Sublevel Sets Research Articles

Stability for convex vector optimization problems

This article deals with the convergence (in the sense of Kuratowski–Painlevé) of the set of the minimal points of An to the set of minimal points of A, whenever {An } is a sequence of closed convex subsets of an Euclidean space, converging in the same sense to the set A. Next, we consider the convex vector optimization problem under the assumption that the objective function f is such that all its sublevel sets, restricted to the feasible region, are bounded. For this problem we investigate the convergence of the solution sets of perturbed (with respect to the feasible region and the objective function) problems both in the image space and in the decision space. We consider also the same topics for a linear problem. Finally, we apply our results to the study of stability for a vector programming problem with convex inequality and linear equality constraints.

The Size of Plurisubharmonic Lemniscates in Terms of Hausdorff–Riesz Measures and Capacities

The main goal of this paper is to establish new uniform estimates on the size of sublevel sets of plurisubharmonic functions (called plurisubharmonic lemniscates) in terms of Hausdorff–Riesz measures and capacities of certain orders.We first prove a new uniform version of Skoda's integrability theorem for a given class of plurisubharmonic functions in terms of Borel measures of Hausdorff–Riesz type of certain orders with a precise estimate of the integrability exponent in terms of Lelong numbers of the class and the order of the measures.Then we present several applications of this result. We first deduce uniform estimates on the size of plurisubharmonic lemniscates associated to functions from some important classes of plurisubharmonic functions in terms of Hausdorff–Riesz measures.We also derive a new comparison inequality between certain Hausdorff–Riesz capacities and the pluricomplex logarithmic capacity for borelean sets of a fixed bounded domain in Cn or more generally in an affine algebraic manifold.Furthermore, using results from classical potential theory, we finally deduce from this comparison inequality new estimates of the size of polynomial lemniscates in terms of Hausdorff contents in the spirit of the famous lemma of H. Cartan. 2000 Mathematics Subject Classification 31C10, 31C15, 32F05, 32F99, 32U05, 32U99.

A survey on error bounds for lower semicontinuous functions

We survey ancient and recent results on global error bounds for the distance to a sublevel set of a lower semicontinuous function defined on a complete metric space. We emphasize the case of a recent characterization of this property which appeared in [CITE]. We also review the convex case and show how the known result on sufficient condition for a global error bound can be derived from the quoted characterization.

Characterization of Quasiconvexity for Locally Lipschitz Vector-Valued Functions

The aim of the article is to characterize the locally Lipschitz vector-valued functions which are K -quasiconvex with respect to a closed convex cone K in the sense that the sublevel sets are convex. Our criteria are written in terms of a K -quasimonotonicity notion of the generalized directional derivative and of Clarke's generalized Jacobian. This work could be compared to Sach's one in which the author gives necessary and sufficient conditions for a locally Lipschitz map f between two Euclidean spaces to be scalarly K -quasiconvex in the sense that, for any continuous linear form of the nonnegative polar cone K + , the composite function f is quasiconvex.

Nontangency-Based Lyapunov Tests for Convergence and Stability in Systems Having a Continuum of Equilibria

This paper focuses on the stability analysis of systems having a continuum of equilibria. Two notions that are of particular relevance to such systems are convergence and semistability. Convergence is the property whereby every solution converges to a limit point that may depend on the initial condition. Semistability is the additional requirement that all solutions converge to limit points that are Lyapunov stable. We give new Lyapunov-function-based results for convergence and semistability of nonlinear systems. These results do not make assumptions of sign definiteness on the Lyapunov function. Instead, our results use a novel condition based on nontangency between the vector field and invariant or negatively invariant subsets of the level or sublevel sets of the Lyapunov function or its derivative and represent extensions of previously known stability results involving semidefinite Lyapunov functions. To illustrate our results we deduce convergence and semistability of the kinetics of the Michaelis--Menten chemical reaction and the closed-loop dynamics of a scalar system under a universal adaptive stabilizing feedback controller.

Characterizing attraction probabilities via the stochastic Zubov equation

A stochastic differential equation with an a.s. locally stable compact set is considered. The attraction probabilities to the set are characterized by the sublevel sets of the limit of a sequence of solutions to $2^{nd}$ order partial differential equations. Two numerical examples illustrating the method are presented.

Volume of Polynomial Lemniscates in C n

Kiselman's method permitting one to estimate the volume of sublevel sets of a plurisubharmonic function f in Cn with the aid of the Lelong number of f is applied to polynomial lemniscates in Cn. In particular, it yields estimates from below of the volume of polynomial lemniscates which completes recent results of Cuyt, Driver and Lubinsky.

Overapproximating Reachable Sets by Hamilton-Jacobi Projections

In earlier work, we showed that the set of states which can reach a target set of a continuous dynamic game is the zero sublevel set of the viscosity solution of a time dependent Hamilton-Jacobi-Is...

Nonlinear tracking control in the presence of state and control constraints: a generalized reference governor

This paper proposes a new approach to reference governor design. As in prior literature, the governor accepts input commands and modifies their evolution so that specified pointwise-in-time constraints on state and control variables are satisfied. The new approach applies to general discrete-time and continuous-time nonlinear systems with uncertainties. It relies on safety properties provided by sublevel sets of equilibria-parameterized functions. These functions need not be Lyapunov functions, and the corresponding sublevel sets need not be positively invariant. Technical conditions that capture the bare essentials of what is needed are identified and the usual desirable properties of reference governors are established. The new approach significantly broadens the class of methods available for constructing the nonlinear function that is required in the implementation of the reference governors. This advantage is illustrated in a nonlinear control problem where off-line, computer-based simulation is the basis for constructing the nonlinear function.

An existence result for a class of quasilinear elliptic boundary value problems with jumping nonlinearities

We establish an existence result for a class of quasilinear elliptic boundary value problems with jumping nonlinearities using variational arguments. First we calculate certain homotopy groups of sublevel sets of the asymptotic part of the variational functional. Then we use these groups to show that the full functional admits a linking geometry and hence a min-max critical point.

A STOCHASTIC REPRESENTATION FOR THE LEVEL SET EQUATIONS

ABSTRACT A Feynman-Kac representation is proved for geometric partial differential equations. This representation is in terms of a stochastic target problem. In this problem the controller tries to steer a controlled process into a given target by judicial choices of controls. The sublevel sets of the unique level set solution of the geometric equation is shown to coincide with the reachability sets of the target problem whose target is the sublevel set of the final data.

Regularity results for some 1-homogeneous functionals

We consider local minimizers for a class of 1-homogeneous integral functionals defined on BV loc (Ω) , with Ω⊂ R 2 . Under general assumptions on the functional, we prove that the boundary of the subgraph of such minimizers is (locally) a lipschitz graph in a suitable direction. The proof of this statement relies on a regularity result holding for boundaries in R 2 which minimize an anisotropic perimeter. This result is applied to the boundary of sublevel sets of a minimizer u∈BV loc (Ω) . We also provide an example which shows that such regularity result is optimal.

On a Crystalline Variational Problem, Part I:¶First Variation and Global L∞ Regularity

Let φ:ℝ n → [0,+∞[ be a given positively one-homogeneous convex function, and let ?φ≔{φ≤ 1 }. Pursuing our interest in motion by crystalline mean curvature in three dimensions, we introduce and study the class ?φ (ℝ n ) of “smooth” boundaries in the relative geometry induced by the ambient Banach space (ℝ n , φ). It can be seen that, even when ?φ is a polytope, ?φ(ℝ n ) cannot be reduced to the class of polyhedral boundaries (locally resembling ∂?φ). Curved portions must be necessarily included and this fact (as well as the nonsmoothness of ∂?φ) is the source of several technical difficulties related to the geometry of Lipschitz manifolds. Given a boundary δE in the class ?φ(ℝ n ), we rigorously compute the first variation of the corresponding anisotropic perimeter, which leads to a variational problem on vector fields defined on δE. It turns out that the minimizers have a uniquely determined (intrinsic) tangential divergence on δE. We define such a divergence to be the φ-mean curvature κφ of δE; the function κφ is expected to be the initial velocity of δE, whenever δE is considered as the initial datum for the corresponding anisotropic mean curvature flow. We prove that κφ is bounded on δE and that its sublevel sets are characterized through a variational inequality.

On a Crystalline Variational Problem, Part II:¶BV Regularity and Structure of Minimizers on Facets

For a nonsmooth positively one-homogeneous convex function φ:ℝ n → [0,+∞[, it is possible to introduce the class ?φ (ℝ n ) of smooth boundaries with respect to φ, to define their φ-mean curvature κφ, and to prove that, for E∈?φ (ℝ n ), κφ∈L ∞(δE) [9]. Based on these results, we continue the analysis on the structure of δE and on the regularity properties of κφ. We prove that a facet F of δE is Lipschitz (up to negligible sets) and that κφ has bounded variation on F. Further properties of the jump set of κφ are inspected: in particular, in three space dimensions, we relate the sublevel sets of κφ on F to the geometry of the Wulff shape ?φ≔{φ≤ 1 }.

The effect of geometry of the domain boundary in an elliptic Neumann problem

In this paper we consider the problem $$ \begin{cases} -\Delta u + \mu u= u^{2^* -1 -{\varepsilon}} \quad \hbox{in } \Omega \\ u>0 \quad \hbox{in } \Omega \ \quad \frac{\partial u}{\partial \nu} =0 \quad \hbox{on } \partial \Omega \end{cases} $$ where $\Omega$ is a bounded, smooth domain in ${\mathbb R}^N (N\geq 3),\ 2^* = \frac{2N}{N-2}, \ \mu>0, \ {\varepsilon} \in [0, \frac{4}{N-2})$, $\nu$ is the unit outward normal vector to $\partial \Omega$. We show the topological effect of superlevel and/or sublevel sets of the function of mean curvature of $\partial \Omega$ on the number of one- or multipeak positive solutions as ${\varepsilon} \to 0$ for fixed $\mu$ and $\mu \to +\infty$ for ${\varepsilon} =0$. The solutions obtained concentrate at some boundary points with negative mean curvature (for ${\varepsilon}>0 $ small) or positive mean curvature (for ${\varepsilon}=0, \ \mu$ large).

Polynomial hulls and $H^\infty$ control for a hypoconvex constraint

We say that a subset of \({\mathbb C}^n \) is hypoconvex if its complement is the union of complex hyperplanes. Let \(\Delta \) be the closed unit disk in \({\mathbb C}\), \(\Gamma=\partial\Delta\). We prove two conjectures of Helton and Marshall. Let \(\rho \) be a smooth function on \(\Gamma\times{\mathbb C}^n\) whose sublevel sets have compact hypoconvex fibers over \(\Gamma\). Then, with some restrictions on \(\rho \), if Y is the set where \(\rho \) is less than or equal to 1, the polynomial convex hull of Y is the union of graphs of analytic vector valued functions with boundary in Y. Furthermore, we show that the infimum \(\inf_{f\in H^\infty(\Delta)^n}\|\rho(z,f(z))\|_\infty\) is attained by a unique bounded analytic f which in fact is also smooth on \(\Gamma\). We also prove that if \(\rho\) varies smoothly with respect to a parameter, so does the unique f just found.

Normal Characterization of the Main Classes of Quasiconvex Functions

In this article we explore the concept of the normal cone to the sublevel sets (or strict sublevel sets) of a function. By slightly modifying the original definition of Borde and Crouzeix, we obtain here a new (but strongly related to the already existent) notion of a normal operator. This technique turns out to be appropriate in Quasiconvex Analysis since it allows us to reveal characterizations of the various classes of quasiconvex functions in terms of the generalized quasimonotonicity of their `normal' multifunctions.

Ensembles de sous-niveau et images inversesdes fonctions plurisousharmoniques

Sublevel sets and pull-backs of plurisubharmonic functions We determine when the Lelong number of the pull-back of a plurisubharmonic function by a holomorphic mapping can be majorized by a constant times the Lelong number of the original function. The method of proof uses estimates of the volume of the sublevel sets of plurisubharmonic functions.

Nearest pair with more nonconstant invariant factors and pseudospectrum

Let (A,B)∈ C n×n× C n×m . Suppose that the number of nonconstant (i.e., ≠1) invariant factors of the polynomial matrix λ[I n,0]−[A,B] is less than k. For all complex number λ denote by σ n−(k−1)(λ[I n,0]−[A,B]) the greatest (n−(k−1))th singular value of the matrix λ[I n,0]−[A,B] . The minimum absolute value of the real function of complex variable λ↦σ n−(k−1)(λ[I n,0]−[A,B]) gives the distance from (A,B) to the set of pairs with more or equal number of nonconstant invariant factors. When k=1, this specializes in the formula of Eising for the distance from a controllable pair (A,B) to the nearest uncontrollable pair. The complex numbers λ lying in the sublevel set {λ∈ C | σ n (λ[ I n , 0]−[ A, B])⩽ε} of the function λ↦σ n(λ[I n,0]−[A,B]), are the uncontrollable modes of all the pairs that are within an ε tolerance of (A,B). All the results of this paper are an immediate consequence of the Singular Value Decomposition of a matrix and of the interpretation of the singular values as the distances to the nearest matrices of lower ranks.

A method of projection onto an acute cone with level control in convex minimization

We study a projection method for convex minimization problems. In each step we construct a certain polyhedral function that approximates the objective function from below. The affine pieces of the approximating function are linearizations of the objective function obtained in previous steps. The linearization are determined by a linearly independent system of subgradients which generates an obtuse cone. The current approximation of the solution is projected onto a sublevel set of the polyhedral function with level which estimates the optimal value and is updated in each iteration. The projection can be easily obtained by solving a system of linear equations and is, actually, the projection onto a translated acute cone dual to the constructed obtuse cone. The construction ensures long steps between succesive approximations of the solution. Such long steps are important for the good behavior of the method.