Non-degeneracy Condition Research Articles

Perturbation analysis of nonlinear semidefinite programming under Jacobian uniqueness conditions

This paper extends a stability results of nonlinear programming problems by Robinson (Math Program 7:1–16, 1974) to nonlinear semidefinite programming problems. We apply an implicit function theorem to a reformulated Karush–Kuhn–Tucker system, while the Robinson’s work is directly related to the conventional Karush–Kuhn–Tucker system. We show that when the Karush–Kuhn–Tucker condition, constraint nondegeneracy condition, strict complementary condition and second order sufficient condition are satisfied at a feasible point of the original problem, the perturbed problem also satisfies these conditions at some stationary point.

On the effective reducibility of a class of Quasi-periodic nonlinear systems near the equilibrium

ABSTRACTIn this paper, we consider the effective reducibility of the following quasi-periodic nonlinear system where A is a constant matrix with the different and nonzero eigenvalues, , and , and are analytic quasi-periodic on with respect to t. Under non-resonance conditions, without any non-degeneracy condition, by a quasi-periodic transformation, the system can be reducible to a quasi-periodic system where and are exponentially small in ϵ.

Reducibility of a class of 2k-dimensional Hamiltonian systems with quasi-periodic coefficients

ABSTRACTIn this paper, we consider the following real analytic Hamiltonian system where A is a constant Hamiltonian matrix with the different eigenvalues , where for are real, and is quasi-periodic with frequencies . Without any non-degeneracy condition with respect to ϵ, we prove that by a quasi-periodic symplectic mapping, then for most of the sufficiently small parameter ϵ, the Hamiltonian system is reducible.

Optimality conditions in convex optimization with locally Lipschitz constraints

In this paper, we consider a convex optimization problem with locally Lipschitz inequality constraints. The KKT optimality conditions (both necessary and sufficient) for quasi $$\epsilon $$ -solutions are established under Slater’s constraint qualification and a non-degeneracy condition. Moreover, we explore the optimality condition for weakly efficient solutions in multiobjective convex optimization involving locally Lipschitz constraints. Some examples are given to illustrate our results.

Quadratic convergence to the optimal solution of second-order conic optimization without strict complementarity

ABSTRACTUnder primal and dual nondegeneracy conditions, we establish the quadratic convergence of Newton's method to the unique optimal solution of second-order conic optimization. Only very few approaches have been proposed to remedy the failure of strict complementarity, mostly based on nonsmooth analysis of the optimality conditions. Our local convergence result depends on the optimal partition of the problem, which can be identified from a bounded sequence of interior solutions. We provide a theoretical complexity bound for identifying the quadratic convergence region of Newton's method from the trajectory of central solutions. By way of experimentation, we illustrate quadratic convergence of Newton's method on some SOCO problems which fail strict complementarity condition.

“Active-set complexity” of proximal gradient: How long does it take to find the sparsity pattern?

Proximal gradient methods have been found to be highly effective for solving minimization problems with non-negative constraints or L1-regularization. Under suitable nondegeneracy conditions, it is known that these algorithms identify the optimal sparsity pattern for these types of problems in a finite number of iterations. However, it is not known how many iterations this may take. We introduce the notion of the "active-set complexity", which in these cases is the number of iterations before an algorithm is guaranteed to have identified the final sparsity pattern. We further give a bound on the active-set complexity of proximal gradient methods in the common case of minimizing the sum of a strongly-convex smooth function and a separable convex non-smooth function.

Local-in-time well-posedness for compressible MHD boundary layer

In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane with no-slip condition on the velocity field, perfectly conducting wall condition on the magnetic field and Dirichlet boundary condition on the temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer which is described by a Prandtl-type system. Under the non-degeneracy condition on the tangential magnetic field instead of monotonicity of velocity, by applying a coordinate transformation in terms of the stream function of magnetic field as motivated by the recent work [27], we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.

Equilibrium statistical mechanics for self-gravitating systems of two species

In this work, we attempt to generalize the statistical mechanics of self-gravitating systems to systems consisting of two species gravitating particles. Under the nondegenerate condition, with the second-order approximation, and with virialization relationships as additional constraints, we obtain the entropy expression for the two species system. Then, by extremizing the constrained entropy with variational calculus, we obtain a series of equations to describe the equilibrium states of the system. Under the assumption that there is no difference of velocity distributions between the two species, from the series of equilibrium-state equations, we obtain the relative ratio of the spatial distributions for the two species. The two species may be mixed homogeneously in space, yet with different choices of the relevant parameters [Formula: see text] and [Formula: see text], two Lagrangian multipliers in the statistical equilibrium equations, their spatial distributions may also differ from each other. The deviation of the spatial distributions between the two species is usually explained as mass segregation, which is confirmed by recent observations in some Galactic stellar clusters. We emphasis, however, that the mass segregation here is not caused by kinetic-energy exchange between the two species, but a new feature of our statistical approach, which is different from Lynden-Bell’s statistics. We believe that these results may have general significance to the statistical mechanics for the general long-range interaction systems.

On global non-degeneracy conditions for chaotic behavior for a class of dynamical systems

For about 25 years, global methods from the calculus of variations have been used to establish the existence of chaotic behavior for some classes of dynamical systems. Like the analytical approaches that were used earlier, these methods require nondegeneracy conditions, but of a weaker nature than their predecessors. Our goal here is study such a nondegeneracy condition that has proved useful in several contexts including some involving partial differential equations, and to show this condition has an equivalent formulation involving stable and unstable manifolds.

An A Posteriori KAM Theorem for Whiskered Tori in Hamiltonian Partial Differential Equations with Applications to some Ill-Posed Equations

The goal of this paper is to develop a KAM theory for tori with hyperbolic directions, which applies to Hamiltonian partial differential equations, even to some ill-posed ones. The main result has an a-posteriori format, that is, we show that if there is an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, then there is a true solution nearby. This allows, besides dealing with the quasi-integrable case, for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi-periodic solutions in degenerate situations. The a-posteriori format also has other automatic consequences (smooth dependence on parameters, bootstrap of regularity, etc.). We emphasize that the non-degeneracy conditions required are just quantities evaluated on the approximate solution (no global assumptions on the system such as twist). Hence, they are readily verifiable in perturbation expansions. We will pay attention to the quantitative relations between the sizes of the approximation and the non-degeneracy conditions. This will allow us to prove what experts call small twist theorems (the non-degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation). The method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant. We also solve functional equations for bundles which imply that are invariant under the linearization. The iterative method does not use transformation theory nor action-angle variables. The main result does not assume that the system is close to integrable. More surprisingly, we do not need that the equations we study define an evolution for all initial conditions and are well posed. Even if the systems we study do not admit solutions for all initial conditions, we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi-periodic. We first develop an abstract theorem. Then, we show how this abstract result applies to some concrete examples. The examples considered in this paper are the scalar Boussinesq equation and the Boussinesq system (both are PDE models that aim to describe water waves in the long wave limit). For these equations we construct small amplitude time quasi-periodic solutions which are even in the spatial variable. The strategy for the abstract theorem is inspired by that in Fontich et al. (Electron Res Announc Math Sci 16:9–22, 2009; J Differ Equ 246(8):3136–3213, 2009). The main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill-posed equations and which is stable under addition of unbounded perturbations. This requires that we assume smoothing properties. We also present very detailed bounds on the change of the splittings under perturbations.

Concentration on Curves for a Neumann Ambrosetti–Prodi-Type Problem in Two-Dimensional Domains

Given a smooth bounded domain $$\Omega \subset \mathbb {R}^2 $$ , we consider the problem $$\begin{aligned} \left\{ \begin{array} {cccccc} - \Delta u = |u|^p - s\,\psi &{}\hbox {in } \ \Omega \\ \\ \dfrac{\partial u}{\partial \nu } = 0 &{}\hbox {on}\ \partial \Omega \end{array}\right. \end{aligned}$$ where $$p > 1$$ , $$s > 0$$ is a large parameter, $$\psi $$ is a positive function and $$\nu $$ denotes the outward normal of $$\partial \Omega $$ . Let $$\Gamma $$ be a curve intersecting orthogonally with $$\partial \Omega $$ at exactly two points and dividing $$\Omega $$ into two parts. Assuming moreover that $$\Gamma $$ satisfies a stationary and non-degeneracy conditions with respect to the functional $$\int _{\Gamma } \psi ^\sigma ,$$ where $$\sigma = \frac{p+3}{2p}$$ , we prove the existence of a solution $$u_s$$ concentrating along the whole of $$\Gamma $$ , exponentially small in s at any positive distance from it, provided that s is large and away from certain critical numbers.

Reducible problem for a class of almost-periodic non-linear Hamiltonian systems

This paper studies the reducibility of almost-periodic Hamiltonian systems with small perturbation near the equilibrium which is described by the following Hamiltonian system: \t\t\tdxdt=J[A+εQ(t,ε)]x+εg(t,ε)+h(x,t,ε).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\frac{dx}{dt} = J \\bigl[{A} +\\varepsilon{Q}(t,\\varepsilon) \\bigr]x+ \\varepsilon g(t,\\varepsilon)+h(x,t,\\varepsilon). $$\\end{document} It is proved that, under some non-resonant conditions, non-degeneracy conditions, the suitable hypothesis of analyticity and for the sufficiently small ε, the system can be reduced to a constant coefficients system with an equilibrium by means of an almost-periodic symplectic transformation.

Multiscale KAM theorem for Hamiltonian systems

In this paper, we study the persistence of invariant tori in nearly integrable multiscale Hamiltonian systems with highorder degeneracy in the integrable part. Such Hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems. We introduce multiscale nondegenerate condition and multiscale Diophantine condition, comparable to the usual Diophantine condition. Using quasilinear KAM method, we prove a multiscale KAM theorem.

Solutions of mountain pass type for double well potential systems

This paper studies a Hamiltonian system possessing a double well potential for which the existence of multitransition heteroclinic and homoclinic solutions that are local minimizers of an associated functional is known. Under an additional mild non-degeneracy condition on the set of all homoclinic and heteroclinic solutions, the existence of further heteroclinic and homoclinic solutions that are of mountain pass type is established. A key tool for the existence arguments is a variant of the Mountain Pass Theorem that is of independent interest.

Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Russmanns non-degeneracy condition whensymplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus, numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shangs previous ones (1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.

Perturbation analysis of a class of conic programming problems under Jacobian uniqueness conditions

We consider the stability of a class of parameterized conic programming problems which are more general than $C^2$-smooth parameterization. We show that when the Karush-Kuhn-Tucker (KKT) condition, the constraint nondegeneracy condition, the strict complementary condition and the second order sufficient condition (named as Jacobian uniqueness conditions here) are satisfied at a feasible point of the original problem, the Jacobian uniqueness conditions of the perturbed problem also hold at some feasible point.

Codimension-3 Flip Bifurcation of a Class of Difference Equations

In this paper, we consider a one-dimensional difference equation with three parameters, the derivative of which at a fixed point has an eigenvalue [Formula: see text] as the parameters are all zero. In the case that both nondegeneracy conditions of the flip bifurcation and the generalized flip bifurcation are not satisfied, by computing normal form, we give the nondegeneracy condition and transversality condition of the codimension-3 flip bifurcation. Moreover, by discussing the number of positive zeros of a cubic function in a neighborhood of the origin, we show the bifurcation scenario and give the parameter conditions, respectively, that the normal form of the equation possesses three 2-cycles, two 2-cycles, only one 2-cycle or none.

Exact augmented Lagrangian functions for nonlinear semidefinite programming

In this paper, we study augmented Lagrangian functions for nonlinear semidefinite programming (NSDP) problems with exactness properties. The term exact is used in the sense that the penalty parameter can be taken appropriately, so a single minimization of the augmented Lagrangian recovers a solution of the original problem. This leads to reformulations of NSDP problems into unconstrained nonlinear programming ones. Here, we first establish a unified framework for constructing these exact functions, generalizing Di Pillo and Lucidi's work from 1996, that was aimed at solving nonlinear programming problems. Then, through our framework, we propose a practical augmented Lagrangian function for NSDP, proving that it is continuously differentiable and exact under the so-called nondegeneracy condition. We also present some preliminary numerical experiments.

Backward SDEs for optimal control of partially observed path-dependent stochastic systems: A control randomization approach

We introduce a suitable backward stochastic differential equation (BSDE) to represent the value of an optimal control problem with partial observation for a controlled stochastic equation driven by Brownian motion. Our model is general enough to include cases with latent factors in mathematical finance. By a standard reformulation based on the reference probability method, it also includes the classical model where the observation process is affected by a Brownian motion (even in presence of a correlated noise), a case where a BSDE representation of the value was not available so far. This approach based on BSDEs allows for greater generality beyond the Markovian case, in particular our model may include path-dependence in the coefficients (both with respect to the state and the control), and does not require any nondegeneracy condition on the controlled equation. We use a randomization method, previously adopted only for cases of full observation, and consisting in a first step, in replacing the control by an exogenous process independent of the driving noise and in formulating an auxiliary (“randomized”) control problem where optimization is performed over changes of equivalent probability measures affecting the characteristics of the exogenous process. Our first main result is to prove the equivalence between the original partially observed control problem and the randomized problem. In a second step, we prove that the latter can be associated by duality to a BSDE, which then characterizes the value of the original problem as well.

Melnikov processes and chaos in randomly perturbed dynamical systems

We consider a wide class of randomly perturbed systems subjected to stationary Gaussian processes and show that chaotic orbits exist almost surely under some nondegenerate condition, no matter how small the random forcing terms are. This result is very contrasting to the deterministic forcing case, in which chaotic orbits exist only if the influence of the forcing terms overcomes that of the other terms in the perturbations. To obtain the result, we extend Melnikov’s method and prove that the corresponding Melnikov functions, which we call the Melnikov processes, have infinitely many zeros, so that infinitely many transverse homoclinic orbits exist. In addition, a theorem on the existence and smoothness of stable and unstable manifolds is given and the Smale–Birkhoff homoclinic theorem is extended in an appropriate form for randomly perturbed systems. We illustrate our theory for the Duffing oscillator subjected to the Ornstein–Uhlenbeck process parametrically.