Problem Of Optimal Partitioning Research Articles

Equivariant Solutions to the Optimal Partition Problem for the Prescribed Q-Curvature Equation

We study the optimal partition problem for the prescribed constant Q-curvature equation induced by the higher-order conformal operators under the effect of cohomogeneity one actions on Einstein manifolds with positive scalar curvature. This allows us to give a precise description of the solution domains and their boundaries in terms of the orbits of the action. We also prove the existence of least energy symmetric solutions to a weakly coupled elliptic system of prescribed Q-curvature equations under weaker assumptions and conclude a multiplicity result of sign-changing solutions to the prescribed constant Q-curvature problem induced by the Paneitz-Branson operator. Moreover, we study the coercivity of GJMS operators on Ricci solitons, compute the Q-curvature of these manifolds, and give a multiplicity result for the sign-changing solutions to the Yamabe problem with a prescribed number of nodal domains on the Koiso–Cao Ricci soliton.

Tinba: Incremental partitioning for efficient trajectory analytics

Applications with mobile and sensing devices have already become ubiquitous. In most of these applications, trajectory data is continuously growing to huge volumes. Existing systems for big trajectory data organize trajectories at distributed block storage systems. Systems like DITA that use block storage (e.g., 128 MB each) are more efficient for analytical queries, but they cannot incrementally maintain the partitioned data and do not support delete operations, resulting in difficulties in trajectory analytics. In this paper, we propose an incremental trajectory partitioning framework Tinba that enables distributed block storage systems to efficiently maintain optimized partitions under incremental updates of trajectories. We employ a data flushing technique to bulk ingest trajectory data for random writing in distributed file system (DFS). We recast the incremental partitioning problem as an optimal partitioning problem and prove its NP-hardness. A cost–benefit model is proposed to address the optimal partitioning problem. Moreover, Tinba supports most of the existing similarity measures to quantify the similarity between trajectories. A heuristic is developed to instantiate the Tinba framework. Comprehensive experiments on real-world and synthetic datasets demonstrate the advancements in ingestion performance and partition quality, as opposed to other trajectory partition methods.

Cascading Failures Mitigation Strategy for Resilient Water Infrastructures*

Critical infrastructures are required to provide high quality of services and operation in present modern societies. Recent incidents of cyber-physical attacks and failures in water distribution systems (WDSs) have necessitated finding new solutions to mitigate large-scale cascading failures. In this paper, we propose a graph partitioning method based on the most vulnerable typed-graphlets to contain the damage impact in the most vulnerable target partition by preventing the failures from propagating outside the said partition. Towards this objective, we first formulate the optimal partitioning problem with the objective of minimizing the water leakages subject to hydraulic constraints, and then we solve it using a low-complexity algorithm. We conduct simulations on a 396-node WDS. Our investigations reveal a 54% average decrease in the overall damage compared to the case without any partitioning.

An optimal partition problem for the localization of eigenfunctions

We study the minimizers of a functional on the set of partitions of a domain $\Omega \subset R^n$ into $N$ subsets $W_j$ of locally finite perimeter in $\Omega$, whose main term is $\sum_{j=1^N} \int_{\Omega \cap \partial W_j} a(x) dH^{n--1}(x)$. Here the positive bounded function $a$ may for instance be related to the Landscape function of some Schr{\o}dinger operator. We prove the existence of minimizers through the equivalence with a weak formulation, and the local Ahlfors regularity and uniform rectifiability of the boundaries $\Omega \cap \partial W_j$.

AAE-Dpeak-SC: A novel unsupervised clustering method for space target ISAR images based on adversarial autoencoder and density peak-spectral clustering

Aiming at the problem of unlabeled ISAR image clustering of space targets, the paper proposes a new unsupervised clustering method based on adversarial autoencoder (AAE) and density peak-spectral clustering (Dpeak-SC). This method introduces the AAE into the unsupervised clustering problem of radar images for the first time and realizes the automatic extraction of the features of unlabeled space target ISAR image data. And on this basis, by combining the density peak algorithm with the spectral clustering algorithm, the ISAR image data is classified into multiple clusters with good clustering accuracy, clustering coincidence, and clustering similarity without providing any prior information. This method first implements the self-supervised training of the AAE model through the adversarial and reconstruction process. Then, it uses the trained encoder to extract the latent features of the ISAR image of the space target in the low-dimensional space. Next, a decision graph is constructed by calculating the local density and relative distance of the proposed features to determine the number of clusters in the hidden feature vector set. Finally, the spectral clustering algorithm is used to transform the problem into the graph's optimal partition problem. The clustering is realized by constructing and segmenting the undirected weight graph of the hidden feature vector set. The simulation test on the unlabeled space target ISAR image data verifies the method's effectiveness in this paper.

Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems

We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous probability measure f mathrm {d}x by a discrete probability measure sum _i m_i delta _{z_i}, subject to a constraint on the particle sizes m_i. The locations z_i of the particles, their sizes m_i, and the number of particles are all unknowns of the problem. We study a one-parameter family of constraints. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, and numerical integration. We establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, the discrete measure whose support is a triangular lattice is asymptotically optimal. In addition, we prove an analogous result for a problem where the constraint is replaced by a penalization. These results can also be viewed as the asymptotic optimality of the hexagonal tiling for an optimal partitioning problem. They generalise the crystallization result of Bourne et al. (Commun Math Phys, 329: 117–140, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitté et al. (J Math Pures Appl, 95:382–419, 2011). Finally, we prove a crystallization result which states that optimal configurations with energy close to that of a triangular lattice are geometrically close to a triangular lattice.

Optimal Partitioning of Multithermal Zone Buildings for Decentralized Control

In this article, we used an optimization-based systematic methodology for the challenging, less studied, and important problem of optimal partitioning of multithermal zone buildings for decentralized control. The proposed methodology consists of: i) construction of a graph-based network to quantitatively characterize the thermal interaction levels between neighboring zones and ii) the application of two different approaches for optimal clustering of the resulting network graph: 1) stochastic optimization and 2) robust optimization. The proposed optimization methods were tested and compared with each other on two case studies: 1) a five-zone building (a small-scale example) which allows one to consider all possible partitions to assess the success rate of the developed methods and 2) a 20-zone building (a large-scale example) for which the developed methods were used to predict the optimal partitioning of the thermal zones. Compared to the existing literature, our approach provides a systematic and potentially optimal solution for the considered problem.

Model abstraction for discrete-event systems by binary linear programming with applications to manufacturing systems.

Model abstraction for finite state automata is helpful for decreasing computational complexity and improving comprehensibility for the verification and control synthesis of discrete-event systems (DES). Supremal quasi-congruence equivalence is an effective method for reducing the state space of DES and its effective algorithms based on graph theory have been developed. In this paper, a new method is proposed to convert the supremal quasi-congruence computation into a binary linear programming problem which can be solved by many powerful integer linear programming and satisfiability (SAT) solvers. Partitioning states to cosets is considered as allocating states to an unknown number of cosets and the requirement of finding the coarsest quasi-congruence is equivalent to using the least number of cosets. The novelty of this paper is to solve the optimal partitioning problem as an optimal state-to-coset allocation problem. The task of finding the coarsest quasi-congruence is equivalent to the objective of finding the least number of cosets. Then the problem can be solved by optimization methods, which are respectively implemented by mixed integer linear programming (MILP) in MATLAB and binary linear programming (BLP) in CPLEX. To reduce the computation time, the translation process is first optimized by introducing fewer decision variables and simplifying constraints in the programming problem. Second, the translation process formulates a few techniques of converting logic constraints on finite automata into binary linear constraints. These techniques will be helpful for other researchers exploiting integer linear programming and SAT solvers for solving partitioning or grouping problems. Third, the computational efficiency and correctness of the proposed method are verified by two different solvers. The proposed model abstraction approach is applied to simplify the large-scale supervisor model of a manufacturing system with five automated guided vehicles. The proposed method is not only a new solution for the coarsest quasi-congruence computation, but also provides us a more intuitive understanding of the quasi-congruence relation in the supervisory control theory. A future research direction is to apply more computationally efficient solvers to compute the optimal state-to-coset allocation problem.

Optimal Matroid Partitioning Problems

This paper studies optimal matroid partitioning problems for various objective functions. In the problem, we are given k weighted-matroids on the same ground set. Our goal is to find a feasible partition that minimizes (maximizes) the value of an objective function. A typical objective is the maximum over all subsets of the total weights of the elements in a subset, which is extensively studied in the scheduling literature. Likewise, as an objective function, we handle the maximum/minimum/sum over all subsets of the maximum/minimum/total weight(s) of the elements in a subset. In this paper, we determine the computational complexity of the optimal partitioning problem with the above-described objective functions. Namely, for each objective function, we either provide a polynomial time algorithm or prove NP-hardness. We also discuss the approximability for the NP-hard cases.

Chance-Constrained Optimal Distribution Network Partitioning to Enhance Power Grid Resilience

This article proposes a method for identifying potential self-adequate sub-networks in the existing power distribution grid. These sub-networks can be equipped with control and protection schemes to form microgrids capable of sustaining local loads during power systems contingencies, thereby mitigating disasters. Towards identifying the best microgrid candidates, this work formulates a chance-constrained optimal distribution network partitioning (ODNP) problem addressing uncertainties in load and distributed energy resources; and presents a solution methodology using the sample average approximation (SAA) technique. Practical constraints like ensuring network radiality and availability of grid-forming generators are considered. Quality of the obtained solution is evaluated by comparison with- a) an upper bound on the probability that the identified microgrids are supply-deficient, and b) a lower bound on the objective value for the true optimization problem. Performance of the ODNP formulation is illustrated through case-studies on a modified IEEE 37-bus feeder. It is shown that the network flexibility is well utilized; the partitioning changes with risk budget; and that the SAA method is able to yield good quality solutions with modest computation cost.

Towards Optimal Matrix Partitioning for Data Parallel Computing on a Hybrid Heterogeneous Server

Optimal partitioning of a square computational domain over several heterogeneous processors, balancing the load of the processors and minimizing the inter-processor communication cost, is crucial for data parallel dense linear algebra and other applications having similar communication pattern on modern hybrid servers. Although a solution has been found for two processors, the cases of three and more processors are still open. The state of-the-art solution for three processors uses an approximation communication cost function which fails to accurately account for the total amount of data moved between processors, leaving thus the question of its global optimality unanswered. In this work, we formulate and solve a mathematical problem of optimal partitioning a real-valued square over three heterogeneous processors using a new cost function, which accurately accounts for the total amount of data communicated between processors. We also develop an original method for accurate experimental evaluation of the communication time of data movement between memories of the compute devices in the hybrid platform during the execution of data parallel applications. We successfully use this method in the experimental validation of our mathematical results. Finally, we propose a communication energy model predicting the dynamic energy consumption of data movement between processors and experimentally validate its accuracy. This model predicts, and the experiments confirm, that the performance-optimal partition is not necessarily energy optimal.

Regularity of all minimizers of a class of spectral partition problems

We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers for a natural relaxed version of the original problem, together with the regularity of eigenfunctions and a universal free boundary condition. Among others, our result covers the cases of the following functional costs $ (\omega_1, \dots, \omega_m) \mapsto \sum\limits_{i = 1}^{m} \left( \sum\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)^{p_i}\right)^{1/p_i}, \quad \prod\limits_{i = 1}^{m} \left( \prod\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)\right), \quad \prod\limits_{i = 1}^{m} \left( \sum\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)\right) $ where $(\omega_1, \dots, \omega_m)$ are the sets of the partition and $\lambda_{j}(\omega_i)$ is the $j$-th Laplace eigenvalue of the set $\omega_i$ with zero Dirichlet boundary conditions.

Construction of a generalized Voronoi diagram with optimal placement of generator points based on the theory of optimal set partitioning

The problem of construction of a generalized Voronoi diagram with optimal placement of a finite number of generator points in a bounded set of \textit{n}-dimensional Euclidean space is considered. A method is proposed for solving such a problem based on the formulation of the corresponding continuous problem of optimal partitioning of a set in \textit{n}-dimensional Euclidean space with a partition quality criterion that provides the corresponding form of the Voronoi diagram. Further, to solve such a problem, the developed mathematical and algorithmic apparatus is used, the part of which is Shor's \textit{r}-algorithm.

Algorithm for Constructing Voronoi Diagrams with Optimal Placement of Generator Points Based on Theory of Optimal Set Partitioning

An algorithm is proposed for constructing a generalized Voronoi diagram with optimal placement of a finite number of generator points in a bounded set of n-dimensional Euclidean space. The algorithm is based on the formulation of the corresponding continuous problem of optimal set partitioning with a partition quality criterion providing the corresponding form of Voronoi diagram, and on the application of the apparatus of the optimal partitioning theory to solve this problem. Herewith, the efficient method of non-differentiable optimization is used for the numerical solution of an auxiliary finite-dimensional optimization problem arising in the development of the method for solving the mentioned infinite-dimensional problem of optimal set partitioning. Namely, that is one of the variants of the generalized gradient descent method with space expansion in the direction of the difference of two successive generalized antigradients (Shor's r-algorithm). The proposed algorithm for constructing a generalized Voronoi diagram with optimal placement of a finite number of generator points in a bounded set of n-dimensional Euclidean space has some advantages compared to those known in a scientific literature. It does not depend on a dimension of Euclidean space containing the initial bounded set; it is applicable for the large-scale problems (over 300 generator points); it works not only for Euclidean metrics, but also for Chebyshev, Manhattan and other ones; the complexity of the algorithm implementation for constructing Voronoi diagram based on the proposed approach does not increase with increasing number of generator points. The results of software implementation of the developed algorithm are presented for constructing a standard Voronoi diagram with an optimal placement of generator points as well as some of its generalizations such as additive, multiplicative and additive-multiplicative Voronoi diagrams with optimal placement of generator points.

Solving a Two-Stage Continuous-Discrete Problem of Optimal Partition–Allocation with a Given Position of the Centers of Subsets

A method and algorithm of solving a two-stage continuous-discrete problem of optimal partition–allocation are proposed. On the one hand, the problem is a generalization of the classical transportation problem in the case where production (storage and recycling) amounts at given points are unknown in advance, and are sought as a solution of the corresponding continuous problem of optimal partitioning of a set of continuously allocated consumers (suppliers) onto their service areas by these points. On the other hand, the problem generalizes discrete two-stage production–transportation problems to the case of a continuously allocated consumer. Operation of the proposed algorithm is demonstrated by solving a model problem.

Solving a two-step continuous-discrete optimal split-distribution problem with fuzzy parameters

The theory of optimal set partitioning from an n-dimensional Euclidean space En is an important part of infinite-dimensional mathematical programming. The mostly reason of high interest in development of the theory of optimal set partitioning is that its results can be applied to solving the classes of different theoretical and applied optimization problems, which are transferred into continuous optimal set partitioning problem. This paper investigates the further development of the theory of optimal set partitioning from En in the case of a two-stage continuous-discrete problem of optimal partitioningdistribution with non-determined input data, which is frequently appear in solving practical problems. The two-stage continuous-discrete problem of optimal partition-distribution under constraints in the form of equations and determined position of centers of subsets is generalized by proposed continuous-discrete problem of optimal partition-distribution in case if some parameters are presented in incomplete, inaccurate or unreliable form. These parameters can be represented as linguistic variables and the method of neurolinguistic identification of unknown complex, nonlinear dependencies can be used in purpose to recovery them. A method for solving the two-stage continuous-discrete optimal partitioning-distribution problem with fuzzy parameters in target functional which based on usage of neurolinguistic identification of unknown dependencies for recovering precise values of fuzzy parameters, methods of the theory of optimal set partitioning and the method of potentials for solving a transportation problem is proposed.

Construction of a multiplicatively weighted diagram of a crow with fuzzy parameters

An algorithm for constructing a multiplicatively weighted Voronoi diagram in the presence of fuzzy parameters with optimal location of a finite number of generator points in a bounded set of n-dimensional Euclidean space En is proposed in the paper. The algorithm is based on the formulation of a continuous set partitioning problem from En into non-intersecting subsets with a partitioning quality criterion providing the corresponding form of Voronoi diagram. Algorithms for constructing the classical Voronoi diagram and its various generalizations, which are based on the usage of the methods of the optimal set partitioning theory, have several advantages over the other used methods: they are out of thedependence of En space dimensions, which containing a partitioned bounded set into subsets, independent of the geometry of the partitioned sets, the algorithm’s complexity is not growing under increasing of number of generator points, it can be used for constructing the Voronoi diagram with optimal location of the points and others. The ability of easily construction not only already known Voronoi diagrams but also the new ones is the result of this general-purpose approach. The proposed in the paper algorithm for constructing a multiplicatively weighted Voronoi diagram in the presence of fuzzy parameters with optimal location of a finite number of generator points in a bounded set of n-dimensional Euclidean space En is developed using a synthesis of methods for solving optimal set partitioning problems, neurofuzzy technologies and modifications of the Shor’s r-algorithm for solving non-smooth optimization problems.

On the Singular Set of Free Interface in an Optimal Partition Problem

AbstractWe study the singular set of free interface in an optimal partition problem for the Dirichlet eigenvalues. We prove that its upper (n − 2)‐dimensional Minkowski content, and consequently its (n − 2)‐dimensional Hausdorff measure, are locally finite. We also show that the singular set is countably (n − 2)‐rectifiable; namely, it can be covered by countably many C1‐manifolds of dimension (n − 2), up to a set of (n − 2)‐dimensional Hausdorff measure zero. Our results hold for optimal partitions on Riemannian manifolds and harmonic maps into homogeneous trees as well. © 2019 Wiley Periodicals, Inc.

Adaptive Capacity Partitioning in Cooperative Computing to Maximize Received Resources

Users in a cooperative computing environment always have a tendency to free-ride. One can employ incentive mechanisms to prevent such behavior. Some of the cooperative computing scenarios have the same access link shared between upload and download. In such a situation, increasing upload capacity decreases the download capacity and vice versa. Optimal partitioning of link capacity between upload and download needs to be done by each user to maximize its gain (i.e., download) from the network. We model this link capacity partitioning problem as a feedback control system, where feedback (resources received) decides the number of resources to be uploaded by a user. The resulting algorithm called adaptive step size (ASZ) dynamically adjusts the partitioning of link capacity to an optimal value. To compare this approach with others, a metric “level of optimality (U)”is introduced. U achieved by the ASZ is closer to the optimal level than the reputation-based resource allocation policy (existing scheme), thus resulting in its better performance. The ASZ is also integrated with BitTorrent, and the simulation results show that it increases the resources received by the users. The ASZ can provide an efficient solution to the problem of optimal partitioning in real-life distributed networks due to its distributed implementation, robustness to changes in network dynamics, and compatibility with the existing partitioning schemes.

Regularity results for segregated configurations involving fractional Laplacian

We study the regularity of segregated profiles arising from competition–diffusion models, where the diffusion process is of nonlocal type and is driven by the fractional Laplacian of power s∈(0,1). Among others, our results apply to the regularity of the densities of an optimal partition problem involving the eigenvalues of the fractional Laplacian. More precisely, we show C0,α∗ regularity of the density, where the exponent α∗ is explicit and is given by α∗=sfors∈(0,1∕2]2s−1fors∈(1∕2,1).Under some additional assumptions, we then show that solutions are C0,s. These results are optimal in the class of Hölder continuous functions. Thus, we find a complete correspondence with known results in case of the standard Laplacian.