Topic
Dense set
About: Dense set is a research topic. Over the lifetime, 1538 publications have been published within this topic receiving 20931 citations.
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Papers
TL;DR: The main result of this paper is that the general MADS framework is flexible enough to allow the generation of an asymptotically dense set of refining directions along which the Clarke derivatives are nonnegative.
Abstract: This paper addresses the problem of minimization of a nonsmooth function under general nonsmooth constraints when no derivatives of the objective or constraint functions are available. We introduce the mesh adaptive direct search (MADS) class of algorithms which extends the generalized pattern search (GPS) class by allowing local exploration, called polling, in an asymptotically dense set of directions in the space of optimization variables. This means that under certain hypotheses, including a weak constraint qualification due to Rockafellar, MADS can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained.
The main GPS convergence result is to identify limit points $\hat{x}$, where the Clarke generalized derivatives are nonnegative in a finite set of directions, called refining directions. Although in the unconstrained case, nonnegative combinations of these directions span the whole space, the fact that there can only be finitely many GPS refining directions limits rigorous justification of the barrier approach to finitely many linear constraints for GPS. The main result of this paper is that the general MADS framework is flexible enough to allow the generation of an asymptotically dense set of refining directions along which the Clarke derivatives are nonnegative.
We propose an instance of MADS for which the refining directions are dense in the hypertangent cone at $\hat{x}$ with probability 1 whenever the iterates associated with the refining directions converge to a single $\hat{x}$. The instance of MADS is compared to versions of GPS on some test problems. We also illustrate the limitation of our results with examples.
1,349 citations
TL;DR: A practical, robust algorithm to locally minimize such functions as f, a continuous function on $\Rl^n$, which is continuously differentiable on an open dense subset, based on gradient sampling is presented.
Abstract: Let f be a continuous function on $\Rl^n$, and suppose f is continuously differentiable on an open dense subset. Such functions arise in many applications, and very often minimizers are points at which f is not differentiable. Of particular interest is the case where f is not convex, and perhaps not even locally Lipschitz, but is a function whose gradient is easily computed where it is defined. We present a practical, robust algorithm to locally minimize such functions, based on gradient sampling. No subgradient information is required by the algorithm.
When f is locally Lipschitz and has bounded level sets, and the sampling radius $\eps$ is fixed, we show that, with probability 1, the algorithm generates a sequence with a cluster point that is Clarke $\eps$-stationary. Furthermore, we show that if f has a unique Clarke stationary point $\bar x$, then the set of all cluster points generated by the algorithm converges to $\bar x$ as $\eps$ is reduced to zero.
Numerical results are presented demonstrating the robustness of the algorithm and its applicability in a wide variety of contexts, including cases where f is not locally Lipschitz at minimizers. We report approximate local minimizers for functions in the applications literature which have not, to our knowledge, been obtained previously. When the termination criteria of the algorithm are satisfied, a precise statement about nearness to Clarke $\eps$-stationarity is available. A MATLAB implementation of the algorithm is posted at http://www.cs.nyu.edu/overton/papers/gradsamp/alg.
546 citations
TL;DR: In this paper, the existence and uniqueness of the Cauchy problem for a wide class of closed linear operators is studied in terms of some probability densities, and an application is given for the theory of integral-partial differential equations of fractional orders.
Abstract: In the present paper, if 0 d α u d t α =Au(t)+B(t)u(t), where A is a closed linear operator defined on a dense set in E into E, which generates a semigroup and {B(t):t⩾0} is a family of a closed linear operators defined on a dense set in E into E. The existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the family of operators {B(t):t⩾0}. The solution is given in terms of some probability densities. An application is given for the theory of integro-partial differential equations of fractional orders.
403 citations
TL;DR: In this paper, the real line up to order isomorphism is characterized by the following properties: R is order complete, order dense, has no first or last elements, and contains a countable dense subset.
Abstract: We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a countable dense subset. (One shows first that the countable dense subset is order isomorphic to the rationals, Q, and then that the ordered set is isomorphic to the Dedekind completion of its dense subset.) Souslin raised the question as to whether the "countable dense subset" condition could be replaced by the following consequence [15]: (*) Every disjoint family of non-empty open intervals is countable.' We use SH (Souslin's Hypothesis) to denote the following proposition: Every order complete order dense linearly ordered set satisfying ( * ) contains a countable dense subset. We use ZFC to denote Zermelo-Fraenkel set theory (including the axiom of choice). In [16], Tennenbaum constructed models of ZFC in which SH is false. Moreover, in one of these models the continuum hypothesis (CH) is false, while in another one, the generalized continuum hypothesis (GCH) is true. Thus SH is independent of the usual axioms of set theory. (This result is due, independently, to Jech [7].)2
344 citations
TL;DR: In this article, the real line up to order isomorphism is characterized by the following properties: R is order complete, order dense, has no first or last elements, and contains a countable dense subset.
Abstract: We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a countable dense subset. (One shows first that the countable dense subset is order isomorphic to the rationals, Q, and then that the ordered set is isomorphic to the Dedekind completion of its dense subset.) Souslin raised the question as to whether the "countable dense subset" condition could be replaced by the following consequence [15]: (*) Every disjoint family of non-empty open intervals is countable.' We use SH (Souslin's Hypothesis) to denote the following proposition: Every order complete order dense linearly ordered set satisfying ( * ) contains a countable dense subset. We use ZFC to denote Zermelo-Fraenkel set theory (including the axiom of choice). In [16], Tennenbaum constructed models of ZFC in which SH is false. Moreover, in one of these models the continuum hypothesis (CH) is false, while in another one, the generalized continuum hypothesis (GCH) is true. Thus SH is independent of the usual axioms of set theory. (This result is due, independently, to Jech [7].)2
323 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 76 |
| 2020 | 76 |
| 2019 | 59 |
| 2018 | 75 |
| 2017 | 65 |
| 2016 | 59 |