Constrained partitioning problems

AbstractWe consider partitioning problems subject to the constraint that the subsets in the partition are independent sets or bases of given matroids. We derive conditions for the functions F and [fnof] such that an optimal partition (S∗1, S∗2,…, S∗k) which minimizes F([fnof](S1),…, [fnof](S k)) has certain order properties. These order properties allow to determine optimal partitions by Greedy-like algorithms. In particular balancing partitioning problems can be solved in this way

Disparity and Optical Flow Partitioning Using Extended Potts Priors

This paper addresses the problems of disparity and optical flow partitioning based on the brightness invariance assumption. We investigate new variational approaches to these problems with Potts priors and possibly box constraints. For the optical flow partitioning, our model includes vector-valued data and an adapted Potts regularizer. Using the notation of asymptotically level stable functions we prove the existence of global minimizers of our functionals. We propose a modified alternating direction method of minimizers. This iterative algorithm requires the computation of global minimizers of classical univariate Potts problems which can be done efficiently by dynamic programming. We prove that the algorithm converges both for the constrained and unconstrained problems. Numerical examples demonstrate the very good performance of our partitioning method

Disparity and optical flow partitioning using extended Potts priors

This paper addresses the problems of disparity and optical flow partitioning based on the brightness invariance assumption. We investigate new variational approaches to these problems with Potts priors and possibly box constraints. For the optical flow partitioning, our model includes vector-valued data and an adapted Potts regularizer. Using the notion of asymptotically level stable (als) functions, we prove the existence of global minimizers of our functionals. We propose a modified alternating direction method of multipliers. This iterative algorithm requires the computation of global minimizers of classical univariate Potts problems which can be done efficiently by dynamic programming. We prove that the algorithm converges both for the constrained and unconstrained problems. Numerical examples demonstrate the very good performance of our partitioning method

Optimization Techniques for Modern Power Systems Planning, Operation and Control

Recent developments in computing, communication and improvements in optimization techniques have piqued interest in improving the current operational practices and in addressing the challenges of future power grids. This dissertation leverages these new developments for improved quasi-static analysis of power systems for applications in power system planning, operation and control. The premise of much of the work presented in this dissertation centers around development of better mathematical modeling for optimization problems which are then used to solve current and future challenges of power grid. To this end, the models developed in this research work contributes to the area of renewable integration, demand response, power grid resilience and constrained contiguous and non-contiguous partitioning of power networks. The emphasis of this dissertation is on finding solutions to system operator level problems in real-time. For instance, multi-period mixed integer linear programming problem for applications in demand response schemes involving more than million variables are solved to optimality in less than 20 seconds of computation time through tighter formulation. A balanced, constrained, contiguous partitioning scheme capable of partitioning 20,000 bus power system in under one minute is developed for use in time sensitive application area such as controlled islanding

A reliable hybrid solver for nonconvex optimization

International audienceNonconvex and highly multimodal optimization problems represent a challenge both for stochastic and deterministic global optimization methods. The former (metaheuristics) usually achieve satisfactory solutions but cannot guarantee global optimality, while the latter (generally based on a spatial branch and bound scheme [1], an exhaustive and non-uniform partitioning method) may struggle to converge toward a global minimum within reasonable time. The partitioning process is exponential in the number of variables, which prevents the resolution of large instances. The performances of the solvers even dramatically deteriorate when using reliable techniques, namely techniques that cope with rounding errors.In this paper, we present a fully reliable hybrid algorithm named Charibde (Cooperative Hybrid Algorithm using Reliable Interval-Based methods and Dierential Evolution) [2] that reconciles stochastic and deterministic techniques. An Evolutionary Algorithm (EA) cooperates with intervalbased techniques to accelerate convergence toward the global minimum and prove the optimality of the solution with user-defined precision. Charibde may be used to solve continuous, nonconvex, constrained or bound-constrained problems involving factorable functions