1,720,981 research outputs found
Global search perspectives for multiobjective optimization
No full textExtending the notion of global search to multiobjective optimization is far than straightforward, mainly for the reason that one almost always has to deal with infinite Pareto optima and correspondingly infinite optimal values. Adopting Stephen Smale's global analysis framework, we highlight the geometrical features of the set of Pareto optima and we are led to consistent notions of global convergence. We formulate then a multiobjective version of a celebrated result by Stephens and Baritompa, about the necessity of generating everywhere dense sample sequences, and describe a globally convergent algorithm in case the Lipschitz constant of the determinant of the Jacobian is known
Extracting optimal datasets for metamodelling and perspectives for incremental samplings
No full textFinite mechanical proxies for a class of reducible continuum systems
Full text linkWe present the exact finite reduction of a class of nonlinearly perturbed wave equations -typically, a non-linear elastic string- based on the Amann-Conley-Zehnder paradigm. By solving an inverse eigenvalue problem, we establish an equivalence between the spectral finite description derived from A-C-Z and a discrete mechanical model, a well definite finite spring-mass system. By doing so, we decrypt the abstract information encoded in the finite reduction and obtain a physically sound proxy for the continuous problem
Singular Continuation: Generating Piecewise Linear Approximations to Pareto Sets via Global Analysis
Full text linkWe propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale [Global analysis and economics. I. Pareto optimum and a generalization of Morse theory, in Dynamical Systems, Academic Press, New York, 1973, pp. 531-544]. The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. The method distinguishes the hierarchy between singular set, Pareto critical set, and stable Pareto critical set, and it can handle the problem of superposition of local Pareto fronts, occurring in the general nonconvex case. Furthermore, a quadratic convergence result in a suitable setwise sense is proven and tested in a number of numerical examples
Discrete structures equivalent to non-linear Dirichlet and wave equations
No full textThe Amann–Conley–Zehnder (ACZ) reduction is a global Lyapunov–Schmidt reduction for PDEs based on spectral decomposition. ACZ has been applied in conjunction to diverse topological methods, to derive existence and multiplicity results for Hamiltonian systems, for elliptic boundary value problems, and for nonlinear wave equations. Recently, the ACZ reduction has been translated numerically for semilinear Dirichlet problems and for modeling molecular dynamics, showing competitive performances with standard techniques. In this paper, we apply ACZ to a class of nonlinear wave equations in Tn , attaining to the definition of a finite lattice of harmonic oscillators weakly nonlinearly coupled exactly equivalent to the continuum model. This result can be thought as a thermodynamic limit arrested at a small but finite scale without resid- uals. Reduced dimensional models reveal the macroscopic scaled features of the continuum, which could be interpreted as collective variables
Implementation of an exact finite reduction scheme for steady-state reaction-diffusion equations
No full textIn this paper we propose the numerical solution of a steady-state reaction-diffusion problem by means of application of a non-local Lyapunov-Schmidt type reduction originally devised for field theory. A numerical algorithm is developed on the basis of the discretization of the differential operator by means of simple finite differences. The eigendecomposition of the resulting matrix is used to implement a discrete version of the reduction process. By the new algorithm the problem is decomposed into two coupled subproblems of different dimensions. A large subproblem is solved by means of a fixed point iteration completely controlled by the features of the original equation, and a second problem, with dimensions that can be made much smaller than the former, which inherits most of the non-linear difficulties of the original system. The advantage of this approach is that sophisticated linearization strategies can be used to solve this small non-linear system, at the expense of a partial eigendecomposition of the discretized linear differential operator. The proposed scheme is used for the solution of a simple nonlinear one-dimensional problem. The applicability of the procedure is tested and experimental convergence estimates are consolidated. Numerical results are used to show the performance of the new algorithm
A Pareto–Pontryagin Maximum Principle for Optimal Control
No full textIn this paper, an attempt to unify two important lines of thought in applied optimization is proposed. We wish to integrate the well-known (dynamic) theory of Pontryagin optimal control with the Pareto optimization (of the static type), involving the maximization/minimization of a non-trivial number of functions or functionals, Pontryagin optimal control offers the definitive theoretical device for the dynamic realization of the objectives to be optimized. The Pareto theory is undoubtedly less known in mathematical literature, even if it was studied in topological and variational details (Morse theory) by Stephen Smale. This reunification, obviously partial, presents new conceptual problems; therefore, a basic review is necessary and desirable. After this review, we define and unify the two theories. Finally, we propose a Pontryagin extension of a recent multiobjective optimization application to the evolution of trees and the related anatomy of the xylems. This work is intended as the first contribution to a series to be developed by the authors on this subject
Microscopic structures from reduction of continuum nonlinear problems
Full text linkWe present an application of the Amann–Zehnder exact finite reduction to a class of nonlinear perturbations of elliptic elasto-static problems. We propose the existence of minmax solutions by applying Ljusternik–Schnirelmann theory to a finite dimensional variational formulation of the problem, based on a suitable spectral cut–off. As a by–product, with a choice of fit variables, we establish a variational equivalence between the above spectral finite description and a discrete mechanical model. By doing so, we decrypt the abstract information encoded in the AZ reduction and give rise to a concrete and finite description of the continuous problem
New activity pattern in human interactive dynamics
No full textWe investigate the response function of human agents as demonstrated by written correspondence, uncovering a new pattern for how the reactive dynamics of individuals is distributed across the set of each agent's contacts. In long-term empirical data on email, we find that the set of response times considered separately for the messages to each different correspondent of a given writer, generate a family of heavy-tailed distributions, which have largely the same features for all agents, and whose characteristic times grow exponentially with the rank of each correspondent. We furthermore show that this new behavioral pattern emerges robustly by considering weighted moving averages of the priority-conditioned response-time probabilities generated by a basic prioritization model. Our findings clarify how the range of priorities in the inputs from one's environment underpin and shape the dynamics of agents embedded in a net of reactive relations. These newly revealed activity patterns might be universal, being present in other general interactive environments, and constrain future models of communication and interaction networks, affecting their architecture and evolution
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