Augmented Lagrangian Algorithm Research Articles

Global convergence of dislocation hyperbolic augmented Lagrangian algorithm for nonconvex optimization

In this work, we present a new augmented Lagrangian type algorithm based on the dislocation hyperbolic augmented Lagrangian function (DHALF). This algorithm is called dislocation hyperbolic augmented Lagrangian algorithm (DHALA). We ensure that DHALA converges to a global solution for the inequality constrained nonconvex optimization problem. We present DHALA considering the safeguards technique and ensure the convergence of this algorithm. Finally, we present a computational experiment to show the performance of our proposed algorithm.

Is order-2 proportionality good enough for approximating the most likely path flow in user equilibrium traffic assignment?

The proportionality condition is a standard approach to dealing with the non-uniqueness issue in the user equilibrium (UE) traffic assignment problems (TAP). Although the proportionality condition can reduce the degree of arbitrariness, it remains unclear how much arbitrariness remains and whether it can meaningfully affect model outcomes and relevant decisions that depend on them. The answers to these questions are impeded by the lack of an efficient algorithm that can find the exact maximum entropy UE path flow solution for networks of practical size. In this paper, we fill this gap by developing a high-performance augmented Lagrangian algorithm that effectively exploits the special problem structure. Our numerical results reveal that there are a considerable number of links with non-negligible arbitrariness in the solution generated by the proportionality condition, and that this problem becomes worse if the level of congestion increases in the network. Since about a decade ago, many practitioners have relied on state-of-the-art traffic assignment tools based on the proportionality condition to perform select link analysis, among other applications. The results reported herein are a reminder that their toolbox may need reevaluation and perhaps an upgrade.

Simultaneous clustering and estimation of networks in multiple graphical models.

Gaussian graphical models are widely used to study the dependence structure among variables. When samples are obtained from multiple conditions or populations, joint analysis of multiple graphical models are desired due to their capacity to borrow strength across populations. Nonetheless, existing methods often overlook the varying levels of similarity between populations, leading to unsatisfactory results. Moreover, in many applications, learning the population-level clustering structure itself is of particular interest. In this article, we develop a novel method, called Simultaneous Clustering and Estimation of Networks via Tensor decomposition (SCENT), that simultaneously clusters and estimates graphical models from multiple populations. Precision matrices from different populations are uniquely organized as a three-way tensor array, and a low-rank sparse model is proposed for joint population clustering and network estimation. We develop a penalized likelihood method and an augmented Lagrangian algorithm for model fitting. We also establish the clustering accuracy and norm consistency of the estimated precision matrices. We demonstrate the efficacy of the proposed method with comprehensive simulation studies. The application to the Genotype-Tissue Expression multi-tissue gene expression data provides important insights into tissue clustering and gene coexpression patterns in multiple brain tissues.

A constraint qualification for the dislocation hyperbolic augmented Lagrangian algorithm

In this paper, we study an augmented Lagrangian-type algorithm called the Dislocation Hyperbolic Augmented Lagrangian Algorithm (DHALA), which solves an inequality nonconvex optimization problem. We show that the sequence generated by DHALA converges to a Karush–Kuhn–Tucker (KKT) point under the Mangasarian–Fromovitz constraint qualification. The contribution of our work is to consider a constraint qualification into this algorithm. Finally, we present some computational illustrations to demonstrate the performance our algorithm works.

The Legacy of Roland Glowinski

Roland Glowinski published 8 books and more than 300 articles. He was also an editor of many very well cited proceedings. Hence this attempt to summarize his scientific work is not likely to do justice to his work. Nevertheless, we will try to extract his major contributions, such as the augmented Lagrangian algorithm, various domain decomposition and fictitious methods and their performance on the Navier–Stokes equations in a moving domain. Roland has created a school of applied mathematicians remarkable by their rigor and efficiency for industrial applications. He marked his time and his books will be authorities as long as computer architectures are similar to their present structures.

Safeguarded augmented Lagrangian algorithms with scaled stopping criterion for the subproblems

Safeguarded augmented Lagrangian algorithms with scaled stopping criterion for the subproblems

Hyperbolic Augmented Lagrangian algorithm for multiobjective optimization problems

Hyperbolic Augmented Lagrangian algorithm for multiobjective optimization problems

Dislocation hyperbolic augmented Lagrangian algorithm in convex programming

The dislocation hyperbolic augmented Lagrangian algorithm (DHALA) is a new approach to the hyperbolic augmented Lagrangian algorithm (HALA). DHALA is designed to solve convex nonlinear programming problems. We guarantee that the sequence generated by DHALA converges towards a Karush-Kuhn-Tucker point. We are going to observe that DHALA has a slight computational advantage in solving the problems over HALA. Finally, we will computationally illustrate our theoretical results.

Solution of the Simultaneous Routing and Bandwidth Allocation Problem in Energy-Aware Networks Using Augmented Lagrangian-Based Algorithms and Decomposition

We discuss several algorithms for solving a network optimization problem of simultaneous routing and bandwidth allocation in green networks in a decomposed way, based on the augmented Lagrangian. The problem is difficult due to the nonconvexity caused by binary routing variables. The chosen algorithms, which are several versions of the Multiplier Method, including the Alternating Direction Method of Multipliers (ADMM), have been implemented in Python and tested on several networks’ data. We derive theoretical formulations for the inequality constraints of the Bertsekas, Tatjewski and SALA methods, formulated originally for problems with equality constraints. We also introduce some modifications to the Bertsekas and Tatjewski methods, without which they do not work for an MINLP problem. The final comparison of the performance of these algorithms shows a significant advantage of the augmented Lagrangian algorithms, using decomposition for big problems. In our particular case of the simultaneous routing and bandwidth allocation problem, these algorithms seem to be the best choice.

A Si IGBT/SiC MOSFET Hybrid Isolated Bidirectional DC–DC Converter for Reducing Losses and Costs of DC Solid State Transformer

The DC solid state transformer (DCSST) is a crucial component for connecting buses of different voltage levels in the DC distribution grid. This paper proposes a Si IGBT/SiC MOSFET hybrid isolated bidirectional DC–DC converter and an optimized modulation strategy (OMS) to reduce the losses and costs of DCSST. Based on the analysis of topology and operating principles, a duty-cycle modulation strategy is proposed and the converter is modeled by the time domain analysis (TDA) method. Through the analysis of switching characteristics, an optimization problem is established, which aims to reduce the conduction losses of switches while ensuring zero-voltage switching (ZVS) for all switches and low-current turn-off for IGBTs simultaneously. The optimization problem is solved by the augmented Lagrangian genetic algorithm (ALGA), and the OMS for the proposed converter is deduced. Finally, a 2 kW experimental prototype with the primary voltage of 405–495 V and the secondary voltage of 150 V is built to verify the effectiveness of the proposed topology and OMS. The switching costs of the proposed converter is reduced by 27.3% and the efficiency is improved by up to 4.04% compared to the existing method.

Analysis of contact creep behaviour of nanofilled composites

An effective modelling method for analysing the contact creep behaviour of composite materials with different nanofillers was proposed. Considering both the contact and creep material nonlinearities, an augmented Lagrangian algorithm was used to treat the unilateral contact constraints, and a time-hardening creep constitutive equation was adopted to describe the deformation characteristics of the nanocomposites. The constitutive parameters were extracted through a series of linear fittings based on the multiaxial creep theory and creep test curves, and the accuracy of the constitutive equation was validated. The applications of the proposed modelling method were demonstrated using a rope-wheel contact system (RWCS), in which the wheels were made of polystyrene (PS) with carbon black (CB), multi-walled carbon nanotubes (MWCNT), or chemically reduced graphene oxide (CRGO). The results indicated that the addition of nanofillers significantly reduced the creep deformation of the contact zone. The CRGO exhibited a better anti-creep performance than that of CB and MWCNT. Compared with pure PS, the maximum deformation on the contact path of the nanocomposite wheel with 5.0 wt% CRGO sheets at 1500 s and 104 s was reduced by approximately 22 % and 52 %, respectively. The coefficient of friction μ had a strong influence on the contact creep deformation when μ < 0.3. Additionally, the anti-creep capability of the added nanofillers became more evident as the loading time increased. The contact creep deformation of the CRGO composite was less than one-tenth that of the matrix material after 30 d.

A truncated test scheme design method for success-failure in-orbit tests

Based on the success-failure test feature of in-orbit tests (IOTs) for typical space equipment, this paper presents a method for designing a truncated test scheme for success-failure in-orbit tests. With this method, a small upper boundary of the sample size for the IOT verification test can be obtained before the test starts. The method introduces the truncated Bayes-sequential mesh test (SMT) method into the design of the IOT verification test scheme and greatly compresses the continuous test area by incorporating optimization theory, resulting in a smaller upper limit of the IOT sample size. First, this paper derives a specific calculation formula for the Bayes-SMT critical line. Second, the Markov chain model is adopted to calculate the occurrence probabilities of each acceptance and rejection point through state transition. Finally, an optimal truncated test optimization algorithm based on the augmented lagrangian genetic algorithm is proposed. Simulation tests show that, compared with the classical single sampling method, the truncated sequential probability ratio test method, the truncated SMT method, and the truncated Bayes-SMT method based on step-by-step calculation, the method presented in this paper can be used to obtain a sequential test scheme with smaller truncated sample size.

Enhancement and Segmentation of Low-Light Images Using Illumination Map Estimation based Level Set (IME-LS) Method

One of the most difficult aspects of image segmentation is that it cannot be successfully segmented if the image is dark or degraded. In this paper, proposes a method for the segmentation of low-quality and degraded images is put forth which is called Illumination Map Estimation based Level Set (IME-LS) Method. This proposed model has been classified into two parts: Firstly, we use an enhancement approach using illumination map approximation to enhance the input image. In this approach, Illumination map is constructed then refined. The refined illumination map undergoes an Augmented Lagrangian algorithm and then we use a sped-up solver for a considerably more efficient outcome. Secondly, Then the segmentation procedure begins once the image is enhanced. The enhanced image is segmented using level set bias method through Fuzzy clustering. In this method we employ the Fuzzy C-Means (FCM) algorithm to categorize data points into clusters. The fuzzy C-Means algorithm separates different entities in the image based on their varied intensities and sorts them into various clusters. The level set bias approach then tracks the variational boundaries of the image. We have designed the integrated algorithm in such a way that the image is classified or grouped into various clusters using a novel fuzzy Clustering algorithm and the variational boundaries of those clusters are tracked by employing the level set algorithm. In this paper, we further perform quantitative and comparative analysis of the suggested technique with respect to other segmentation techniques to illustrate the efficiency and flexibility of the suggested model.

On the paper “Augmented Lagrangian algorithms for solving the continuous nonlinear resource allocation problem”

In the paper Torrealba et al. (2021) an augmented Lagrangian algorithm was proposed for resource allocation problems with the intriguing characteristic that instead of solving the box-constrained augmented Lagrangian subproblem, they propose projecting the solution of the unconstrained subproblem onto such box. A global convergence result for the quadratic case was provided, however, this is somewhat counterintuitive, as in usual augmented Lagrangian theory, this strategy can fail in solving the augmented Lagrangian subproblems. In this note we investigate further this algorithm and we show that the proposed method may indeed fail when the Hessian of the quadratic is not a multiple of the identity. In the paper, it is not clear enough that two different projections are being used: one for obtaining their convergence results and other in their implementation. However, despite the lack of theoretical convergence, their strategy works remarkably well in some classes of problems; thus, we propose a hybrid method which uses their idea as a starting point heuristics, switching to a standard augmented Lagrangian method under certain conditions. Our contribution consists in presenting an efficient way of determining when the heuristics is failing to improve the KKT residual of the problem, suggesting that the heuristic procedure should be abandoned. Numerical results are provided showing that this strategy is successful in accelerating the standard method.

Correlation challenges for North Sea offshore wind power: a Norwegian case study

Offshore wind power projects are currently booming around the North Sea. However, there are inherent correlation challenges between wind farms in this area, which has implications for the optimal composition of locations and the scale-up of installed capacities. This paper is aimed at addressing the correlation problem by minimizing the variance of total wind power accumulated around the North Sea. We show that this nonlinear convex optimization problem can be solved by applying the Augmented Lagrangian Algorithm (ALA). The premise of the study is that more interconnections between the EU countries will be prioritized in order to optimize and smooth out the wind power production patterns. A publicly available dataset with historical hour-by-hour data spanning over 20 years was used for the analysis. We explore two distinct scenarios for Norwegian offshore wind development. In the first scenario, we consider the ongoing activities on the European continental side of the North Sea and their implications for Norway. Here, we illustrate the advantages of focusing on expanding wind power capacity in the northern regions of Norway to enhance the overall value of the generated wind power. In contrast, the second reference scenario neglects these interconnections, resulting in a significantly greater concentration of offshore wind development in the southern parts of Norway, particularly in Sørlige Nordsjø II. Additionally, our work estimates the wind power correlation coefficient in the North Sea as a function of distance. Furthermore, we analyze deviations and intermittencies in North Sea wind power over various time intervals, emphasizing that the perceived integration challenges are highly dependent on the chosen time resolution in the analysis.

An Inexact Semismooth Newton-Based Augmented Lagrangian Algorithm for Multi-Task Lasso Problems

This paper is concerned with the [Formula: see text]-norm ball constrained multi-task learning problem, which has received extensive attention in many research areas such as machine learning, cognitive neuroscience, and signal processing. To address the challenges of solving large-scale multi-task Lasso problems, this paper develops an inexact semismooth Newton-based augmented Lagrangian (Ssnal) algorithm. When solving the inner problems in the Ssnal algorithm, the semismooth Newton (Ssn) algorithm with superlinear or even quadratic convergence is applied. Theoretically, this paper presents the global and asymptotically superlinear local convergence of the Ssnal algorithm under standard conditions. Computationally, we derive an efficient procedure to construct the generalized Jacobian of the projector onto [Formula: see text]-norm ball, which is an important component of the Ssnal algorithm, making the computational cost in the Ssn algorithm very cheap. Comprehensive numerical experiments on the multi-task Lasso problems demonstrate that the Ssnal algorithm is more efficient and robust than several existing state-of-the-art first-order algorithms.

Dislocation hyperbolic augmented Lagrangian algorithm for nonconvex optimization

The dislocation hyperbolic augmented Lagrangian algorithm (DHALA) solves the nonconvex programming problem considering an update rule for its penalty parameter and considering a condition to ensure the complementarity condition. in this work, we ensure that the sequence generated by DHALA converges to a Karush-Kuhn-Tucker (KKT) point, and we present computational experiments to demonstrate the performance of our proposed algorithm.

Distributionally Robust Decentralized Scheduling Between the Transmission Market and Local Energy Hubs

The cross impacts between transmission and distribution systems have drawn extensive attention, where multi-energy carriers become increasingly dominant in local market operations. Local energy hubs, integrated with multi-energy carriers, e.g., electricity, gas, heat, present great potentials to provide adequate power and reserve support for the transmission system, which could mitigate issues such as boundary mismatches in the upstream market. However, energy hubs are typically equipped with abundant distributed renewable resources. Such local uncertainties can adversely affect the well-being of the coordinated multi-energy market hierarchy. This paper presents a novel multi-period scheduling framework that considers the coordination between the transmission network and distribution energy hubs. Each agent performs local scheduling operations capturing independent uncertainties via a distributionally robust formulation. We then apply a tailored accelerated augmented Lagrangian algorithm embedded with the column-and-constraint method to decentralize the overall operation with agents' privacy preserved. The fast and convergent feature of the algorithm ensures its scalability and reliability in real-world applications of the energy hub design with transmission coordination. Numerical experiments confirm the efficacy of the proposed method.

Bi-level optimization for customized bus routing serving passengers with multiple-trips based on state–space–time network

The emerging customized bus (CB) can serve personalized trip requests in a more flexible and convenient way, especially for passengers who have multiple-trip requests in a short time period. Considering the CB service for passengers with multiple trips (CSPMT), a loading-state-oriented state–space–time network-based bi-level programming model is proposed to optimize the routing of CBs, with the objectives of maximizing the operational profit and minimizing the travel cost, while considering the characteristics of multiple trips, time windows, capacity and mixed loads. Besides, a nested algorithm combining the genetic algorithm (GA) and the augmented Lagrangian relaxation-based dynamic programming algorithm is proposed. Then, the proposed model and algorithm are verified and analyzed through a Sioux Falls network and a Beijing sketch network. It can be found from the results that the method can optimize a CB routing plan for passengers with multiple-trip requests and for different network scales. The proposed bi-level model and corresponding algorithm can better adapt to passengers’ personalized trip requests, and promote a higher level of public transport service which will attract more residents from private cars to public transportation, ultimately reducing energy consumption and exhaust emissions, and promoting the sustainable development of modern metropolises.

An SQP-based multiple shooting algorithm for large-scale PDE-constrained optimal control problems

Multiple shooting methods for solving optimal control problems have been developed rapidly in the past decades and are widely considered a promising direction to speed up the optimization process. Here we propose and analyze a new multiple shooting algorithm based on a sequential quadratic programming (SQP) method that is suitable for optimal control problems governed by large-scale time-dependent partial-differential equations (PDEs). We investigate the structure of the KKT matrix and solve the large-scale KKT system by a preconditioned conjugate gradient algorithm. A simplified block Schur complement preconditioner is proposed, that allows for the parallelization of the method in the time domain. The proposed algorithm is first validated for an optimal control problem constrained by the Nagumo equation. The results indicate that considerable accelerations can be achieved for multiple shooting approaches with appropriate starting guesses and scaling of the matching conditions. We further apply the proposed algorithm to a two-dimensional velocity tracking problem governed by the Navier–Stokes equations. We find algorithmic speed-ups of up to 12 versus single shooting on up to 50 shooting windows. We also compare results with earlier work that uses an augmented Lagrangian algorithm instead of SQP, showing better performance of the SQP method for most of the cases.