Decomposition Method For Problems Research Articles

Code and Data Repository for Penalty Decomposition Methods for Second-Best Congestion Pricing Problems on Large-Scale Networks

he second-best congestion pricing (SBCP) problem is one of the most challenging problems in transportation due to its two-level hierarchical structure. In spite of various intriguing attempts for solving SBCP, existing solution methods are either heuristic without convergence guarantee or suitable for solving SBCP on small networks only. In this paper, we first reveal some convexity-based structural properties of the marginal value function reformation of SBCP and then, by effectively exploiting these structural properties, we propose two dedicated decomposition methods for solving SBCP on large-scale networks which are different from existing methods in that they avoid linearizing nonconvex functions. We establish the convergence of the two decomposition methods under commonly used conditions and provide the maximum number of iterations for deriving an approximate stationary solution. The computational experiments based on a collection of real road networks show that in comparison with three existing popular methods, the two proposed methods are capable of solving SBCP on larger-scale networks; and for instances that can be solved by existing methods, the two proposed methods are substantially faster.

Recent advances in domain decomposition methods for large-scale saddle point problems

Scalability of parallel solvers for problems with high heterogeneities relies on adaptive coarse spaces built from generalized eigenvalue problems in the subdomains. The corresponding theory is powerful and flexible but the development of an efficient parallel implementation is challenging. We report here on recent advances in adaptive coarse spaces and on their open source implementations in domain specific languages such as FreeFem, focusing on a new domain decomposition for saddle point formulations with some numerical tests.

New Time Domain Decomposition Methods for Parabolic Optimal Control Problems I: Dirichlet–Neumann and Neumann–Dirichlet Algorithms

New Time Domain Decomposition Methods for Parabolic Optimal Control Problems I: Dirichlet–Neumann and Neumann–Dirichlet Algorithms

Similarity-based decomposition algorithm for two-stage stochastic scheduling

This paper presents a novel decomposition method for two-stage stochastic mixed-integer optimization problems. The algorithm builds upon the idea of similarity between finite sample sets to measure how similar the first-stage decisions are among the uncertainty realization scenarios. Using such a Similarity Index, the non-anticipative constraints are removed from the problem formulation so that the original problem becomes block-separable on a scenario basis. Then, a term for maximizing the Similarity Index is included in all the sub-problems objective functions. Such sub-problems are solved iteratively in parallel so that their solutions are used to update the weighting parameter for maximizing the Similarity Index. The algorithm obtains a feasible solution when full similarity among scenario first stages is reached, that is, when the incumbent solution is non-anticipative. The proposal is tested in four instances of different sizes of an industrial-like scheduling problem. Comparison results show that the Similarity Index Decomposition provides significant speed-ups compared with the monolithic problem formulation, and provides simpler tuning and improved convergence over the Progressive Hedging Algorithm.

Objective contribution decomposition method and multi-population optimization strategy for large-scale multi-objective optimization problems

As problem complexity and data dimensionality increase, practical problems tend to have a large number of variables. Large-scale multi-objective optimization problems (LSMOPs) are a complex and difficult class of problems with numerous decision variables to be optimized. In addressing these problems using traditional evolutionary algorithms, achieving convergence is often difficult. The decomposition of decision variables and optimization is a more intuitive way to deal with this problem, which reduces the size of the problem to some extent. Thus, in this study, an objective contribution decomposition method and multi-population optimization strategy are proposed for handling LSMOPs. First, the decision variables are decomposed according to their contribution to the objectives. Further decomposition is performed based on the interactions among the decision variables. Then, multiple populations are generated to optimize the variable group and determine the information interaction among the subpopulations. In experimental results, the performance of the proposed algorithm was competitive in comparisons with six state-of-the-art algorithms.

A GenEO Domain Decomposition method for Saddle Point problems

We introduce an adaptive element-based domain decomposition (DD) method for solving saddle point problems defined as a block two by two matrix. The algorithm does not require any knowledge of the constrained space. We assume that all sub matrices are sparse and that the diagonal blocks are spectrally equivalent to a sum of positive semi definite matrices. The latter assumption enables the design of adaptive coarse space for DD methods that extends the GenEO theory (Spillane et al., 2014) to saddle point problems. Numerical results on three dimensional elasticity problems for steel-rubber structures discretized by a finite element with continuous pressure are shown for up to one billion degrees of freedom.

Robust domain decomposition methods for high-contrast multiscale problems on irregular domains with virtual element discretizations

Our research focuses on the development of domain decomposition preconditioners tailored for second-order elliptic partial differential equations. Our approach addresses two major challenges simultaneously: i) effectively handling coefficients with high-contrast and multiscale properties, and ii) accommodating irregular domains in the original problem, the coarse mesh, and the subdomain partition. The robustness of our preconditioners is crucial for real-world applications, such as the efficient and accurate modeling of subsurface flow in porous media and other important domains.The core of our method lies in the construction of a suitable partition of unity functions and coarse spaces utilizing local spectral information. Leveraging these components, we implement a two-level additive Schwarz preconditioner. We demonstrate that the condition number of the preconditioned systems is bounded with a bound that is independent of the contrast. Our claims are further substantiated through selected numerical experiments, which confirm the robustness of our preconditioners.

Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls

We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method.

Deflated domain decomposition method for structural problems

The paper presents a fast and stable solver algorithm for structural problems. The point is the distance between the eigenvector of the constrained stiffness matrix and the unconstrained matrix. The coarse motions are close to the kernel of the unconstrained matrix. We use lower-frequency deformation modes to construct an iterative solver algorithm through domain decomposition expressing near-rigid-body motions, deflation algorithms, and two-level algorithms. We remove the coarse space from the solution space and hand over the iteration space to the fine space. Our solver is parallelized, and the solver thus has two sets of domain decomposition. One decomposition generates the coarse space, and the other is for parallelization. The basic framework of the solver is the parallel conjugate gradient (CG) method on the fine space. We use the CG method for the basic framework instead of the (simplest) domain decomposition method. We conducted benchmark tests using elastic static analysis for thin plate models. A comparison with the standard CG solver results shows the new solver’s high-speed performance and remarkable stability.

Dynamic fluorescence molecular tomography metabolic parameters solution based on problem decomposition and prior refactor.

Dynamic fluorescence molecular tomography (DFMT), as a noninvasive optical imaging method, can quantify metabolic parameters of living animal organs and assist in the diagnosis of metabolic diseases. However, existing DFMT methods do not have a high capacity to reconstruct abnormal metabolic regions, and require additional prior information and complicated solution methods. This paper introduces a problem decomposition and prior refactor (PDPR) method. The PDPR decomposes the metabolic parameters into two kinds of problems depending on their temporal coupling, which are solved using regularization and parameter fitting. Moreover, PDPR introduces the idea of divide-and-conquer to refactor prior information to ensure discrimination between metabolic abnormal regions and normal tissues. Experimental results show that PDPR is capable of separating abnormal metabolic regions of the liver and has the potential to quantify metabolic parameters and diagnose liver metabolic diseases in small animals.

Domain Decomposition Methods for Diffusion Problems with Discontinuous Coefficients Revisited

Domain Decomposition Methods for Diffusion Problems with Discontinuous Coefficients Revisited

Karhunen–Loeve orthogonal decomposition method for problems of EEG assessment of patients with migraine

The purpose of this work is to identify patterns in the recordings of electroencephalograms of patients with migraine using the Karhunen–Loeve orthogonal decomposition method. The work examines the main features of electroencephalographic dynamics, and the impact on these features of сhronic migraine severity. Methods. To collect experimental data, the method of recording electroencephalograms during the modified multiple sleep latency test was used. During the experiment, studies were conducted of the subjects´reaction to the presented visual stimulus. The obtained data were processed using the Karhunen–Loeve transformation, which allows one to interpret the complex dynamics of the system from the point of view of the coexistence and interaction of coherent orthogonal space-time structures. Results. Studies have shown that the energy distribution of modes in active and sleep states can differ significantly. The character of this distribution depends on the brain zone of signal recordings, on the duration of the experiment, and on at what time point in the experiment certain stages of the subject’s reaction were recorded. It has been shown that the greatest response in the form of evoked potentials in people with migraine is most often localized in the occipital lobe of the brain, and there is a correlation of this effect with the frequency of migraine attacks. For some groups of patients, there is a connection between the severity of evoked potentials in the brain and the energy of the first, most energetic, Karhunen–Loeve mode. Conclusion. It has been shown that there is a relationship between the number of significant modes and the power of the alpha rhythm in electroencephalography signals, and the spatial localization of this effect in the occipital region of the brain can be traced. For the frontal lobe of the brain, significant differences in the distribution of the first mode were demonstrated, assessed for groups of patients with rare and frequent migraine attacks.

Multiscale hierarchical decomposition methods for ill-posed problems

The multiscale hierarchical decomposition method (MHDM) was introduced in Tadmor et al (2004 Multiscale Model. Simul. 2 554–79; 2008 Commun. Math. Sci. 6 281–307) as an iterative method for total variation (TV) regularization, with the aim of recovering details at various scales from images corrupted by additive or multiplicative noise. Given its success beyond image restoration, we extend the MHDM iterates in order to solve larger classes of linear ill-posed problems in Banach spaces. Thus, we define the MHDM for more general convex or even nonconvex penalties, and provide convergence results for the data fidelity term. We also propose a flexible version of the method using adaptive convex functionals for regularization, and show an interesting multiscale decomposition of the data. This decomposition result is highlighted for the Bregman iteration method that can be expressed as an adaptive MHDM. Furthermore, we state necessary and sufficient conditions when the MHDM iteration agrees with the variational Tikhonov regularization, which is the case, for instance, for one-dimensional TV denoising. Finally, we investigate several particular instances and perform numerical experiments that point out the robust behavior of the MHDM.

An asynchronous parallel benders decomposition method for stochastic network design problems

Benders decomposition (BD) is one of the most popular solution algorithms for stochastic integer programs. The BD method decomposes stochastic problems into one master problem and multiple disjoint subproblems. It thus lends itself readily to parallelization. In almost all studies on the parallelization of this algorithm, the master problem remains idle until every subproblem is solved and vice versa. This can clearly result in an extremely inefficient parallel algorithm due to excessive idle times. On the other hand, relaxing the synchronization requirement can yield a nonconvergent or less efficient algorithm that may even underperform when compared to the sequential version. Addressing these issues, we introduce an asynchronous parallel BD method for stochastic network design problems. We show that the proposed algorithm converges to the global optimum and suggest various acceleration strategies to enhance its performance. We conduct an extensive numerical study on benchmark instances from a generic stochastic network design problem. The results indicate that using up to 20 processors, our asynchronous algorithm is on average 1.4 times faster than the conventional low-level parallel methods.

A Deep Reinforcement Learning Based Real-Time Solution Policy for the Traveling Salesman Problem

The rapid development of logistics and navigation has led to increasing demand for solving route optimization problems in real-time. The traveling salesman problem (TSP) tends to require fast and reliable online solutions, which may not be met by traditional iterative optimization algorithms. In this work, a real-time solution policy is proposed for TSP. The idea is to build a mapping between city information and optimal solutions using deep neural networks. Therefore, when given a new set of city coordinates, the optimal route can be directly and quickly calculated without iteration. Considering the recent advancement in computer vision with deep convolutional neural networks (DCNNs), an image representation is proposed to convert TSP to a computer vision problem. A problem decomposition method is introduced to reduce the mapping complexity. Taking advantage of the powerful fitting capabilities of DCNN, a deep reinforcement learning method is designed without any labeling requirement. The proposed method is superior for real-time applications compared with other algorithms.

On proximal augmented Lagrangian based decomposition methods for dual block-angular convex composite programming problems

On proximal augmented Lagrangian based decomposition methods for dual block-angular convex composite programming problems

Inexact penalty decomposition methods for optimization problems with geometric constraints

This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some situations where parts of the constraints are nonconvex and complicated, like cardinality constraints, disjunctive programs, or matrix problems involving rank constraints. By a variable duplication and decomposition strategy, the method presented here explicitly handles these difficult constraints, thus generating iterates which are feasible with respect to them, while the remaining (standard and supposingly simple) constraints are tackled by sequential penalization. Inexact optimization steps are proven sufficient for the resulting algorithm to work, so that it is employable even with difficult objective functions. The current work is therefore a significant generalization of existing papers on penalty decomposition methods. On the other hand, it is related to some recent publications which use an augmented Lagrangian idea to solve optimization problems with geometric constraints. Compared to these methods, the decomposition idea is shown to be numerically superior since it allows much more freedom in the choice of the subproblem solver, and since the number of certain (possibly expensive) projection steps is significantly less. Extensive numerical results on several highly complicated classes of optimization problems in vector and matrix spaces indicate that the current method is indeed very efficient to solve these problems.

MIMO Unmanned Surface Vessels Enabled Maritime Wireless Network Coexisting With Satellite Network: Beamforming and Trajectory Design

Due to the flexible deployment, unmanned surface vessels (USVs) have attracted much interest recently. To solve the resource scarcity problem at sea, USV needs to leverage existing terrestrial and satellite systems for efficient backhaul and spectrum sharing. In this case, the multiple input multiple output (MIMO) technology can be applied for diversity gain improvement and interference coordination. However, how to adopt MIMO technology into maritime networks with a sparse scattering environment is still an open issue. In this paper, we employ a multi-antenna USV to support on-demand communications. Utilizing the two-ray channel, we aim to maximize the sum throughput over all USV intended users, by jointly optimizing the cooperative beamforming and trajectory, subject to several practical constraints, including the USV kinetics, quality of service requirement and backhaul capacity. Different from existing whole period designs, we decompose the problem into sequential one-slot problems. Within each slot, the non-convex problem is solved iteratively by using problem decomposition and successive convex optimization methods. Then, channel estimation errors are considered to investigate a robust beamforming scheme. Numerical simulations validate that the USV coexists well with the satellite network and show that the beamforming scheme and trajectory design complement each other for performance improvement.

Subdomain solution decomposition method for nonstationary problems

The reduction of computational costs in the numerical solution of nonstationary problems is achieved through splitting schemes. In this case, solving a set of less computationally complex problems provides the transition to a new level in time. The traditional construction approach of splitting schemes is based on an additive representation of the problem operator(s) and uses explicit-implicit approximations for individual terms. Recently (Efendiev and Vabishchevich (2021) [22]), a new class of methods of approximate solution of nonstationary problems has been introduced based on decomposition not of operators but of the solution itself. This new approach with subdomain solution selection is used in this paper to construct domain decomposition schemes. The boundary value problem for a second-order parabolic equation in a rectangle with a difference approximation in space is typical. Two and three-level schemes for decomposition of the domain with and without overlapping subdomains are investigated. Our numerical experiments complement the theoretical results.

A decomposition method for lasso problems with zero-sum constraint

In this paper, we consider lasso problems with zero-sum constraint, commonly required for the analysis of compositional data in high-dimensional spaces. A novel algorithm is proposed to solve these problems, combining a tailored active-set technique, to identify the zero variables in the optimal solution, with a 2-coordinate descent scheme. At every iteration, the algorithm chooses between two different strategies: the first one requires to compute the whole gradient of the smooth term of the objective function and is more accurate in the active-set estimate, while the second one only uses partial derivatives and is computationally more efficient. Global convergence to optimal solutions is proved and numerical results are provided on synthetic and real datasets, showing the effectiveness of the proposed method. The software is publicly available.