Weakly Convex Research Articles

High probability bounds on AdaGrad for constrained weakly convex optimization

In this paper, we study the high probability convergence of AdaGrad-Norm for constrained, non-smooth, weakly convex optimization with bounded noise and sub-Gaussian noise cases. We also investigate a more general accelerated gradient descent (AGD) template (Ghadimi and Lan, 2016) encompassing the AdaGrad-Norm, the Nesterov's accelerated gradient descent, and the RSAG (Ghadimi and Lan, 2016) with different parameter choices. We provide a high probability convergence rate O˜(1/T) without knowing the information of the weak convexity parameter and the gradient bound to tune the step-sizes.

Computing the hull and interval numbers in the weakly toll convexity

A walk u0u1…uk−1uk of a graph G is a weakly toll walk if u0uk∉E(G), u0ui∈E(G) implies ui=u1, and ujuk∈E(G) implies uj=uk−1. The weakly toll interval of a set S⊆V(G), denoted by I(S), is formed by S and the vertices belonging to some weakly toll walk between two vertices of S. Set S is weakly toll convex if I(S)=S. The weakly toll convex hull of S, denote by H(S), is the minimum weakly toll convex set containing S. The weakly toll interval number of G is the minimum cardinality of a set S⊆V(G) such that I(S)=V(G); and the weakly toll hull number of G is the minimum cardinality of a set S⊆V(G) such that H(S)=V(G). In this work, we show how to compute the weakly toll interval and the weakly toll hull numbers of a graph in polynomial time. In contrast, we show that determining the weakly toll convexity number of a graph G (the size of a maximum weakly toll convex set different from V(G)) is NP-hard.

Stochastic Subgradient Descent Escapes Active Strict Saddles on Weakly Convex Functions

In nonsmooth stochastic optimization, we establish the nonconvergence of the stochastic subgradient descent (SGD) to the critical points recently called active strict saddles by Davis and Drusvyatskiy. Such points lie on a manifold M, where the function f has a direction of second-order negative curvature. Off this manifold, the norm of the Clarke subdifferential of f is lower-bounded. We require two conditions on f. The first assumption is a Verdier stratification condition, which is a refinement of the popular Whitney stratification. It allows us to establish a strengthened version of the projection formula of Bolte et al. for Whitney stratifiable functions and which is of independent interest. The second assumption, termed the angle condition, allows us to control the distance of the iterates to M. When f is weakly convex, our assumptions are generic. Consequently, generically, in the class of definable weakly convex functions, SGD converges to a local minimizer. Funding: The work of Sholom Schechtman was supported by “Région Ile-de-France”.

Weakly Convex Hop Dominating Sets in Graphs

Let G be an undirected connected graph with vertex and edge sets V (G) and E(G), respectively. A set C ⊆ V (G) is called weakly convex hop dominating if for every two vertices x, y ∈ C, there exists an x-y geodesic P(x, y) such that V (P(x, y)) ⊆ C and for every v ∈ V (G)\C, there exists w ∈ C such that dG(v, w) = 2. The minimum cardinality of a weakly convex hop dominating set of G, denoted by γwconh(G), is called the weakly convex hop domination number of G. In this paper, we introduce and initially investigate the concept of weakly convex hop domination. We show that every two positive integers a and b with 3 ≤ a ≤ b are realizable as the weakly convex hop domination number and convex hop domination number of some connected graph. Furthermore, we characterize the weakly convex hop dominating sets in some graphs under some binary operations.

The $$Q_k$$ flow on complete non-compact graphs

We establish the long time existence of complete non-compact weakly convex and smooth hypersurfaces \(\Sigma _t\) evolving by the \(Q_k\)-flow. We show that the maximum existence time T depends on the dimension \(d_W\) of the vector space \(W{:}{=}\{w \in \mathbb {R}^{n+1}: \sup _{X\in \Sigma _0} |\langle X,w\rangle | = +\infty \}\) which contains each direction in which our initial data \(\Sigma _0\) is infinite. If \(d_W=\text {dim}(W) \ge n-k+1\), then the solution \(\Sigma _t\) exists for all time \(t \in (0,+\infty )\); if \(d_W=\text {dim}(W) \le n-k\), then the solution \(\Sigma _t\) exsist up to some finite time \(T < +\infty \). In the latter case, the trace at infinity \(\Gamma _t\) of the solution \(\Sigma _t\) is a closed convex viscosity solution of the \((n-d_W)\)-dimensional \(Q_k\) flow on \(t \in (0,T)\).

A zeroth order method for stochastic weakly convex optimization

In this paper, we consider stochastic weakly convex optimization problems, however without the existence of a stochastic subgradient oracle. We present a derivative free algorithm that uses a two point approximation for computing a gradient estimate of the smoothed function. We prove convergence at a similar rate as state of the art methods, however with a larger constant, and report some numerical results showing the effectiveness of the approach.

Characterization of solutions of strong-weak convex programming problems

Finite-dimensional problems of minimizing a strongly or weakly convex function on a strongly or weakly convex set are considered. Necessary and sufficient conditions for solutions of such problems are presented, which are based on estimates for the behaviour of the objective function on the feasible set taking account of the parameters of strong or weak convexity, as well as certain local features of the set and the function. The mathematical programming problem is considered separately for strongly and weakly convex functions. In addition, necessary conditions for a global and a local solution with differentiable objective function are found, in which a ‘strong’ stationarity condition is assumed to hold. Bibliography: 33 titles.

Weakly convex hypersurfaces of pseudo-Euclidean spaces satisfying the condition LkHk+1 = λHk+1

In this paper, we try to give a classification of spacelike hypersurfaces of the Lorentz-Minkowski space-time E1n+1, whose mean curvature vector field of order (k+ 1) is an eigenvector of the kth linearized operator Lk, for a non-negative integer k less than n. The operator Lk is defined as the linear part of the first variation of the (k + 1)th mean curvature of a hypersurface arising from its normal variations. We show that any spacelike hypersurface of E1n+1 satisfying the condition LkHk+1 = λHk+1 (where 0 ≤ k ≤ n − 1) belongs to the class of Lk-biharmonic, Lk-1-type or Lk-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of spacelike hypersurfaces of Lorentz-Minkowski spaces, named the weakly convex hypersurfaces (i.e. on which all of principle curvatures are nonnegative). We prove that, on any weakly convex spacelike hypersurface satisfying the above condition for an integer k (where, 0 ≤ r ≤ n−1), the (k + 1)th mean curvature will be constant. As an interesting result, any weakly convex spacelike hypersurfaces, having assumed to be Lk-biharmonic, has to be k-maximal.

Invariant weakly convex cocompact subspaces for surface groups in $A_2$-buildings

This paper deals with non-Archimedean representations of punctured surface groups in $\operatorname {PGL}_3$, associated actions on (not necessarily discrete) Euclidean buildings of type $A_2$, and degenerations of convex real projective structures on surfaces. The main result is that, under good conditions on Fock-Goncharov generalized shear parameters, non-Archimedean representations acting on the Euclidean building preserve a cocompact weakly convex subspace, which is part flat surface and part tree. In particular the eigenvalue and length(s) spectra are given by an explicit finite $A_2$-complex. We use this result to describe degenerations of convex real projective structures on surfaces for an open cone of parameters. The main tool is a geometric interpretation of Fock-Goncharov parameters in $A_2$-buildings.

Variable smoothing incremental aggregated gradient method for nonsmooth nonconvex regularized optimization

In this paper, we focus on the problem of minimizing the sum of nonconvex smooth component functions and a nonsmooth weakly convex function composed with a linear operator. One specific application is logistic regression problems with weakly convex regularizers that introduce better sparsity than the standard convex regularizers. Based on the Moreau envelope with a decreasing sequence of smoothing parameters as well as incremental aggregated gradient method, we propose a variable smoothing incremental aggregated gradient (VS-IAG) algorithm. We also prove a complexity of $${\mathcal {O}}(\epsilon ^{-3})$$ to achieve an $$\epsilon $$ -approximate solution.

加边对图的弱凸控制数和凸控制数的影响

加边对图的弱凸控制数和凸控制数的影响

On the Gradient Projection Method for Weakly Convex Functions on a Proximally Smooth Set

Let a weakly convex function (in the general case, nonconvex and nonsmooth) satisfy the quadratic growth condition. It is proved that the gradient projection method for minimizing such a function on a set converges with linear rate on a proximally smooth (nonconvex) set of special form (for example, on a smooth manifold), provided that the constant of weak convexity of the function is less than the constant in the quadratic growth condition and the constant of proximal smoothness for the set is sufficiently large. The connection between the quadratic growth condition on the function and other conditions is discussed.

A regularization interpretation of the proximal point method for weakly convex functions

<p style='text-indent:20px;'>Empirical evidence and theoretical results suggest that the proximal point method can be computed approximately and still converge faster than the corresponding gradient descent method, in both the stochastic and exact gradient case. In this article we provide a perspective on this result by interpreting the method as gradient descent on a regularized function. This perspective applies in the case of weakly convex functions where proofs of the faster rates are not available. Using this analysis we find the optimal value of the regularization parameter in terms of the weak convexity.

First record of the genus Brachyzapus Gauld Dubois, 2006 (Hymenoptera: Ichneumonidae: Pimplinae) from the Afrotropical region, with descriptions of two new species.

Brachyzapus Gauld Dubois, 2006 is a small genus belonging to the Polysphincta genus-group of the subfamily Pimplinae, with nine known species worldwide (Uchida Momoi, 1958; Yu et al., 2012; Pham et al., 2012). The genus can be recognized by twisted mandibles, a very weakly convex and rather long clypeus with quite well-defined angulation laterally, elongate lower face, sparse pubescence on the eyes, hind coxal cavities separated from the metasoma insertion by a very strong sclerotized bridge, and the lack of a basal lobe on the ovipositor (Gauld Dubois, 2006).

Strong and Weak Convexity in Nonlinear Differential Games

Abstract We obtain sufficient conditions for the values of the Minkowski operators to be weakly convex and smooth. These operators play the same role in nonlinear differential games as the Minkowski sum and the Minkowski difference do in linear differential games: they are basic operators in algorithms of computing reachable sets and optimal strategies. We also prove that the signed distance to convex sets is a Lipschitz continuous function of the set with respect to the Hausdorff distance.

Nonconvex Sparse Logistic Regression With Weakly Convex Regularization

In this work we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem. The idea is based on the finding that a weakly convex function as an approximation of the $\ell_0$ pseudo norm is able to better induce sparsity than the commonly used $\ell_1$ norm. For a class of weakly convex sparsity inducing functions, we prove the nonconvexity of the corresponding sparse logistic regression problem, and study its local optimality conditions and the choice of the regularization parameter to exclude trivial solutions. Despite the nonconvexity, a method based on proximal gradient descent is used to solve the general weakly convex sparse logistic regression, and its convergence behavior is studied theoretically. Then the general framework is applied to a specific weakly convex function, and a necessary and sufficient local optimality condition is provided. The solution method is instantiated in this case as an iterative firm-shrinkage algorithm, and its effectiveness is demonstrated in numerical experiments by both randomly generated and real datasets.

On the classification of hypersurfaces in Euclidean spaces satisfying LrHr+1 = AHr+1

In this paper, we study isometrically immersed hypersurfaces of the Euclidean space En+1 satisfying the condition LrH r+i = λHr+1for an integer r ( 0 ≤ r ≤ n — 1), where Hr+iis the (r + 1)th mean curvature vector field on the hypersurface, Lr is the linearized operator of the first variation of the (r + 1) th mean curvature of hypersurface arising from its normal variations. Having assumed that on a hypersurface x : Mn → En+1, the vector field Hr+i be an eigenvector of the operator Lr with a constant real eigenvalue λ, we show that, Mn has to be an Lr-biharmonic, Lr-1-type, or Lr-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of hypersurfaces, named the weakly convex hypersurfaces (i.e. on which principal curvatures are nonnegative). We prove that, any weakly convex Euclidean hypersurface satisfying the condition Lr Hr+i = λ Hr+i for an integer r ( 0 ≤ r ≤ n — 1), has constant mean curvature of order (r + 1). As an interesting result, we have that, the Lr-biharmonicity condition on the weakly convex Euclidean hypersurfaces implies the r-minimality.

A maximal element theorem in FWC-spaces and its applications.

A maximal element theorem is proved in finite weakly convex spaces (FWC-spaces, in short) which have no linear, convex, and topological structure. Using the maximal element theorem, we develop new existence theorems of solutions to variational relation problem, generalized equilibrium problem, equilibrium problem with lower and upper bounds, and minimax problem in FWC-spaces. The results represented in this paper unify and extend some known results in the literature.

Generalized selection theorems for fuzzy mappings and their applications

We develop generalized selection theorems for fuzzy mappings in finite weakly convex spaces (in short, FWC-spaces) without any linear, convex, and topological structure. Applying the generalized selection theorems in noncompact FWC-spaces, we obtain new fuzzy collective coincidence point theorems and fuzzy collectively fixed point theorems. These results generalize many known theorems in the literature.

Weak Convex Decomposition by Lines‐of‐sight

AbstractWe define the convexity rank of a set of points to be the portion of mutually visible pairs of points out of the total number of pairs. Based on this definition of weak convexity, we introduce a spectral method that decomposes a given shape into weakly convex regions. The decomposition is applied without explicitly measuring the convexity rank. The method merely amounts to a spectral clustering of a matrix representing the all‐pairs line of sight. Our method can be directly applied on an oriented point cloud and does not require any topological information, nor explicit concavity or convexity measures. We demonstrate the efficiency of our algorithm on a large number of examples and compare them qualitatively with competitive approaches.