One-parameter families of optimization problems: Equality constraints

In this paper, we introduce generalized critical points and discuss their relationship with other concepts of critical points [resp., stationary points]. Generalized critical points play an important role in parametric optimization. Under generic regularity conditions, we study the set of generalized critical points, in particular, the change of the Morse index. We focus our attention on problems with equality constraints only and provide an indication of how the present theory can be extended to problems with inequality constraints as well.     

Bemerkung zu einer Arbeit von H. Schlosser

In [5], [7] is was shown that the operator of KLEIN-GORDON type is a spectral operator with critical points. In this paper we estimate where the critical points can be found. We are especially interested in finding sufficient conditions for the absence of singular critical points.     

A local minimax characterization for computing multiple nonsmooth saddle critical points

This paper is concerned with characterizations of nonsmooth saddle critical points for numerical algorithm design. Most characterizations for nonsmooth saddle critical points in the literature focus on existence issue and are converted to solve global minimax problems. Thus they are not helpful for numerical algorithm design. Inspired by the results on computational theory and methods for finding multiple smooth saddle critical points in [14, 15, 19, 21, 23], a local minimax characterization for multiple nonsmooth saddle critical points in either a Hilbert space or a reflexive Banach space is established in this paper to provide a mathematical justification for numerical algorithm design. A local minimax algorithm for computing multiple nonsmooth saddle critical points is presented by its flow chart. Dedicated to Terry Rockafellar on his 70th birthday     

Nonsymmetric bifurcations of solutions of the 2D Navier–Stokes system

We consider the 2D Navier–Stokes system written for the stream function with periodic boundary conditions and construct a set of initial data such that initial critical points bifurcate from 1 to 2 and then to 3 critical points in finite time. The bifurcation takes place in a small neighborhood of the origin. Our construction does not require any symmetry assumptions or the existence of special fixed points. For another set of initial data we show that 3 critical points merge into 1 critical point in finite time. We also construct a set of initial data so that bifurcation can be generated by the Navier–Stokes flow and do not require the existence of an initial critical point.     

ON THE BERLINSKII''''S THEOREM FOR CUBIC SYSTEMS

1 IntroductionIn [1]-[31, the Berlinskii's theorem of the distribution of critical pointsfor quadratic differential systems is extended to the general n--th differentialsystems with nZ finite critical points. For cubic systems with 9 critical pointswhich must be all elementary, i.e., saddles or antisaddles, there are possiblysix kinds of distributions of critical points, such as 6 -- 3, 5 -- 4, 5 -- 3 1, 4 4 1, 4 -- 3 2 and 3 -- 3 3. For example, "5 -- 3 1" means that 5 outmostcriti…     

Level set methods for finding critical points of mountain pass type

Computing mountain passes is a standard way of finding critical points. We describe a numerical method for finding critical points that is convergent in the nonsmooth case and locally superlinearly convergent in the smooth finite dimensional case. We apply these techniques to describe a strategy for addressing the Wilkinson problem of calculating the distance from a matrix to a closest matrix with repeated eigenvalues. Finally, we relate critical points of mountain pass type to nonsmooth and metric critical point theory.     

Critical points of the integral map of the charged three-body problem

This is the first in a series of three papers where we study the integral manifolds of the charged three-body problem. The integral manifolds are the fibers of the map of integrals. Their topological type may change at critical values of the map of integrals. Due to the non-compactness of the integral manifolds one has to take into account besides ‘ordinary’ critical points also critical points at infinity. In the present paper we concentrate on ‘ordinary’ critical points and in particular elucidate their connection to central configurations. In a second paper we will study critical points at infinity. The implications for the Hill regions, i.e. the projections of the integral manifolds to configuration space, are the subject of a third paper.     

A Note on Critical p-adic L-functions

We study the adjunction property of the Jacquet–Emerton functor in certain neighborhoods of critical points in the eigencurve. As an application, we construct two-variable p-adic L-functions around critical points via Emerton's representation theoretic approach.     

On the result equivalence of constraints of an asymptotic nature

The topological classification is discussed for real polynomials of degree 4 in two real independent variables whose critical points and critical values are all different. It is proved that among the 17 746 topological types of smooth functions with the same number of critical points, at most 426 types are realizable by polynomials of degree 4.     

Stability of Critical Values and Isolated Critical Continua

We extend the study of critical points in [4]. We show that isolated components of critical points lying on a levelset can be described by an integer which is a lower bound to the “number” of critical points of any function near to the original one in C¹-sup-norm. We also derive a global theorem about continua of critical values similar to that given by Rabinowitz for continua of solutions of certain nonlinear eigenvalue problems. We give a simple application of our abstract results to the problem of bifurcation for gradient systems when the linearization is not completely continuous.     

Periodic solutions of asymptotically linear Hamiltonian systems

We prove existence and multiplicity results for periodic solutions of time dependent and time independent Hamiltonian equations, which are assumed to be asymptotically linear. The periodic solutions are found as critical points of a variational problem in a real Hilbert space. By means of a saddle point reduction this problem is reduced to the problem of finding critical points of a function defined on a finite dimensional subspace. The critical points are then found using generalized Morse theory and minimax arguments.     

Critical points and values of complex polynomials

This paper is primarily concerned with complex polynomials which have critical points which are also fixed points. We show that certain perturbations of a critical fixed point satisfy an inequality. This inequality permits us to prove a local version of Smale's mean value conjecture. We also use Thurston's topological characterization of critically finite rational mappings to enumerate explicitly as branched mappings the set of complex polynomials which have all their critical points fixed.     

Data loci in algebraic optimization

We consider parametric optimization problems from an algebraic viewpoint. The idea is to find all of the critical points of an objective function thereby determining a global optimum. For generic parameters (data) in the objective function the number of critical points remains constant. This number is known as the algebraic degree of an optimization problem. In this article, we go further by considering the inverse problem of finding parameters of the objective function so it gives rise to critical points exhibiting a special structure. For example if the critical point is in the singular locus, has some symmetry, or satisfies some other algebraic property. Our main result is a theorem describing such parameters.     

一个分歧定理

本文用 Morse 理论给出 Krasnoselski Rabinowitz 关于位算子分歧点定理的一个新的证明.设 H 是一个 Hilbert 空间,Ω 是 H 中零元θ的一个邻域.又设 L 是 H 上的一个有界自伴算子,而 G∈C(Ω,H)满足 G(u)=o(‖u‖),当 u→θ.倘若 G 还是一个位算子,即存在 g∈C~1(Ω,R~1),使得 dg=G.求解带参数 λ∈R~1的方程     

Multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Sturm-Liouville boundary conditions

This paper consider the multiple solutions for even Hamiltonian systems satisfying Sturm-Liouville boundary conditions. The gradient of Hamiltonian function is generalized asymptotically linear. The solutions obtained are shown to coincide with the critical points of a dual functional. Thanks to the index theory for linear Hamiltonian systems by Dong (2010) [1], we find critical points of this dual functional by verifying the assumptions of a lemma about multiple critical points given by Chang (1993) [2].     

Regularity for critical points of a non local energy

We study the regularity of critical points of an energy which stems from micromagnetism theory. First we show that in dimension two critical points are smooth in B ² . In the three dimensional case we prove that the stationary critical points of the energy are smooth except in a subset of one dimensional Hausdorff measure zero. The particularity of this work is the non local character of one term of the energy. Received July 10, 1995 / Accepted April 15, 1996     

Sign-changing critical points via Sandwich Pair theorems

The Sandwich Pair theorems have presented very efficient ways to determine the existence of critical points or critical sequences for nonlinear differentiable functionals. In this paper, under rather weak hypotheses new relationships are established between sign-changing critical points and Sandwich Pairs or Linking Sandwich Pairs. The abstract results are demonstrated by applications on semi-linear elliptic equations.     

Multiple solutions for indefinite functionals and applications to asymptotically linear problems

In this paper, by means of Morse theory of isolated critical points (orbits) we study further the critical points theory of asymptotically quadratic functionals and give some results concerning the existence of multiple critical points (orbits) which generalize a series of previous results due to Amann, Conley, Zehnder and K.C. Chang. As applications, the existence of multiple periodic solutions for asymptotically linear Hamiltonian systems is investigated. And our results generalize some recent ones due to Coti-Zelati, J.Q.Liu, S.Li, etc.This research was supported in part by the National Postdoctoral Science Fund.     

The phase transition and classification of critical points in the multistability chemical reactions

IntroductionThe multistability in chemical reaction has been paid much attention to at all timesll--3].As chemical multistability acts at the non-linear area, and its critical phenomenon is part ofnon-equilibrium statistical mechanics of non-linearity, we usually take the critical phenomenaand phase transition in the equilibrium thermodynamic for -reference when we discuss the critical phenomena and phase transition in non-linear systemsl'--6]. As we all know, Ehrenfest (see,for example, [7]) …