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### A smoothing Newton algorithm for the LCP with a sufficient matrix that terminates finitely at a maximally complementary solution

Jie Sun, Zheng-Hai Huang
<span title="">2006</span> <i title="Informa UK Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hoxn7ulnz5btnj2efs2t2mj7f4" style="color: black;">Optimization Methods and Software</a> </i> &nbsp;
We show that, for the LCP with a sufficient matrix, the iteration sequence generated by the proposed algorithm is bounded as long as the LCP has a solution.  ...  Moreover, we show that the proposed algorithm can find a maximally complementary solution to the LCP in a finite number of iterations.  ...  This together with Theorem 4.8 implies that if Algorithm 3.1 terminates at the k-th iteration with k being sufficiently large, then the solution obtained is a maximally complementary solution of LCP(M,  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1080/10556780600627727">doi:10.1080/10556780600627727</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tkdl36nxwzfvlmzfuxvd4ibvo4">fatcat:tkdl36nxwzfvlmzfuxvd4ibvo4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190222113741/http://pdfs.semanticscholar.org/37ec/c5eff08ef5b5fb67b9d76f492e5d0dbb0fbc.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/37/ec/37ecc5eff08ef5b5fb67b9d76f492e5d0dbb0fbc.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1080/10556780600627727"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> tandfonline.com </button> </a>

### A Non-Interior Continuation Algorithm for the P0 or P* LCP with Strong Global and Local Convergence Properties

Zheng-Hai Huang\$, Jie Sun
<span title="2005-07-08">2005</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/x6rgcktxnfe5be2rcega3a7j7u" style="color: black;">Applied Mathematics and Optimization</a> </i> &nbsp;
We propose a non-interior continuation algorithm for the solution of the linear complementarity problem (LCP) with a P 0 matrix.  ...  If the matrix in the LCP is a P * matrix, then the above results can be strengthened to include global linear and local quadratic convergence under a strict complementary condition without the nonsingularity  ...  We are very grateful to the two referees for their valuable comments, which have considerably improved the paper.  ...
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### Solving piecewise linear systems in abs-normal form

Andreas Griewank, Jens-Uwe Bernt, Manuel Radons, Tom Streubel
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wsx3rzhpingfvewcn5nwhfkq3e" style="color: black;">Linear Algebra and its Applications</a> </i> &nbsp;
With the ultimate goal of iteratively solving piecewise smooth (PS) systems, we consider the solution of piecewise linear (PL) equations.  ...  PL models can be derived in the fashion of automatic or algorithmic differentiation as local approximations of PS functions with a second order error in the distance to a given reference point.  ...  Acknowledgements The proof of Proposition 4.2 and Lemma 4.3 was thankfully provided by our colleague Dorothee Schüth of Humboldt University. The authors are also indebted  ...
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### A projected–gradient interior–point algorithm for complementarity problems

Roberto Andreani, Joaquim J. Júdice, José Mario Martínez, Joao Patrício
<span title="2010-11-27">2010</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/xu6z5syhq5ablpzs2jemy3kxfu" style="color: black;">Numerical Algorithms</a> </i> &nbsp;
Complementarity and optimization problems are discussed, for which the algorithm is able to process by either finding a solution or showing that no solution exists.  ...  A modification of the algorithm for dealing with infeasible linear complementarity problems is introduced  ...  If this solution is complementary, then a solution of the LCP (optimal solution of the LP or convex QP) is at hand.  ...
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### A convex, smooth and invertible contact model for trajectory optimization

Emanuel Todorov
<span title="">2011</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nytnrt3gtzbixld5r6a4talbky" style="color: black;">2011 IEEE International Conference on Robotics and Automation</a> </i> &nbsp;
The resulting model has a parameter which sets the amount of contact smoothing, facilitating continuation methods for optimization.  ...  We implemented the proposed contact solver in our new physics engine (MuJoCo). A full Newton step of trajectory optimization for a 3D walking gait takes only 160 msec, on a 12-core PC.  ...  Acknowledgements This work was supported by the US National Science Foundation.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/icra.2011.5979814">doi:10.1109/icra.2011.5979814</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/icra/Todorov11.html">dblp:conf/icra/Todorov11</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pui3sa4hsbdotphwbfbrrij2iy">fatcat:pui3sa4hsbdotphwbfbrrij2iy</a> </span>
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### SLIC-a simulator for linear integrated circuits

T.E. Idleman, F.S. Jenkins, W.J. McCalla, D.O. Pederson
<span title="">1971</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cz5rf4o3ezafnl4kjpq643g32e" style="color: black;">IEEE Journal of Solid-State Circuits</a> </i> &nbsp;
<note that a proper labellinq alone, is already a sufficient condition for this). -A completely labelled simplex provides an approximation for the solution of the LCP.  ...  Therefor cycling cannot occure, and the algorithm must terminate in a finite number of steps, due to the finite number of different states (i.e. at most 2n with n the dimension of the problem).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/jssc.1971.1050168">doi:10.1109/jssc.1971.1050168</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ywih5o6ab5burj6mmdzwi57pae">fatcat:ywih5o6ab5burj6mmdzwi57pae</a> </span>
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### A Smoothing Newton-Type Algorithm of Stronger Convergence for the Quadratically Constrained Convex Quadratic Programming

Zheng-Hai Huang, Defeng Sun, Gongyun Zhao
<span title="2006-06-05">2006</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jkj6zja2ubhybmybu4bm5sgkwe" style="color: black;">Computational optimization and applications</a> </i> &nbsp;
In this paper we propose a smoothing Newton-type algorithm for the problem of minimizing a convex quadratic function subject to finitely many convex quadratic inequality constraints.  ...  (ii) (Slater Constraint Qualification) There is a point x 0 such that f (x 0 ) < 0.  ...  We are very grateful to the two referees for their valuable comments on the paper, which have considerably improved the paper.  ...
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### A Mathematical Model and Descent Algorithm for Bilevel Traffic Management

Michael Patriksson, R. Tyrrell Rockafellar
<span title="">2002</span> <i title="Institute for Operations Research and the Management Sciences (INFORMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/tl73bfx6jngkfcsigtc5zxofea" style="color: black;">Transportation Science</a> </i> &nbsp;
For this model, we also propose and analyze a descent algorithm.  ...  W e provide a new mathematical model for strategic traffic management, formulated and analyzed as a mathematical program with equilibrium constraints (MPEC).  ...  We recall that a function p → ∪ is proper if x > − for every x ∈ dom , and finite for at least one x. Let lev = x ∈ p x ≤ denote the lower level set of for the level .  ...
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### On Solving Contact Problems with Coulomb Friction: Formulations and Numerical Comparisons [chapter]

Vincent Acary, Maurice Brémond, Olivier Huber
<span title="">2018</span> <i title="Springer International Publishing"> Advanced Topics in Nonsmooth Dynamics </i> &nbsp;
Acknowledgements The authors are grateful to Pierre Alart, Paul Armand, Florent Cadoux, Frédéric Dubois, Claude Lémaréchal, Jérôme Malick and Mathieu Renouf for stimulating discussions.  ...  t → r(t) a parametrization such that lim t→0 r(t) = r 0 with r(t) in the smooth domain for all t.  ...  The resulting SOCCP is then solved with a semismooth Newton method, and a smoothing parameter can be added.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-75972-2_10">doi:10.1007/978-3-319-75972-2_10</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vibbxjbvdbdh5oozigujqb6a54">fatcat:vibbxjbvdbdh5oozigujqb6a54</a> </span>
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### A New Path-Following Algorithm for Nonlinear P*Complementarity Problems

Y. B. Zhao, D. Li
<span title="2006-03-23">2006</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jkj6zja2ubhybmybu4bm5sgkwe" style="color: black;">Computational optimization and applications</a> </i> &nbsp;
It turns out that the algorithm is globally convergent for any P * problem provided that its solution set is nonempty.  ...  Different from most existing interior-point algorithms that are based on the central path, this algorithm tracks the "regularized central path" which exists for any continuous P * problem.  ...  Acknowledgments: We are grateful to the three anonymous referees and the associate editor for their constructive comments.  ...
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### QC [chapter]

<span title="">2013</span> <i title="Springer US"> Encyclopedia of Operations Research and Management Science </i> &nbsp;
But the complementary pivot algorithm is a finite path following method for finding a solution, when one exists, to a class of LCPs which includes the KKT systems corresponding to convex QP.  ...  Lemke formulated the KKT system for a QP as an LCP and developed a beautiful algorithm for it called the complementary pivot algorithm.  ...  Open Jackson networks have a product-form solution based on birth-and-death queues for the steady-state probability p(i), with arrival rate l j used at node j.  ...
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### Identity Matrix [chapter]

<span title="">2013</span> <i title="Springer US"> Encyclopedia of Operations Research and Management Science </i> &nbsp;
It appraises the system at selected time points which allow the system to be analyzed via a discrete-parameter Markov chain.  ...  An analysis technique used to analyze a queueing system that is not a continuous-time Markov chain.  ...  so that a Vice-President would be able to represent North America.  ...
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### On self-regular IPMs

Maziar Salahi, Renata Sotirov, Tamás Terlaky
<span title="">2004</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/zwln3qnyojcidnd44j6flcqryy" style="color: black;">TOP - An Official Journal of the Spanish Society of Statistics and Operations Research</a> </i> &nbsp;
However, at present there is still a gap between the practical behavior of these algorithms and their theoretical worst-case complexity results, with respect to the strategies of updating the duality gap  ...  In this paper we discuss the new algorithmic variants and improved complexity results with respect to the new family of Self-Regular proximity based IPMs for Linear Optimization problems, and their generalizations  ...  The authors give two examples of families of functions that satisfy the self-regularity conditions, state a long step interior method, and prove an  ...
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20060222081353/http://www.ms.unimelb.edu.au:80/~rsotirov/top122-IPMs.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/4b/8d/4b8dfb5191be59d43eadc3ee79f5c82a26b6cec2.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/bf02578956"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

### Numerical Stabilization of the Melt Front for Laser Beam Cutting [chapter]

Torsten Adolph, Willi Schönauer, Markus Niessen, Wolfgang Schulz
<span title="">2010</span> <i title="Springer Berlin Heidelberg"> Numerical Mathematics and Advanced Applications 2009 </i> &nbsp;
It is well known that high order numerical schemes exhibit oscillations around shocks but are very efficient for smooth solutions.  ...  The solution is obtained by means of Finite Element Method (FEM).  ...  It is proved that partial upper bidiagonalization of the extended matrix (b,A) determines a core approximation problem A 11 x 1~b1 , with the necessary and sufficient information for solving the original  ...
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180721182541/http://www.it.uu.se/research/tdb/enumath2009/final/enumath_abstracts.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/4c/34/4c343b1ce275f64d03689c4505c2c1011604403d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-11795-4_6"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

### Differential variational inequalities

Jong-Shi Pang, David E. Stewart
<span title="2007-01-24">2007</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/o653fqsowvcyrhw6p4gdmpdkwm" style="color: black;">Mathematical programming</a> </i> &nbsp;
For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional The work of J.-S.  ...  The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic  ...  Acknowledgments The authors are very grateful to two referees and the Associate Editor for their many insightful and constructive comments.  ...
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20140801161949/http://php.math.unifi.it:80/users/cime/Courses/2014/03/part_4.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/50/e2/50e2cb2ad2002f2de72b43f90eaf33b2ce512bfa.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10107-006-0052-x"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>
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