Diagonal Constraints Research Articles

Distributed Synchronous and Asynchronous Algorithms for Semidefinite Programming With Diagonal Constraints

This article develops distributed synchronous and asynchronous algorithms for the large-scale semidefinite programming with diagonal constraints, which has wide applications in combinatorial optimization, image processing, and community detection. The information of the semidefinite programming is allocated to multiple interconnected agents such that each agent aims to find a solution by communicating to its neighbors. Based on the low-rank property of solutions and the Burer–Monteiro factorization, we transform the original problem into a distributed optimization problem over unit spheres to reduce variable dimensions and ensure positive semidefiniteness without involving semidefinite projections, which are computationally expensive. For the distributed optimization problem, we propose distributed synchronous and asynchronous algorithms, both of which reduce computational burden and storage space compared with existing centralized algorithms. Specifically, the distributed synchronous algorithm almost surely escapes strict saddle points and converges to the set of optimal solutions to the optimization problem. In addition, the proposed distributed asynchronous algorithm allows communication delays and converges to critical points to the optimization problem under mild conditions. By applying the proposed algorithms to image segmentation, we illustrate the efficiency and convergence performance of the two proposed algorithms.

Block Diagonal Least Squares Regression for Subspace Clustering

Least squares regression (LSR) is an effective method that has been widely used for subspace clustering. Under the conditions of independent subspaces and noise-free data, coefficient matrices can satisfy enforced block diagonal (EBD) structures and achieve good clustering results. More importantly, LSR produces closed solutions that are easier to solve. However, solutions with block diagonal properties that have been solved using LSR are sensitive to noise or corruption as they are fragile and easily destroyed. Moreover, when using actual datasets, these structures cannot always guarantee satisfactory clustering results. Considering that block diagonal representation has excellent clustering performance, the idea of block diagonal constraints has been introduced into LSR and a new subspace clustering method, which is named block diagonal least squares regression (BDLSR), has been proposed. By using a block diagonal regularizer, BDLSR can effectively reinforce the fragile block diagonal structures of the obtained matrices and improve the clustering performance. Our experiments using several real datasets illustrated that BDLSR produced a higher clustering performance compared to other algorithms.

Convergence rate of block-coordinate maximization Burer–Monteiro method for solving large SDPs

Semidefinite programming (SDP) with diagonal constraints arise in many optimization problems, such as Max-Cut, community detection and group synchronization. Although SDPs can be solved to arbitrary precision in polynomial time, generic convex solvers do not scale well with the dimension of the problem. In order to address this issue, Burer and Monteiro (Math Program 95(2):329–357, 2003) proposed to reduce the dimension of the problem by appealing to a low-rank factorization and solve the subsequent non-convex problem instead. In this paper, we present coordinate ascent based methods to solve this non-convex problem with provable convergence guarantees. More specifically, we prove that the block-coordinate maximization algorithm applied to the non-convex Burer–Monteiro method globally converges to a first-order stationary point with a sublinear rate without any assumptions on the problem. We further show that this algorithm converges linearly around a local maximum provided that the objective function exhibits quadratic decay. We establish that this condition generically holds when the rank of the factorization is sufficiently large. Furthermore, incorporating Lanczos method to the block-coordinate maximization, we propose an algorithm that is guaranteed to return a solution that provides \(1-{\mathcal {O}}\left( 1/r\right) \) approximation to the original SDP without any assumptions, where r is the rank of the factorization. This approximation ratio is known to be optimal (up to constants) under the unique games conjecture, and we can explicitly quantify the number of iterations to obtain such a solution.

Robust subspace clustering based on automatic weighted multiple kernel learning

Multiple kernel learning (MKL), which combines a set of prespecified basic kernels to improve the clustering performance, has become an important research topic. Unfortunately, the current methods have the following defects in noisy circumstances. 1) Their clustering performance may be significantly reduced due to the noise in the kernel, which is caused by the lack of a reliable discriminant guideline for basic kernel combinations. 2) The noise from corrupted data or occlusion may destroy the block-diagonal structures of the affinity matrices they obtained, which will affect the clustering performance when using spectral clustering. In this work, to solve the above problems, we propose an automatic weighted multikernel learning-based robust subspace clustering (AWLKSC) algorithm. The model integrates multikernel learning strategies, the Correntropy-Induced Metric (CIM), low rank approximation technology and block diagonal constraints. In addition, an effective AM&GST algorithm, which is integrated by alternating minimization and generalized soft-thresholding, is developed to optimize the AWLKSC. Seven types of noise are considered in the experiments, and the experimental results illustrate that AWLKSC is more effective and robust than several up-to-date single kernel and multiple kernel clustering methods.

Configurable verification of timed automata with\xa0discrete\xa0variables

Algorithms and protocols with time dependent behavior are often specified formally using timed automata. For practical real-time systems, besides real-valued clock variables, these specifications typically contain discrete data variables with nontrivial data flow. In this paper, we propose a configurable lazy abstraction framework for the location reachability problem of timed automata that potentially contain discrete variables. Moreover, based on our previous work, we uniformly formalize in our framework several abstraction refinement strategies for both clock and discrete variables that can be freely combined, resulting in many distinct algorithm configurations. Besides the proposed refinement strategies, the configurability of the framework allows the integration of existing efficient lazy abstraction algorithms for clock variables based on {textit{LU}}-bounds. We demonstrate the applicability of the framework and the proposed refinement strategies by an empirical evaluation on a wide range of timed automata models, including ones that contain discrete variables or diagonal constraints.

CONVECTIVE HEAT TRANSFER DISCRETE MODELING

The article presents the investigation results of the heat transfer process in a moving continuous medium. The lattice Boltzmann model (LBM) was used. This model is based on the principles of the cellular automata system. For the study, an orthogonal spatial lattice with diagonal constraints was adopted, by means of which the particle distribution functions were introduced from a discrete set of allowed velocities. These distribution functions were described by a discrete analogue of the Boltzmann kinetic equation. This approach made it possible to study the evolution of the distribution functions of particles in a continuous medium as a fuction of discrete-time steps. The model used made it possible to describe both mechanisms of thermal energy transfer in a moving medium – macroscopic and microscopic. In this case, the macroscopic transfer due to the motion of a continuous medium was determined by a change in the density of the particle distribution, and the microscopic (molecular) transport was determined by the relaxation heat exchange operator. This operator of the mathematical model characterized the redistribution of heat in a discrete region due to the collision of particles, that is, it takes into account the thermal conductivity of the medium. Since not only the moving continuous medium but also the boundary surfaces (walls, obstacles) participate in the heat exchange process, the elements taking into account this fact were included in the model. To test an adequacy of the approach used a software application was developed. It was used to simulate and visualize the process of heat transfer by a moving continuous medium. The application also allowed setting confining surfaces of various forms. An analysis of the computer experiment results allows us to state that the obtained data do not contradict to the real concepts of the processes of heat transfer in a moving fluid. The advantage of the proposed discrete approach is the possibility of describing hydrodynamic and thermal processes within the framework of a single model, which makes it quite convenient to use. In addition, this method makes it possible to solve heat transfer problems for objects that have a complex geometric configuration of the interfaces. Forcitation:Chernyavskaya A.S., Bobkov S.P. Convective heat transfer discrete modeling. Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 2018. V. 61. N 2. P. 86-90

Implicit Block Diagonal Low-Rank Representation

While current block diagonal constrained subspace clustering methods are performed explicitly on the original data space, in practice, it is often more desirable to embed the block diagonal prior into the reproducing kernel Hilbert feature space by kernelization techniques, as the underlying data structure in reality is usually nonlinear. However, it is still unknown how to carry out the embedding and kernelization in the models with block diagonal constraints. In this paper, we shall take a step in this direction. First, we establish a novel model termed implicit block diagonal low-rank representation (IBDLR), by incorporating the implicit feature representation and block diagonal prior into the prevalent low-rank representation method. Second, mostly important, we show that the model in IBDLR could be kernelized by making use of a smoothed dual representation and the specifics of a proximal gradient-based optimization algorithm. Finally, we provide some theoretical analyses for the convergence of our optimization algorithm. Comprehensive experiments on synthetic and real-world data sets demonstrate the superiorities of our IBDLR over state-of-the-art methods.

Diagonal latent block model for binary data

This paper addresses the problem of co-clustering binary data in the latent block model framework with diagonal constraints for resulting data partitions. We consider the Bernoulli generative mixture model and present three new methods differing in the assumptions made about the degree of homogeneity of diagonal blocks. The proposed models are parsimonious and allow to take into account the structure of a data matrix when reorganizing it into homogeneous diagonal blocks. We derive algorithms for each of the presented models based on the classification expectation-maximization algorithm which maximizes the complete data likelihood. We show that our contribution can outperform other state-of-the-art (co)-clustering methods on synthetic sparse and non-sparse data. We also prove the efficiency of our approach in the context of document clustering, by using real-world benchmark data sets.

Improved Mainlobe Interference Suppression Based on Blocking Matrix Preprocess

For the problem of mainlobe direction shifting that is caused by the mainlobe interference suppression based on blocking matrix preprocess, an effective method is proposed which is based on the combination of diagonal loading and linear constraints. Therein, the reason for mainlobe direction shifting is analyzed and found to be that the covariance matrix mismatch exists in the realization of the adaptive beamforming. Therefore, the diagonal loading processing is used to overcome the mismatch and correct the mainlobe direction shifting, and the linear constraints are used to make sure of the beam pattern nulling in the interference directions; then the desired performance of adaptive beamforming is obtained. Simulation results attest the correctness and effectiveness of the proposed method, and they also show that the proposed method is insensitive to the selection of diagonal loading level, which means the loading factor is easy to choose.

Sparsity Inducing Prior Distributions for Correlation Matrices of Longitudinal Data

For longitudinal data, the modeling of a correlation matrix R can be a difficult statistical task due to both the positive definite and the unit diagonal constraints. Because the number of parameters increases quadratically in the dimension, it is often useful to consider a sparse parameterization. We introduce a pair of prior distributions on the set of correlation matrices for longitudinal data through the partial autocorrelations (PACs), which vary independently over (−1,1). The first prior shrinks each of the PACs toward zero with increasingly aggressive shrinkage in lag. The second prior (a selection prior) is a mixture of a zero point mass and a continuous component for each PAC, allowing for a sparse representation. The structure implied under our priors is readily interpretable for time-ordered responses because each zero PAC implies a conditional independence relationship in the distribution of the data. Selection priors on the PACs provide a computationally attractive alternative to selection on the elements of R or R− 1 for ordered data. These priors allow for data-dependent shrinkage/selection under an intuitive parameterization in an unconstrained setting. The proposed priors are compared to standard methods through a simulation study and illustrated using a multivariate probit data example. Supplemental materials for this article (appendix, data, and R code) are available online.

Decentralized linear quadratic power system stabilizers for multi-machine power systems

Linear quadratic stabilizers are well-known for their superior control capabilities when compared to the conventional lead–lag power system stabilizers. However, they have not seen much of practical importance as the state variables are generally not measurable; especially the generator rotor angle measurement is not available in most of the power plants. Full state feedback controllers require feedback of other machine states in a multi-machine power system and necessitate block diagonal structure constraints for decentralized implementation. This paper investigates the design of Linear Quadratic Power System Stabilizers using a recently proposed modified Heffron–Phillip’s model. This model is derived by taking the secondary bus voltage of the step-up transformer as reference instead of the infinite bus. The state variables of this model can be obtained by local measurements. This model allows a coordinated linear quadratic control design in multi machine systems. The performance of the proposed controller has been evaluated on two widely used multi-machine power systems, 4 generator 10 bus and 10 generator 39 bus systems. It has been observed that the performance of the proposed controller is superior to that of the conventional Power System Stabilizers (PSS) over a wide range of operating and system conditions.

Flavor symmetric sectors and collider physics

We discuss the phenomenology of effective field theories with new scalar or vector representations of the Standard Model quark flavor symmetry group, allowing for large flavor breaking involving the third generation. Such field content can have a relatively low mass scale \lesssim TeV and O(1) couplings to quarks, while being naturally consistent with both flavor violating and flavor diagonal constraints. These theories therefore have the potential for early discovery at LHC, and provide a flavor safe "tool box" for addressing anomalies at colliders and low energy experiments. We catalogue the possible flavor symmetric representations, and consider applications to the anomalous Tevatron t-tbar forward backward asymmetry and B_s mixing measurements, individually or concurrently. Collider signatures and constraints on flavor symmetric models are also studied more generally. In our examination of the t-tbar forward backward asymmetry we determine model independent acceptance corrections appropriate for comparing against CDF data that can be applied to any model seeking to explain the t-tbar forward backward asymmetry.

Time–Frequency Cepstral Features and Heteroscedastic Linear Discriminant Analysis for Language Recognition

The shifted delta cepstrum (SDC) is a widely used feature extraction for language recognition (LRE). With a high context width due to incorporation of multiple frames, SDC outperforms traditional delta and acceleration feature vectors. However, it also introduces correlation into the concatenated feature vector, which increases redundancy and may degrade the performance of backend classifiers. In this paper, we first propose a time-frequency cepstral (TFC) feature vector, which is obtained by performing a temporal discrete cosine transform (DCT) on the cepstrum matrix and selecting the transformed elements in a zigzag scan order. Beyond this, we increase discriminability through a heteroscedastic linear discriminant analysis (HLDA) on the full cepstrum matrix. By utilizing block diagonal matrix constraints, the large HLDA problem is then reduced to several smaller HLDA problems, creating a block diagonal HLDA (BDHLDA) algorithm which has much lower computational complexity. The BDHLDA method is finally extended to the GMM domain, using the simpler TFC features during re-estimation to provide significantly improved computation speed. Experiments on NIST 2003 and 2007 LRE evaluation corpora show that TFC is more effective than SDC, and that the GMM-based BDHLDA results in lower equal error rate (EER) and minimum average cost (Cavg) than either TFC or SDC approaches.

Correlators of vertex operators for circular strings with winding numbers in AdS 5 × S 5

We compute semiclassically the two-point correlator of the marginal vertex operators describing the rigid circular spinning string state with one large spin and one windining number in AdS_5 and three large spins and three winding numbers in S^5. The marginality condition and the conformal invariant expression for the two-point correlator obtained by using an appropriate vertex operator are shown to be associated with the diagonal and off-diagonal Virasoro constraints respectively. We evaluate semiclassically the three-point correlator of two heavy circular string vertex operators and one zero-momentum dilaton vertex operator and discuss its relation with the derivative of the dimension of the heavy circular string state with respect to the string tension.

Simplicity in simplicial phase space

A key point in the spin foam approach to quantum gravity is the implementation of simplicity constraints in the partition functions of the models. Here, we discuss the imposition of these constraints in a phase space setting corresponding to simplicial geometries. On the one hand, this could serve as a starting point for a derivation of spin foam models by canonical quantization. On the other, it elucidates the interpretation of the boundary Hilbert space that arises in spin foam models. More precisely, we discuss different versions of the simplicity constraints, namely, gauge-variant and gauge-invariant versions. In the gauge-variant version, the primary and secondary simplicity constraints take a similar form to the reality conditions known already in the context of (complex) Ashtekar variables. Subsequently, we describe the effect of these primary and secondary simplicity constraints on gauge-invariant variables. This allows us to illustrate their equivalence to the so-called diagonal, cross and edge simplicity constraints, which are the gauge-invariant versions of the simplicity constraints. In particular, we clarify how the so-called gluing conditions arise from the secondary simplicity constraints. Finally, we discuss the significance of degenerate configurations, and the ramifications of our work in a broader setting.

Non-orthogonal joint diagonalization with diagonal constraints

Joint diagonalization has attracted much attention and many algorithms have been presented so far. However, some ambiguities still exist in the objective functions for joint diagonalization. In this paper, some criterions are proposed to eliminate the above ambiguities at first, and then a new objective function which satisfies these criterions is presented. The new objective function introduces diagonal constraints for joint diagonalization, thus the trivial and unbalance solutions are excluded easily. Exactly jointly diagonalizable theorem is built to interpret the reasonableness of the new objective function and the conjugate gradient method is used to provide fast and reliable convergence. Finally, non-orthogonal joint diagonalization algorithm with diagonal constraints (DDiag) is developed. Simulations show that DDiag is efficient and robust.

Markovian credit transition probabilities under inequality constraints: the US portfolio 1984–2004

Credit risk transition probabilities between aggregate portfolio classes constitute a very useful tool when individual transition data is not available. Jones (2005) estimates Markovian credit transition matrices using an adjusted least squares method. Given the arguments of Judge and Takayama (1966), a least squares estimator under inequality constraints is consistent but has unknown distribution, thus parameter testing is essentially not immediately available. In this paper, we view transition probabilities as parameters from a Bayesian perspective, which allows us to impose the non-negativity and diagonal dominance constraints to transition probabilities using prior densities and then estimate the model through Monte Carlo integration (MCI). This approach reveals the empirical posterior distribution of transition probabilities and makes statistical inference readily available. Our empirical results as compared with Jones (2005) suggest that least squares-based methods tend to overestimate non-transition and underestimate transition probabilities, especially toward upgrades. Furthermore, in-sample forecast evaluation statistics indicate that our estimator tends to slightly over-predict (under-predict) non-performing (performing) loan proportions consistent with prudent asymmetric preferences and is substantially more accurate in all cases.

Concurrency in timed automata

We introduce Concurrent Timed Automata ( CTAs) where automata running in parallel are synchronized. We consider the subclasses of CTAs obtained by admitting, or not, diagonal clock constraints and constant updates, and by letting, or not, sequential automata to update the same clocks. We prove that such subclasses recognize the same languages but differ w.r.t. the succinctness of the models. Moreover, we distinguish between subclasses that are polynomially closed w.r.t. finite union and finite intersection, and subclasses that do not have such properties.

A specially structured nonlinear integer resource allocation problem

AbstractWe present an algorithm for solving a specially structured nonlinear integer resource allocation problem. This problem was motivated by a capacity planning study done at a large Health Maintenance Organization in Texas. Specifically, we focus on a class of nonlinear resource allocation problems that involve the minimization of a convex function over one general convex constraint, a set of block diagonal convex constraints, and bounds on the integer variables. The continuous variable problem is also considered. The continuous problem is solved by taking advantage of the structure of the Karush‐Kuhn‐Tucker (KKT) conditions. This method for solving the continuous problem is then incorporated in a branch and bound algorithm to solve the integer problem. Various reoptimization results, multiplier bounding results, and heuristics are used to improve the efficiency of the algorithms. We show how the algorithms can be extended to obtain a globally optimal solution to the nonconvex version of the problem. We further show that the methods can be applied to problems in production planning and financial optimization. Extensive computational testing of the algorithms is reported for a variety of applications on continuous problems with up to 1,000,000 variables and integer problems with up to 1000 variables. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 770–792, 2003.

Primal-dual methods for sustainable harvest scheduling problems

Timber harvest schedules form the heart of forest resource management. The operations research methodology is applied to resolving the multi-product, multi-period sustainable timber harvest scheduling problems, where area constraints, harvest flow constraints, and ending inventory constraints are imposed. By taking advantage of the model characterization of block diagonal constraints with multiple sets of network sub-problems and the set of coupling constraints, an efficient algorithm is explored. Specifically, a primal-dual method of closed-form solutions is first developed to solve the network sub-problems on the individual basis. Then, a primal-dual steepest-edge algorithm that achieves the global optimum is presented. A numerical example illustrating steps of the solution procedure is presented. The proposed algorithm is implemented, and its performance is compared with that of the AMPL-CPLEX package. The proposed primal-dual algorithm achieves, on the average, an approximately four to one reduction in iteration numbers and an about eight to one reduction in the CPU execution time. Scope and purpose Timber harvest schedules form the heart of forest resource management. This paper presents the model formulation, structure identification, and algorithm explored for resolving the multi-product, multi-period harvest scheduling problem, where area constraints, harvest flow constraints, and ending inventory constraints serve as the basis of a sustainable forest policy. The work applies the mathematical programming methodology to the forest management regime. To streamline the solution procedure, we first identify the block diagonal constraint structure, the network sub-problems, and a set of coupling constraints. By employing a primal-dual algorithm using only closed-form solutions that resolves the network sub-problems, we explore a primal-dual steepest-edge algorithm that achieves the optimal harvest schedule. The proposed algorithm is coded in the Borland C++ programming language and run on a personal computer. For comparison, the package of AMPL-CPLEX and the proposed method are applied to solving a set of arbitrarily generated sample problems. The proposed primal-dual algorithm outperforms the general-purpose package in both iteration numbers and the CPU execution time.