We report on recent results that show that the pair correlation function of systems with exponentially decaying interactions can fail to exhibit Ornstein-Zernike asymptotics at all sufficiently high temperatures and all sufficiently small densities.
This turns out to be related to a lack of analyticity of the correlation length as a function of temperature and/or density and even occurs for one-dimensional systems.
The key element of the approach to the theory of necessary conditions in optimal control discussed in the paper is reduction of the original constrained problem to unconstrained minimization with subsequent application of a suitable mechanism of loca
l analysis to characterize minima of (necessarily nonsmooth) functionals that appear after reduction. Using unconstrained minimization at the crucial step of obtaining necessary conditions definitely facilitates studies of new phenomena and allows to get more transparent and technically simple proofs of known results. In the paper we offer a new proof of the maximum principle for a nonsmooth optimal control problem (in the standard Pontryagin form) with state constraints and then prove a new second order condition for a strong minimum in the same problem but with data differentiable with respect to the state and control variables. The role of variational analysis is twofold. Conceptually, the main considerations behind the reduction are connected with metric regularity and Ekelands principle. On the other hand, technically, calculation of subdifferentials of components of the functionals that appear after the reduction is an essential part of the proofs.
We consider optimization algorithms that successively minimize simple Taylor-like models of the objective function. Methods of Gauss-Newton type for minimizing the composition of a convex function and a smooth map are common examples. Our main result
is an explicit relationship between the step-size of any such algorithm and the slope of the function at a nearby point. Consequently, we (1) show that the step-sizes can be reliably used to terminate the algorithm, (2) prove that as long as the step-sizes tend to zero, every limit point of the iterates is stationary, and (3) show that conditions, akin to classical quadratic growth, imply that the step-sizes linearly bound the distance of the iterates to the solution set. The latter so-called error bound property is typically used to establish linear (or faster) convergence guarantees. Analogous results hold when the step-size is replaced by the square root of the decrease in the models value. We complete the paper with extensions to when the models are minimized only inexactly.
The regularity theory for variational inequalities over polyhedral sets developed in a series of papers by Robinson, Ralph and Dontchev-Rockafellar in the 90s has long become classics of variational analysis. But in the available proofs of almost all
main results, fairly nontrivial as they are, techniques of variational analysis do not play a significant part. In the paper we develop a new approach that allows to obtain some generalizations of the the mentioned results without invoking anything beyond elementary geometry of convex polyhedra and some basic facts of the theory of metric regularity.
We present a theorem of Sard type for semi-algebraic set-valued mappings whose graphs have dimension no larger than that of their range space: the inverse of such a mapping admits a single-valued analytic localization around any pair in the graph, fo
r a generic value parameter. This simple result yields a transparent and unified treatment of generic properties of semi-algebraic optimization problems: typical semi-algebraic problems have finitely many critical points, around each of which they admit a unique active manifold (analogue of an active set in nonlinear optimization); moreover, such critical points satisfy strict complementarity and second-order sufficient conditions for optimality are indeed necessary.
We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear convergence. Wh
en the two sets are semi-algebraic and bounded, but not necessarily transversal, we nonetheless prove subsequence convergence.
Steepest descent is central in variational mathematics. We present a new transparent existence proof for curves of near-maximal slope --- an influential notion of steepest descent in a nonsmooth setting. We moreover show that for semi-algebraic funct
ions --- prototypical nonpathological functions in nonsmooth optimization --- such curves are precisely the solutions of subgradient dynamical systems.
Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to the domain
and limits of gradients generate the entire Clarke subdifferential. The characterization formula we obtain unifies various apparently disparate results that have appeared in the literature. Our techniques also yield a simplified proof that closed semialgebraic functions on $R^n$ have a limiting subdifferential graph of uniform local dimension $n$.
We consider the Bernoulli bond percolation process $mathbb{P}_{p,p}$ on the nearest-neighbor edges of $mathbb{Z}^d$, which are open independently with probability $p<p_c$, except for those lying on the first coordinate axis, for which this probabilit
y is $p$. Define [xi_{p,p}:=-lim_{ntoinfty}n^{-1}log mathbb{P}_{p,p}(0leftrightarrow nmathbf {e}_1)] and $xi_p:=xi_{p,p}$. We show that there exists $p_c=p_c(p,d)$ such that $xi_{p,p}=xi_p$ if $p<p_c$ and $xi_{p,p}<xi_p$ if $p>p_c$. Moreover, $p_c(p,2)=p_c(p,3)=p$, and $p_c(p,d)>p$ for $dgeq 4$. We also analyze the behavior of $xi_p-xi_{p,p}$ as $pdownarrow p_c$ in dimensions $d=2,3$. Finally, we prove that when $p>p_c$, the following purely exponential asymptotics holds: [mathbb {P}_{p,p}(0leftrightarrow nmathbf {e}_1)=psi_de^{-xi_{p,p}n}bigl(1+o(1)bigr)] for some constant $psi_d=psi_d(p,p)$, uniformly for large values of $n$. This work gives the first results on the rigorous analysis of pinning-type problems, that go beyond the effective models and dont rely on exact computations.
In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit
times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.
