The Experts below are selected from a list of 12663 Experts worldwide ranked by ideXlab platform
Emil Ernst - One of the best experts on this subject based on the ideXlab platform.
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The image of a Closed Convex Set under a Fredholm operator
Journal of Functional Analysis, 2014Co-Authors: Emil ErnstAbstract:Abstract The purpose of this article is two-fold. In the first place, we prove that a Set is the image of a non empty Closed Convex subSet of a real Banach space under an onto Fredholm operator of positive index if and only if it can be written as the union of { D n : n ∈ N } , a non-decreasing family of non empty, Closed, Convex and bounded Sets such that D n + D n + 2 ⊆ 2 D n + 1 for every n ∈ N . The second part of this article proves that in every infinite dimensional real Banach space there is a Convex Set which can be expressed as the union of countably many Closed Sets, but not as the union of countably many Closed and Convex Sets. Accordingly, every infinite dimensional real Banach space contains a Convex F σ Set which is not the image of a Closed Convex Set under a Fredholm operator.
Yiran He - One of the best experts on this subject based on the ideXlab platform.
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subdifferentials of a perturbed minimal time function in normed spaces
Optimization Letters, 2014Co-Authors: Yongle Zhang, Yiran He, Yi JiangAbstract:In a general normed vector space, we study the perturbed minimal time function determined by a bounded Closed Convex Set \(U\) and a proper lower semicontinuous function \(f(\cdot )\). In particular, we show that the Frechet subdifferential and proximal subdifferential of a perturbed minimal time function are representable by virtue of corresponding subdifferential of \(f(\cdot )\) and level Sets of the support function of \(U\). Some known results is a special case of these results.
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subdifferentials of a minimal time function in normed spaces
Journal of Mathematical Analysis and Applications, 2009Co-Authors: Yi Jiang, Yiran HeAbstract:Abstract In a general normed vector space, we study the minimal time function determined by a differential inclusion where the Set-valued mapping involved has constant values of a bounded Closed Convex Set U and by a Closed target Set S. We show that proximal and Frechet subdifferentials of a minimal time function are representable by virtue of corresponding normal cones of sublevel Sets of the function and level or suplevel Sets of the support function of U. The known results in the literature require the Set U to have the origin as an interior point or U be compact. (In particular, if the Set U is the unit Closed ball, the results obtained reduce to the subdifferential of the distance function defined by S.)
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subdifferentials of a minimum time function in banach spaces
Journal of Mathematical Analysis and Applications, 2006Co-Authors: Yiran He, Kung Fu NgAbstract:In general Banach space Setting, we study the minimum time function determined by a Closed Convex Set K and a Closed Set S (this function is simply the usual Minkowski function of K if S is the singleton consisting of the origin). In particular we show that various subdifferentials of a minimum time function are representable by virtue of corresponding normal cones of sublevel Sets of the function.
Christian Wagner - One of the best experts on this subject based on the ideXlab platform.
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inequalities for the lattice width of lattice free Convex Sets in the plane
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, 2012Co-Authors: Gennadiy Averkov, Christian WagnerAbstract:A Closed, Convex Set K in \({\mathbb{R}^2}\) with non-empty interior is called lattice-free if the interior of K is disjoint with \({\mathbb{Z}^2}\). In this paper we study the relation between the area and the lattice width of a planar lattice-free Convex Set in the general and centrally symmetric case. A correspondence between lattice width on the one hand and covering minima on the other, allows us to reformulate our results in terms of covering minima introduced by Kannan and Lovasz (Ann Math (2) 128(3):577–602, 1988). We obtain a sharp upper bound for the area for any given value of the lattice width. The lattice-free Convex Sets satisfying the upper bound are characterized. Lower bounds are studied as well. Parts of our results are applied in Averkov et al. (Maximal lattice-free polyhedra: finiteness and an explicit description in dimension three, http://arxiv.org/abs/1010.1077, 2010) for cutting plane generation in mixed integer linear optimization, which was the original inducement for this paper. We further rectify a result of Kannan and Lovasz (Ann Math (2) 128(3):577–602, 1988) with a new proof.
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inequalities for the lattice width of lattice free Convex Sets in the plane
arXiv: Metric Geometry, 2010Co-Authors: Gennadiy Averkov, Christian WagnerAbstract:A Closed, Convex Set $K$ in $\mathbb{R}^2$ with non-empty interior is called lattice-free if the interior of $K$ is disjoint with $\mathbb{Z}^2$. In this paper we study the relation between the area and the lattice width of a planar lattice-free Convex Set in the general and centrally symmetric case. A correspondence between lattice width on the one hand and covering minima on the other, allows us to reformulate our results in terms of covering minima introduced by Kannan and Lov\'asz. We obtain a sharp upper bound for the area for any given value of the lattice width. The lattice-free Convex Sets satisfying the upper bound are characterized. Lower bounds are studied as well. Parts of our results are applied in a paper by the authors and Weismantel for cutting plane generation in mixed integer linear optimization, which was the original inducement for this paper. We further rectify a result of Kannan and Lov\'asz with a new proof.
Vy Khoi Le - One of the best experts on this subject based on the ideXlab platform.
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on variational inequalities with maximal monotone operators and multivalued perturbing terms in sobolev spaces with variable exponents
Journal of Mathematical Analysis and Applications, 2012Co-Authors: Vy Khoi LeAbstract:Abstract We are concerned in this paper with variational inequalities of the form: { 〈 A ( u ) , v − u 〉 + 〈 F ( u ) , v − u 〉 ⩾ 〈 L , v − u 〉 , ∀ v ∈ K , u ∈ K , where A is a maximal monotone operator, F is an integral multivalued lower order term, and K is a Closed, Convex Set in a Sobolev space of variable exponent. We study both coercive and noncoercive inequalities. In the noncoercive case, a sub-supersolution approach is followed to obtain the existence and some other qualitative properties of solutions between sub- and supersolutions.
Quanyi Liang - One of the best experts on this subject based on the ideXlab platform.
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distributed optimization with Closed Convex Set for multi agent networks over directed graphs
Journal of The Franklin Institute-engineering and Applied Mathematics, 2018Co-Authors: Tianrong Weng, Lei Wang, Zhikun She, Quanyi LiangAbstract:Abstract In this paper, a distributed projection algorithm based on the subgradient method is presented to solve the distributed optimization problem with a constrained Set over a directed multi-agent network, where the designed protocol is scaled by the left eigenvector associated with the weighted adjacency matrix. By using the property of the projection operation and nonnegative almost supermartingales, we give the convergence analysis of our algorithm and show that the optimal solution is the ultimate consensus state of all agents to be reached. A numerical simulation for a specific optimization problem is given to verify the effectiveness of our algorithm.