In this paper we discuss the application of a certain class of Monte Carlo methods to stochastic optimization problems. Particularly, we study variable-sample techniques, in which the objective function is replaced, at each iteration, by a sample average approximation. We first provide general results on the schedule of sample sizes, under which variable-sample methods yield consistent estimators as well as bounds on the estimation error. Because the convergence analysis is performed sample-path wise, we are able to obtain our results in a flexible setting, which includes the possibility of using different sampling distributions along the algorithm, without making strong assumptions on the underlying distributions. In particular, we allow the distributions to depend on the decision variables x. We illustrate these ideas by studying a modification of the wellknown simulated annealing method, adapting it to the variable-sample scheme, and show conditions for convergence of the algorithm.

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Abstract. It is known that in two-dimensional systems of ODEs of the form ˙x i = x i f i (x) with ∂f i /∂x j < 0 (strongly competitive systems), boundaries of the basins of repulsion of equilibria consist of invariant Lipschitz curves, unordered with respect to the coordinatewise (partial) order. We prove that such curves are in fact of class C 1. (S) A two-dimensional system of C 1 ordinary differential equations (ODEs) ˙x i = x i f i (x), where f = (f 1, f 2) : K → R 2, K: = {x = (x 1, x 2) ∈ R 2: x i ≥ 0 for i = 1, 2} is called strongly competitive if ∂f i (x) < 0 ∂xj for all x ∈ K, i, j = 1, 2, i ̸ = j (see M. W. Hirsch’s papers [4] and [5]). Systems of the form (S) describe communities of two interacting biological species: the function f i represents the per capita growth rate of the ith species. Strong competitiveness means that the growth of each species inhibits the growth of the other. An important subclass of strongly competitive systems consists of strongly competitive Lotka–Volterra systems of the form ˙x i = x i( 2∑ bi + aijx j) j=1

Abstract. We introduce the total s-energy of a multiagent system with time-dependent links. This provides a new analytical perspective on bidirectional agreement dynamics, which we use to bound the convergence rates of dynamical systems for synchronization, flocking, opinion dynamics, and social epistemology.

Minimal pseudo-Anosov translation lengths on the complex of curves

Abstract. The essential spectral radius of a sub-Markovian process is defined as the infimum of the spectral radiuses of all local perturbations of the process. When the family of rescaled processes satisfies sample path large deviation principle, the spectral radius and the essential spectral radius are expressed in terms of the rate function. The paper is motivated by applications to reflected diffusions and jump Markov processes describing stochastic networks for which the sample path large deviation principle has been established and the rate function has been identified while essential spectral radius has not been calculated. 1.

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