Lagrange Multiplier Theory Research Articles

Lagrange Multiplier-Based Optimization for Hybrid Energy Management System with Renewable Energy Sources and Electric Vehicles

The issues of energy scarcity and environmental harm have become major priorities for both business and human progress. Hence, it is important and useful to focus on renewable energy research and efficient utilization of distributed energy sources (DERs). A microgrid (MG) is a self-managed system that encompasses these energy resources as well as interconnected consumers. It has the flexibility to function in both isolated and grid-connected configurations. This study aims to design an effective method of power management for a MG in the two operating modes. The proposed optimization model seeks to strike a balance between energy usage, protecting the life of batteries, and maximizing economic benefits for users in the MG, with consideration of the real-time electricity price and constraints of the power grid. Furthermore, in order to accurately account for the dynamic nature of not only the stationary battery banks used as the energy storage systems (ESS) but also the built-in batteries of electric vehicles (EVs), the model is presented as a multi-objective, multiparametric and constrained problem. The solution is proposed to be found using the Lagrange multiplier theory, which helps to achieve good performance with less computational burden. Lastly, simulation results from both the isolated and grid-connected modes also demonstrate the effectiveness of the designed method.

Phase retrieval via Lagrange programming neural network

In this paper, the problem of phase retrieval is addressed. Its solution is based on the Lagrange programming neural network (LPNN), which is an analog neural computational technique for solving nonlinear constrained optimization problems according to the Lagrange multiplier theory. The local stability of the proposed algorithm is also investigated. Furthermore, we extend the LPNN based approach to more challenging array signal processing applications, namely, when multiple vector signals with full column rank or other constraints are required to be recovered from measured magnitudes. One of the key difficulties for these challenges is pairing the separately estimated parameters, while the parameters estimated by the extended method are automatically paired. The performance of the developed algorithms is demonstrated via computer simulations.

A novel neural emulator identification of nonlinear dynamical systems using Lyapunov stability theory

This paper deals with a new weight-updating algorithm using Lyapunov stability theory (LST) for the training of a neural emulator (NE), of nonlinear systems, connected by an autonomous algorithm inspired from the real-time recurrent learning (RTRL). The proposed method is formulated by an inequality-constraint optimization problem where the Lagrange multiplier theory is used as the optimization tool. The contribution of this paper is the integration of the LST into the Lagrange constraint function to synthesize a new analytical adaptation gain rate satisfying the asymptotic stability of the NE and providing good emulation performances. To confirm the good performances and the convergence ability of the proposed adaptation algorithm, a numerical example and an experimental validation on a chemical reactor are proposed.

Probabilistic definition of the clinical target volume—implications for tumor control probability modeling and optimization

Evidence has been presented that moving beyond the binary definition of clinical target volume (CTV) towards a probabilistic CTV can result in better treatment plans. The probabilistic CTV takes the likelihood of disease spread outside of the gross tumor into account. An open question is: how to optimize tumor control probability (TCP) based on the probabilistic CTV. We derive expressions for TCP under the assumptions of voxel independence and dependence. For the dependent case, we make the assumption that tumors grow outward from the gross tumor volume. We maximize the (non-convex) TCP under convex dose constraints for all models. For small numbers of voxels, and when a dose-influence matrix is not used, we use exhaustive search or Lagrange multiplier theory to compute optimal dose distributions. For larger cases we present (1) a multi-start strategy using linear programming with a random cost vector to provide random feasible starting solutions, followed by a local search, and (2) a heuristic strategy that greedily selects which subvolumes to dose, and then for each subvolume assignment runs a convex approximation of the optimization problem. The optimal dose distributions are in general different for the independent and dependent models even though the probabilities of each voxel being tumorous are set to the same in both cases. We observe phase transitions, where a subvolume is either dosed to a high level, or it gets ‘sacrificed’ by not dosing it at all. The greedy strategy often yields solutions indistinguishable from the multi-start solutions, but for the 2D case involving organs-at-risk and the dependent TCP model, discrepancies of around 5% (absolute) for TCP are observed. For realistic geometries, although correlated voxels is a more reasonable assumption, the correlation function is in general unknown. We demonstrate a tractable heuristic that works very well for the independent models and reasonably well for the dependent models. All data are provided.

Optimal Control of State Constrained Age-Structured Problems

The aim of this work is to study an optimal control problem with state constraints where the state is given by an age-structured, abstract parabolic differential equation. We prove the existence and uniqueness of solution for the state equation and provide first and second parabolic estimates. We analyze the differentiability of the cost function and, based on the general theory of Lagrange multipliers, we give a first order optimality condition. We also define and analyze the regularity of the costate.

Lagrange Programming Neural Network Approach for Target Localization in Distributed MIMO Radar

In this paper, the problem of source localization in distributed multiple-input multiple-output (MIMO) radar using bistatic range measurements, which correspond to the sum of transmitter-to-target and target-to-receiver distances, is addressed. Our solution is based on the Lagrange programming neural network (LPNN), which is an analog neural computational technique for solving nonlinear constrained optimization problems according to the Lagrange multiplier theory. The local stability of the proposed positioning algorithm is also investigated. Furthermore, we have extended the LPNN based approach to more challenging scenarios, namely, when time synchronization among all antennas cannot be fulfilled, and there are position uncertainties in the MIMO radar transmit and receive elements. The optimality of the developed algorithms is demonstrated by comparing with the Cramer-Rao lower bound via computer simulations.

A Sequential Method for a Class of Stable Mathematical Programming Problems

A sequential programming method of first order for constrained optimization in infinite dimensional spaces is presented. It is referred to as the SP method. It relies on Lipschitz stability of the minimizers with respect to perturbations in an essential way, and is motivated by optimal control problems with partial differential equations as constraints. Convergence and convergence rate are analyzed based on solutions of first order optimality conditions and on Lagrange multiplier theory. The first order SP method does not rely on an approximation of the Hessian of the Lagrange functional and consequently it avoids instabilities due to possible indefiniteness far from a local minimum. It is especially well suited for bilinear control problems and can be extended to certain classes of nonsmooth and convex problems. It can efficiently be implemented by the use of saddle point solvers, with complexity which is between that of gradient and SQP methods. For the important class of bilinear control problems the m...

Algebraic Error Based Triangulation and Metric of Lines.

Line triangulation, a classical geometric problem in computer vision, is to determine the 3D coordinates of a line based on its 2D image projections from more than two views of cameras with known projection matrices. Compared to point features, line segments are more robust to matching errors, occlusions, and image uncertainties. In addition to line triangulation, a better metric is needed to evaluate 3D errors of line triangulation. In this paper, the line triangulation problem is investigated by using the Lagrange multipliers theory. The main contributions include: (i) Based on the Lagrange multipliers theory, a formula to compute the Plücker correction is provided, and from the formula, a new linear algorithm, LINa, is proposed for line triangulation; (ii) two optimal algorithms, OPTa-I and OPTa-II, are proposed by minimizing the algebraic error; and (iii) two metrics on 3D line space, the orthogonal metric and the quasi-Riemannian metric, are introduced for the evaluation of line triangulations. Extensive experiments on synthetic data and real images are carried out to validate and demonstrate the effectiveness of the proposed algorithms.

Two Lagrange Optimization Theory Based Methods for Solving Economic Load Dispatch Problems

The optimal generation dispatch problem with only one fuel option for each generating unit has been solved for many recent years. However, it is more realistic to represent the fuel cost function for each fossil fired plant as a segmented piece-wise quadratic functions. This is because of development of technology in thermal plants to reach maximum fuel save. Those units are faced with the difficulty of determining which the most economical fuel to burn is. This paper presents two effective methods for solving economic load dispatch problem with multiple fuel options. An advantage of the methods is to formulate Lagrange mathematical function easily based on the Lagrange multiplier theory. The proposed methods are tested on one test system consisting of ten generating units with various load demands and compared to other methods. The simulation results show that the methods are very efficient for the optimal generation dispatch problem with multiple fuel options

A novel adaptive filter design using Lyapunov stability theory

This paper presents a new approach to design an adaptive lter using Lyapunov stability theory. The design procedure is formulated as an inequality constrained optimization problem. Lagrange multiplier theory is used as an optimization tool. Lyapunov stability theory is integrated into the constraint function to satisfy the asymptotic stability of the proposed ltering system. The tracking capability is improved by using a new analytical adaptation gain rate, which has the ability to adaptively adjust itself depending on a sequential tracking square error rate. The fast and robust convergence ability of the proposed algorithm is comparatively examined by simulation examples.

Optimal Spatial Matrix Filter Design for Array Signal Preprocessing

An efficient technique of designing spatial matrix filter for array signal preprocessing based on convex programming was proposed. Five methods were considered for designing the filter. In design method 1, we minimized the passband fidelity subject to the controlled overall stopband attenuation level. In design method 2, the objective function and the constraint in the design method 1 were reversed. In design method 3, the optimal matrix filter which has the general mean square error was considered. In design method 4, the left stopband and the right stopband were constrained with specific attenuation level each, and the minimized passband fidelity was received. In design method 5, the optimization objective function was the sum of the left stopband and the right stopband attenuation levels with the weighting factors 1 andγ, respectively, and the passband fidelity was the constraints. The optimal solution of the optimizations above was derived by the Lagrange multiplier theory. The relations between the optimal solutions were analyzed. The generalized singular value decomposition was introduced to simplify the optimal solution of design methods 1 and 2 and enhanced the efficiency of solving the Lagrange multipliers. By simulations, it could be found that the proposed method was effective for designing the spatial matrix filter.

Abstract theory of variational inequalities with Lagrange multipliers and application to nonlinear PDEs

Recently, we established some generalizations of the theory of Lagrange multipliers arising from nonlinear programming in Banach spaces, which enable us to treat not only elliptic problems but also parabolic problems in the same generalized framework. The main objective of the present paper is to discuss a typical time-dependent double obstacle problem as a new application of the above mentioned generalization. Actually, we describe it as a usual parabolic variational inequality and then characterize it as a parabolic inclusion by using the Lagrange multiplier and the nonlinear maximal monotone operator associated with the time differential under time-dependent double obstacles.

A variational approach to sparsity optimization based on Lagrange multiplier theory

Sparsity optimization for linear least squares problems formulated as non-smooth regularization problems are considered in infinite-dimensional sequence spaces ℓp with p ∈ [0, 1]. Necessary optimality conditions in the format of a complementarity system are obtained. A monotonically convergent scheme is developed for the case p ∈ (0, 1]. For the case p = 0 a primal dual active set strategy based on the Lagrange multiplier rule is proposed and analyzed for special cases.

Solving the problem of combination of Multi-fuel electric generator units using lagrange multiplier theory

This paper presents an approach to solve the unit commitment problem with multifuel options in the thermal power plants. Traditionally, each generator unit is used to each fuel option with the segmented piecewise quadratic functions, so that it is not difficult to solve them. However, it is more realistic to represent the fuel cost function for each fossil fired plant as the segmented piecewise quadratic functions. Those units are faced with the difficulty of determining which fuel is the most economical to burn. Therefore, this paper presents an approach to solve the unit commitment problem with multi-fuel options. An advantage of the method is to formulate Lagrange mathematical function easily based on the Lagrange multiplier theory. The simulation result for 10 generator systems are compared with others methods to show that the approach is a new method and an effective method to solve the minimizing of electricity production cost of generator units with multi-fuel option.

Variational Properties of Value Functions

Regularization plays a key role in a variety of optimization formulations of inverse problems. A recurring theme in regularization approaches is the selection of regularization parameters, and their effect on the solution and on the optimal value of the optimization problem. The sensitivity of the value function to the regularization parameter can be linked directly to the Lagrange multipliers. This paper characterizes the variational properties of the value functions for a broad class of convex formulations, which are not all covered by standard Lagrange multiplier theory. An inverse function theorem is given that links the value functions of different regularization formulations (not necessarily convex). These results have implications for the selection of regularization parameters, and the development of specialized algorithms. Numerical examples illustrate the theoretical results.

Minimum Outage Probability for the Relay Channel under Individual Power Constraints

In this letter, minimum outage probability for a fading relay channel is considered for long-term individual power constraints (IPC) at the source and the relay. By utilizing Lagrange multiplier theory, it is shown that solving the IPC problem is equivalent to solving a weighted sum power constrained (wSPC) problem. Opportunistic decode and forward (ODF), and simplified ODF (SODF) protocol performances are investigated and compared with a lower bound on outage probability, under both individual and sum power constraints (SPC). It is shown that, under IPC, the source or the relay does not necessarily deplete its average power constraint, but some power can remain unutilized for ODF or SODF.

Minimal Effort Problems and Their Treatment by Semismooth Newton Methods

The paper introduces minimum effort control problems. These provide an answer to the question of the smallest possible control bound which still allows us to drive the system to a target within a fixed time T. This is a counterpart to the time optimal control problem which minimizes the time required to drive the system to the target, given a control bound. The problem is formulated as an optimal control problem with pointwise constraint on the control. The necessary conditions of optimality are derived by Lagrange multiplier theory. The semismooth Newton method is applied to a properly regularized problem. Well-posedness and superlinear convergence of the semismooth Newton method are proved for linear control systems under a controllability condition. Numerical results are presented for demonstrating the applicability and feasibility of the proposed method.

Optimal matrix filter design with controlled mean-square sidelobe level

In this study, a simple approach for designing a matrix filter with controlled mean-square sidelobe level is proposed. Two methods are used for designing the matrix filter. In design method 1, the authors minimise the normalised mean-square error between the designed and the desired filter responses in the passband subject to the different normalised mean-square error constraints in the left and right stopbands. In design method 2, the objective function and the constraints in the design method 1 are reversed. By using Lagrange multiplier theory, the optimal solution and the optimal value of the optimisations are given, the equations of finding the optimal Lagrange multipliers are offered. In addition, the optimal matrix lowpass filters with symmetric stopband region and equally sidelobe level are given, the matrix codiagonalisation method is used for improving the design efficiency. Numerical results show that the proposed approach is effective for designing matrix filter and filtering short data records.

A revised particle swarm optimization based discrete Lagrange multipliers method for nonlinear programming problems

In this paper, a new algorithm for solving constrained nonlinear programming problems is presented. The basis of our proposed algorithm is none other than the necessary and sufficient conditions that one deals within a discrete constrained local optimum in the context of the discrete Lagrange multipliers theory. We adopt a revised particle swarm optimization algorithm and extend it toward solving nonlinear programming problems with continuous decision variables. To measure the merits of our algorithm, we provide numerical experiments for several renowned benchmark problems and compare the outcome against the best results reported in the literature. The empirical assessments demonstrate that our algorithm is efficient and robust.

Lagrange Multipliers and the Fundamental Theorem of Algebra

(2009). Lagrange Multipliers and the Fundamental Theorem of Algebra. The American Mathematical Monthly: Vol. 116, No. 9, pp. 828-830.