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International audience; Simulating frictional contacts remains a challenging research topic in robotics. Recently, differentiable physics emerged and has proven to be a key element in modelbased Reinforcement Learning (RL) and optimal control fields. However, most of the current formulations deploy coarse approximations of the underlying physical principles. Indeed, the classic simulators lose precision by casting the Nonlinear Complementarity Problem (NCP) of frictional contact into a Linear Complementarity Problem (LCP) to simplify computations. Moreover, such methods deploy non-smooth operations and cannot be automatically differentiated. In this paper, we propose (i) an extension of the staggered projections algorithm for more accurate solutions of the problem of contacts with friction. Based on this formulation, we introduce (ii) a differentiable simulator and an efficient way to compute the analytical derivatives of the involved optimization problems. Finally, (iii) we validate the proposed framework with a set of experiments to present a possible application of our differentiable simulator. In particular, using our approach we demonstrate accurate estimation of friction coefficients and object masses both in synthetic and real experiments.

With increasing penetration of renewable energy, multi-energy systems constitute an effective mechanism to optimize energy distribution and improve social welfare. However, a centralized operation of the multi-energy system might not be appropriate under the existing energy markets. Therefore, this paper proposes an equilibrium model for improving the operation of the electricity, gas and district heating subsystems of a district or urban area. The proposed model allows each energy subsystem to pursue its own objective (i.e., maximum social welfare), while considering the interconnection with other subsystems. More specifically, this model represents the behavior of each subsystem and reflects the interactions of the multi-energy system in a practical way. This equilibrium problem is formulated as a nonlinear complementarity problem. An illustrative case study is analyzed to show the relevance of the proposed approach.

In this paper, we present a new smoothing approach to solve general nonlinear complementarity problems. Under the $P_0$ condition on the original problems, we prove some existence and convergence results . We also present an error estimate under a new and general monotonicity condition. The numerical tests confirm the efficiency of our proposed methods., 16 pages, 1 figures

The modeling of migration of hydrogen produced by the corrosion of the nuclear waste packages in an underground storage including the dissolution of hydrogen involves a set of nonlinear partial differential equations with nonlinear complementarity constraints. This article shows how to apply a modern and efficient solution strategy, the Newton-min method, to this geoscience problem and investigates its applicability and efficiency. In particular, numerical experiments show that the Newton-min method is quadratically convergent for this problem., Comment: Accepted for Publication in Mathematics and Computers in Simulation. Available online 6 August 2013, Mathematics and Computers in Simulation (2013)

The nonlinear complementarity problem can be reformulated as a nonlinear programming whose objective function may be nonsmooth. For this case, we use decomposition strategy to decompose the nonsmooth function into a smooth one and a nonsmooth one. Together with filter method, we present an improved filter algorithm with decomposition strategy for solving nonlinear complementarity problem, which has better numerical results compared to the method that without the filter technique. Under mild conditions, the global convergent property is shown. In the end, the numerical example is reported.

The global error bound estimation for the generalized nonlinear complementarity problem over a closed convex cone (GNCP) is considered. To obtain a global error bound for the GNCP, we first develop an equivalent reformulation of the problem. Based on this, a global error bound for the GNCP is established. The results obtained in this paper can be taken as an extension of previously known results.

Based on the smoothing function of penalized Fischer-Burmeister NCP-function, we propose a new smoothing inexact Newton algorithm with non-monotone line search for solving the generalized nonlinear complementarity problem. We view the smoothing parameter as an independent variable. Under suitable conditions, we show that any accumulation point of the generated sequence is a solution of the generalized nonlinear complementarity problem. We also establish the local superlinear (quadratic) convergence of the proposed algorithm under the BD-regular assumption. Preliminary numerical experiments indicate the feasibility and efficiency of the proposed algorithm.

Dynamical systems include measuring sensor inputs of phenomena to yield accurate predictions of the evolving sensor outputs or to determine optimal control management policies based on sensor data. The input and output sets of the system may be generalized and transformed with respect to the sets of sensors available and formal deductive methods and chaos theory may be formulated to obtain Dynamical Inquiring Systems over a horizon to yield solutions which will be precise and be certainty equivalent to the future results of the phenomenon.The aim of this chapter is to present a formalization of Mathematical Systems Theory to demonstrate the theoretical basis of nonlinear dynamical chaotic systems solved by simultaneous estimation and optimal control processes and to present accurate predictions based on generalized sensor data of many forms both in input and output such as dynamic malfunctioning of systems including engineering, medical, economic, and environmental inquiring systems.

We describe the solution of a bound constrained convex quadratic problem with limited memory resources. The problem arises from physical simulations occurring within video games. The motivating problem is outlined, along with a simple interior point approach for its solution. Various linear algebra issues arising in the implementation are explored, including preconditioning, ordering and a number of ways of solving an equivalent augmented system. Alternative approaches are briefly surveyed, and some recommendations for solving these types of problems are given. © EDP Sciences, ROADEF, SMAI 2007.

In this paper an eigenvalue complementarity problem (EiCP) is studied, which finds its origins in the solution of a contact problem in mechanics. The EiCP is shown to be equivalent to a Nonlinear Complementarity Problem, a Mathematical Programming Problem with Complementarity Constraints and a Global Optimization Problem. A finite Reformulation–Linearization Technique (Rlt)-based tree search algorithm is introduced for processing the EiCP via the lattermost of these formulations. Computational experience is included to highlight the efficacy of the above formulations and corresponding techniques for the solution of the EiCP. http://dx.doi.org/10.1007/s10589-007-9017-0

It is known that a nonlinear complementarity problem (NCP) can be reformulated as a semismooth system of nonlinear equations by using a so-called NCP-function. Global Newton-type methods for solving NCP via semismooth reformulation need to use a merit function, which is usually required to be continuously differentiable. In this paper we present a global Newton-type method which does not require the differentiability for the merit function used in the line-search procedure. The method is used to numerically compare the effectiveness of two NCP-functions widely discussed in literature, the minimum function and the Fischer-Burmeister function. The results on several examples allow to gain some new acquaintance of the respective numerical advantages of the two functions.

Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:19:44Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:45:10Z : No. of bitstreams: 1 2-s2.0-0001702409.pdf: 293520 bytes, checksum: 1c0cf3d39e35c7481e9fabb60fc1badf (MD5) Made available in DSpace on 2014-05-27T11:19:44Z (GMT). No. of bitstreams: 0 Previous issue date: 1999-07-01 A bounded-level-set result for a reformulation of the box-constrained variational inequality problem proposed recently by Facchinei, Fischer and Kanzow is proved. An application of this result to the (unbounded) nonlinear complementarity problem is suggested. © 1999 Elsevier Science Ltd. All rights reserved. Dept. of Comp. Sci. and Statistics University of the State of S. Paulo, C.P. 136, Sao Jose do Rio Preto-SP Department of Applied Mathematics IMECC-UNICAMP University of Campinas, CP 6065, 13081-970 Campinas SP

QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel [14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously differentiable, pseudo-monotone mixed nonlinear complementarity problem. QPCOMP also extends the NE/SQP method for the nonlinear complementarity problem to the more general mixed nonlinear complementarity problem. Computational results are provided, which demonstrate the effectiveness of the algorithm.

The traditional traffic equilibrium problem (TEP) is mainly based on the additivity assumption that the route cost is simply the sum of the link costs on that route. However, there are many situations where this assumption on the route costs is inappropriate, and thus it is neccesary to have to explicitly formulate and solve the TEP in the route space instead of link space. In this paper, the authors firstly reformulate the TEP with nonadditive route cost function to a nonlinear complementarity problem (NCP), and then the NCP is converted to an equivalent least square problem (LSP) with a new NCP function; then the authors propose a modified Levenberg- Marquardt algorithm to solve the LSP, and also, the quadratic convergence and the equivalent condition of the proposed Levenberg–Marquardt (L-M) algorithm are proved under some assumptions. Finally, a numerical example is presented in the paper. As the results shown, the proposed method has the capability to converge to a high level accuracy with reasonable computational efforts.

In accordance with the principle of using sufficiently the delayed information, and by making use of the nonlinear multisplitting and the nonlinear relaxation techniques, we present in this paper a class of asynchronous parallel nonlinear multisplitting accelerated overrelaxation (AOR) methods for solving the large sparse nonlinear complementarity problems on the high-speed MIMD multiprocessor systems. These new methods, in particular, include the so-called asynchronous parallel nonlinear multisplitting AOR-Newton method, the asynchronous parallel nonlinear multisplitting AOR-chord method and the asynchronous parallel nonlinear multisplitting AOR-Steffensen method. Under suitable constraints on the nonlinear multisplitting and the relaxation parameters, we establish the local convergence theory of this class of new methods when the Jacobi matrix of the involved nonlinear mapping at the solution point of the nonlinear complementarity problem is an H-matrix.

In this paper, we reformulate a nonlinear complementarity problem or a mixed complementarity problem as a system of piecewise almost linear equations. The problems arise, for example, from the obstacle problems with a nonlinear source term or some contact problems. Based on the reformulated systems of the piecewise almost linear equations, we propose a class of semi-iterative algorithms to find the exact solution of the problems. We prove that the semi-iterative algorithms enjoy a nice monotone convergence property in the sense that subsets of the indices consisting of the indices, for which the corresponding components of the iterates violate the constraints, become smaller and smaller. Then the algorithms converge monotonically to the exact solutions of the problems in a finite number of steps. Some numerical experiments are presented to show the effectiveness of the proposed algorithms.

In this paper, we propose a new Newton-type method for solving the nonlinear complementarity problem (NCP) based on a class of one-parametric NCP-functions, where an approximate Newton direction can be obtained by solving a modified Newton equation in each iteration. The method is shown to be globally convergent without any additional assumption. To investigate the fast convergence of this class of methods, we propose a modified version of the proposed method and show the method is globally and locally superlinearly convergent. The preliminary numerical results show the effectiveness of the modified method.