Lagrangian Optimization Method Research Articles

A Barrier Lagrangian Dual Method for Multi-stage Stochastic Convex Semidefinite Optimization

A Barrier Lagrangian Dual Method for Multi-stage Stochastic Convex Semidefinite Optimization

A three-stage evaluation model for biomass co-firing combined with carbon capture and storage (in oil wells) retrofit in coal-fired power plants based on multi-criteria decision-making: An example from Hebei Province, China

Coal-fired power plants can be retrofitted by biomass co-firing combined with CO2 capture and storage, reducing carbon emissions significantly. Site selection is an important task for the entire project, but little research has been done on this. Therefore, this paper proposed a three-level evaluation model. Firstly, alternative power plants and carbon sequestration sites (set oil wells as an example) were screened by feasibility evaluations. Secondly, source-sink pairs were performed based on the shortest distance between power plants and oil wells. Finally, source-sink pairs were evaluated with suitability evaluation. Entropy weight method (EWM) and Analytic hierarchy process (AHP) were used to calculate objective and subjective weights, respectively. Lagrangian optimization method was used to combine the two weights. The ranking results of alternative source-sink pairs were obtained through Multi-Objective Optimization on the basis of Ratio Analysis (MOORA) improved by combined weight. An example of in-service power plants in Hebei Province was conducted. Engineering geology and reservoir characteristics were the most important considerations. D5 (source: Langfang thermal power plant, sink: Bieguzhuang - Jing 11 oil well) was the best choice of the five given candidates. The results of the study provided an evaluation framework for governments and investors in retrofitting power plants.

A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization

A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization

An adaptive sampling augmented Lagrangian method for stochastic optimization with deterministic constraints

An adaptive sampling augmented Lagrangian method for stochastic optimization with deterministic constraints

Application of Natural Language Processing-based Emotional Semantic Analysis in the “One Core, Three Integrations” Vocal Music Teaching Model

Abstract To be able to better improve musical literacy and artistic aesthetics, this paper presents a study on the application of sentiment analysis for vocal music teaching under natural language processing. Firstly, for the training text that has been labeled with categories, a suitable classification model is built for the unlabeled test text using the model category prediction. The sum of the identified sentiment scores is calculated, and the different results are compared to set a reasonable threshold to determine the sentiment polarity. According to the mutual information statistic value of each feature item, the feature items with a mutual information statistic value higher than the threshold are retained. The weights of each feature term are normalized to the interval to which the range of weights is normalized, and the feature weights are obtained. The optimal classification surface is calculated by converting the Lagrangian optimization method into a problem of finding the pair, which makes the low-dimensional problem transformed into a high-dimensional space and combines the advantages of Bayes’ ability to automatically adjust the weight coefficients in training and the huge training capacity of neural networks to improve the flexibility and robustness of the training model. Feature extraction is performed by convolutional layers, and the dimensionality of the feature vector is reduced by the maximum pooling method to realize an emotional-semantic vocal teaching model. The results show that the proposed method can improve the flexibility and accuracy of the semantic analysis model, and also enhance the training robustness of the network, with the F1 value reaching 91.65%, which can accurately mine the application data of “one core and three integrations” vocal teaching model and improve the teaching model in universities.

Reducing the Complexity of Two Classes of Optimization Problems by Inexact Accelerated Proximal Gradient Method

We propose a double-loop inexact accelerated proximal gradient (APG) method for a strongly convex composite optimization problem with two smooth components of different smoothness constants and computational costs. Compared to APG, the inexact APG can reduce the time complexity for finding a near-stationary point when one smooth component has higher computational cost but a smaller smoothness constant than the other. The strongly convex composite optimization problem with this property arises from subproblems of a regularized augmented Lagrangian method for affine-constrained composite convex optimization and also from the smooth approximation for bilinear saddle-point structured nonsmooth convex optimization. We show that the inexact APG method can be applied to these two problems and reduce the time complexity for finding a near-stationary solution. Numerical experiments demonstrate significantly higher efficiency of our methods over an optimal primal-dual first-order method by Hamedani and Aybat [SIAM J. Optim., 31 (2021), pp. 1299–1329] and the gradient sliding method by Lan, Ouyang, and Zhou [arXiv2101.00143, 2021].

A semismooth Newton based augmented Lagrangian method for nonsmooth optimization on matrix manifolds

This paper is devoted to studying an augmented Lagrangian method for solving a class of manifold optimization problems, which have nonsmooth objective functions and nonlinear constraints. Under the constant positive linear dependence condition on manifolds, we show that the proposed method converges to a stationary point of the nonsmooth manifold optimization problem. Moreover, we propose a globalized semismooth Newton method to solve the augmented Lagrangian subproblem on manifolds efficiently. The local superlinear convergence of the manifold semismooth Newton method is also established under some suitable conditions. We also prove that the semismoothness on submanifolds can be inherited from that in the ambient manifold. Finally, numerical experiments on compressed modes and (constrained) sparse principal component analysis illustrate the advantages of the proposed method.

Variasi Nilai Tegangan Percik Arrester Akibat Pengaruh Kapasitas Trafo Daya dan Jarak Arrester

The ability of each substation to withstand lightning and overvoltage affects the power distribution of the power system. In order to make the best operation of the substations, it is necessary to install Arresters which are used to protect the transformers in each substation from overvoltage caused by lightning surges. In this study, the effect of capacity on the placement distance of the arrester is discussed with 30 MVA and 60 MVA transformers. The purpose of this study was to determine the effect of the power transformer capacity of the four substations on the value of the arrester spark voltage and to obtain the optimal arrester installation technique at each substation using the Lagrange optimization method. Lagrange optimization method is used to determine the optimal distance between the arrester and the power transformer. The calculation results show that in a transformer with a capacity of 60 MVA, the Arrester spark voltage occurs at t = 20 microseconds with a voltage drop of 390.49 kV. Meanwhile, in a transformer with a capacity of 30 MVA, the Arrester spark time is at t = 13.9 microseconds with a voltage drop of 266.3 kV. This shows the capacity of the transformer that affects the magnitude of the Arrester's spark voltage and the spark time. Thus, the smaller the transformer capacity, the smaller the arrester spark time with the large voltage drop

A manifold inexact augmented Lagrangian method for nonsmooth optimization on Riemannian submanifolds in Euclidean space

Abstract We develop a manifold inexact augmented Lagrangian framework to solve a family of nonsmooth optimization problem on Riemannian submanifold embedding in Euclidean space, whose objective function is the sum of a smooth function (but possibly nonconvex) and a nonsmooth convex function in Euclidean space. By utilizing the Moreau envelope, we get a smoothing Riemannian minimization subproblem at each iteration of the proposed method. Consequentially, each iteration subproblem is solved by a Riemannian Barzilai–Borwein gradient method. Theoretically, the convergence to critical point of the proposed method is established under some mild assumptions. Numerical experiments on compressed modes problems in physic and sparse principal component analysis demonstrate that the proposed method is a competitive method compared with some state-of-the-art methods.

A two-stage framework for site selection of underground pumped storage power stations using abandoned coal mines based on multi-criteria decision-making method: An empirical study in China

With the continued transformation of the energy structure, more and more coal mines have been abandoned. The construction of underground pumped storage power stations using abandoned coal mines not only solves the problem of renovating abandoned coal mines, but also ensures a high level of photovoltaic and wind integration. However, the most basic site selection problem of underground pumped storage power plants using waste coal mines has rarely been studied due to the complexity of the technical scheme and the uncertainty of the decision environment. Therefore, a two-stage fuzzy evaluation model is proposed in this paper. Firstly, two sets of site selection index systems are established, namely the veto assessment system and the comprehensive evaluation criteria system. Triangular intuitionistic fuzzy numbers are used for information representation. Then, the veto assessment system is applied for the initial screening of the alternatives. Afterward, the alternatives are evaluated by the comprehensive evaluation criteria system. The entropy method and step-wise weight assessment ratio analysis are utilized to calculate the objective and subjective weights respectively, and the Lagrange optimization method is used to obtain the combined weights. The sorting results of the alternatives are obtained through multi-attributive border approximation area comparison. Finally, an empirical analysis based on eight abandoned coal mines in Datong, Shanxi Province, proves the rationality of the decision-making framework. The weights of natural condition, society, resources, and economy are 29.52%, 23.83%, 28.42% and 18.23% respectively. Natural condition is the most important factor to consider when choosing the site for underground pumped storage power stations. The ranking results of the alternatives is A5 > A2 > A3 > A8 > A7. Therefore, A5 is the optimal location. The results have a guiding role for both governments and investors in the construction of underground pumped storage power stations and transformation of abandoned coal mines.

A Lagrangian dual method for two-stage robust optimization with binary uncertainties

This paper presents a new exact method to calculate worst-case parameter realizations in two-stage robust optimization problems with categorical or binary-valued uncertain data. Traditional exact algorithms for these problems, notably Benders decomposition and column-and-constraint generation, compute worst-case parameter realizations by solving mixed-integer bilinear optimization subproblems. However, their numerical solution can be computationally expensive not only due to their resulting large size after reformulating the bilinear terms, but also because decision-independent bounds on their variables are typically unknown. We propose an alternative Lagrangian dual method that circumvents these difficulties and is readily integrated in either algorithm. We specialize the method to problems where the binary parameters switch on or off constraints as these are commonly encountered in applications, and discuss extensions to problems that lack relatively complete recourse and to those with integer recourse. Numerical experiments provide evidence of significant computational improvements over existing methods.

Lyapunov Optimization-Based Trade-Off Policy for Mobile Cloud Offloading in Heterogeneous Wireless Networks

In order to improve mobile users’ service experience, mobile cloud computing (MCC) is promoted. Although MCC can alleviate the burdens of Smart mobile devices (SMDs) by offloading computation-intensive applications to the cloud, it also aggravates computing and storage overheads in cloud centers and bandwidth overhead on wireless links for offloading workloads of mobile users. Therefore, we should carefully design the offloading policy to decrease these overheads while easing the burdens of SMDs. To this end, we investigate the offloading policy in heterogeneous wireless networks. In this paper, a queue model is built to formulate the mobile users’ workload offloading problem and Lyapunov optimization framework is proposed to make trade-off between system offloading utility and queue backlog. For deterministic WiFi connections, a Lagrangian optimization method is proposed to decide the optimal offloading workloads. Furthermore, considering random WiFi connection durations, a multi-stage stochastic programming method is proposed. The experimental results show effectiveness of the Lagrangian optimization offloading method for deterministic WiFi connection and the multi-stage stochastic programming method for random WiFi connection.

Smooth augmented Lagrangian method for twin bounded support vector machine

<p style='text-indent:20px;'>In this paper, we propose a method for solving the twin bounded support vector machine (TBSVM) for the binary classification. To do so, we use the augmented Lagrangian (AL) optimization method and smoothing technique, to obtain new unconstrained smooth minimization problems for TBSVM classifiers. At first, the augmented Lagrangian method is recruited to convert TBSVM into unconstrained minimization programming problems called as AL-TBSVM. We attempt to solve the primal programming problems of AL-TBSVM by converting them into smooth unconstrained minimization problems. Then, the smooth reformulations of AL-TBSVM, which we called AL-STBSVM, are solved by the well-known Newton's algorithm. Finally, experimental results on artificial and several University of California Irvine (UCI) benchmark data sets are provided along with the statistical analysis to show the superior performance of our method in terms of classification accuracy and learning speed.</p>

Energy-Efficient Resource Allocation in Multi-UAV Networks With NOMA

This paper investigates the energy efficiency (EE) optimization in a wireless communication network where multiple UAVs serve different types of devices, namely, information receivers (IRs) and energy receivers (ERs). The UAVs transmit power signals towards the ERs, and then enable data transmission to IRs on the downlink and from ERs on the uplink with non-orthogonal multiple access (NOMA). The optimization problem to maximize the overall EE is formulated and solved using Lagrangian optimization and gradient-descent methods. The optimization is decomposed into two sub-problems. Firstly, by connecting the path loss of the devices’ channels with their rate demands, the UAVs’ optimal positions are obtained. Then, based on the obtained UAVs’ optimal positions and a closed-form expression for the EE, a resource allocation aiming to maximize EE is developed. For the simulations, two main scenarios for single and multiple UAVs are considered. Numerical results and comparisons are provided. In particular, for the single-UAV scenario, the results show an enhancement in EE for the operation with NOMA compared with OMA. For the multiple-UAV scenario, several cases depending on different combinations of the devices’ rate requirements are considered. The results show the superiority of NOMA over OMA in all use cases. The results also reveal the effect of considering the devices’ rate requirements on the EE, where the case with equal rate requirements has the best performance.

Optimized routing algorithm for maximizing transmission rate in D2D network

<span style="font-size: 9pt; font-family: 'Times New Roman', serif;">Wireless devices have been equiping extensive services over recent years. Since most of these devices are randomly distributed, a fundamental trade-off to be addressed is the transmission rate, latency, and packet loss of the ad hoc route selection in device to device (D2D) networks. Therefore, this paper introduces a notion of weighted transmission rate and total delay, as well as the probability of packet loss. By designing optimal transmission algorithms, this proposed algorithm aims to select the best path for device-to-device communication that maximizes the transmission rate while maintaining minimum delay and packet loss. Using the Lagrange optimization method, the lagrangian optimization of rate, delay, and the probability of packet loss algorithm (LORDP) is modeled. For practical designation, we consider the fading effect of the wireless channels scenario. The proposed optimal algorithm is modeled to compute the optimal cost objective function and represents the best possible solution for the corresponding path. Moreover, a simulation for the optimized algorithm is presented based on optimal cost objective function. Simulation results establish the efficiency of the proposed LORDP algorithm</span><span>.</span><span style="font-size: 9pt; font-family: 'Times New Roman', serif;">Wireless devices have been equiping extensive services over recent years. Since most of these devices are randomly distributed, a fundamental trade-off to be addressed is the transmission rate, latency, and packet loss of the ad hoc route selection in device to device (D2D) networks. Therefore, this paper introduces a notion of weighted transmission rate and total delay, as well as the probability of packet loss. By designing optimal transmission algorithms, this proposed algorithm aims to select the best path for device-to-device communication that maximizes the transmission rate while maintaining minimum delay and packet loss. Using the Lagrange optimization method, the lagrangian optimization of rate, delay, and the probability of packet loss algorithm (LORDP) is modeled. For practical designation, we consider the fading effect of the wireless channels scenario. The proposed optimal algorithm is modeled to compute the optimal cost objective function and represents the best possible solution for the corresponding path. Moreover, a simulation for the optimized algorithm is presented based on optimal cost objective function. Simulation results establish the efficiency of the proposed LORDP algorithm</span>

Crashworthiness Design of Thin-Walled Tubes Reinforced by Triply Periodic Minimal Surfaces.

This article aims to propose a kind of internally reinforced square tube based on triply periodic minimal surfaces (TPMSs), which can be obtained by computer-assisted techniques and additive manufacturing. To achieve this goal, a finite element simulation model is exploited to simulate the collision situation of reinforced thin-walled tubes. The design of reinforced tubes can be formulated into a multiparameter and multiobjective optimization problem, which can be solved using the well-known non-linear programming by quadratic Lagrangian (NLPQL) optimization method. Three types of TPMS-based tubes and two multicell tubes are compared under the same conditions to show the effectiveness of our method. Through the methods mentioned above, TPMS-reinforced tubes are found to be superior in crashworthiness. This means that the safety performance of the automobiles with lighter weight can be effectively improved. The optimal parameters of three types of TPMS-reinforced tubes under different conditions were obtained, providing the foundations and references for subsequent related studies. In addition, TPMS is first explored in the design of crashworthiness for automobiles in this article. Due to the controllability and implicit functional expression of TPMSs, TPMS-reinforced tubes can be easily controlled and optimized. Meanwhile, it is easy to manufacture them by three-dimensional printing technologies.

Numerical and experimental research on damage shape recognition of aluminum alloy plate based on Lamb wave

The structure health monitoring (SHM) system with Lamb waves technique is studied numerically and experimentally in this paper. A damage shape recognition algorithm (DSRA) is developed for the hole on aluminum alloy plate specimen. The proposed DSRA is established based on the two arrival time difference method (2/ATDM) and Lagrangian optimization method. This method captures the damage shape by describing the coordinates of the reflection point from the piezoelectric ceramic lead zirconate titanate (PZT) transducers to the damage edge. 2/ATDM provides a way to obtain coordinate points by the arrival time which is determined by Lamb waves. And the coordinate points are optimized by the Lagrangian optimization method. A numerical model is established to simulate the proposed experiment process. Its convergence rate of mesh size and time steps is investigated, and the numerical results are verified by these of experiment. Hence, the shape recognition of damage occurring at an arbitrary position and that of irregular damage are studied by the proposed numerical and experiment methods.

Optimization and sensitivity of TMD parameters for mitigating bridge maximum vibration response under moving forces

In order to effectively reduce the vertical vibration response of the bridge, this paper proposes a frequency-domain analysis method of bridge vibration that can consider the influence of the tuned mass damper (TMD) device. In this method, the time-domain motion equation of bridge-TMD under moving concentrated loads’ excitation is first established. Then, the equation of motion is Fourier transformed to obtain the displacement response spectrum of the bridge with the TMD device installed. Further, the 2-norm of the displacement response spectrum of the bridge is optimized by the augmented Lagrangian optimization method to obtain the optimal TMD parameters. Finally, numerical simulations verify the superiority of this method over the traditional optimization method (Den Hartog’s method) in terms of maximum displacement control. And, the effects of bridge damping ratio, moving load number, and axle-span ratio and moving load speed on the optimal parameters of TMD to control the maximum displacement response of the bridge were studied. The results show that using the augmented Lagrangian optimization method to optimize the 2-norm of the displacement response spectrum of the bridge can not only effectively analyze the influence of each bridge and load parameters on the selection of TMD optimal parameters, but also it can reasonably configure TMD parameters for all kinds of simply supported beam bridges, which can provide references for practical engineering applications.

On the real stability radius of sparse systems

In this paper, we study robust stability of sparse LTI systems using the stability radius (SR) as a robustness measure. We consider real perturbations with an arbitrary and pre-specified sparsity pattern of the system matrix and measure their size using the Frobenius norm. We formulate the SR problem as an equality-constrained minimization problem. Using the Lagrangian method for optimization, we characterize the optimality conditions of the SR problem, thereby revealing the relation between an optimal perturbation and the eigenvectors of an optimally perturbed system. Further, we use the Sylvester equation based parametrization to develop a penalty based gradient/Newton descent algorithm which converges to the local minima of the optimization problem. Finally, we illustrate how our framework provides structural insights into the robust stability of sparse networks.

An Alternating Augmented Lagrangian method for constrained nonconvex optimization

ABSTRACTWe consider the problem of minimizing a smooth nonconvex function over a structured convex feasible set, that is, defined by two sets of constraints that are easy to treat when considered separately. In order to exploit the structure of the problem, we define an equivalent formulation by duplicating the variables and we consider the augmented Lagrangian of this latter formulation. Following the idea of the Alternating Direction Method of Multipliers (ADMM), we propose an algorithm where a two-blocks decomposition method is embedded within an augmented Lagrangian framework. The peculiarities of the proposed algorithm are the following: (1) the computation of the exact solution of a possibly nonconvex subproblem is not required; (2) the penalty parameter is iteratively updated once an approximated stationary point of the augmented Lagrangian is determined. Global convergence results are stated under mild assumptions and without requiring convexity of the objective function. Although the primary aim of the paper is theoretical, we perform numerical experiments on a nonconvex problem arising in machine learning, and the obtained results show the practical advantages of the proposed approach with respect to classical ADMM.