Compactness Assumption Research Articles

An abstract Logvinenko-Sereda type theorem for spectral subspaces

We provide an abstract framework for a Logvinenko-Sereda type theorem, where the classical compactness assumption on the support of the Fourier transform is replaced by the assumption that the functions under consideration belong to a spectral subspace associated with a finite energy interval for some lower semibounded self-adjoint operator on a Euclidean L2-space. Our result then provides a bound for the L2-norm of such functions in terms of their L2-norm on a thick subset with a constant explicit in the geometric and spectral parameters. This recovers previous results for functions on the whole space, hyperrectangles, and infinite strips with compact Fourier support and for finite linear combinations of Hermite functions and allows to extend them to other domains. The proof follows the approach by Kovrijkine and is based on Bernstein-type inequalities for the respective functions, complemented with a suitable covering of the underlying domain.

Nonlinear Young Differential Equations: A Review

Nonlinear Young integrals have been first introduced in Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016) and provide a natural generalisation of classical Young ones, but also a versatile tool in the pathwise study of regularisation by noise phenomena. We present here a self-contained account of the theory, focusing on wellposedness results for abstract nonlinear Young differential equations, together with some new extensions; convergence of numerical schemes and nonlinear Young PDEs are also treated. Most results are presented for general (possibly infinite dimensional) Banach spaces and without using compactness assumptions, unless explicitly stated.

Integro-differential optimality equations for the risk-sensitive control of piecewise deterministic Markov processes

In this paper we study the minimization problem of the infinite-horizon expected exponential utility total cost for continuous-time piecewise deterministic Markov processes with the control acting continuously on the jump intensity $$\lambda $$ and on the transition measure Q of the process. The action space is supposed to depend on the state variable and the state space is considered to have a frontier such that the process jumps whenever it touches this boundary. We characterize the optimal value function as the minimal solution of an integro-differential optimality equation satisfying some boundary conditions, as well as the existence of a deterministic stationary optimal policy. These results are obtained by using the so-called policy iteration algorithm, under some continuity and compactness assumptions on the parameters of the problem, as well as some non-explosive conditions for the process.

Pullback and forward attractors of contractive difference equations

The construction of attractors of a dissipative difference equation is usually based on compactness assumptions. In this paper, we replace them with contractivity assumptions under which the pullba...

Strong and Weak Equilibria for Time-Inconsistent Stochastic Control in Continuous Time

A new definition of continuous-time equilibrium controls is introduced. As opposed to the standard definition, which involves a derivative-type operation, the new definition parallels how a discrete-time equilibrium is defined and allows for unambiguous economic interpretation. The terms “strong equilibria” and “weak equilibria” are coined for controls under the new and standard definitions, respectively. When the state process is a time-homogeneous continuous-time Markov chain, a careful asymptotic analysis gives complete characterizations of weak and strong equilibria. Thanks to the Kakutani–Fan fixed-point theorem, the general existence of weak and strong equilibria is also established under an additional compactness assumption. Our theoretic results are applied to a two-state model under nonexponential discounting. In particular, we demonstrate explicitly that there can be incentive to deviate from a weak equilibrium, which justifies the need for strong equilibria. Our analysis also provides new results for the existence and characterization of discrete-time equilibria under infinite horizon.

Rabinowitz Floer Homology for Tentacular Hamiltonians

Abstract This paper extends the definition of Rabinowitz Floer homology to non-compact hypersurfaces. We present a general framework for the construction of Rabinowitz Floer homology in the non-compact setting under suitable compactness assumptions on the periodic orbits and the moduli spaces of Floer trajectories. We introduce a class of hypersurfaces arising as the level sets of specific Hamiltonians: strongly tentacular Hamiltonians for which the compactness conditions are satisfied, cf. [ 21], thus enabling us to define the Rabinowitz Floer homology for this class. Rabinowitz Floer homology in turn serves as a tool to address the Weinstein conjecture and establish existence of closed characteristics for non-compact contact manifolds.

A New Iterative Algorithm for Multivalued Nonexpansive Mappping and Equlibruim Problems with Applications

In this paper, we introduce two iterative schemes by a modified Krasnoselskii-Mann algorithm for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of multivalued nonexpansive mappings in Hilbert space. We prove that the sequence generated by the proposed method converges strongly to a common element of the set of solutions of equilibruim problems and the set of fixed points of multivalued nonexpansive mappings which is also the minimum-norm element of the above two sets. Finally, some applications of our results to optimization problems with constraint and the split feasibility problem are given. No compactness assumption is made. The methods in the paper are novel and different from those in early and recent literature.

Equilibrium Non-Existence in Generalized Games

A generalized game is a strategic situation in which agents’ behavior restricts their opponents’ available action choices, giving rise to interdependencies in terms of what strategy profiles remain mutually feasible. This paper proposes a novel example of a simple generalized game in which the well-known convexity, compactness, continuity, and concavity assumptions are satisfied, but no Nash equilibrium exists. This finding reinforces that certain additional conditions appearing in the literature (primarily, the lower hemicontinuity of feasibility correspondences) are indispensable for equilibrium existence and must be considered as supplemental desiderata beyond the usual regularity conditions. Implications for institutional design are discussed.

Subdifferential of the Supremum via Compactification of the Index Set

We give new characterizations for the subdifferential of the supremum of an arbitrary family of convex functions, dropping out the standard assumptions of compactness of the index set and upper semi-continuity of the functions with respect to the index (J. Convex Anal. 26, 299–324, 2019). We develop an approach based on the compactification of the index set, giving rise to an appropriate enlargement of the original family. Moreover, in contrast to the previous results in the literature, our characterizations are formulated exclusively in terms of exact subdifferentials at the nominal point. Fritz–John and KKT conditions are derived for convex semi-infinite programming.

Generalizing dyadicity and Esenin-Vol'pin's Theorem

For a pair (G,X), where G is a topological group and X is a subspace of G, we consider continuous mappings f of G onto a topological space Y such that f(G)=Y=f(X). If, in addition, X is compact (σ-compact), then Y is called a gc-image of G (respectively, a gσc-image of G). For example, all dyadic compacta are gc-images of compact topological groups. Below Esenin-Vol'pin's Theorem on metrizability of first-countable dyadic compacta [10] is extended, in several ways, to gc-images and gσc-images, and to certain generalizations of them. Essential new features of these generalizations are: the compactness assumption and the topological group assumption are completely separated, compactness is weakened to σ-compactness, and first-countability is replaced by the Gδ-points requirement. In particular, in Section 2 the following Theorem 2.2 is proved. Suppose that G is a topological group, X is a Lindelöf Σ subspace of G, and f is a continuous mapping of G onto a space Y such that f(G)=f(X)=Y. Suppose also that Z is a closed subset of Y such that every point of Z is a Gδ-point in Y. Then Z has a countable network. Hence, if in addition Z is compact, or a p-space, then Z is separable and metrizable. See also Theorems 3.6, 4.6, where the last statement is a generalization of the main result to the case of paratopological groups.

General iterative algorithm for demicontractive-type mapping in real Hilbert spaces

In this paper, we investigate the problem of finding a solution to fixed point problem involving demicontractive mappings in the framework of Hilbert spaces. Inspired by general iterative algorithm, a new iterative method for solving the problem is introduced. Strong convergence theorem of the proposed method is established without any compactness assumption. Our theorems are significant improvements on several important recent results.

Convergence Rate of $\mathcal{O}(1/k)$ for Optimistic Gradient and Extragradient Methods in Smooth Convex-Concave Saddle Point Problems

We study the iteration complexity of the optimistic gradient descent-ascent (OGDA) method and the extragradient (EG) method for finding a saddle point of a convex-concave unconstrained min-max problem. To do so, we first show that both OGDA and EG can be interpreted as approximate variants of the proximal point method. This is similar to the approach taken in (A. Nemirovski (2004), SIAM J. Optim., 15, pp. 229--251) which analyzes EG as an approximation of the “conceptual mirror prox.” In this paper, we highlight how gradients used in OGDA and EG try to approximate the gradient of the proximal point method. We then exploit this interpretation to show that both algorithms produce iterates that remain within a bounded set. We further show that the primal-dual gap of the averaged iterates generated by both of these algorithms converge with a rate of $\mathcal{O}(1/k)$. Our theoretical analysis is of interest as it provides the first convergence rate estimate for OGDA in the general convex-concave setting. Moreover, it provides a simple convergence analysis for the EG algorithm in terms of function value without using a compactness assumption.

Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints

In this paper we focus on a general optimal control problem involving a dynamical system described by a nonlinear Caputo fractional differential equation of order 0 < α≤ 1, associated to a general Bolza cost written as the sum of a standard Mayer cost and a Lagrange cost given by a Riemann-Liouville fractional integral of order β ≥α. In addition the present work handles general control and mixed initial/final state constraints. Adapting the standard Filippov's approach based on appropriate compactness assumptions and on the convexity of the set of augmented velocities, we give an existence result for at least one optimal solution. Then, the major contribution of this paper is the statement of a Pontryagin maximum principle which provides a first-order necessary optimality condition that can be applied to the fractional framework considered here. In particular, Hamiltonian maximization condition and transversality conditions on the adjoint vector are derived. Our proof is based on the sensitivity analysis of the Caputo fractional state equation with respect to needle-like control perturbations and on Ekeland's variational principle. The paper is concluded with two illustrating examples and with a list of several perspectives for forthcoming works.

Markov decision processes under ambiguity

We consider statistical Markov Decision Processes where the decision maker is risk averse against model ambiguity. The latter is given by an unknown parameter which influences the transition law and the cost functions. Risk aversion is either measured by the entropic risk measure or by the Average Value at Risk. We show how to solve these kind of problems using a general minimax theorem. Under some continuity and compactness assumptions we prove the existence of an optimal (deterministic) policy and discuss its computation. We illustrate our results using an example from statistical decision theory.

A Modified Thakur Three-Step Iterative Algorithm to Garcia-Falset Mappings and Variational Inequalities

In this paper, we suggest and analyze a modified Thakur three-step iterative algorithm to approximate a common element of the set of common fixed points of Garcia-Falset mappings and the set of solutions of some variational inequalities in Banach spaces. We also establish strong convergence theorems for a common solution of the above-said problems by the proposed iterative algorithm without the compactness assumption. The methods in this paper are novel and different from those given in many other papers. And the results are the extension and improvement of the recent results announced by many others.

Bounding Extreme Events in Nonlinear Dynamics Using Convex Optimization

We study a convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs). This framework bounds from above the largest value of an observable along trajectories that start from a chosen set and evolve over a finite or infinite time interval. The approach needs no explicit trajectories. Instead, it requires constructing suitably constrained auxiliary functions that depend on the state variables and possibly on time. Minimizing bounds over auxiliary functions is a convex problem dual to the non-convex maximization of the observable along trajectories. This duality is strong, meaning that auxiliary functions give arbitrarily sharp bounds, for sufficiently regular ODEs evolving over a finite time on a compact domain. When these conditions fail, strong duality may or may not hold; both situations are illustrated by examples. We also show that near-optimal auxiliary functions can be used to construct spacetime sets that localize trajectories leading to extreme events. Finally, in the case of polynomial ODEs and observables, we describe how polynomial auxiliary functions of fixed degree can be optimized numerically using polynomial optimization. The corresponding bounds become sharp as the polynomial degree is raised if strong duality and mild compactness assumptions hold. Analytical and computational ODE examples illustrate the construction of bounds and the identification of extreme trajectories, along with some limitations. As an analytical PDE example, we bound the maximum fractional enstrophy of solutions to the Burgers equation with fractional diffusion.

Metric methods for heteroclinic connections in infinite dimensional spaces

We consider the minimal action problem min \int\_R 1/2 |$\gamma$'|^2 + W($\gamma$) dt among curves lying in a non-locally-compact metric space and connecting two given zeros of W $\ge$ 0. For this problem, the optimal curves are usually called heteroclinic connections. We reduce it, following a standard method, to a geodesic problem of the form min \int\_0^1 K($\gamma$)|$\gamma$'| dt with K = (2W)^(1/2). We then prove existence of curves minimizing this new action under some suitable compactness assumptions on K, which are minimal. The method allows to solve some PDE problems in unbounded domains, in particular in two variables x, y, when y = t and when the metric space is an L^2 space in the first variable x, and the potential W includes a Dirichlet energy in the same variable. We then apply this technique to the problem of connecting, in a functional space, two different heteroclinic connections between two points of the Euclidean space, as it was previously studied by Alama-Bronsard-Gui and by Schatzman more than fifteen years ago. With a very different technique, we are able to recover the same results, and to weaken some assumptions.

Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties

This is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space (X,d) that can be characterized by Evolution Variational Inequalities (EVI). We present new results concerning the structural properties of solutions to the EVI formulation, such as contraction, regularity, asymptotic expansion, precise energy identity, stability, asymptotic behavior and their link with the geodesic convexity of the driving functional.Under the crucial assumption of the existence of an EVI gradient flow, we will also prove two main results:–the equivalence with the De Giorgi variational characterization of curves of maximal slope;–the convergence of the Minimizing Movement-JKO scheme to the EVI gradient flow, with an explicit and uniform error estimate of order 1/2 with respect to the step size, independent of any geometric hypothesis (as upper or lower curvature bounds) on d. In order to avoid any compactness assumption, we will also introduce a suitable relaxation of the Minimizing Movement algorithm obtained by the Ekeland variational principle, and we will prove its uniform convergence as well.

Belief Change and Non-Monotonic Reasoning Sans Compactness

Belief change and non-monotonic reasoning are arguably different perspectives on the same phenomenon, namely, jettisoning of currently held beliefs in response to some incompatible evidence. Investigations in this area typically assume, among other things, that the underlying (background) logic is compact, that is, whatever can be inferred from a set of sentences X can be inferred from a finite subset of X. Recent research in the field shows that this compactness assumption can be dispensed without inflicting much damage on the AGM paradigm of belief change. In this paper we investigate the impact of such relaxation on non-monotonic logics instead. In particular, we show that, when compactness is not guaranteed, while the bridge from the AGM paradigm of belief change to expectation logics remains unaffected, the “return trip” from expectation logics to AGM paradigm is no longer guaranteed. We finally explore the conditions under which such guarantee can be given.

Existence of Fixed Points in a Class of Convex Sets

There are many well-known results about existence of fixed points for nonexpansive mappings defined on convex sets of Banach spaces which are \tau -compact, where \tau is usually a topology coarser than the norm topology. In this paper, we prove that it is possible to remove the \tau -compactness assumption and still obtain some fixed point results in a large framework. As particular cases, we can consider certain families of non- w^* -compact subsets in \ell_1 , the James space J_p , the trace space \mathcal{C}_1(\ell_2) and some other dual nonreflexive Banach spaces and we prove that „local" stability results for nonexpansive mappings can be achieved. In order to reach our purposes, we will make use of two new geometric coefficients a_\tau(X) and t_\tau(C) . Since the problem of the existence of fixed points for a nonexpansive mapping can equivalently be redefined assuming that the mapping is asymptotically regular, we will first focus on asymptotically regular mappings which satisfy a rather weak Lipschitz condition according to the value of the fraction \frac{a_\tau(X)}{t_\tau(C)} . Furthermore, we will show some examples proving that our results are sharp and can be used to obtain „global" stability results for the fixed point property.