5,447 research outputs found
On orbital allotments for geostationary satellites
The following satellite synthesis problem is addressed: communication satellites are to be allotted positions on the geostationary arc so that interference does not exceed a given acceptable level by enforcing conservative pairwise satellite separation. A desired location is specified for each satellite, and the objective is to minimize the sum of the deviations between the satellites' prescribed and desired locations. Two mixed integer programming models for the satellite synthesis problem are presented. Four solution strategies, branch-and-bound, Benders' decomposition, linear programming with restricted basis entry, and a switching heuristic, are used to find solutions to example synthesis problems. Computational results indicate the switching algorithm yields solutions of good quality in reasonable execution times when compared to the other solution methods. It is demonstrated that the switching algorithm can be applied to synthesis problems with the objective of minimizing the largest deviation between a prescribed location and the corresponding desired location. Furthermore, it is shown that the switching heuristic can use no conservative, location-dependent satellite separations in order to satisfy interference criteria
Strongly Polynomial Primal-Dual Algorithms for Concave Cost Combinatorial Optimization Problems
We introduce an algorithm design technique for a class of combinatorial
optimization problems with concave costs. This technique yields a strongly
polynomial primal-dual algorithm for a concave cost problem whenever such an
algorithm exists for the fixed-charge counterpart of the problem. For many
practical concave cost problems, the fixed-charge counterpart is a well-studied
combinatorial optimization problem. Our technique preserves constant factor
approximation ratios, as well as ratios that depend only on certain problem
parameters, and exact algorithms yield exact algorithms.
Using our technique, we obtain a new 1.61-approximation algorithm for the
concave cost facility location problem. For inventory problems, we obtain a new
exact algorithm for the economic lot-sizing problem with general concave
ordering costs, and a 4-approximation algorithm for the joint replenishment
problem with general concave individual ordering costs
A Lagrangean Relaxtion Based Algorithm for Solving Set Partitioning Problems
In this paper we discuss a solver that is developed to solve set partitioning problems.The methods used include problem reduction techniques, lagrangean relaxation and primal and dual heuristics.The optimal solution is found using a branch and bound approach.In this paper we discuss these techniques.Furthermore, we present the results of several computational experiments and compare the performance of our solver with the well-known mathematical optimization solver Cplex.algorithm;integer programming
Adapting the interior point method for the solution of linear programs on high performance computers
In this paper we describe a unified algorithmic framework for the interior point method (IPM) of solving Linear Programs (LPs) which allows us to adapt it over a range of high performance computer architectures. We set out the reasons as to why IPM makes better use of high performance computer architecture than the sparse simplex method. In the inner iteration of the IPM a search direction is computed using Newton or higher order methods. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system and the design of data structures to take advantage of coarse grain parallel and massively parallel computer architectures are considered in detail. Finally, we present experimental results of solving NETLIB test problems on examples of these architectures and put forward arguments as to why integration of the system within sparse simplex is beneficial
Improving Optimization Bounds using Machine Learning: Decision Diagrams meet Deep Reinforcement Learning
Finding tight bounds on the optimal solution is a critical element of
practical solution methods for discrete optimization problems. In the last
decade, decision diagrams (DDs) have brought a new perspective on obtaining
upper and lower bounds that can be significantly better than classical bounding
mechanisms, such as linear relaxations. It is well known that the quality of
the bounds achieved through this flexible bounding method is highly reliant on
the ordering of variables chosen for building the diagram, and finding an
ordering that optimizes standard metrics is an NP-hard problem. In this paper,
we propose an innovative and generic approach based on deep reinforcement
learning for obtaining an ordering for tightening the bounds obtained with
relaxed and restricted DDs. We apply the approach to both the Maximum
Independent Set Problem and the Maximum Cut Problem. Experimental results on
synthetic instances show that the deep reinforcement learning approach, by
achieving tighter objective function bounds, generally outperforms ordering
methods commonly used in the literature when the distribution of instances is
known. To the best knowledge of the authors, this is the first paper to apply
machine learning to directly improve relaxation bounds obtained by
general-purpose bounding mechanisms for combinatorial optimization problems.Comment: Accepted and presented at AAAI'1
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