Set Of Critical Points Research Articles

Normalized penalty gradient flow: A continuous-time approach for solving constrained nonconvex nonsmooth optimization problems

To avoid calculating penalty parameters, this paper introduces a continuous-time approach that combines the normalized gradient flow with the penalty method to solve the nonconvex nonsmooth optimization problem with a convex constraint set. Subsequently, this approach is extended to solve nonconvex nonsmooth optimization with a box constraint set and affine equality constraints. Compared with the current methods, the proposed approach does not require calculating the penalty parameters, allows the objective function to be nonconvex or nonsmooth, and allows the initial point to be arbitrarily selected. It is proved that the solution trajectory with any initial point converges to the critical point set of the optimization problem. Finally, the effectiveness of the proposed approach is authenticated through several numerical experiments.

Optimal mass of structure with motion described by Sturm-Liouville operator: design and predesign

We find an optimal design of a structure described by a Sturm-Liouville (S-L) problem with a spectral parameter in the boundary conditions. Using an approach from calculus of variations, we determine a set of critical points of a corresponding mass functional. However, these critical points - which we call predesigns - do not necessarily themselves represent meaningful solutions: it is of course natural to expect a mass to be real and positive. This represents a generalization of previous work on the topic in several ways. First, previous work considered only boundary conditions and S-L coefficients under certain simplifying assumptions. Principally, we do not assume that one of the coefficients vanishes as in the previous work. Finally, we introduce a set of solvability conditions on the S-L problem data, confirming that the corresponding critical points represent meaningful solutions we refer to as designs. Additionally, we present a natural schematic for testing these conditions, as well as suggesting a code and several numerical examples.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/08/abstr.html

Energy Function for Direct Products of Discrete Dynamical Systems

This paper is devoted to the construction of an energy function, i.e. a smooth Lyapunov function, whose set of critical points coincides with the chain-recurrent set of a dynamical system — for a cascade that is a direct product of two systems. One of the multipliers is a structurally stable diffeomorphism given on a two-dimensional torus, whose non-wandering set consists of a zero-dimensional non-trivial basic set without pairs of conjugated points and without fixed source and sink, and the second one is an identical mapping on a real axis. It was previously proved that if a non-wandering set of a dynamical system contains a zero-dimensional basic set, as the diffeomorphism under consideration has, then such a system does not have an energy function, namely, any Lyapunov function will have critical points outside the chain-recurrent set. For an identical mapping, the energy function is a constant on the entire real line. In this paper, it is shown that the absence of an energy function for one of the multipliers is not a sufficient condition for the absence of such a function for the direct product of dynamical systems, that is, in some cases it is possible to select the second cascade in such a way that the direct product will have an energy function.

Convergence of a Class of Nonmonotone Descent Methods for Kurdyka–Łojasiewicz Optimization Problems

This note is concerned with a class of nonmonotone descent methods for minimizing a proper lower semicontinuous Kurdyka–Łojasiewicz (KL) function , which generates a sequence satisfying a nonmonotone decrease condition and a relative error tolerance. Under suitable assumptions, we prove that the whole sequence converges to a limiting critical point of , and, when is a KL function of exponent , the convergence admits a linear rate if and a sublinear rate associated to if . These assumptions are shown to be sufficient and necessary if in addition is weakly convex on a neighborhood of the set of critical points. Our results not only extend the convergence results of monotone descent methods for KL optimization problems but also resolve the convergence problem on the iterate sequence generated by a class of nonmonotone line search algorithms for nonconvex and nonsmooth problems.

Distributed Proximal Gradient Algorithm for Nonconvex Optimization Over Time-Varying Networks

This paper studies the distributed non-convex optimization problem with non-smooth regularization, which has wide applications in decentralized learning, estimation and control. The objective function is the sum of local objective functions, which consist of differentiable (possibly non-convex) cost functions and non-smooth convex functions. This paper presents a distributed proximal gradient algorithm for the non-smooth non-convex optimization problem. Over time-varying multi-agent networks, the proposed algorithm updates local variable estimates with a constant step-size at the cost of multiple consensus steps, where the number of communication rounds increases over time. We prove that the generated local variables achieve consensus and converge to the set of critical points. Finally, we verify the efficiency of the proposed algorithm by numerical simulations.

Irrational pencils and Betti numbers

We study irrational pencils with isolated critical points on compact aspherical complex manifolds. We prove that if the set of critical points is nonempty, the homology of the kernel of the morphism induced by the pencil on fundamental groups is not finitely generated. This generalizes a result by Dimca, Papadima and Suciu. By considering self-products of the Cartwright-Steger surface, this allows us to build new examples of smooth projective varieties whose fundamental group has a non-finitely generated homology.

Bicritical Rational Maps With a Common Iterate

Abstract Let $f$ be a degree $d$ bicritical rational map with critical point set $\mathcal{C}_f$ and critical value set $\mathcal{V}_f$. Using the group $\textrm{Deck}(f^k)$ of deck transformations of $f^k$, we show that if $g$ is a bicritical rational map that shares an iterate with $f$, then $\mathcal{C}_f = \mathcal{C}_g$ and $\mathcal{V}_f = \mathcal{V}_g$. Using this, we show that if two bicritical rational maps of even degree $d$ share an iterate, then they share a second iterate, and both maps belong to the symmetry locus of degree $d$ bicritical rational maps.

Local uniqueness of concentrated solutions and some applications on nonlinear Schrödinger equations with very degenerate potentials

We revisit the following nonlinear Schrödinger equation−ε2Δu+V(x)u=up−1,u>0,u∈H1(RN), where ε>0 is a small parameter, N≥2 and 2<p<2⁎. We obtain a more accurate location for the concentrated points, the existence and the local uniqueness for positive k-peak solutions when V(x) possesses non-isolated critical points by using the modified finite dimensional reduction method based on local Pohozaev identities. Moreover, for several special potentials, with its critical point set being a low-dimensional ellipsoid, or a part of hyperboloid of one sheet or two sheets, we obtain the number and symmetry of k-peak solutions by using local uniqueness of concentrated solutions. Here the main difficulty comes from the different degenerate rate along different directions at the critical points of V(x).

Challenges toward Evidence-Based Policymaking Using Agent-Based Modeling for Federal Sports Grants: A Self-Reflection from a Transdisciplinary Project

Despite their importance, federal grant systems often need more clarity regarding cost-effectiveness, lack of transparency, and slow feedback cycles. Sports funding systems aimed at improving child health and contributing to sustainable development goals are incredibly challenging due to their heterogeneity in stakeholders and regional aspects. Here, we analyze how we tackled these challenges in a transdisciplinary EU project in Austria, targeting the use of agent-based modeling for evidence-based policymaking in a co-creation process with policy stakeholders in the domain of federal sports grants to improve the health and well-being of children and youth. The initial and executed set of procedures is described, along with lessons learned during the project’s lifetime. These lessons derive a framework that provides an adapted set of processes, supporting methods, and critical decision points for an improved use of transdisciplinarity. In addition, the steps of the developed framework are combined with essential aspects of knowledge integration, following the main phases of the policy cycle and providing suggestions for required skills and competencies for capacity building concerning implementing the developed framework in the public sector. Our results show that the combination of transdisciplinarity, human-centered policymaking, and sports, supported by cutting-edge technologies such as agent-based modeling, can achieve significantly better results than a pure disciplinary approach and generate positive spill-over effects.

Association between nonextraction clear aligner therapy and alveolar bone dehiscences and fenestrations in adults with mild-to-moderate crowding

This study aimed to assess the association between nonextraction clear aligner therapy (CAT) and the presence of alveolar bone dehiscences (ABDs) and fenestrations (ABFs) in adults with mild-to-moderate crowding. Cone-beam computed tomography images from 29 adults were obtained before and immediately after nonextraction CAT. Total root lengths were evaluated in axial and cross-sectional slices. Linear measurement for dehiscence (LM-D) was defined as the distance between the alveolar crest to the cementoenamel junction of each root (critical point set at 2 mm). Linear measurement for fenestration (LM-F) was recorded when the defect involved only the apical one-third of a root (critical point set at 2.2 mm). Counts of ABDs/ABFs and magnitudes of LM-Ds/LM-Fs were recorded before and immediately after nonextraction CAT at buccal and lingual root surfaces. Binary logistic regression analyses and repeated measures analyses of variance were performed. Counts of ABDs/ABFs and magnitudes of LM-Ds/LM-Fs increased at most jaw locations and root surfaces. Nonextraction CAT was associated with an increased presence of ABDs and ABFs. Nonextraction CAT was associated with a higher magnitude of LM-Ds but not LM-Fs. Immediate posttreatment cone-beam computed tomography scans showed that nonextraction CAT is associated with increased ABDs and ABFs in adults with mild-to-moderate crowding.

A neurodynamic optimization approach to nonconvex resource allocation problem

A neurodynamic optimization approach characterized by a negative subgradient flow (NSF) is proposed to solve distributed nonconvex resource allocation problem (DNRAP). Under some mild assumptions, it is proved that the state solutions of the proposed approach converge to the critical point set of the considered DNRAP. Moreover, benefiting from the well-known nonsmooth Lojasiewicz inequality, analysis results show that the state solutions converge toward a critical point (not set) of the considered DNRAP. The convergence rate of the state solution, including exponential or finite-time convergence, can be evaluated through the quantitative calculation of Lojasiewicz exponent. Furthermore, we assess the related Lojasiewicz exponent of presented NSF in some special situation, for instance, in nonconvex quadratic programming. Finally, simulation illustrates the well performance of presented neurodynamic optimization approach.

A continuous-time neurodynamic algorithm for distributed nonconvex nonsmooth optimization problems with affine equality and nonsmooth convex inequality constraints

In this paper, a distributed nonsmooth nonconvex optimization (DNNO) problem with affine inequality and nonsmooth convex inequality constraints is studied. A continuous-time distributed neurodynamic algorithm is proposed to solve this problem. Under the assumed conditions, for any initial state, the solution of distributed neurodynamic algorithm is bounded and globally exists, and will converge to the critical point set of distributed problems in a finite time. Compared with other DNNO algorithms, distributed neurodynamic algorithm has a lower dimension and does not need to satisfy the assumption that the feasible region is bounded. Finally, a series of numerical examples are given to verify the effectiveness of the proposed algorithm.

Resolution of Singularities of the Orbit Spaces $$G_{n,2}/T^n$$

The problem of the description of the orbit space $X_{n} = G_{n,2}/T^n$ for the standard action of the torus $T^n$ on a complex Grassmann manifold $G_{n,2}$ is widely known and it appears in diversity of mathematical questions. A point $x\in X_{n}$ is said to be a critical point if the stabilizer of its corresponding orbit is nontrivial. In this paper, the notion of singular points of $X_n$ is introduced which opened the new approach to this problem. It is showed that for $n>4$ the set of critical points $\text{Crit}X_n$ belongs to our set of singular points $\text{Sing}X_{n}$, while the case $n=4$ is somewhat special for which $\text{Sing}X_4\subset \text{Crit}X_4$, but there are critical points which are not singular. The central result of this paper is the construction of the smooth manifold $U_n$ with corners, $\dim U_n = \dim X_n$ and an explicit description of the projection $p_{n} : U_{n}\to X_{n}$ which in the defined sense resolve all singular points of the space $X_n$. Thus, we obtain the description of the orbit space $G_{n,2}/T^n$ combinatorial structure. Moreover, the $T^n$-action on $G_{n,2}$ is a seminal example of complexity $(n-3)$ - action. Our results demonstrate the method for general description of orbit spaces for torus actions of positive complexity.

Critical points of solutions to a kind of linear elliptic equations in multiply connected domains

In this paper, we mainly study the critical points and critical zero points of solutions u to a kind of linear elliptic equations with nonhomogeneous Dirichlet boundary conditions in a multiply connected domain Ω in ℝ2. Based on the delicate analysis about the distributions of connected components of the super-level sets {x ∈ Ω: u(x) > t} and sub-level sets {x ∈ Ω: u(x) < t} for some t, we obtain the geometric structure of interior critical point sets of u. Precisely, let Ω be a multiply connected domain with the interior boundary γI and the external boundary γE, where u∣γI = ψ1(x), u∣γE = ψ2(x). When ψ1(x) and ψ2(x) have N1 and N2 local maximal points on γI and γE respectively, we deduce that \(\sum\nolimits_{i = 1}^k {{m_i} \le {N_1} + {N_2}} \), where m1,…,mk are the respective multiplicities of interior critical points x1,…,xk of u. In addition, when \({\min _{{\gamma _E}}}{\psi _2}\left( x \right) \ge {\max _{{\gamma _I}}}{\psi _1}\left( x \right)\) and u has only N1 and N2 equal local maxima relative to \(\overline \Omega \) on γI and γe respectively, we develop a new method to show that one of the following three results holds \(\sum\nolimits_{i = 1}^k {{m_i} = {N_1} + {N_2}} \) or \(\sum\nolimits_{i = 1}^k {{m_i} + 1 = {N_1} + {N_2}} \) or \(\sum\nolimits_{i = 1}^k {{m_i} + 2 = {N_1} + {N_2}} \). Moreover, we investigate the geometric structure of interior critical zero points of u. We obtain that the sum of multiplicities of the interior critical zero points of u is less than or equal to the half of the number of its isolated zero points on the boundaries.

The Dynamics of Complex Box Mappings

In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is:To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues.To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus, we call such complex box mappings dynamically natural. Having such box mappings is the first step in tackling many problems in one-dimensional dynamics.Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting, some of these tools are:the Enhanced Nest (a nest of puzzle pieces around critical points) from Kozlovski, Shen, van Strien (Ann Math 165:749–841, 2007), referred to below as KSS;the Covering Lemma (which controls the moduli of pullbacks of annuli) from Kahn and Lyubich (Ann Math 169(2):561–593, 2009);the QC-Criterion and the Spreading Principle from KSS. The purpose of this paper is to make these tools more accessible so that they can be used as a ‘black box’, so one does not have to redo the proofs in new settings.To give an intuitive, but also rather detailed, outline of the proof from KSS and Kozlovski and van Strien (Proc Lond Math Soc (3) 99:275–296, 2009) of the following results for non-renormalizable dynamically natural complex box mappings:puzzle pieces shrink to points,(under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate.We prove the fundamental ergodic properties for dynamically natural box mappings. This leads to some necessary conditions for when such a box mapping supports a measurable invariant line field on its filled Julia set. These mappings are the analogues of Lattès maps in this setting.We prove a version of Mañé’s Theorem for complex box mappings concerning expansion along orbits of points that avoid a neighborhood of the set of critical points.

Accumulation set of critical points of the multipliers in the quadratic family

AbstractA parameter $c_{0}\in {\mathbb {C}}$ in the family of quadratic polynomials $f_{c}(z)=z^{2}+c$ is a critical point of a period n multiplier if the map $f_{c_{0}}$ has a periodic orbit of period n, whose multiplier, viewed as a locally analytic function of c, has a vanishing derivative at $c=c_{0}$ . We study the accumulation set ${\mathcal X}$ of the critical points of the multipliers as $n\to \infty $ . This study complements the equidistribution result for the critical points of the multipliers that was previously obtained by the authors. In particular, in the current paper, we prove that the accumulation set ${\mathcal X}$ is bounded, connected, and contains the Mandelbrot set as a proper subset. We also provide a necessary and sufficient condition for a parameter outside of the Mandelbrot set to be contained in the accumulation set ${\mathcal X}$ and show that this condition is satisfied for an open set of parameters. Our condition is similar in flavor to one of the conditions that define the Mandelbrot set. As an application, we get that the function that sends c to the Hausdorff dimension of $f_{c}$ does not have critical points outside of the accumulation set ${\mathcal X}$ .

Dynamical properties of direct products of discrete dynamical systems

A natural way for creating new dynamical systems is to consider direct products of already known systems. The paper studies some dynamical properties of direct products of homeomorphisms and diffeomorphisms. In particular, authors prove that a chain-recurrent set of the direct product of homeomorphisms is a direct product of the chain-recurrent sets. Another result established in the paper is that the direct product of diffeomorphisms holds hyperbolic structure on the direct product of hyperbolic sets. It is known that if a diffeomorphism has a hyperbolic chain-recurrent set, then this mapping is Ω-stable. Therefore, it follows from the results of the paper that the direct product of Ω-stable diffeomorphisms is also Ω-stable. Another question which is raised in the article concerns the existence of an energy function for the direct product of diffeomorphisms which already have such functions (recall that energy function is a smooth Lyapunov function whose set of critical points coincides with the chain-recurrent set of the system). Authors show that in this case the function can be found as a weighted sum of energy functions of initial diffeomorphisms.

Towards understanding theoretical advantages of complex-reaction networks

Complex-valued neural networks have attracted increasing attention in recent years, while it remains open on the advantages of complex-valued neural networks in comparison with real-valued networks. This work takes one step on this direction by introducing the complex-reaction network with fully-connected feed-forward architecture. We prove the universal approximation property for complex-reaction networks, and show that a class of radial functions can be approximated by a complex-reaction network using the polynomial number of parameters, whereas real-valued networks need at least exponential parameters to reach the same approximation level. For empirical risk minimization, we study the landscape and convergence of complex gradient descents. Our theoretical result shows that the critical point set of complex-reaction networks is a proper subset of that of real-valued networks, which may show some insights on finding the optimal solutions more easily for complex-reaction networks.

Prescribing Morse scalar curvatures: Critical points at infinity

Abstract The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this problem consist in subcritical approximations or negative pseudogradient flows. We show under a mild nondegeneracy assumption the equivalence of both approaches with respect to zero weak limits, in particular a one-to-one correspondence of zero weak limit finite energy subcritical blow-up solutions, zero weak limit critical points at infinity of negative type and sets of critical points with negative Laplacian of the function to be prescribed.

Optimization flows landing on the Stiefel manifold⋆

We study a continuous-time system that solves optimization problems over the set of orthonormal matrices, which is also known as the Stiefel manifold. The resulting optimization flow follows a path that is not always on the manifold but asymptotically lands on the manifold. We introduce a generalized Stiefel manifold to which we extend the canonical metric of the Stiefel manifold. We show that the vector field of the proposed flow can be interpreted as the sum of a Riemannian gradient on a generalized Stiefel manifold and a normal vector. Moreover, we prove that the proposed flow globally converges to the set of critical points, and any local minimum and isolated critical point is asymptotically stable.