Parallelization of an interior point algorithm for linear programming

Ankara : Department of Computer Engineering and Information Science and the Institute of Engineering of Bilkent University, 1994.Thesis (Master's) -- -Bilkent University, 1994.Includes bibliographical references leaves 71-73.In this study, we present the parallelization of Mehrotra’s predictor-corrector interior point algorithm, which is a Karmarkar-type optimization method for linear programming. Computation types needed by the algorithm are identified and parallel algorithms for each type are presented. The repeated solution of large symmetric sets of linear equations, which constitutes the major computational effort in Karmarkar-type algorithms, is studied in detail. Several forward and backward solution algorithms are tested, and buffered backward solution algorithm is developed. Heurustic bin-packing algorithms are used to schedule sparse matrix-vector product and factorization operations. Algorithms having the best performance results are used to implement a system to solve linear programs in parallel on multicomputers. Design considerations and implementation details of the system are discussed, and performance results are presented from a number of real problems.Simitçi, HüseyinM.S

Review of Optimization Problems in Wireless Sensor Networks

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Global Optimisation for Energy System

The goal of global optimisation is to find globally optimal solutions, avoiding local optima and other stationary points. The aim of this thesis is to provide more efficient global optimisation tools for energy systems planning and operation. Due to the ongoing increasing of complexity and decentralisation of power systems, the use of advanced mathematical techniques that produce reliable solutions becomes necessary. The task of developing such methods is complicated by the fact that most energy-related problems are nonconvex due to the nonlinear Alternating Current Power Flow equations and the existence of discrete elements. In some cases, the computational challenges arising from the presence of non-convexities can be tackled by relaxing the definition of convexity and identifying classes of problems that can be solved to global optimality by polynomial time algorithms. One such property is known as invexity and is defined by every stationary point of a problem being a global optimum. This thesis investigates how the relation between the objective function and the structure of the feasible set is connected to invexity and presents necessary conditions for invexity in the general case and necessary and sufficient conditions for problems with two degrees of freedom. However, nonconvex problems often do not possess any provable convenient properties, and specialised methods are necessary for providing global optimality guarantees. A widely used technique is solving convex relaxations in order to find a bound on the optimal solution. Semidefinite Programming relaxations can provide good quality bounds, but they suffer from a lack of scalability. We tackle this issue by proposing an algorithm that combines decomposition and linearisation approaches. In addition to continuous non-convexities, many problems in Energy Systems model discrete decisions and are expressed as mixed-integer nonlinear programs (MINLPs). The formulation of a MINLP is of significant importance since it affects the quality of dual bounds. In this thesis we investigate algebraic characterisations of on/off constraints and develop a strengthened version of the Quadratic Convex relaxation of the Optimal Transmission Switching problem. All presented methods were implemented in mathematical modelling and optimisation frameworks PowerTools and Gravity

Anytime Replanning of Robot Coverage Paths for Partially Unknown Environments

In this paper, we propose a method to replan coverage paths for a robot operating in an environment with initially unknown static obstacles. Existing coverage approaches reduce coverage time by covering along the minimum number of coverage lines (straight-line paths). However, recomputing such paths online can be computationally expensive resulting in robot stoppages that increase coverage time. A naive alternative is greedy detour replanning, i.e., replanning with minimum deviation from the initial path, which is efficient to compute but may result in unnecessary detours. In this work, we propose an anytime coverage replanning approach named OARP-Replan that performs near-optimal replans to an interrupted coverage path within a given time budget. We do this by solving linear relaxations of mixed-integer linear programs (MILPs) to identify sections of the interrupted path that can be optimally replanned within the time budget. We validate our approach in simulation using maps of real-world environments and compare our approach against a greedy detour replanner and other state-of-the-art approaches.Comment: 14 pages, 15 figures, Paper submitted to T-R