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Linear complementarity, linear and nonlinear programming
01 Jan 1988-
About: The article was published on 1988-01-01 and is currently open access. It has received 1012 citations till now. The article focuses on the topics: Mixed complementarity problem & Complementarity theory.
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TL;DR: The theoretical results allow us to conjecture that local methods for box constrained optimization applied to the associated problem are efficient tools for solving linear complementarity problems.
Abstract: We define a minimization problem with simple bounds associated to the horizontal linear complementarity problem (HLCP). When the HLCP is solvable, its solutions are the global minimizers of the associated problem. When the HLCP is feasible, we are able to prove a number of properties of the stationary points of the associated problem. In many cases, the stationary points are solutions of the HLCP. The theoretical results allow us to conjecture that local methods for box constrained optimization applied to the associated problem are efficient tools for solving linear complementarity problems. Numerical experiments seem to confirm this conjecture.
35 citations
TL;DR: A new optimization algorithm for the classical STRUCTURE model in a maximum likelihood framework is introduced and it is shown that the new method finds solutions with higher likelihoods than the state‐of‐the‐art method in the same computational time.
Abstract: Motivation Structure methods are highly used population genetic methods for classifying individuals in a sample fractionally into discrete ancestry components. Contribution We introduce a new optimization algorithm for the classical STRUCTURE model in a maximum likelihood framework. Using analyses of real data we show that the new method finds solutions with higher likelihoods than the state-of-the-art method in the same computational time. The optimization algorithm is also applicable to models based on genotype likelihoods, that can account for the uncertainty in genotype-calling associated with Next Generation Sequencing (NGS) data. We also present a new method for estimating population trees from ancestry components using a Gaussian approximation. Using coalescence simulations of diverging populations, we explore the adequacy of the STRUCTURE-style models and the Gaussian assumption for identifying ancestry components correctly and for inferring the correct tree. In most cases, ancestry components are inferred correctly, although sample sizes and times since admixture can influence the results. We show that the popular Gaussian approximation tends to perform poorly under extreme divergence scenarios e.g. with very long branch lengths, but the topologies of the population trees are accurately inferred in all scenarios explored. The new methods are implemented together with appropriate visualization tools in the software package Ohana. Availability and implementation Ohana is publicly available at https://github.com/jade-cheng/ohana . In addition to source code and installation instructions, we also provide example work-flows in the project wiki site. Contact jade.cheng@birc.au.dk. Supplementary information Supplementary data are available at Bioinformatics online.
34 citations
TL;DR: A concave minimization algorithm for solving (AVE) that consists of solving a few linear programs, typically two by solving 2 or less linear programs per LCP problem.
Abstract: We consider the linear complementarity problem (LCP): $$Mz+q\ge 0, z\ge 0, z^{\prime }(Mz+q)=0$$
as an absolute value equation (AVE): $$(M+I)z+q=|(M-I)z+q|$$
, where $$M$$
is an $$n\times n$$
square matrix and $$I$$
is the identity matrix. We propose a concave minimization algorithm for solving (AVE) that consists of solving a few linear programs, typically two. The algorithm was tested on 500 consecutively generated random solvable instances of the LCP with $$n=10, 50, 100, 500$$
and 1,000. The algorithm solved $$100\,\%$$
of the test problems to an accuracy of $$10^{-8}$$
by solving 2 or less linear programs per LCP problem.
34 citations
Cites background from "Linear complementarity, linear and ..."
...Keywords: linear complementarity, absolute value equation, linear programming...
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01 Jan 2008
TL;DR: This thesis deals with the time integration of rigid mechanical systems with unilateral contacts and/or friction, which are modeled within this thesis in a non-smooth way, i.e. the time evolution of the displacements and of the velocities is not requested to be smooth anymore.
Abstract: This thesis deals with the time integration of rigid mechanical systems with unilateral contacts and/or friction. Such systems are modeled within this thesis in a non-smooth way, i.e. the time evolution of the displacements and of the velocities is not requested to be smooth anymore. Due to possible impacts, the velocities are even allowed to un¬ dergo jumps at certain time instances in order to fulfill the kinematical restrictions. As a consequence, the accelerations and the contact forces are not defined anymore at such impact points. The behaviour of the unilateral contacts and/or friction is described by set-valued force laws, which are linked to the equations of motion to obtain an analytic representation of the mechanical system for the impact-free case. If impacts have to be considered, then separate impact equations and set-valued impact laws are stated which relate the impulsive contact forces to the admissible post-impact velocities. A compre¬ hensive description of the mechanical system which is applicable for both the impulsive and non-impulsive case can be obtained by using so-called equalities of measures with associated set-valued laws. This formulation in terms of measures incorporates also the classical smooth mechanical system as most simple special case. Set-valued laws can be expressed as normal cone inclusions. Doing so, a broad scope of non-smooth interactions can be described by the same mathematical structure, which allows amongst others for a unified treatment of arbitrary friction laws, i.e. classical Coulomb friction, Coulomb-Contensou friction or anisotropic friction. The thesis uses an augmented Lagrangian approach to solve the occurring normal cone inclusion problems. Doing so, the inclusions which are associated with the individual set-valued laws can be transformed into projective equations. These equations can be solved iteratively by a Ja¬ cobi or Gauss-Seidel like approach, i.e. an underlying system of linear equations is solved in the classical Jacobi or Gauss-Seidel way, and additional projections are applied at each step to guarantee admissibility. The different set-valued laws are taken into account by different instructions for these additional projections. A discretization of the equality of measures and of the associated set-valued laws leads to the so-called time-stepping schemes. These schemes are closely related to the dis¬ cretization of differential algebraic equations, which describe smooth mechanical systems subjected to bilateral constraints. The thesis gives an overview on existing time-stepping schemes and categorizes them. Special notice is given to the time-stepping method of Moreau, which has proven to be a simple and robust integrator for arbitrary kind of nonsmooth systems. Switching points are time instances at which the system configuration
34 citations
TL;DR: The theoretical and practical background of complementarity is discussed, which states that for unilateral contacts either relative kinematics is zero and the corresponding constraint forces are not zero, or vice versa, which leads to a complementarity problem which is related to linear programming problems.
Abstract: Considered in a straightforward manner multibody systems with many multiple unilateral contacts involve a combinatorial problem of huge dimensions, which can be solved reasonably only by the introduction of the complementarity idea. It states that for unilateral contacts either relative kinematics is zero and the corresponding constraint forces are not zero, or vice versa. This leads to a complementarity problem which is related to linear programming problems. This paper discusses the theoretical and practical background of complementarity.
34 citations