Limit point
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On the convergence of a linesearch based proximal-gradient method for nonconvex optimization Open
We consider a variable metric line-search based proximal gradient method for the minimization of the sum of a smooth, possibly nonconvex function plus a convex, possibly nonsmooth term. We prove convergence of this iterative algorithm to a…
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Global Analysis of Expectation Maximization for Mixtures of Two Gaussians Open
Expectation Maximization (EM) is among the most popular algorithms for estimating parameters of statistical models. However, EM, which is an iterative algorithm based on the maximum likelihood principle, is generally only guaranteed to fin…
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Global analysis of Expectation Maximization for mixtures of two Gaussians Open
Expectation Maximization (EM) is among the most popular algorithms for estimating parameters of statistical models. However, EM, which is an iterative algorithm based on the maximum likelihood principle, is generally only guaranteed to fin…
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Evaluation Complexity for Nonlinear Constrained Optimization Using Unscaled KKT Conditions and High-Order Models Open
The evaluation complexity of general nonlinear, possibly nonconvex, constrained optimization is analyzed. It is shown that, under suitable smoothness conditions, an $\epsilon$-approximate first-order critical point of the problem can be co…
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Experimental Tracking of Limit-Point Bifurcations and Backbone Curves Using Control-Based Continuation Open
Control-based continuation (CBC) is a means of applying numerical continuation directly to a physical experiment for bifurcation analysis without the use of a mathematical model. CBC enables the detection and tracking of bifurcations direc…
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Minmax Optimization: Stable Limit Points of Gradient Descent Ascent are Locally Optimal. Open
Minmax optimization, especially in its general nonconvex-nonconcave
formulation, has found extensive applications in modern machine learning
frameworks such as generative adversarial networks (GAN), adversarial training
and multi-agent rei…
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Optimality and Complexity for Constrained Optimization Problems with Nonconvex Regularization Open
In this paper, we consider a class of constrained optimization problems where the feasible set is a general closed convex set, and the objective function has a nonsmooth, nonconvex regularizer. Such a regularizer includes widely used SCAD,…
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What is topological about topological dynamics? Open
We consider various notions from the theory of dynamical systems from a topological point of view. Many of these notions can be sensibly defined either in terms of (finite) open covers or uniformities. These Hausdorff or uniform versions c…
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THE PROX-TIKHONOV-LIKE FORWARD-BACKWARD METHOD AND APPLICATIONS Open
It is known, by Rockafellar [SIAM J. Control Optim., 14 (1976), 877-898], that\nthe proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone\noperator in a Hilbert space, but it fails to converge strongly. Lehdili an…
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Sequential optimality conditions for cardinality-constrained optimization problems with applications Open
Recently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed. Following this approach, we derive a problem-tailored sequential optimality condition, which …
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The structure of spaces with Bakry–Émery Ricci curvature bounded below Open
We explore the structure of limit spaces of sequences of Riemannian manifolds with Bakry–Émery Ricci curvature bounded below in the Gromov–Hausdorff topology. By extending the techniques established by Cheeger and Cloding for Riemannian ma…
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On Rough ℐ<sub>2</sub>-Convergence of Double Sequences Open
In this study, we introduce the notion of rough ℐ_2-convergence and the set of rough ℐ_2-limit points of a double sequence and obtained two rough ℐ_2-convergence criteria associated with this set. Later, we proved that this set is closed a…
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Spectral estimates for unreduced symmetric KKT systems arising from Interior Point methods Open
Summary We consider symmetrized Karush–Kuhn–Tucker systems arising in the solution of convex quadratic programming problems in standard form by Interior Point methods. Their coefficient matrices usually have 3 × 3 block structure, and unde…
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On rough convergence in 2-normed spaces and some properties Open
In this study, we investigated relationships between rough convergence and classical convergence and studied some properties about the notion of rough convergence, the set of rough limit points and rough cluster points of a sequence in 2-n…
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On a Family of Strictly Non-Volterra Quadratic Stochastic Operators Open
In this paper we consider a class of strictly non-Volterra quadratic stochastic operators defined on the two-dimensional simplex. We show that such operators have a unique fixed point and the set of limit points is either a single point or…
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Eigenvibrations of a bar with load Open
\nThe differential eigenvalue problem describing eigenvibrations of an elastic bar with load is investigated. The problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalue…
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The Switch Point Algorithm Open
The Switch Point Algorithm is a new approach for solving optimal control\nproblems whose solutions are either singular or bang-bang or both singular and\nbang-bang, and which possess a finite number of jump discontinuities in an\noptimal c…
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A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs Open
With the help of a logarithmic barrier augmented Lagrangian function, we can obtain closed-form solutions of slack variables of logarithmicbarrier problems of nonlinear programs. As a result, a two-parameter primaldual nonlinear system is …
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Beyond the fold: experimentally traversing limit points in nonlinear structures Open
Recent years have seen a paradigm shift regarding the role of nonlinearities and elastic instabilities in engineering science and applied physics. Traditionally viewed as unwanted aberrations, when controlled to be reversible and well beha…
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Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption Open
Consider a Markov chain $(X_n)_{n\\geqslant 0}$ with values in the state space\n$\\mathbb X$. Let $f$ be a real function on $\\mathbb X$ and set $S_0=0,$ $S_n =\nf(X_1)+\\cdots + f(X_n),$ $n\\geqslant 1$. Let $\\mathbb P_x$ be the probabil…
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Global bifurcation of solitary waves for the Whitham equation Open
The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of the KdV equation with the linear dispersion of the full water wave problem. Whitham conjectured the existence of a highest, cusped, tr…
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The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization Open
Motivated by applications in Optimization, Game Theory, and the training of Generative Adversarial Networks, the convergence properties of first order methods in min-max problems have received extensive study. It has been recognized that t…
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A convergent iterative hard thresholding for nonnegative sparsity optimization Open
The iterative hard thresholding (IHT) algorithm is a popular greedy-type method in (linear and nonlinear) compressed sensing and sparse optimization problems.In this paper, we give an improved iterative hard thresholding algorithm for solv…
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Rough statistical cluster points Open
In this paper, we define the concepts of rough statistical cluster point and rough statistical limit point of a sequence in a finite dimensional normed space. Then we obtain an ordinary statistical convergence criteria associated with roug…
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From deterministic cellular automata to coupled map lattices Open
A general mathematical method is presented for the systematic construction of coupled map lattices (CMLs) out of deterministic cellular automata (CAs). The entire CA rule space is addressed by means of a universal map for CAs that we have …
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Convergence of Laplacian spectra from random samples Open
Eigenvectors and eigenvalues of discrete graph Laplacians are often used for manifold learning and nonlinear dimensionality reduction. It was previously proved by Belkin and Niyogi that the eigenvectors and eigenvalues of the graph Laplaci…
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Fixed Point Theorems and Generalizations of Dislocated Metric Spaces Open
In this note we discuss some topological properties and generalizations of dislocated metric space and establish some fixed point theorems. Mathematics Subject Classification: 47H10, 54H25. Keywords: Fixed Point, Ld-Closed, Ld-Convergent, …
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F-evolution algebra Open
We consider the evolution algebra of a free population generated by an F-quadratic stochastic operator. We prove that this algebra is commutative, not associative and necessarily power-associative. We show that this algebra is not conserva…
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Nonconvex Sparse Spectral Clustering by Alternating Direction Method of Multipliers and Its Convergence Analysis Open
Spectral Clustering (SC) is a widely used data clustering method which first learns a low-dimensional embedding U of data by computing the eigenvectors of the normalized Laplacian matrix, and then performs k-means on UT to get the final cl…
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An adaptive discretization method solving semi-infinite optimization\n problems with quadratic rate of convergence Open
Semi-infinite programming can be used to model a large variety of complex\noptimization problems. The simple description of such problems comes at a\nprice: semi-infinite problems are often harder to solve than finite nonlinear\nproblems. …