Добірка наукової літератури з теми "Nonlinear complementarity constraints"

It is well-known that the linear rate of convergence can be established for the classical augmented Lagrangian method for constrained optimization problems without strict complementarity. Whether this result is still valid for other nonlinear Lagrangian methods (NLM) is an interesting problem. This paper proposes a nonlinear Lagrangian function based on Fischer–Burmeister (F–B) nonlinear complimentarity problem (NCP) function for constrained optimization problems. The rate of convergence of this NLM is analyzed under the linear independent constraint qualification and the strong second-order sufficient condition without strict complementarity when subproblems are assumed to be solved exactly and inexactly, respectively. Interestingly, it is demonstrated that the Lagrange multipliers associating with inactive inequality constraints at the local minimum point converge to zeros superlinearly. Several illustrative examples are reported to show the behavior of the NLM.

Abstract Against the shortcomings that many existing algorithms for solving the standard smoothing nonlinear programming would fail if they were used directly to solve the mathematical programs with complementary constraints( MPCC). By using a complementarity function and the idea of smoothing approximation method, the MPCC problem was transformed into a smoothing nonlinear programming. Combined with the supermemory gradient idea, a generalized projection gradient algorithm is proposed and its global convergence is obtained.

In this article, we aim to find the most effective reformulation techniques to solve the MPCC (mathematical program with complementarity constraints) model that we proposed recently for continuous network design problems under asymmetric user equilibria. The MPCC model is based on a link-node nonlinear complementarity formulation for asymmetric user equilibria. By applying various reformulation techniques for the lower level nonlinear complementarity, the original bilevel formulation can be converted to a single level nonlinear programming problem. We show that certain reformulations are more effective than others to solve the proposed MPCC model. Recommendations are thus provided on how to choose a reformulation of the continuous network design problem that can be solved effectively and/or efficiently.

Environmental sustainability is a significant aspect in the sustainable development of modern urban cities, especially in the road transport system. As traffic demands increase, public transport requires more promotion to accommodate the increasing travel demands while maintaining the environmental quality. Public transport, however, is less attractive in vast suburb areas mainly due to its longer travel distance and waiting time. Therefore, this paper proposes a rail-based Park-and-Ride (RPR) scheme to promote public transport in the multimodal transport network. To remedy the heterogeneous distribution of vehicle pollutants in the network, regulations in environmental sensitive districts are required and studied in this paper. To quantitatively evaluate and analyse this joint RPR and environmental regulation strategy in multimodal transport systems, this paper develops an environmental constrained combined modal split and traffic assignment (EC-CMSTA) model. The proposed formulation adopts the concept of fix-point to reformulate the nonlinear complementarity conditions associated with the combined modal split and user equilibrium conditions, which is subsequently incorporated into a VI formulated nonlinear complementarity conditions associated with environmental constraints. The proposed VI formulation can handle a general constraint structure, which enhances the modelling adaptability and flexibility. The strictly monotone and Lipschitz continuity properties of this model are rigorously proved, giving rise to efficient algorithms for the model. A customized projection based self-adaptive gradient projection (SAGP) algorithm is then developed. Numerical studies demonstrate that the EC-MSTA model could enhance the behavioural modelling of network users’ travel decisions and assist in quantitatively evaluating the effectiveness of RPR schemes and environmental regulations.

Nous considérons des inégalités variationnelles écrites sous forme d'équations aux dérivées partielles avec contraintes de complémentarité non linéaires. La discrétisation de tels problèmes conduit à des systèmes discrets non linéaires et non différentiables qui peuvent être résolus en employant une méthode de linéarisation itérative de type semi-lisse. Notre objectif est de concevoir une approche de régularisation qui approxime le problème par un système d'équations non linéaires différentiables. Une application directe des méthodes classiques de type Newton est ainsi possible. Nous construisons des estimations d'erreur a posteriori qui sont à la base d'un algorithme de Newton régularisé, inexact et adaptatif, pour une solution des problèmes considérés. Dans le chapitre 1, dans un cadre discret, nous nous intéressons aux systèmes algébriques non linéaires avec des contraintes de complémentarité provenant de discrétisations numériques d'EDP avec problèmes de complémentarité. Nous produisons une approximation différentiable d'une fonction non différentiable, en reformulant les conditions de complémentarité. Le système non linéaire qui en résulte est résolu par la méthode de Newton, ainsi qu'un solveur algébrique linéaire itératif. Nous établissons une borne supérieure sur le résidu du système considéré et concevons des estimateurs d'erreur a posteriori identifiant les composantes d'erreur de régularisation, de linéarisation et algébrique. Ces ingrédients sont utilisés pour formuler des critères d'arrêt efficaces pour les solveurs non linéaires et algébriques. Avec la même méthodologie, une méthode adaptative de points intérieurs est proposée. Nous appliquons notre algorithme au système algébrique d'inégalités variationnelles décrivant le contact entre deux membranes et à un problème d'écoulement diphasique. Nous fournissons une comparaison numérique de notre approche avec une méthode de Newton semi-lisse, éventuellement combinée avec une stratégie de path-following, et une méthode non-paramétrique de points intérieurs. Dans le chapitre 2, en dimension infinie, nous considérons le problème de contact entre deux membranes. Nous utilisons une discrétisation par la méthode des volumes finis et appliquons l'approche de régularisation proposée dans le chapitre 1 pour lisser la non-différentiabilité dans les contraintes de complémentarité. La résolution du système régularisé non linéaire qui en résulte est réalisée grâce à la méthode de Newton, en combinaison avec un solveur algébrique itératif. Nous concevons des reconstructions de potentiel H1-conformes et des reconstructions de flux équilibrés discrets H(div)-conformes. Nous prouvons une borne supérieure pour l'erreur totale par la norme d'énergie et concevons des estimateurs reflétant les erreurs provenant de la discrétisation en volumes finis, du lissage de la non-différentiabilité, de la linéarisation par la méthode de Newton et du solveur algébrique, respectivement. Cela nous permet d'établir des critères d'arrêt adaptatifs pour arrêter les différents solveurs dans l'algorithme proposé et de concevoir un algorithme adaptatif pilotant ces quatre composantes. Dans le chapitre 3, nous introduisons une application à un modèle industriel d’écoulement multiphasique compositionnel avec transitions de phase en milieu poreux. Une discrétisation par la méthode des volumes finis produit un système algébrique non linéaire et non différentiable que nous résolvons en utilisant notre technique de Newton régularisé et inexacte. En suivant le processus du chapitre 1, nous construisons des estimateurs a posteriori en majorant la norme du résidu du système discret, ce qui résulte des critères adaptatifs que nous incorporons dans l'algorithme employé. Des expériences numériques confirment l'efficacité de nos estimations. En particulier, nous montrons que les algorithmes adaptatifs développés réduisent significativement le nombre global d'itérations par rapport aux méthodes existantes
We consider variational inequalities written in the form of partial differential equations with nonlinear complementarity constraints. The discretization of such problems leads to nonlinear non-differentiable discrete systems that can be solved employing an iterative linearization method of semismooth type like, e.g., the Newton-min algorithm. Our goal in this thesis is to conceive a simple smoothing approach that involves approximating the problem as a system of nonlinear smooth (differentiable) equations. In this setting, a direct application of classical Newton-type methods is possible. We construct a posteriori error estimates that lie at the foundation of an adaptive inexact smoothing Newton algorithm for a solution of the considered problems. We first present the strategy in a discrete framework. Then, we develop the method for the model problem of contact between two membranes. Last, an application to a compositional multiphase flow industrial model is introduced. In Chapter 1, we are concerned about nonlinear algebraic systems with complementarity constraints arising from numerical discretizations of PDEs with nonlinear complementarity problems. We produce a smooth approximation of a nonsmooth function, reformulating the complementarity conditions. The ensuing nonlinear system is solved employing the Newton method, together with an iterative linear algebraic solver to approximately solve the linear system. We establish an upper bound on the considered system’s residual and design a posteriori error estimators identifying the smoothing, linearization, and algebraic error components. These ingredients are used to formulate efficient stopping criteria for the nonlinear and algebraic solvers. With the same methodology, an adaptive interior-point method is proposed. We apply our algorithm to the algebraic system of variational inequalities describing the contact between two membranes and a two-phase flow problem. We provide numerical comparison of our approach with a semismooth Newton method, possibly combined with a path-following strategy, and a nonparametric interior-point method. In Chapter 2, in an infinite-dimensional framework, we consider as a model problem the contact problem between two membranes. We employ a finite volume discretization and apply the smoothing approach proposed in Chapter 1 to smooth the non-differentiability in the complementarity constraints. The resolution of the arising nonlinear smooth system is again realized thanks to the Newton method, in combination with an iterative algebraic solver for the solution of the resulting linear system. We design H1-conforming potential reconstructions as well as H(div)-conforming discrete equilibrated flux reconstructions. We prove an upper bound for the total error in the energy norm and conceive discretization, smoothing, linearization, and algebraic estimators reflecting the errors stemming from the finite volume discretization, the smoothing of the non-differentiability, the linearization by the Newton method, and the algebraic solver, respectively. This enables us to establish adaptive stopping criteria to stop the different solvers in the proposed algorithm and design adaptive algorithm steering all these four components. In Chapter 3, we consider a compositional multiphase flow (oil, gas, and water) with phase transitions in a porous media. A finite volume discretization yields a nonlinear non-differentiable algebraic system which we solve employing our inexact smoothing Newton technique. Following the process of Chapter 1, we build a posteriori estimators by bounding the norm of the discrete system’s residual, resulting in adaptive criteria that we incorporate in the employed algorithm. Throughout this thesis, numerical experiments confirm the efficiency of our estimates. In particular, we show that the developed adaptive algorithms considerably reduce the overall number of iterations in comparison with the existing methods

Este trabalho apresenta uma nova abordagem para a modelagem e resolução de problemas de fluxo de carga em sistemas elétricos de potência. O modelo proposto é formado simultaneamente pelo conjunto de equações não lineares que representam as restrições de carga do problema e por restrições de complementaridade associadas com as restrições de operação da rede, as quais propiciam o controle implícito das tensões nas barras com controle de geração. Também é proposta uma técnica para a obtenção dos valores discretos dos taps de tranformadores, de maneira que o ajuste dessas variáveis possa ser realizado em passos discretos. A metodologia desenvolvida consiste em tratar o sistema misto de equações e inequações não lineares como um problema de factibilidade não linear e transformá-lo em um problema de mínimos quadrados não lineares, o qual é resolvido por uma sequência de subproblemas linearizados dentro de uma região de confiança. Para a obtenção de soluções aproximadas desse subproblema foi adotado o método do gradiente conjugado de Steihaug, combinando estratégias de região de confiança e filtros multidimensionais para analisar a qualidade das soluções fornecidas. Foram realizados testes numéricos com os sistemas de 14, 30, 57, 118 e 300 barras do IEEE, e com um sistema brasileiro equivalente CESP 53 barras, os quais indicaram boa flexibilidade e robustez do método proposto.
This work presents a new approach to the load flow problem in electrical power systems and develops a methodology for its resolution. The proposed model is simultaneously composed by nonlinear equations and inequations which represent the load and operational restrictions of the system, where a set of complementarity constraints model the relationship between voltage and reactive power generation in controled buses. It is also proposed a new technique to obtaining a discrete solution for the transformer taps, allowing their discrete adjustment. The method developed treats the mixed system of equations and inequations of the load flow problem as a nonlinear feasibility problem and converts it in a nonlinear least squares problem, which is solved by minimizing a sequence of linearized subproblems, whitin a trust region. To obtain approximate solutions at every iteration, we use the Steihaug conjugate gradient method, combining trust region and multidimensional filters techniques to analyse the quality of the provided solution. Numerical results using 14, 30, 57, 118 and 300-bus IEEE power systems, and a real brazilian equivalent system CESP 53-bus, indicate the flexibility and robustness of the proposed method.

Este trabalho propõe um novo modelo e uma nova abordagem para resolução do problema de fluxo de potência ótimo reativo com variáveis de controle discretas e restrições de atuação de dispositivos de controle de tensão. Matematicamente, esse problema é formulado como um problema de programação não linear com variáveis contínuas e discretas e restrições de complementaridade, cuja abordagem para resolução proposta neste trabalho se baseia na resolução de uma sequência de problemas modificados pelo algoritmo da função Lagrangiana barreira modificada-penalidade-discreto. Nessa abordagem, o problema original é modificado da seguinte forma: 1) as variáveis discretas são tratadas como contínuas por funções senoidais incorporadas na função objetivo do problema original; 2) as restrições de complementaridade são transformadas em restrições de desigualdade equivalentes; e 3) as restrições de desigualdade são transformadas em restrições de igualdade a partir do acréscimo de variáveis de folga não negativas. Para resolver o problema modificado, a condição de não negatividade das variáveis de folga é tratada por uma função barreira modificada com extrapolação quadrática. O problema modificado é transformado em um problema Lagrangiano, cuja solução é determinada a partir da aplicação das condições necessárias de otimalidade. No algoritmo da função Lagrangiana barreira modificada-penalidade-discreto, uma sequência de problemas modificados é resolvida até que todas as variáveis do problema modificado associadas às variáveis discretas do problema original assumam valores discretos. Para demonstrar a eficácia do modelo proposto e a robustez dessa abordagem para resolução de problemas de fluxo de potência ótimo reativo, foram realizados testes com os sistemas elétricos IEEE de 14, 30, 57 e 118 barras e com o sistema equivalente CESP 440 kV de 53 barras. Os resultados mostram que a abordagem para resolução de problemas de programação não linear proposta é eficaz no tratamento de variáveis discretas e restrições de complementaridade.
This work proposes a novel model and a new approach for solving the reactive optimal power flow problem with discrete control variables and voltage-control actuation constraints. Mathematically, such problem is formulated as a nonlinear programming problem with continuous and discrete variables and complementarity constraints, whose proposed resolution approach is based on solving a sequence of modified problems by the discrete penalty-modified barrier Lagrangian function algorithm. In this approach, the original problem is modified in the following way: 1) the discrete variables are treated as continuous by sinusoidal functions incorporated into the objective function of the original problem; 2) the complementarity constraints are transformed into equivalent inequality constraints; and 3) the inequality constraints are transformed into equality constraints by the addition of non-negative slack variables. To solve the modified problem, the non-negativity condition of the slack variables is treated by a modified barrier function with quadratic extrapolation. The modified problem is transformed into a Lagrangian problem, whose solution is determined by the application of the first-order necessary optimality conditions. In the discrete penalty- modified barrier Lagrangian function algorithm, a sequence of modified problems is successively solved until all the variables of the modified problem that are associated with the discrete variables of the original problem assume discrete values. The efectiveness of the proposed model and the robustness of this approach for solving reactive optimal power flow problems were verified with the IEEE 14, 30, 57 and 118-bus test systems and the 440 kV CESP 53-bus equivalent system. The results show that the proposed approach for solving nonlinear programming problems successfully handles discrete variables and complementarity constraints.

博士
國立臺灣師範大學
數學系
105
This dissertation focuses on two types of optimization problems, nonlinear complementarity problem (NCP for short) and quadratic programming with second-order cone constraints (SOCQP for short). Based on NCP-function and SOC-complementarity function, we propose suitable neural networks for each of them, respectively. For the NCP-function, we propose new one which is the generalization of natural residual function for NCP. It is a discrete generalization of natural residual function phinr, denoted as phinrp. Besides being a NCP-function, we also show its twice dierentiability and present the geometric view. In addition, we utilize neural network approach to solving nonlinear complementarity problems and quadratic programming problems with second-order cone constraints. By building neural networks based on dierent families of smooth NCP or SOCCP-functions. Our goal is to study the stability of the equilibrium with respect to dierent neural network models. Asymptotical stability are built in most neural network models. Under suitable conditions, we show the equilibrium being exponentially stable. Finally, the simulation results are reported to demonstrate the effectiveness of the proposed neural network.

In this article, we aim to find the most effective reformulation techniques to solve the MPCC (mathematical program with complementarity constraints) model that we proposed recently for continuous network design problems under asymmetric user equilibria. The MPCC model is based on a link-node nonlinear complementarity formulation for asymmetric user equilibria. By applying various reformulation techniques for the lower level nonlinear complementarity, the original bilevel formulation can be converted to a single level nonlinear programming problem. We show that certain reformulations are more effective than others to solve the proposed MPCC model. Recommendations are thus provided on how to choose a reformulation of the continuous network design problem that can be solved effectively and/or efficiently.

The optimal design of hybrid power generation systems (HPGS) can significantly improve the economical and technical performance of power supply. However, the discrete-time simulation with logical disjunctions involved in HPGS design usually leads to a nonsmooth optimization model, to which well established techniques for smooth nonlinear optimization could not be directly applied. This paper proposes a multistage design optimization problem with complementarity constraints approach for HPGS design, which introduces a complementarity formulation of the nonsmooth logical disjunction, as well as a multistage decomposition framework, to ensure a fast local solution. A numerical study of a stand-alone hybrid photovoltaic (PV)/wind power generation system is presented to demonstrate the effectiveness of the proposed approach.

In this paper, a simulation method is proposed for a sub-category of compressor vanes showing nonlinear behavior due to an adjustable upstream flow angle. The proposed algorithm computes the forced response of a single vane based on the New-mark time stepping scheme after reducing the structural matrices using the Craig-Bampton method. The contacts are modeled by Coulomb friction and Newton impact constraints. Contact forces are determined using linear complementarity conditions with decoupled orthogonal friction force directions. Different discretization methods for the cylindrical contact partners are proposed. Finally, numerical results are shown in order to validate the proposed algorithms.

In the area of robotics simulation, multibody dynamics plays an important role in designing and controlling robots, especially when the robot contacts the environment. Contacts give rise to non-penetration and friction constraints, which are nonsmooth and nonlinear. One way to simulate such systems is through the use of a discrete-time multibody dynamics model in the form of a nonlinear complementarity problem (NCP), for which, finding a solution is known to be NP-hard [1]. In situations where analytical solutions don’t exist, a suite of numerical solutions accessible through a benchmarking framework is useful to fairly evaluate performance of different computer algorithms. However, many algorithm designers don’t have easy access to test data from physical simulators. Under such circumstances, randomized data are used to test the performance of solution algorithms. In this paper, we present our Benchmark Problems of Multibody Dynamics (BPMD) framework and database, with the data sets from different physics engines, as a benchmarking platform. Then we compare the performance of several solvers on synthetic data and simulation data, to show the superiority of testing solution algorithms with simulation data over testing with synthetic data. We will show that algorithm tested only on synthetic data often fail to solve problem obtained from physics simulations, to demonstrate the benefit of BPMD database.

We recently developed a time-stepping method for simulating rigid multi-body systems with intermittent contact that is implicit in the geometric information [1]. In this paper, we extend this formulation to quasi-rigid or locally compliant objects, i.e., objects with a rigid core surrounded by a compliant layer, similar to Song et al. [2]. The difference in our compliance model from existing quasi-rigid models is that, based on physical motivations, we assume the compliant layer has a maximum possible normal deflection beyond which it acts as a rigid body. Therefore, we use an extension of the Kelvin-Voigt (i.e. linear spring-damper) model for obtaining the normal contact forces by incorporating the thickness of the compliant layer explicitly in the contact model. We use the Kelvin-Voigt model for the tangential forces and assume that the contact forces and moment satisfy an ellipsoidal friction law. We model each object as an intersection of convex inequalities and write the contact constraint as a complementarity constraint between the contact force and a distance function dependent on the closest points and the local deformation of the body. The closest points satisfy a system of nonlinear algebraic equations and the resultant continuous model is a Differential Complementarity Problem (DCP). This enables us to formulate a geometrically implicit time-stepping scheme for solving the DCP which is more accurate than a geometrically explicit scheme. The discrete problem to be solved at each time-step is a mixed nonlinear complementarity problem.

The paper presents a new constitutive model for iron-based shape memory alloys (Fe-SMAs) adapted from the ZM model initially proposed for Nitinol by Zaki and Moumni [JMPS2007]. The model introduces nonlinear hardening terms to account for interactions between the grains, martensite variants and slip systems that may exist within a volume element of the material. The expressions used for the hardening terms are similar to those in (Khalil et al. [JIMSS2012]). The equations of the model are derived from the expression of a Helmholtz free energy potential, with complementary loading conditions obtained within the framework of generalized standard materials with internal constraints. A detailed derivation of the implicit algorithm used for the integration of the model is provided and used for numerical simulations that are shown to agree with experimental data.

Abstract Mobile bearing total knees avoid the conformity/constraint tradeoff of fixed bearing total knees. However, a recent in vivo fluoroscopic study of the most popular mobile bearing total knee in the U.S. showed that bearing motion failed to occur in half of the patients observed. A nonlinear, multiple-surface contact finite element model of a rotating platform total knee was therefore developed to investigate the interaction at the “mobile” interface (contact between the tibial tray and the polyethylene insert) under physiologically relevant loads (1–4 BW) and rotations (10° endorotation). The data showed that there was a linear relationship between axial load and the torque resisting endorotation. Peak contact stresses were located on the medial and lateral peripheral edges of the polyethylene insert. All relative rotation occurred at the “mobile” interface. The same trends were seen in a complementary experimental study of the same components, suggesting that the finite element model is valid under these loading conditions.

Feedback control system design, for general single-in-single-out (SISO) applications, requires accurate knowledge of the loop transfer function. Active combustion control design is usually implemented using such SISO architectures, but is quite challenging because the thermoacoustic response results from a relatively unknown, self-excited system and nonlinear processes that must be understood before learning the gain/phase relationship of the system precisely at the instability frequency. However, recent experiments have shown that it is possible to obtain accurate measurements of the relevant loop transfer (frequency response) functions at frequencies adjacent to the instability frequency. Using a simple tube combustor, operating with a premixed, gaseous, burner-stabilized flame, the loop frequency response measurements have been used to develop a methodology that leads to ‘test-based predictions’ of the absolute phase settings and ‘best’ gain settings for a proportional, phase-shifting controller commanding an acoustic actuator in the combustor. The contributions of this methodology are twofold. First, it means that a manual search for the required phase setting of the controller is no longer necessary. In fact, this technique allows the absolute value of controller phase to be determined without running the controller. To the authors’ knowledge, this has not been previously reported in the literature. In addition, the ‘best’ gain setting of the controller, based on this new design approach, can be defined as one that eliminates or reduces the limit cycle amplitude as much as possible within the constraint of avoiding generation of any controller-induced instabilities. (This refers to the generation of ‘new’ peaks in the controlled acoustic pressure spectrum.) It is shown that this tradeoff in limit cycle suppression and avoidance of controller-induced instabilities is a manifestation of the well-known tradeoff in the sensitivity/complementary sensitivity function for feedback control solutions. The focus of this article is limited to the presentation of the design method and does not discuss the detailed nonlinear phenomena that must be understood to determine the optimal gain/phase settings at the limit cycle frequency for a real (versus theoretical) combustor system. A companion paper describes how the proposed design method can be used to generate an AI controller that maintains stabilizing control for a range of changing operating conditions.