Sublevel Sets Research Articles

Formal Abstraction of Linear Systems via Polyhedral Lyapunov Functions

In this paper we present an abstraction algorithm that produces a finite bisimulation quotient for an autonomous discrete-time linear system. We assume that the bisimulation quotient is required to preserve the observations over an arbitrary, finite number of polytopic subsets of the system state space. We generate the bisimulation quotient with the aid of a sequence of contractive polytopic sublevel sets obtained via a polyhedral Lyapunov function. The proposed algorithm guarantees that at iteration i, the bisimulation of the system within the i-th sublevel set of the Lyapunov function is completed. We then show how to use the obtained bisimulation quotient to verify the system with respect to arbitrary Linear Temporal Logic formulas over the observed regions.

Functions Which Are Quasiconvex under Linear Perturbations

A quasiconvex function is a function which has convex sublevel sets. This paper studies robustly quasiconvex functions, that is, quasiconvex functions which remain quasiconvex under small linear perturbations. Relations to pseudoconvexity and other generalized convexity concepts and necessary and sufficient first-order conditions for robust quasiconvexity of smooth functions are presented. Convex-analytic properties and convexification of robustly quasiconvex functions are studied. Supporting robustly quasiconvex functions by simpler functions is discussed, with duality theory as motivation. Through the use of a second-order condition for robust quasiconvexity of nonsmooth functions, nontrivial examples of such functions are given.

Existence Conditions for an Optimal Affine Leader Function in the Reverse Stackelberg Game

We investigate the solvability of the reverse Stackelberg game. Here, a leader player acts first by presenting a leader function that maps the follower decision space into the leader decision space. Subsequently, the follower acts by determining his optimal decision variable. Such a game setting can be adopted within a multi-level optimization approach for large-scale control problems like road tolling. However, due to the complexity of the general game, results often rely on specific examples. As a starting point towards developing a systematic approach for the use of reverse Stackelberg games in control, a characterization of cases is given in which the desired leader equilibrium can be achieved by an affine leader function. Here, we focus on the single-leader single-follower deterministic, static (one-shot) case. This characterization follows a geometric approach and extends the special cases considered in the existing literature to also incorporate the more general case in which nonconvex and nonsmooth sublevel sets apply.

Edge-Preserving Image Regularization Based on Morphological Wavelets and Dyadic Trees

Despite the tremendous success of wavelet-based image regularization, we still lack a comprehensive understanding of the exact factor that controls edge preservation and a principled method to determine the wavelet decomposition structure for dimensions greater than 1. We address these issues from a machine learning perspective by using tree classifiers to underpin a new image regularizer that measures the complexity of an image based on the complexity of the dyadic-tree representations of its sublevel sets. By penalizing unbalanced dyadic trees less, the regularizer preserves sharp edges. The main contribution of this paper is the connection of concepts from structured dyadic-tree complexity measures, wavelet shrinkage, morphological wavelets, and smoothness regularization in Besov space into a single coherent image regularization framework. Using the new regularizer, we also provide a theoretical basis for the data-driven selection of an optimal dyadic wavelet decomposition structure. As a specific application example, we give a practical regularized image denoising algorithm that uses this regularizer and the optimal dyadic wavelet decomposition structure.

Density Estimates for a Nonlocal Variational Model via the Sobolev Inequality

We consider the minimizers of the energy $\|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)\,dx,$ with $s \in (0,1/2)$, where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of u, and W is a double-well potential. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer u in a large ball $B_R$ occupy a volume comparable with the volume of $B_R$.

A coercive James’s weak compactness theorem and nonlinear variational problems

We discuss a perturbed version of James’s sup theorem for weak compactness that not only properly generalizes that classical statement, but also some recent extensions of this central result: the sublevel sets of an extended real valued and coercive function whose subdifferential is surjective are relatively weakly compact. Furthermore, we apply it to generalize and unify some facts in mathematical finance and to prove that the unique possible framework in the development of an existence theory for a wide class of nonlinear variational problems is the reflexive one.

Uniform Sublevel Radon-like Inequalities

This paper is concerned with establishing uniform weighted Lp–Lq estimates for a class of operators generalizing both Radon-like operators and sublevel set operators. Such estimates are shown to hold under general circumstances whenever a sublevel-type inequality is satisfied by certain associated measures (the inequality is of the sort studied by Oberlin (Math. Proc. Camb. Philos. Soc. 129(3):517–526, 2000), relating measures of parallelepipeds to powers of their Euclidean volumes). These ideas lead to previously unknown, weighted affine-invariant estimates for Radon-like operators as well as new Lp-improving estimates for degenerate Radon-like operators with folding canonical relations which satisfy an additional curvature condition of Greenleaf and Seeger (J. Reine Angew. Math. 455:35–56, 1994) for FIOs (building on the ideas of Sogge (Invent. Math. 104(2):349–376, 1991) and Mockenhaupt et al. (J. Am. Math. Soc. 6(1):65–130, 1993)); these new estimates fall outside the range of estimates which are known to hold in the generality of the FIO context.

A conic scalarization method in multi-objective optimization

This paper presents the conic scalarization method for scalarization of nonlinear multi-objective optimization problems. We introduce a special class of monotonically increasing sublinear scalarizing functions and show that the zero sublevel set of every function from this class is a convex closed and pointed cone which contains the negative ordering cone. We introduce the notion of a separable cone and show that two closed cones (one of them is separable) having only the vertex in common can be separated by a zero sublevel set of some function from this class. It is shown that the scalar optimization problem constructed by using these functions, enables to characterize the complete set of efficient and properly efficient solutions of multi-objective problems without convexity and boundedness conditions. By choosing a suitable scalarizing parameter set consisting of a weighting vector, an augmentation parameter, and a reference point, decision maker may guarantee a most preferred efficient or properly efficient solution.

Non-commutative morphology: Shapes, filters, and convolutions

Group morphology is a generalization of mathematical morphology which makes an explicit distinction between shapes and filters. Shapes are modeled as point sets, for example binary images or 3D solid objects, while filters are collections of transformations (such as translations, rotations or scalings). The action of a filter on a shape generalizes the basic morphological operations of dilation and erosion. This shift in perspective allows us to compose filters independent of shapes, and leads to a non-commutative generalization of the Minkowski sum and difference which we call the Minkowski product and quotient respectively. We show that these operators are useful for unifying, formulating and solving a number of important problems, including translational and rotational configuration space problems, mechanism workspace computation, and symmetry detection. To compute these new operators, we propose the use of group convolution algebras, which extend classical convolution and the Fourier transform to non-commutative groups. In particular, we show that all Minkowski product and quotient operations may be represented implicitly as sublevel sets of the same real-valued convolution function.

Uniform geometric estimates of sublevel sets

This paper reconsiders the uniform sublevel set estimates of Carbery, Christ, and Wright [7], Phong, Stein, and Sturm [23], and Carbery and Wright [8] from a geometric perspective. This perspective leads one to consider a natural collection of homogeneous, nonlinear differential operators, which generalize mixed derivatives in ℝ d . As a consequence, it is shown that, in comparison to these previous works, improved uniform estimates are possible in all but certain explicitly “flat” situations.

Verification of continuous dynamical systems by timed automata

This paper presents a method for abstracting continuous dynamical systems by timed automata. The abstraction is based on partitioning the state space of a dynamical system using positive invariant sets, which form cells that represent locations of a timed automaton. The abstraction is intended to enable formal verification of temporal properties of dynamical systems without simulating any system trajectory, which is currently not possible. Therefore, conditions for obtaining sound, complete, and refinable abstractions are set up. The novelty of the method is the partitioning of the state space, which is generated utilizing sub-level sets of Lyapunov functions, as they are positive invariant sets. It is shown that this partition generates sound and complete abstractions. Furthermore, the complete abstractions can be composed of multiple timed automata, allowing parallelization of the verification process. The proposed abstraction is applied to two examples, which illustrate how sound and complete abstractions are generated and the type of specification we can check. Finally, an example shows how the compositionality of the abstraction can be used to analyze a high-dimensional system.

Strongly Consistent Estimation of the Sample Distribution of Noisy Continuous-Parameter Fields

The general problem of defining and determining the sample distribution in the case of continuous-parameter random fields is addressed. Defining a distribution in the case of deterministic functions is straightforward, based on measures of sublevel sets. However, the fields we consider are the sum of a deterministic component (nonrandom multidimensional function) and an i.i.d. random field; an attempt to extend the same notion to the stochastic case immediately raises some fundamental difficulties. We show that by “uniformly sampling” such random fields the difficulties may be avoided and a sample distribution may be compatibly defined and determined. Not surprisingly, the obtained result resembles the known fact that the probability distribution of the sum of two independent random variables is the convolution of their distributions. Finally, we apply the results to derive a solution to the problem of deformation estimation of one- and multidimensional signals in the presence of measurement noise.

A saddle-point theorem for strongly and weakly convex functions

We prove a theorem on the existence, uniqueness, and continuous dependence on parameters for a saddle point in a type of minimax problem that arises, for example, in differential game theory. Our theorem on the existence of a saddle point does not follow from the well-known theorems of von Neumann, Ky Fan, Sion and others since the intersection of sublevel sets of the function considered may be disconnected and non-empty. The hypotheses of our theorem are stated in terms of the strong and weak convexity of functions defined on a Banach space. We study properties of strongly and weakly convex functions related to the operations of minimization and maximization. We obtain unimprovable estimates of convexity parameters for the infimal convolution (episum) and epidifference of functions. This results in the construction of a calculus of convexity parameters of functions with respect to epioperations. We give typical examples and show that the hypotheses of our theorems are essential.

Optimality conditions under relaxed quasiconvexity assumptions using star and adjusted subdifferentials

A set-constrained optimization problem and a mathematical programming problem are considered. We assume that the sublevel sets of the involving functions are convex only at the point under question and hence these functions are not assumed quasiconvex. Using the two star subdifferentials and the adjusted subdifferential, we establish optimality conditions for usual minima and strict minima. Our results contain and improve some recent ones in the literature. Examples are provided to explain the advantages of each of our results.

Algorithmic Approach to Abstracting Linear Systems by Timed Automata

AbstractThis paper proposes an LMI-based algorithm for abstracting dynamical systems by timed automata, which enables automatic formal verification of linear systems.The proposed abstraction is based on partitioning the state space of the system using positive invariant sets, generated by Lyapunov functions. This partitioning ensures that the vector field of the dynamical system is transversal to all facets of the cells, which induces some desirable properties of the abstraction.The algorithm is based on identifying intersections of level sets of quadratic Lyapunov functions, and determining the minimum and maximum time that a trajectory of the system can stay in a set, defined as the set-difference of sub-level sets of Lyapunov functions. The proposed algorithm applies for linear systems and can therefore be efficiently implemented using LMI-based tools.

Minimizing Irregular Convex Functions: Ulam Stability for Approximate Minima

The main concern of this article is to study Ulam stability of the set of e-approximate minima of a proper lower semicontinuous convex function bounded below on a real normed space X, when the objective function is subjected to small perturbations (in the sense of Attouch & Wets). More precisely, we characterize the class all proper lower semicontinuous convex functions bounded below such that the set-valued application which assigns to each function the set of its e-approximate minima is Hausdorff upper semi-continuous for the Attouch–Wets topology when the set \(\mathcal{C}(X)\) of all the closed and nonempty convex subsets of X is equipped with the Hausdorff topology. We prove that a proper lower semicontinuous convex function bounded below has Ulam-stable e-approximate minima if and only if the boundary of any of its sublevel sets is bounded.

Weak regularity of Gauss mass transport

Given two probability measures μ and ν we consider a mass transportation mapping T satisfying 1) T sends μ to ν, 2) T has the form T=φ∇φ|∇φ|, where φ is a function with convex sublevel sets. We prove a change of variables formula for T. We also establish Sobolev estimates for φ, and a new form of the parabolic maximum principle. In addition, we discuss relations to the Monge–Kantorovich problem, curvature flows theory, and parabolic non-linear PDE's.

Mean curvature flow via convex functions on Grassmannian manifolds

Using the convex functions on Grassmannian manifolds, the authors obtain the interior estimates for the mean curvature flow of higher codimension. Confinable properties of Gauss images under the mean curvature flow have been obtained, which reveal that if the Gauss image of the initial submanifold is contained in a certain sublevel set of the υ-function, then all the Gauss images of the submanifolds under the mean curvature flow are also contained in the same sublevel set of the υ-function. Under such restrictions, curvature estimates in terms of υ-function composed with the Gauss map can be carried out.

Single-Directional Property of Multivalued Maps and Variational Systems

Dontchev and Hager [Math. Oper. Res., 19 (1994), pp. 753-768] have shown that a monotone set-valued map defined from a Banach space to its dual which satisfies the Aubin property around a point $(x,y)$ of its graph is actually single-valued in a neighborhood of $x$. We prove a result which is the counterpart of the above for quasi-monotone set-valued maps, based on the concept of single-directional property. As applications, we provide sufficient conditions for this single-valued property to hold for the solution map of general variational systems and quasi-variational inequalities. We also investigate the single-directionality property for the normal operator to the sublevel sets of a quasi-convex function.

R-composition of quadratic Lyapunov functions for stabilizability of linear differential inclusions

A novel state feedback control technique to stabilise linear differential inclusions through composite quadratic Lyapunov functions is presented. By using a gradient-based control technique, the minimum effort control is composed through intersection and union operations, derived from the theory of R-functions. While conventional min and max compositions are recovered as a special case, it is shown that smoother sublevel sets and everywhere differentiability are obtained tuning the composition parameter. Examples of both intersection and union compositions are provided to show that intermediate control performances in terms of convergence time are obtained, while improved performances in the control signal can be achieved.