Existence and Uniqueness of Solutions for Homogeneous Cone Complementarity Problems

Kong, Lingchen; Tunçel, Levent; Xiu, Naihua

doi:10.1007/s10957-011-9971-7

Cited by 3 publications

(4 citation statements)

References 28 publications

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“…Kong et al, 2010). A simple model after Horn (2012) is applied to determine the corresponding changes in the number concentrations and to ensure a reduction of the cloud droplet number density to zero if there is no cloud water present.…”

Section: Two-moment Warm Cloud Microphysics Schemementioning

confidence: 99%

ASAM v2.7: a compressible atmospheric model with a Cartesian cut cell approach

et al. 2015

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Abstract. In this work, the fully compressible, threedimensional, nonhydrostatic atmospheric model called All Scale Atmospheric Model (ASAM) is presented. A cut cell approach is used to include obstacles and orography into the Cartesian grid. Discretization is realized by a mixture of finite differences and finite volumes and a state limiting is applied. Necessary shifting and interpolation techniques are outlined. The method can be generalized to any other orthogonal grids, e.g., a lat-long grid. A linear implicit Rosenbrock time integration scheme ensures numerical stability in the presence of fast sound waves and around small cells. Analyses of five two-dimensional benchmark test cases from the literature are carried out to show that the described method produces meaningful results with respect to conservation properties and model accuracy. The test cases are partly modified in a way that the flow field or scalars interact with cut cells. To make the model applicable for atmospheric problems, physical parameterizations like a Smagorinsky subgrid-scale model, a two-moment bulk microphysics scheme, and precipitation and surface fluxes using a sophisticated multi-layer soil model are implemented and described. Results of an idealized three-dimensional simulation are shown, where the flow field around an idealized mountain with subsequent gravity wave generation, latent heat release, orographic clouds and precipitation are modeled.

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Section: Two-moment Warm Cloud Microphysics Schemementioning

confidence: 99%

ASAM v2.7: a compressible atmospheric model with a Cartesian cut cell approach

et al. 2015

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“…We can thus invoke Propositions 1 (i) and (ii) to conclude that D N satisfies both the P 0 and R 0 property. Using now [26,Theorem 3.7] we obtain that the cone complementarity problem (53) has a nonempty and bounded solution set. Thus, from Lemma 2 so does (52).…”

Section: Proofmentioning

confidence: 99%

“…Assume that the cone QΥ has a strictly convex barrier function I QΥ (η), which is three times continuously differentiable for η in the interior of QΥ, and which satisfies I QΥ (η) → ∞ if η approaches the boundary of QΥ. As in the context discussed here, all cones are rotated or otherwise scaled second-order cones in three dimensions barriers can be easily found [26]. We then solve the parametric nonlinear equation for α ∈ (0, 1].…”

Section: Algorithmic Considerationsmentioning

confidence: 99%

A compliant visco-plastic particle contact model based on differential variational inequalities

Tasora¹,

Anitescu²,

Negrini³

et al. 2013

International Journal of Non-Linear Mechanics

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This work describes an approach to simulate contacts between three dimensional shapes with compliance and damping using the framework of the Differential Variational Inequality (DVI) theory. Within the context of nonsmooth dynamics, we introduce an extension to the classical set-valued model for frictional contacts between rigid bodies, allowing contacts to experience local compliance, viscosity and plasticization. Different types of yield surfaces can be defined for various types of contact, a versatile approach that contains the classic dry Coulomb friction as a special case. The resulting problem is a differential variational inequality that can be solved, at each integration time step, as a variational inequality over a convex set.

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“…Decompositions such as the above have potential applications in linear and nonlinear complementarity problems over homogeneous cones and in the design of algorithms and theories utilizing Moreau decompositions; see for instance Kong, Tunçel and Xiu (2012).…”

Section: The Adjoints Lmentioning

confidence: 99%

Linear optimization over homogeneous matrix cones

Tunçel¹,

Vandenberghe²

2023

Acta Numerica

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A convex cone is homogeneous if its automorphism group acts transitively on the interior of the cone. Cones that are homogeneous and self-dual are called symmetric. Conic optimization problems over symmetric cones have been extensively studied, particularly in the literature on interior-point algorithms, and as the foundation of modelling tools for convex optimization. In this paper we consider the less well-studied conic optimization problems over cones that are homogeneous but not necessarily self-dual.We start with cones of positive semidefinite symmetric matrices with a given sparsity pattern. Homogeneous cones in this class are characterized by nested block-arrow sparsity patterns, a subset of the chordal sparsity patterns. Chordal sparsity guarantees that positive define matrices in the cone have zero-fill Cholesky factorizations. The stronger properties that make the cone homogeneous guarantee that the inverse Cholesky factors have the same zero-fill pattern. We describe transitive subsets of the cone automorphism groups, and important properties of the composition of log-det barriers with the automorphisms.Next, we consider extensions to linear slices of the positive semidefinite cone, and review conditions that make such cones homogeneous. An important example is the matrix norm cone, the epigraph of a quadratic-over-linear matrix function. The properties of homogeneous sparse matrix cones are shown to extend to this more general class of homogeneous matrix cones.We then give an overview of the algebraic theory of homogeneous cones due to Vinberg and Rothaus. A fundamental consequence of this theory is that every homogeneous cone admits a spectrahedral (linear matrix inequality) representation.We conclude by discussing the role of homogeneous structure in primal–dual symmetric interior-point methods, contrasting this with the well-developed algorithms for symmetric cones that exploit the strong properties of self-scaled barriers, and with symmetric primal–dual methods for general convex cones.

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Existence and Uniqueness of Solutions for Homogeneous Cone Complementarity Problems

Cited by 3 publications

References 28 publications

ASAM v2.7: a compressible atmospheric model with a Cartesian cut cell approach

ASAM v2.7: a compressible atmospheric model with a Cartesian cut cell approach

A compliant visco-plastic particle contact model based on differential variational inequalities

Linear optimization over homogeneous matrix cones

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