Lagrange Multiplier Research Articles

A progressive hedging-based matheuristic for the stochastic production routing problem with adaptive routing

The production routing problem (PRP) arises in the context where a manufacturing facility manages its production schedule, the delivery of goods to customers by a fleet of vehicles, and the inventory levels both at the plant and at the customers. The presence of uncertainty often complicates the planning process. In particular, production and distribution costs may significantly increase if demand uncertainty is ignored in the planning phase. Nevertheless, only a few studies have considered demand uncertainty in the PRP. In this article, we propose a two-stage stochastic programming approach for a one-to-many PRP with a single product and demand uncertainty. Unlike previous studies in the literature, we consider the case where routing decisions are made in the second stage after customer demands become known. This offers more flexibility, which can decrease transportation costs by preventing unnecessary customer visits. In addition to the static–dynamic case, in which setup decisions are made first and production quantities are decided in the second stage, we also consider the static–static setting where both sets of decisions must be made prior to the demand realization. A progressive hedging algorithm combined with a matheuristic is developed to solve the problem. The role of the progressive hedging algorithm is to decompose the stochastic problem into more tractable scenario-specific subproblems and lead the first-stage variables towards convergence by modifying their Lagrangean multipliers. However, solving the subproblems remains challenging since they include routing decisions, and we thus propose a matheuristic to exploit the structural characteristics of the subproblems. First, a Traveling Salesman Problem (TSP) is solved to find the optimal tour for all customers regardless of demand and capacity. Utilizing the sequence obtained from the TSP, we then solve a restricted PRP while taking into account the other constraints of the original problem. Finally, for each period and scenario, a vehicle routing problem is solved to improve the quality of the solutions. Computational experiments are conducted to analyze the algorithm’s efficiency and to assess the benefits that can be achieved by handling routing in the second stage.

Analysis and Optimization of a Novel Compact Compliant 2-DOF Positioner for Positioning to Assess Bio-Specimen Characteristics

A novel compact 2-DOF compliant positioner is developed for the purpose of achieving good characteristics such as high natural frequency, high displacement amplification ratio, good linear motion, and compact structure based on its symmetrical structure. To be specific, the developed stage is proposed according to an advanced six-lever displacement amplifier arranged at an inclination angle of the rigid bar utilizing right circular hinges and a parallel guiding mechanism with integrated flexure leaf hinges to attain the above-mentioned characteristics and reduce the decoupling mobility error. First, to quickly assess the initial quality response, an integration method of kinetostatic analysis, the Lagrange method, and finite element analysis was applied to evaluate and verify the quality characteristic of the stage. The experimental result showed that the error between the analytical method and the FEA method was 1.3%, which was relatively small and reliable for quickly assessing the primary quality response of the proposed positioner. Next, to boost the important output characteristics of the developed positioner, the integration approach of the response surface method and NSGA-II algorithm was utilized to find the optimal design variables. Finally, a prototype was manufactured based on the CNC milling method to validate the experimental and FEA analysis results. The attained results show that the optimal results of safety factor and output displacement were 2.4025 and 248.9 µm. Moreover, the FEA verification results were 2.4989 and 242.16 µm, with errors for safety factor and output displacement between the optimal result and the FEA result of 3.86% and 2.78%, respectively. In addition, the simulation and experimental results of the first natural frequency were 371.83 Hz and 329.59 Hz, respectively, and the error between the FEA result and experimental result for the first natural frequency was 11.36%. Furthermore, the achieved results show that the relationship between input displacement and output displacement of the experimental result and the FEA result of the developed structure achieved a good linear connection. These results suggest that the proposed positioner will be a potential structure employed in precise positioning systems and nanoindentation testing positioning systems for checking bio-specimens’ behaviors.

A Delaunay refinement algorithm for the particle finite element method applied to free surface flows

AbstractThis article proposes two contributions to the calculation of free‐surface flows using the particle finite element method (PFEM). The PFEM is based upon a Lagrangian approach: a set of particles defines the fluid and each particle is associated with a velocity vector. Then, unlike a pure Lagrangian method, all the particles are connected by a triangular mesh. The difficulty lies in locating the free surface from this mesh. It is a matter of deciding which of the elements in the mesh are part of the fluid domain, and to define a boundary—the free surface. Then, the incompressible Navier–Stokes equations are solved on the fluid domain and the particle position is updated using the velocity vector from the finite element solver. Our first contribution is to propose an approach to adapt the mesh with theoretical guarantees of quality: the mesh generation community has acquired a lot of experience and understanding about mesh adaptation approaches with guarantees of quality on the final mesh. The approach we use here is based on a Delaunay refinement strategy, allowing to insert and remove nodes while gradually improving mesh quality. We show that what is proposed allows to create stable and smooth free surface geometries. One characteristic of the PFEM is that only one fluid domain is modeled, even if its shape and topology change. It is nevertheless necessary to apply conditions on the domain boundaries. When a boundary is a free surface, the flow on the other side is not modeled, it is represented by an external pressure. On the external free surface boundary, atmospheric pressure can be imposed. Nevertheless, there may be internal free surfaces: the fluid can fully encapsulate cavities to form bubbles. The pressure required to maintain the volume of those bubbles is a priori unknown. For example, the atmospheric pressure would not be sufficient to prevent the bubbles from deflating and eventually disappearing. Our second contribution is to propose a multi‐point constraint approach to enforce global incompressibility of those empty bubbles. We show that this approach allows to accurately model bubbly flows that involve two fluids with large density differences, for instance water and air, while only modeling the heavier fluid.

Solution of the Elliptic Interface Problem by a Hybrid Mixed Finite Element Method

This paper addresses the elliptic interface problem involving jump conditions across the interface. We propose a hybrid mixed finite element method on the triangulation where the interfaces are aligned with the mesh. The second-order elliptic equation is initially decomposed into two equations by introducing a gradient term. Subsequently, weak formulations are applied to these equations. Scheme continuity is enforced using the Lagrange multiplier technique. Finally, we derive an explicit formula for the entries of the matrix equation representing Lagrange multiplier unknowns resulting from hybridization. The method yields approximations of all variables, including the solution and gradient, with optimal order. Furthermore, the matrix representing the final linear algebra systems is not only symmetric but also positive definite. Numerical examples convincingly demonstrate the effectiveness of the hybrid mixed finite element method in addressing elliptic interface problems.

Active Distribution Network Expansion Planning Based on Wasserstein Distance and Dual Relaxation

In the future, a high proportion of distributed generations (DG) will be integrated into the distribution network. The existing active distribution network (ADN) planning methods have not fully considered multiple uncertainties, differentiated regulation modes or the cost of multiple types of interconnection switches. Meanwhile, it is difficult to solve large-scale problems at small granularity. Therefore, an expansion planning method of ADN considering the selection of multiple types of interconnection switches is proposed. Firstly, a probability distribution ambiguity set of DG output and electrical-load consumption based on the Wasserstein distance is established for dealing with the issue of source-load uncertainty. Secondly, a distributionally robust optimization model for collaborative planning of distribution network lines and multiple types of switches based on the previously mentioned ambiguity set is established. Then, the original model is transformed into a mixed integer second-order cone programming (SOCP) model by using the convex relaxation method, the Lagrangian duality method and the McCormick relaxation method. Finally, the effectiveness of the proposed method is systematically verified using the example of Portugal 54. The results indicate that the proposed method raises the annual net profit by nearly 5% compared with the traditional planning scheme and improves the reliability and low-carbon nature of the planning scheme.

Optimizing Conical Intersections without Explicit Use of Non-Adiabatic Couplings.

We present two alternative methods for optimizing minimum energy conical intersection (MECI) molecular geometries without knowledge of the derivative coupling (DC). These methods are based on the utilization of Lagrange multipliers: (i) one method uses an approximate calculation of the DC, while the other (ii) do not require the DC. Both methods use the fact that information on the DC is contained in the Hessian of the squared energy difference. Tests done on a set of small molecular systems, in comparison with other methods, show the ability of the proposed methods to optimize MECIs. Finally, we apply the methods to the furimamide molecule, to optimize and characterize its S1/S2 MECI, and to optimizing the S0/S1 MECI of the silver trimer.

Electromagnetic breathing dromion-like structures in an anisotropic ferromagnetic medium

The influence of Gilbert damping on the propagation of electromagnetic waves (EMWs) in an anisotropic ferromagnetic medium is investigated theoretically. The interaction of the magnetic field component of the electromagnetic wave with the magnetization of a ferromagnetic medium has been studied by solving the associated Maxwell’s equations coupled with the Landau–Lifshitz–Gilbert (LLG) equation. When small perturbations are made on the magnetization of the ferromagnetic medium and magnetic field along the direction of propagation of electromagnetic wave by using the reductive perturbation method, the associated nonlinear dynamics is governed by a time-dependent damped derivative nonlinear Schrödinger (TDDNLS) equation. The Lagrangian density function is constructed by using the variational method to solve the TDDNLS equation to understand the dynamics of the system under consideration. The propagation of EMW in a ferromagnetic medium with inherent Gilbert damping admits very interesting nonlinear dynamical structures. These structures include Gilbert damping-managing symmetrically breathing solitons, localized erupting electromagnetic breathing dromion-like modes of excitations, breathing dromion-like soliton, decaying dromion-like modes and an unexpected creation-annihilation mode of excitations in the form of growing-decaying dromion-like modes.

Optimizing the Location of Supports under a Monolithic Floor Slab

Monolithic reinforced concrete floor slabs are one of the most common types of building structures, and their optimization is an urgent task. The article presents the methodology for finding the optimal position of point supports under a reinforced concrete floor slab of arbitrary configuration at arbitrary load. The slab is considered thin, elastic and isotropic, with constant over-the-area stiffness, that is, the reinforcement is not taken into account or is constant. The solution is performed using the finite element method in combination with the nonlinear optimization methods. Finite element analysis is implemented by authors in MATLAB (R2024a) environment in such a way that the location of the columns may not coincide with the nodes of the finite element mesh of the slab. This allows to significantly increase the efficiency of solving the optimization problem compared to previously used algorithms, including the Monte Carlo method. Boundary conditions are taken into account using the Lagrange multiplier method. As an optimization criterion, the maximum deflection value is used, as well as the value of the potential strain energy. The effectiveness of six nonlinear optimization methods is compared in the example of a square slab under the action of a uniformly distributed load. For solutions obtained using the pattern search, simulated annealing and internal point methods, the maximum deflections are at least 1.2 times higher than for solutions obtained using the particle swarm method and genetic algorithm. An example of real object optimization is also presented. By changing the position of seven columns, it was possible to reduce the maximum deflection of the floor slab by 1.6 times.

Optimization of thruster configuration and control allocation for a spherical magnetic coupling thrusters actuated underwater robot

Propeller-based reconfigurable magnetic coupling thrusters are a promising propulsion system for underwater robots, due to their vectored thrust and watertightness. This paper presents a lightweight and compact 3-DOF spherical reconfigurable magnetic coupling thruster (SRMCT-II), and focuses on its thruster configuration and control allocation, considering its intrinsic characteristics. To enhance robot agility, we develop an optimization problem for the configuration of SRMCT-II without relying on the engineers' experience or intuition and use a genetic algorithm to determine the optimal layout of three SRMCT-IIs for an underwater robot prototype. To assign motion control commands to the SRMCT-IIs, we formulate the control allocation as a constrained convex optimization problem and solve it using a practical augmented Lagrangian method. In the experiments, a customized force-sensing testbed is fabricated to evaluate the performance of the SRMCT-II and the robot's control allocation with its optimal thruster layout. The results verify the vectored characteristics of the SRMCT-II and demonstrate the computational efficiency and accuracy of the control allocation. The validation of thrust attainability reveals that the optimized thruster configuration achieves complete decoupling of the generalized control forces and torques. This indicates that the underwater robot can control its thrust and torque independently in all three dimensions.

Optimizing resource allocation for cluster D2D-assisted fog computing networks: A three-layer Stackelberg game approach

Resource management in fog computing is a highly challenging issue. Resource constraints of fog devices, the dynamic and diverse nature of workloads such as data-intensive, computationally intensive, and latency-sensitive applications, and the unpredictability of fog computing environments make it more challenging. How to collaboratively schedule edge, and end multi-layer ubiquitous computing resources to break through single node scale and resource bottlenecks to further improve computing performance also needs to be addressed. In this paper, we leverage idle resources on the user side, and study the resource allocation problems in the D2D-assisted fog computing networks. Firstly, we use an improved Louvain algorithm to cluster users and select the D2D device with the highest priority in each cluster as the cluster head to assist fog computing, where cluster heads can act as both servers and relays. Secondly, considering that fog nodes and D2D devices have limited resources and are not obligated to share their resources for free, we propose a price incentive method to construct a three-layer Stackelberg game model that maximizes the benefits of fog nodes and cluster heads while minimizing user costs. Then, we use the Lagrange multiplier method and the Karush–Kuhn–Tucker (KKT) conditions to solve the equilibrium point of the model to achieve our goal. Finally, the simulation results show that our proposed scheme markedly decreases the overall system cost in comparison to the four benchmark schemes. Moreover, in various scenarios the total cost of our scheme consistently remains lower than the other four schemes, affirming the efficiency of the scheme proposed in this paper.

Normalized ground state solutions for the Schrödinger systems with critical exponential growth in [formula omitted]

In this paper, we study the existence of normalized solutions to the following nonlinear Schrödinger systems with critical exponential growth:{−Δu+λ1u=f1(u)+βr1|u|r1−2u|v|r2,inR2,−Δv+λ2v=f2(v)+βr2|u|r1|v|r2−2v,inR2,∫R2u2dx=a2and∫R2v2dx=b2,u,v∈H1(R2), where 0<a,b<1,β>0,r1,r2>1 and r1+r2∈(4,+∞), f1,f2∈C(R,R) have critical exponential growth in the sense of Trudinger-Moser inequality. λ1,λ2∈R will arise as Lagrange multipliers. Under some suitable assumptions on f1,f2, we prove the existence of positive normalized ground state solutions for the problem when β>0 is sufficiently large via variational method. Our results improve and extend the previous results.

When and where to step: Terrain-aware real-time footstep location and timing optimization for bipedal robots

Online footstep planning is essential for bipedal walking robots, allowing them to walk in the presence of disturbances and sensory noise. Most of the literature on the topic has focused on optimizing the footstep placement while keeping the step timing constant. In this work, we introduce a footstep planner capable of optimizing footstep placement and step time online. The proposed planner, consisting of an Interior Point Optimizer (IPOPT) and an optimizer based on Augmented Lagrangian (AL) method with analytical gradient descent, solves the full dynamics of the Linear Inverted Pendulum (LIP) model in real time to optimize for footstep location as well as step timing at the rate of 200 Hz. We show that such asynchronous real-time optimization with the AL method (ARTO-AL) provides the required robustness and speed for successful online footstep planning. Furthermore, ARTO-AL can be extended to plan footsteps in 3D, allowing terrain-aware footstep planning on uneven terrains. Compared to an algorithm with no footstep time adaptation, our proposed ARTO-AL demonstrates increased stability in simulated walking experiments as it can resist pushes on flat ground and on a 10° ramp up to 120 N and 100 N respectively. Videos22https://youtu.be/2pwDTEJzmjI. and open-source code33https://github.com/WangKeAlchemist/ARTO-AL/tree/master. are released.

Stabilised finite element method for Stokes problem with nonlinear slip condition

This work introduces a stabilised finite element formulation for the Stokes flow problem with a nonlinear slip boundary condition of friction type. The boundary condition is enforced with the help of an additional Lagrange multiplier representing boundary traction and the stabilised formulation is based on simultaneously stabilising both the pressure and the traction. We establish the stability and the a priori error analyses, and perform a numerical convergence study in order to verify the theory.

Derivation, characterization, and application of complete orthonormal sequences for representing general three-dimensional states of residual stress

Residual stresses are self-equilibrated stresses on unloaded bodies. Owing to their complex origins, it is useful to develop functions that can be linearly combined to represent any sufficiently regular residual stress field. In this work, we develop orthonormal sequences that span the set of all square-integrable residual stress fields on a given three-dimensional region. These sequences are obtained by extremizing the most general quadratic, positive-definite functional of the stress gradient on the set of all sufficiently regular residual stress fields subject to a prescribed normalization condition; each such functional yields a sequence. For the special case where the sixth-order coefficient tensor in the functional is homogeneous and isotropic and the fourth-order coefficient tensor in the normalization condition is proportional to the identity tensor, we obtain a three-parameter subfamily of sequences. Upon a suitable parameter normalization, we find that the viable parameter space corresponds to a semi-infinite strip. For a further specialized spherically symmetric case, we obtain analytical expressions for the sequences and the associated Lagrange multipliers. Remarkably, these sequences change little across the entire parameter strip. To illustrate the applicability of our theoretical findings, we employ three such spherically symmetric sequences to accurately approximate two standard residual stress fields. Our work opens avenues for future exploration into the implications of different sequences, achieved by altering both the spatial distribution and the material symmetry class of the coefficient tensors, toward specific objectives.

Comprehensive consensus representation learning for incomplete multiview subspace clustering

Incomplete multiview clustering (IMVC) aims to address clustering problems caused by missing views. While some progress has been made, the current approaches still have some problems. First, the existing IMVC methods excessively rely on the original high-dimensional multiview data, leading to suboptimal results when the data are heavily contaminated. Second, the utilization of high-dimensional data often overlooks the complementarity of and consistency across multiple views. Third, the cluster structure of the input data is not adequately considered, which is detrimental to the resulting clustering performance. To address these problems, we propose a novel method: comprehensive consensus representation learning for incomplete multiview subspace clustering (CLR-IMVC). Specifically, CLR-IMVC assumes that all views are generated from a latent representation space. We model the original incomplete multiple views and the latent space to explore their correlations. Subsequently, we use the latent space to generate complete views and construct a subspace self-representation matrix for each view. Moreover, we innovatively treat the latent representation itself as new a view and likewise extract its self-representation matrix. Furthermore, we construct a third-order tensor by using these self-representation matrices and impose low-rank constraints on the tensor to explore the high-order correlations among the matrices. In addition, a consensus representation learning term is added to explore the consistency information between the views and the latent space, ultimately generating a robust clustering structure. Finally, the proposed method can be effectively solved by the augmented Lagrangian multiplier (ALM). Extensive experiments conducted on diverse datasets demonstrate the effectiveness of CLR-IMVC.

Lagrangian relaxation for continuous‐time optimal control of coupled hydrothermal power systems including storage capacity and a cascade of hydropower systems with time delays

AbstractThis work considers a short‐term, continuous time setting characterized by a coupled power supply system controlled exclusively by a single provider and comprising a cascade of hydropower systems (dams), fossil fuel power stations, and a storage capacity modeled by a single large battery. Cascaded hydropower generators introduce time‐delay effects in the state dynamics, which are modeled with differential equations, making it impossible to use classical dynamic programming. We address this issue by introducing a novel Lagrangian relaxation technique over continuous‐time constraints, constructing a nearly optimal policy efficiently. This approach yields a convex, nonsmooth optimization dual problem to recover the optimal Lagrangian multipliers, which is numerically solved using a limited memory bundle method. At each step of the dual optimization, we need to solve an optimization subproblem. Given the current values of the Lagrangian multipliers, the time delays are no longer active, and we can solve a corresponding nonlinear Hamilton–Jacobi–Bellman (HJB) Partial Differential Equation (PDE) for the optimization subproblem. The HJB PDE solver provides both the current value of the dual function and its subgradient, and is trivially parallelizable over the state space for each time step. To handle the infinite‐dimensional nature of the Lagrange multipliers, we design an adaptive refinement strategy to control the duality gap. Furthermore, we use a penalization technique for the constructed admissible primal solution to smooth the controls while achieving a sufficiently small duality gap. Numerical results based on the Uruguayan power system demonstrate the efficiency of the proposed mathematical models and numerical approach.

Duality between the Maxwell-Chern-Simons and self-dual models in very special relativity

This work investigates the classical and quantum duality between the SIM(1)-Maxwell-Chern-Simons (MCS) model and its self-dual counterpart. Initially, we focus on free-field cases to establish equivalence through two distinct approaches: comparing the equations of motion and utilizing the master Lagrangian method. In both instances, the classical correspondence between the self-dual and MCS dual fields undergoes modifications due to very special relativity (VSR). Specifically, the duality is established when the associated VSR-mass parameters are identical, and the dual field is introduced through a non-local VSR correction. Furthermore, we analyze the duality when the self-dual model is minimally coupled to fermions. As a result, we demonstrate that Thirring-like interactions, corrected for non-local VSR contributions, are included in the MCS model. Additionally, we establish the quantum equivalence of the models by performing a functional integration of the fields and comparing the resulting effective Lagrangians.

Non-negative consistency affinity graph learning for unsupervised feature selection and clustering

Feature selection plays a crucial role in data mining and pattern recognition tasks. This paper proposes an efficient method for robust unsupervised feature selection, called joint local preserving and low-rank representation with nonnegative and symmetric constraint (JLPLRNS). This approach utilizes an indicator matrix instead of the row sparsity of the projection matrix to directly select some significant features from the original data and adaptively preserves the local geometric structure of original data into the low-dimensional embedded feature subspace via learned indicator matrix. By simultaneously imposing nonnegative symmetric and low-rank constraints on the representation coefficient matrix, it cannot only make this matrix discriminative, sparse and weight consistency for each pair of data, but also uncover the global structure of original data. These effectively will improve clustering performance. An algorithm based on the augmented Lagrange multiplier method with an alternating direction strategy is designed to resolve this model. Experimental results on various real datasets show that the proposed method can effectively identify some important features in data and outperforms many state-of-the-art unsupervised feature selection methods in terms of clustering performance.

Delay effects on distributed constrained optimization over double‐integrator multi‐agent systems

AbstractCommunication plays a pivotal role in distributed optimization problems, where unavoidable communication delays are presented. This research studies the distributed constrained optimization problem concerning second‐order multi‐agent systems with double‐integrator under time‐varying communication delays. An adaptive distributed optimization algorithm using multi‐agent system consensus technique and Karush–Kuhn–Tucker conditions is developed to deal with this problem. The local constraint term is solved adaptively through local dual Lagrange multipliers. When the cost function is strongly convex, and the communication topology is undirected and connected, we employ the Lasalle invariance principle to analyze the delay effects on convergence analysis. Moreover, we give an upper bound on communication delay. Finally, the provided numerical simulation examples demonstrate that the developed method is robust for the limited communication delay and the derived results are conservative.

Hidden BPS states of electroweak monopoles and a new bound estimate

Using the BPS Lagrangian method, we obtain a distinct set of Bogomolny equations for the Cho–Maison monopoles from the bosonic sector of a regularized electroweak theory. In the limit of n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow \\infty $$\\end{document} of the permittivity regulator, ϵρn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon \\left( \\rho ^n\\right) $$\\end{document}, the mass of the monopole can be estimated as MW∼3.56\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_W\\sim 3.56$$\\end{document} TeV. This value is within the latest theoretical window, 2.98–3.75 TeV. We also discuss some possible regularization mechanisms of the electroweak monopole in the Yang–Mills sector and the existence of its BPS state.