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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: In this paper, the authors give a complete description of the linear maps that preserve the Lorentz spectrum in the space of real matrices and the subspace of symmetric matrices.
Abstract: In this paper we give a complete description of the linear maps $\phi:W_{n}\rightarrow W_{n}$ that preserve the Lorentz spectrum, when $n=2$ and $W_{n}$ is the space $M_{n}$ of $n\times n$ real matrices or the subspace $S_{n}$ of $M_{n}$ formed by the symmetric matrices. In both cases, we have shown that $\phi(A)=PAP^{-1}$ for all $A\in W_{2}$, where $P$ is a matrix with a certain structure. The results in this paper extend to $n=2$ those for $n\geq 3$ obtained by Bueno, Furtado, and Sivakumar (2021). The case $n=2$ has some specificities, when compared to the case $n\geq3,$ due to the fact that the Lorentz cone in $\mathbb{R}^{2}$ is polyedral, contrary to what happens when it is contained in $\mathbb{R}^{n}$ with $n\geq3.$
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TL;DR: Universal Decision Models (UDM) as discussed by the authors is a formalism based on category theory that allows humans to reason causally to understand the world, act competitively to gain advantage in commerce, games, and war, and learn to make better decisions through trial and error.
Abstract: Humans are universal decision makers: we reason causally to understand the world; we act competitively to gain advantage in commerce, games, and war; and we are able to learn to make better decisions through trial and error. In this paper, we propose Universal Decision Model (UDM), a mathematical formalism based on category theory. Decision objects in a UDM correspond to instances of decision tasks, ranging from causal models and dynamical systems such as Markov decision processes and predictive state representations, to network multiplayer games and Witsenhausen's intrinsic models, which generalizes all these previous formalisms. A UDM is a category of objects, which include decision objects, observation objects, and solution objects. Bisimulation morphisms map between decision objects that capture structure-preserving abstractions. We formulate universal properties of UDMs, including information integration, decision solvability, and hierarchical abstraction. We describe universal functorial representations of UDMs, and propose an algorithm for computing the minimal object in a UDM using algebraic topology. We sketch out an application of UDMs to causal inference in network economics, using a complex multiplayer producer-consumer two-sided marketplace.
TL;DR: In this paper, the quality of inequality constrained least-squares estimates is described by reconstructing the existing approaches that describe the quality and quality of least-square estimates, which are not sufficient to meet the demands of the inequality constrained problems.
Abstract: Existing approaches that describe the quality of inequality constrained estimates are not sufficient to meet the demands of inequality constrained least-square problems. We reconstruct the existing...
Cites methods from "Linear complementarity, linear and ..."
...Lemke’s algorithm (Murty 1988) is used to solve ICLSproblems in this paper....
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TL;DR: In this article , generic interconnections of a class of passive nonsmooth nonlinear dynamical systems, namely linear cone complementarity systems (LCCS), are studied. And the conditions for passivity of the interconnection are given.
Abstract: This paper is largely concerned with generic interconnections of a class of passive nonsmooth nonlinear dynamical systems, namely linear cone complementarity systems (LCCS). Criteria which guarantee the passivity of the interconnection are given. Asymptotic stability is studied in a particular case. Examples from nonsmooth circuits and switching DAEs (Differential Algebraic Equations) illustrate the theory.