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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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01 Sep 2022
TL;DR: In this paper , the authors studied the linear preservers of the Lorentz spectrum on the space of real matrices for the eigenvalue complementarity problem and proved that all such linear presers take the form of (Q \oplus [1])A(Q^T \plus [2] ).
Abstract: For a given $3 \times 3$ real matrix $A$, the eigenvalue complementarity problem relative to the Lorentz cone consists of finding a real number $\lambda$ and a nonzero vector $x \in \mathbb{R}^3$ such that $x^T(A-\lambda I)x=0$ and both $x$ and $(A-\lambda I)x$ lie in the Lorentz cone, which is comprised of all vectors in $\mathbb{R}^3$ forming a $45^\circ$ or smaller angle with the positive $z$-axis. We refer to the set of all solutions $\lambda$ to this eigenvalue complementarity problem as the Lorentz spectrum of $A$. Our work concerns the characterization of the linear preservers of the Lorentz spectrum on the space $M_3$ of $3 \times 3$ real matrices, that is, the linear maps $\phi: M_3 \to M_3$ such that the Lorentz spectra of $A$ and $\phi(A)$ are the same for all $A$. We have proven that all such linear preservers take the form $\phi(A) = (Q \oplus [1])A(Q^T \oplus [1])$, where $Q$ is an orthogonal $2 \times 2$ matrix.
26 Jul 2019
TL;DR: In this paper, a technique based on the Lagrange equations of the first kind and the object representation of models is proposed for the design and computation of a landing gear damping system.
Abstract: Takeoff and landing are the most important stages of the aircraft flight. The flight safety depends on the landing gear at these stages. Therefore, the design and computation of landing gear damping system is one of the most important tasks of the aircraft development. When computing the landing gear damping system, it is required to use methods of numerical simulation of the motion of a system of rigid bodies with holonomic constraints. The most appropriate approach is to use a technique based on the Lagrange equations of the first kind and the object representation of models. The considered technique allows the use of a formalized approach to the description of models, the computation of arbitrary kinematics, as well as the introduction of automatic control algorithms into the system, which is important in the design of adaptive systems. Adaptive damping is one of the promising areas in the development of the landing gear. The effect of the use of adaptive landing gear damping system is expressed in reducing the maximum dynamic loads on the airframe, and as a consequence, through extending life, increasing comfort, and improving operating conditions.
TL;DR: This work proposes a new LCP formulation with the aim of making the underlying matrix belong to the classes R and E and shows that it is an R $$_{0}$$ -matrix, which would imply Lemke processibility.
Abstract: Schultz (J Optim Theory Appl 73(1):89–99, 1992) formulated 2-person, zero-sum, discounted switching control stochastic games as a linear complementarity problem (LCP) and discussed computational results. It remained open to prove or disprove Lemke-processibility of this LCP. We settle this question by providing a counter example to show that Lemke’s algorithm does not always successfully process this LCP.We propose a new LCP formulation with the aim of making the underlying matrix belong to the classes R
$$_{0}$$
and E
$$_{0}$$
, which would imply Lemke processibility. While the underlying matrix in the new formulation is not
$$E_0$$
, we show that it is an R
$$_{0}$$
-matrix. Successful processing of Lemke’s algorithm depends on the choice of the initial vector d. Because of the special structure of the LCP in our context, we may, in fact, be able to find a suitable d such that our LCPs are processible by Lemke’s algorithm. We leave this open.
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TL;DR: This paper addresses the problem of finding the closest generalized essential matrix from a given 6x6 matrix, with respect to the Frobenius norm by converting the original problem into a new one, involving only orthogonal constraints, and proposing an efficient algorithm of steepest descent-type to find the solution.
Abstract: This paper addresses the problem of finding the closest generalized essential matrix from a given 6x6 matrix, with respect to the Frobenius norm. To the best of our knowledge, this nonlinear constrained optimization problem has not been addressed in the literature yet. Although it can be solved directly, it involves a large amount of constraints, and any optimization method to solve it will require much computational time. Then, we start by converting the original problem into a new one, involving only orthogonal constraints, and propose an efficient algorithm of steepest descent-type to find the solution. To test our algorithm, we start by evaluating our method with synthetic data, and conclude that the proposed method is much faster than applying general optimization techniques to the original problem with 33 constraints. To further motivate the relevance of our method, we apply our technique in two pose problems (relative and absolute) using synthetic and real data.
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TL;DR: In this paper, a bilinear algorithm for multiagent planning and coordination problems is presented. But the algorithm is more general and simpler to implement, and it can be terminated at any time and facilitates the derivation of a useful online performance bound.
Abstract: Multiagent planning and coordination problems are common and known to be computationally hard. We show that a wide range of two-agent problems can be formulated as bilinear programs. We present a successive approximation algorithm that significantly outperforms the coverage set algorithm, which is the state-of-the-art method for this class of multiagent problems. Because the algorithm is formulated for bilinear programs, it is more general and simpler to implement. The new algorithm can be terminated at any time and-unlike the coverage set algorithm-it facilitates the derivation of a useful online performance bound. It is also much more efficient, on average reducing the computation time of the optimal solution by about four orders of magnitude. Finally, we introduce an automatic dimensionality reduction method that improves the effectiveness of the algorithm, extending its applicability to new domains and providing a new way to analyze a subclass of bilinear programs.