactive travel behavior, often described. through a spatial "gravity" model. By using techniques from separable programming, we can derive an exact formulation of the problem, i.e. without using any approximation of the function that describes the spatially interactive travel behavior. The resulting model is a pure zero-one location model. We also present a dual ascent and adjustment procedure for the linear programming relaxation of the exact formulation of the simple plant location problem with spatial interaction. Furthermore, we present a solution method based on Lagrangean relaxation and subgradient optimization. The two dual search procedures are computationally tested on problems with a maximal size of 50 x 100.
We describe a weighted version of the k-Chinese or k-rural postman problem that occurs in the con... more We describe a weighted version of the k-Chinese or k-rural postman problem that occurs in the context of snow removal. The problem concerns the questions of which vehicle shall do each task and how the vehicles shall travel between tasks. We also consider different numbers of vehicles, in view of a fixed cost for each vehicle. We describe and discuss heuristic solution approaches, based on usable substructures, such as Chinese/rural postman problems, meta-heuristics, k-means clustering and local search improvements by moving cycles. The methods have been implemented and tested on real life examples.
Planning snow removal is a difficult, infrequently occurring optimization problem, concerning com... more Planning snow removal is a difficult, infrequently occurring optimization problem, concerning complicated routing of vehicles. Clearing a street includes several different activities, and the tours must be allowed to contain subtours. The streets are classified into different types, each type requiring different activities. We address the problem facing a single vehicle, including details such as precedence requirements and turning penalties. We describe a solution approach based on a reformulation to an asymmetric traveling salesman problem in an extended graph, plus a heuristic for finding feasible solutions. The method have been implemented and tested on real life examples, and the solution times are short enough to allow online usage. We compare two different principles for the number of sweeps on a normal street, encountered in discussions with snow removal contractors. A principle using a first sweep in the middle of the street around the block, in order to quickly allow usage of the streets, is found to yield interesting theoretical and practical difficulties.
We present optimization models for the problem of simultaneous routing of connections through a s... more We present optimization models for the problem of simultaneous routing of connections through a symmetric Clos network, and for the problem of minimal rerouting of previously routed connections when a new connection is to be routed. The models can be used as base for solution methods, such as heuristics for rerouting combined with Lagrangean relaxation. These approaches can together give bounds on the optimal number of rearrangements needed. This is done for Clos networks with three stages, five stages, seven stages, and for an arbitrary number of stages.
Real life optimization often concerns difficult objective functions, in two aspects, namely that ... more Real life optimization often concerns difficult objective functions, in two aspects, namely that gradients are unavailable, and that evaluation of the objective function takes a long time. Such pro ...
In a democratic proportional election system, it is vital that the mandates in the parliament are... more In a democratic proportional election system, it is vital that the mandates in the parliament are allocated as proportionally as possible to the number of votes the parties got in the election. We ...
We investigate the effect of different reductions when importing networksfrom OpenStreetMap data.... more We investigate the effect of different reductions when importing networksfrom OpenStreetMap data. We describe the network reductions and report computationaltests for doing the network extraction a ...
In many public facility location problems the clients are free to make their own choice of facili... more In many public facility location problems the clients are free to make their own choice of facility, which might yield different results than minimizing the transportation costs. In order to better model such situations, spatial interaction has been introduced in location models. Simple plant location problems with spatial interaction between the travelers have been treated in [2, 3, 5, 14, 15, 16, 17], modelled as a nonlinear, mixed integer programming problem.
Page 1. Oprimization, 1996, Vol. 36. pp. 139-152 1996 OPA (Overseas Publishers Association) Repri... more Page 1. Oprimization, 1996, Vol. 36. pp. 139-152 1996 OPA (Overseas Publishers Association) Reprints available directly from the publisher Amsterdam BV Published in The Netherlands under Photocopying permitted by license ...
We consider network design and routing for Internet Protocol (IP) traffic. The design problem con... more We consider network design and routing for Internet Protocol (IP) traffic. The design problem concerns ca-pacity dimensioning of communication links, where the design cost consists of fixed charges and linear capacity expansion costs. The optimization problem also con-cerns ...
In this article we present methods based on Lagrangian duality and decomposition techniques for t... more In this article we present methods based on Lagrangian duality and decomposition techniques for the generalized knapsack problem with variable coefficients. The Lagrangian dual is solved with subgradient optimization or interval bisection. We also describe a heuristic that yields primal feasible solutions. Combining the Lagrangian relaxation with a primal (Benders) subproblem yields the subproblem phase in cross decomposition. By using averages in this procedure, we get the new mean-value cross-decomposition method. Finally, we describe how to insert this into a globally convergent generalized Benders decomposition framework, in the case that there is a duality gap. Encouraging computational results for the optimal generating unit commitment problem are presented.
Page 1. Oprimization, 1996, Vol. 36. pp. 139-152 1996 OPA (Overseas Publishers Association) Repri... more Page 1. Oprimization, 1996, Vol. 36. pp. 139-152 1996 OPA (Overseas Publishers Association) Reprints available directly from the publisher Amsterdam BV Published in The Netherlands under Photocopying permitted by license ...
When designing a telecommunication network, one often wish to include some kind of survivability ... more When designing a telecommunication network, one often wish to include some kind of survivability requirement, for example that the network should be two-connected. A two-connected network fulfills the requirement that there should be at least two paths with no links in common between all pairs of nodes. One form of design model is to prescribe that the network should be composed of connected rings of links. The network design problem is then to choose links from a given network, and compose them into a number of rings. A ring is reliable in the sense that there always exist two ways of sending traffic, clockwise or counter-clockwise, which means that a ring fulfills the two-connectivity requirement. There is often a number of requirements on a ring, such as a limited length and limited number of nodes connected to the ring. This means that a ring network will include a number of rings, and traffic between rings must be possible. The traffic between rings is usually made at certain nodes, called transit nodes. Therefore all rings should be connected to at least one of the transit nodes. We focus on the case where we have two transit nodes in the network.Each possible ring is associated with a certain fixed cost, and all links in a certain ring are given the same capacity. Reserve capacity is allocated according to certain principles. The number of possible rings in a network is an exponential function of the number of nodes in the network, so for larger networks is it impossible to a priori generate all possible rings.We describe the problem, and model it as a linear integer programming problem, where a set of rings are assumed to be known. The usage of rings, i.e., the allocation of demand to rings, is determined. In practice, too many rings can not be included in the model. Instead we must be able to generate useful rings. A Lagrangean relaxation of the model is formulated, and the dual solution is used in order to derive reduced costs which can be used to generate new better rings. The information generated describes only the physical structure of the ring, not the usage of it. The ring generation problem is a modified traveling salesman subtour problem, which is known to be difficult to solve. Therefore, we focus on heuristic solution methods for this problem.We also presents a column generation approach where the problem is modeled as a set covering problem. Here, a column describes both the topology of the ring and the exact usage of it. A similar ring generation problem appears as a subproblem, in order to generate new rings.All methods are computationally tested on both real life data and randomly generated data, similar to real life problems.
When designing a telecommunication network, one often wish to include some kind of survivability ... more When designing a telecommunication network, one often wish to include some kind of survivability requirement, for example that there should be at least two paths between every pair of nodes in the ...
We discuss the problem of projecting points on their convex hull. Points in the interior of the c... more We discuss the problem of projecting points on their convex hull. Points in the interior of the convex hull are moved outwards to the boundary of the convex hull. While finding the convex hull is a ...
Data from OpenStreetMap can be a valuable tool in route optimization of many kinds. With GPS data... more Data from OpenStreetMap can be a valuable tool in route optimization of many kinds. With GPS data, analyses of trips can be made. In this paper, we discuss how to extract such data and transform it ...
We present an exact formulation of the simple plant location problem with spatial interaction. Th... more We present an exact formulation of the simple plant location problem with spatial interaction. The formulation is based on the derivation of the spatially interactive travel behavior, often described. through a spatial "gravity" model. By using techniques from separable programming, we can derive an exact formulation of the problem, i.e. without using any approximation of the function that describes the spatially interactive travel behavior. The resulting model is a pure zero-one location model. We also present a dual ascent and adjustment procedure for the linear programming relaxation of the exact formulation of the simple plant location problem with spatial interaction. Furthermore, we present a solution method based on Lagrangean relaxation and subgradient optimization. The two dual search procedures are computationally tested on problems with a maximal size of 50 x 100.
We describe a weighted version of the k-Chinese or k-rural postman problem that occurs in the con... more We describe a weighted version of the k-Chinese or k-rural postman problem that occurs in the context of snow removal. The problem concerns the questions of which vehicle shall do each task and how the vehicles shall travel between tasks. We also consider different numbers of vehicles, in view of a fixed cost for each vehicle. We describe and discuss heuristic solution approaches, based on usable substructures, such as Chinese/rural postman problems, meta-heuristics, k-means clustering and local search improvements by moving cycles. The methods have been implemented and tested on real life examples.
Planning snow removal is a difficult, infrequently occurring optimization problem, concerning com... more Planning snow removal is a difficult, infrequently occurring optimization problem, concerning complicated routing of vehicles. Clearing a street includes several different activities, and the tours must be allowed to contain subtours. The streets are classified into different types, each type requiring different activities. We address the problem facing a single vehicle, including details such as precedence requirements and turning penalties. We describe a solution approach based on a reformulation to an asymmetric traveling salesman problem in an extended graph, plus a heuristic for finding feasible solutions. The method have been implemented and tested on real life examples, and the solution times are short enough to allow online usage. We compare two different principles for the number of sweeps on a normal street, encountered in discussions with snow removal contractors. A principle using a first sweep in the middle of the street around the block, in order to quickly allow usage of the streets, is found to yield interesting theoretical and practical difficulties.
We present optimization models for the problem of simultaneous routing of connections through a s... more We present optimization models for the problem of simultaneous routing of connections through a symmetric Clos network, and for the problem of minimal rerouting of previously routed connections when a new connection is to be routed. The models can be used as base for solution methods, such as heuristics for rerouting combined with Lagrangean relaxation. These approaches can together give bounds on the optimal number of rearrangements needed. This is done for Clos networks with three stages, five stages, seven stages, and for an arbitrary number of stages.
Real life optimization often concerns difficult objective functions, in two aspects, namely that ... more Real life optimization often concerns difficult objective functions, in two aspects, namely that gradients are unavailable, and that evaluation of the objective function takes a long time. Such pro ...
In a democratic proportional election system, it is vital that the mandates in the parliament are... more In a democratic proportional election system, it is vital that the mandates in the parliament are allocated as proportionally as possible to the number of votes the parties got in the election. We ...
We investigate the effect of different reductions when importing networksfrom OpenStreetMap data.... more We investigate the effect of different reductions when importing networksfrom OpenStreetMap data. We describe the network reductions and report computationaltests for doing the network extraction a ...
In many public facility location problems the clients are free to make their own choice of facili... more In many public facility location problems the clients are free to make their own choice of facility, which might yield different results than minimizing the transportation costs. In order to better model such situations, spatial interaction has been introduced in location models. Simple plant location problems with spatial interaction between the travelers have been treated in [2, 3, 5, 14, 15, 16, 17], modelled as a nonlinear, mixed integer programming problem.
Page 1. Oprimization, 1996, Vol. 36. pp. 139-152 1996 OPA (Overseas Publishers Association) Repri... more Page 1. Oprimization, 1996, Vol. 36. pp. 139-152 1996 OPA (Overseas Publishers Association) Reprints available directly from the publisher Amsterdam BV Published in The Netherlands under Photocopying permitted by license ...
We consider network design and routing for Internet Protocol (IP) traffic. The design problem con... more We consider network design and routing for Internet Protocol (IP) traffic. The design problem concerns ca-pacity dimensioning of communication links, where the design cost consists of fixed charges and linear capacity expansion costs. The optimization problem also con-cerns ...
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Papers by Kaj Holmberg