Dual Algorithm Research Articles

Predicting normalized difference vegetation index using a deep attention network with bidirectional GRU: a hybrid parametric optimization approach

AbstractScalable and accurate normalized difference vegetation index (NDVI) prediction is necessary to track the status of vegetation and the environment and to support proper ecological management. Herein, we present an innovative deep-learning approach to improve NDVI prediction performances by considering enhanced temporal modeling and hybrid optimization processes. The analysis is based on a core model that integrates a Bidirectional Gated Recurrent Unit (BiGRU) with the profound attention feature since the primary research incorporates the capability of complex temporal in addition to NDVI-time series value. The model performs better through a dual algorithm combining the waterwheel plant algorithm (WWPA) and statistical fractal search (SFS) named WWPASFS-BiGRU. The proposed approach is evaluated using real-world NDVI datasets, demonstrating its capability to outperform traditional models and state-of-the-art deep learning methods. Key performance metrics highlight the model’s accuracy, with a root mean square error (RMSE) as low as 0.00011, reflecting its superior predictive ability. Comparative experiments showcase the robustness of our model across different environmental conditions and geographical settings, affirming its applicability in diverse ecological forecasting scenarios. Additionally, extensive statistical validation, including ANOVA and Wilcoxon tests, confirms the model’s consistency and reliability. The effectiveness of the WWPASFS-BiGRU model is illustrated through applications in predicting NDVI trends across regions in Saudi Arabia, providing critical insights for ecosystem management and sustainable development planning.

Convergence analysis of a primal–dual optimization-by-continuation algorithm

We present a numerical iterative optimization algorithm for the minimization of a cost function consisting of a linear combination of three convex terms, one of which is differentiable, a second one is prox-simple and the third one is the composition of a linear map and a prox-simple function. The algorithm’s special feature lies in its ability to approximate, in a single iteration run, the minimizers of the cost function for many different values of the parameters determining the relative weight of the three terms in the cost function. A proof of convergence of the algorithm, based on an inexact variable metric approach, is also provided. As a special case, one recovers a generalization of the primal–dual algorithm of Chambolle and Pock, and also of the proximal-gradient algorithm. Finally, we show how it is related to a primal–dual iterative algorithm based on inexact proximal evaluations of the non-smooth terms of the cost function.

Mixed finite element method for multi-layer elastic contact systems

With the development of multi-layer elastic systems in the field of engineering mechanics, the corresponding variational inequality theory and algorithm design have received more attention and research. In this study, a class of equivalent saddle point problems with interlayer Tresca friction conditions and the mixed finite element method are proposed and analyzed. Then, the convergence of the numerical solution of the mixed finite element method is theoretically proven, and the corresponding algebraic dual algorithm is provided. Finally, through numerical experiments, the mixed finite element method is not only compared with the layer decomposition method, but also its convergence relationship with respect to the spatial discretization parameter H is verified.

A NEW DUAL HEURISTIC ALGORITHM FOR FINDING THE INITIAL BASIC FEASIBLE SOLUTION FOR A TRANSPORTATION PROBLEM

Because determining the best initial basic feasible solution (IBFS) for a transportation problem is so crucial, numerous authors have expended a great deal of energy developing effective algorithms that will result in the lowest possible cost of moving products from a given source to a destination. The goal of this work was to develop an efficient dual algorithm for finding an initial basic feasible solution to a transportation problem (TP). Two distinct algorithms that produce the same IBFS make up our suggested approach. Compared to some popular methods in the literature, Using four numerical examples, the Row Minimum Method (RMM), Column Minimum Method (CMM), Least Cost Method (LCM), Extremum Difference Method (EDM), Northwest Corner Method (NWCM), Vogel's Approximation Method (VAM), etc. In comparison to the other heuristic techniques compared with the optimal dictate solution modified distribution (MODI), the proposed heuristic approach (PS-DESPAN) approximation was shown to provide a better starting solution (a solution that is extremely close to the optimal solution)

Distributed event-triggered algorithm for convex optimization with coupled constraints

This paper develops a distributed primal–dual algorithm via an event-triggered mechanism to solve a class of convex optimization problems subject to local set constraints, coupled equality and inequality constraints. Different from some existing distributed algorithms with the diminishing step-sizes, our algorithm uses the constant step-sizes, and is shown to achieve an exact convergence to an optimal solution with an ergodic convergence rate of O(1/k) for general convex objective functions, where k>0 is the iteration number. Based on the event-triggered communication mechanism, the proposed algorithm can effectively reduce the communication cost without sacrificing the convergence rate. Finally, a numerical example is presented to verify the effectiveness of the proposed algorithm.

Non-ergodic convergence rate of an inertial accelerated primal–dual algorithm for saddle point problems

In this paper, we design an inertial accelerated primal–dual algorithm to address the convex–concave saddle point problem, which is formulated as minxmaxyf(x)+〈Kx,y〉−g(y). Remarkably, both functions f and g exhibit a composite structure, combining “nonsmooth” ＋ “smooth” components. Under the assumption of partially strong convexity in the sense that f is convex and g is strongly convex, we introduce a novel inertial accelerated primal–dual algorithm based on Nesterov’s extrapolation. This algorithm can be reduced to two classical accelerated forward–backward methods for unconstrained optimization problem. We show that the proposed algorithm achieves a non-ergodic O(1/k2) convergence rate, where k represents the number of iterations. Several numerical experiments validate the efficiency of our proposed algorithm.

The weighted Euclidean one-center problem in $${\mathbb {R}}^n$$

AbstractGiven a finite set of distinct points in $${\mathbb {R}}^n$$ R n and a positive weight for each point, primal and dual algorithms are developed for finding the Euclidean ball of minimum radius so that the weighted Euclidean distance between the center of the ball and each point is less than or equal to the radius of the ball. Each algorithm is based on a directional search method in which the search path at each iteration is either a ray or a two-dimensional circular arc in $${\mathbb {R}}^n$$ R n . At each iteration, a search path is constructed by intersecting bisectors of pairs of points, where the bisectors are either hyperplanes or n-dimensional spheres. Each search path preserves complementary slackness and primal (dual) feasibility for the primal (dual) algorithm. The step size along each search path is determined explicitly. Test problems up to 1000 dimensions and 10,000 points were solved to optimality by both primal and dual algorithms. Computational results also show that these algorithms outperform several open-source SOCP codes.

A stochastic primal–dual algorithm for composite constrained optimization

This paper studies the decentralized stochastic optimization problem over an undirected network, where each agent owns its local private functions made up of two non-smooth functions and an expectation-valued function. A decentralized stochastic primal–dual algorithm is proposed, by combining the variance-reduced method and the stochastic approximation method. The local gradients are estimated by using the mean of a variable number of sample gradients and the stochastic error decreases with the number of samples in the stochastic approximation process. The highlight of this paper is the extension of the primal–dual algorithm to the stochastic optimization problems. The effectiveness of the proposed algorithm and the correctness of the theory are verified by numerical experiments.

An Efficient and Versatile Variational Method for High-Dimensional Data Classification

High-dimensional data classification is a fundamental task in machine learning and imaging science. In this paper, we propose an efficient and versatile multi-class semi-supervised classification method for classifying high-dimensional data and unstructured point clouds. To begin with, a warm initialization is generated by using a fuzzy classification method such as the standard support vector machine or random labeling. Then an unconstraint convex variational model is proposed to purify and smooth the initialization, followed by a step which is to project the smoothed partition obtained previously to a binary partition. These steps can be repeated, with the latest result as a new initialization, to keep improving the classification quality. We show that the convex model of the smoothing step has a unique solution and can be solved by a specifically designed primal–dual algorithm whose convergence is guaranteed. We test our method and compare it with the state-of-the-art methods on several benchmark data sets. Thorough experimental results demonstrate that our method is superior in both the classification accuracy and computation speed for high-dimensional data and point clouds.

A general solution to the quasi linear screening problem

We provide an algorithm for solving multidimensional screening problems which are intractable analytically. The algorithm is a primal–dual algorithm which alternates between optimizing the primal problem of the surplus extracted by the principal and the dual problem of the optimal assignment to deliver to the agents for a given surplus. We illustrate the algorithm by solving (i) the generic monopolist price discrimination problem and (ii) an optimal tax problem covering income and savings taxes when citizens differ in multiple dimensions.

Purely periodic continued fractions and graph-directed iterated function systems

We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P1R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{P}\\,}}^1\\mathbb {R}$$\\end{document}, determines a dual pair of graph-directed iterated function systems, whose attractors contain intervals and constitute the domains of a dual pair of Gauss-type maps. Our framework covers many continued fraction algorithms (such as Farey fractions, Ceiling, Even and Odd, Nearest Integer, …\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ldots $$\\end{document}) and provides explicit dual algorithms and characterizations of those quadratic irrationals having a purely periodic expansion.

An Optimal ADMM for Unilateral Obstacle Problems

We propose a new alternating direction method of multipliers (ADMM) with an optimal parameter for the unilateral obstacle problem. We first use the five-point difference scheme to discretize the problem. Then, we present an augmented Lagrangian by introducing an auxiliary unknown, and an ADMM is applied to the corresponding saddle-point problem. Through eliminating the primal and auxiliary unknowns, a pure dual algorithm is then used. The convergence of the proposed method is analyzed, and a simple strategy is presented for selecting the optimal parameter, with the largest and smallest eigenvalues of the iterative matrix. Several numerical experiments confirm the theoretical findings of this study.

Distributed online constrained convex optimization with event-triggered communication

This paper focuses on the distributed online convex optimization problem with time-varying inequality constraints over a network of agents, where each agent collaborates with its neighboring agents to minimize the cumulative network-wide loss over time. To reduce communication overhead between the agents, we propose a distributed event-triggered online primal–dual algorithm over a time-varying directed graph. With several classes of appropriately chose decreasing parameter sequences and non-increasing event-triggered threshold sequences, we establish dynamic network regret and network cumulative constraint violation bounds. Finally, a numerical simulation example is provided to verify the theoretical results.

Edge-based distributed primal–dual algorithms for seeking generalized Nash equilibria

This paper introduces two novel distributed algorithms aimed at finding the generalized Nash equilibria (GNE) in noncooperative games. To tackle this, we utilize the variational inequality framework, transforming the noncooperative game into the challenge of identifying the zeros of a sum of monotone operators. Within noncooperative games, the decisions made by individual players are interlinked via shared affine constraints. Our proposed approach involves two edge-based distributed algorithms, differentiated by their access to players’ actions, whether it is full-decision information or partial-decision information. We establish the parameter range and demonstrate the convergence of both algorithms, showcasing their efficacy under local constant step-sizes. Additionally, we validate the effectiveness and advantages of these algorithms through two numerical experiments.

New Dual Algorithm to Placement the Data Aggregation Point for Smart Grid Meters

New Dual Algorithm to Placement the Data Aggregation Point for Smart Grid Meters

Stochastic Primal–Dual Hybrid Gradient Algorithm with Adaptive Step Sizes

In this work, we propose a new primal–dual algorithm with adaptive step sizes. The stochastic primal–dual hybrid gradient (SPDHG) algorithm with constant step sizes has become widely applied in large-scale convex optimization across many scientific fields due to its scalability. While the product of the primal and dual step sizes is subject to an upper-bound in order to ensure convergence, the selection of the ratio of the step sizes is critical in applications. Up-to-now there is no systematic and successful way of selecting the primal and dual step sizes for SPDHG. In this work, we propose a general class of adaptive SPDHG (A-SPDHG) algorithms and prove their convergence under weak assumptions. We also propose concrete parameters-updating strategies which satisfy the assumptions of our theory and thereby lead to convergent algorithms. Numerical examples on computed tomography demonstrate the effectiveness of the proposed schemes.

Linearisation techniques and the dual algorithm for a class of mixed singular/continuous control problems in reinsurance. Part II: Numerical aspects

This paper is intended as a companion to [12] and focuses on the numerical implementations of the method in the context of insurance problems with mixed controls: reinsurance acting as continuous control, and capital injection and dividend payment acting as singular ones. The aim is twofold. On the one hand, we provide a comparison of the output of our method in terms of the optimal value against two benchmarks. The first benchmark is provided by [4] in the no-reinsurance setting and when the claims are exponentially distributed. The second benchmark refers to [8] where capital injections are not permitted, but different claim distributions can be employed. On the other hand, we illustrate the improvement of the models in [8] when capital injection is allowed. We also look into the effect of reinsurance in the model [4]. Several cases for the equity cost parameter are discussed as are the boundary (or limit) situations.

Optimal control problem with nonlinear fractional system constraint applied to image restoration

This study aims to investigate a novel nonlinear optimization problem that incorporate a partial differential equation (PDE) constraint for image denoising in the context of mixed noise removal. Based on ‐norm and decomposition approach, we develop a nonlinear system that involves the fractional Laplacian operator. Based on Schauder's fixed point theorem, we establish the existence and uniqueness of weak solution for the direct problem. Furthermore, we study the well‐posedness of the optimal control, and we also prove the existence of the weak solutions for the adjoint problem by using Galerkin's method. In order to numerically compute the solution of the proposed model, we introduce the numerical discretization scheme and the primal–dual algorithm used to solve our problem. Finally, we provide comparative numerical experiments to evaluate the efficiency and effectiveness of our proposed model.

A variational PDNet network using a learning reaction–diffusion equation

Due to their high performance in modeling and forecasting a large amount of real-world complex phenomena, deep convolutional neural networks have received a great deal of attention over the past ten years. It has been extensively utilized in several recent vision problems, such as image denoising and deblurring. By incorporating recent developments in learning-based nonlinear diffusion models and highlighting the connection between partial differential equations (PDEs) and networks, especially residual networks, we propose a flexible learning reaction–diffusion system with spatially computed convolution kernels. This trained PDE is employed to prevent the low-level features from deteriorating, which is needed for tasks involving denoising. In fact, we introduce an improved network architecture for a denoising network entitled VPDNet through an optimal control problem. More precisely, the network architecture is governed by an explicit Euler discretization of a reaction–diffusion system. We give the main algorithm to identify the kernel parameters using the Alternating Direction Method of Multipliers (ADMM) combined with a Primal–Dual algorithm. Finally, numerical simulations are provided with appropriate comparisons to demonstrate the efficiency of the elaborated network.

Prediction-correction pedestrian flow by means of minimum flow problem

We study a new variant of mathematical prediction-correction model for crowd motion. The prediction phase is handled by a transport equation where the vector field is computed via an eikonal equation [Formula: see text], with a positive continuous function [Formula: see text] connected to the speed of the spontaneous travel. The correction phase is handled by a new version of the minimum flow problem. This model is flexible and can take into account different types of interactions between the agents, from gradient flow in Wassersetin space to granular type dynamics like in sandpile. Furthermore, different boundary conditions can be used, such as non-homogeneous Dirichlet (e.g. outings with different exit-cost penalty) and Neumann boundary conditions (e.g. entrances with different rates). Combining finite volume method for the transport equation and Chambolle–Pock’s primal dual algorithm for the eikonal equation and minimum flow problem, we present numerical simulations to demonstrate the behavior in different scenarios.