1,720,965 research outputs found
The gap between a variational problem and its occupation measure relaxation
Get PDFRecent works have proposed linear programming relaxations of variational
optimization problems subject to nonlinear PDE constraints based on the
occupation measure formalism. The main appeal of these methods is the fact that
they rely on convex optimization, typically semidefinite programming. In this
work we close an open question related to this approach. We prove that the
classical and relaxed minima coincide when the dimension of the codomain of the
unknown function equals one, both for calculus of variations and for optimal
control problems, thereby complementing analogous results that existed for the
case when the dimension of the domain equals one. In order to do so, we prove a
generalization of the Hardt-Pitts decomposition of normal currents applicable
in our setting. We also show by means of a counterexample that, if both the
dimensions of the domain and of the codomain are greater than one, there may be
a positive gap. The example we construct to show the latter serves also to show
that sometimes relaxed occupation measures may represent a more
conceptually-satisfactory "solution" than their classical counterparts, so that
-- even though they may not be equivalent -- algorithms rendering accessible
the minimum in the larger space of relaxed occupation measures remain extremely
valuable. Finally, we show that in the presence of integral constraints, a
positive gap may occur at any dimension of the domain and of the codomain.Comment: 46 pages, 10 figure
A closed-measure approach to stochastic approximation
No full textInternational audienceThis paper introduces a new method to tackle the issue of the almost sure convergence of stochastic approximation algorithms defined from a differential inclusion. Under the assumption of slowly decaying step-sizes, we establish that the set of essential accumulation points of the iterates belongs to the Birkhoff center associated with the differential inclusion. Unlike previous works, our results do not rely on the notion of asymptotic pseudotrajectories introduced by Benaı̈m–Hofbauer–Sorin, which is the predominant technique to address the convergence problem. They follow as a consequence of Young’s superposition principle for closed measures. This perspective bridges the gap between Young’s principle and the notion of invariant measure of set-valued dynamical systems introduced by Faure and Roth. Also, the proposed method allows to obtain sufficient conditions under which the velocities locally compensate around any essential accumulation point
A closed-measure approach to stochastic approximation
Get PDFThis paper introduces a new method to tackle the issue of the almost sure
convergence of stochastic approximation algorithms defined from a differential
inclusion. Under the assumption of slowly decaying step-sizes, we establish
that the set of essential accumulation points of the iterates belongs to the
Birkhoff center associated with the differential inclusion. Unlike previous
works, our results do not rely on the notion of asymptotic pseudotrajectories
introduced by Bena\"im--Hofbauer--Sorin, which is the predominant technique to
address the convergence problem. They follow as a consequence of Young's
superposition principle for closed measures. This perspective bridges the gap
between Young's principle and the notion of invariant measure of set-valued
dynamical systems introduced by Faure and Roth. Also, the proposed method
allows to obtain sufficient conditions under which the velocities locally
compensate around any essential accumulation point.Comment: 20 page
Convergence rates for sums-of-squares hierarchies with correlative sparsity
Get PDFThis work derives upper bounds on the convergence rate of the
moment-sum-of-squares hierarchy with correlative sparsity for global
minimization of polynomials on compact basic semialgebraic sets. The main
conclusion is that both sparse hierarchies based on the Schm\"udgen and Putinar
Positivstellens\"atze enjoy a polynomial rate of convergence that depends on
the size of the largest clique in the sparsity graph but not on the ambient
dimension. Interestingly, the sparse bounds outperform the best currently
available bounds for the dense hierarchy when the maximum clique size is
sufficiently small compared to the ambient dimension and the performance is
measured by the running time of an interior point method required to obtain a
bound on the global minimum of a given accuracy.Comment: 23 page
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Sufficient conditions for the absence of relaxation gaps in state-constrained optimal control
No full text9 page
Young measure relaxation gaps for controllable systems with smooth state constraints
No full textIn this article, we tackle the problem of the existence of a gap corresponding to Young measure relaxations for state-constrained optimal control problems. We provide a counterexample proving that a gap may occur in a very regular setting, namely for a smooth controllable system state constrained to the closed unit ball, provided that the Lagrangian density (i.e., the running cost) is non-convex in the control variables. The example is constructed in the setting of sub-Riemannian geometry with the core ingredient being an unusual admissible curve that exhibits a certain form of resistance to state-constrained approximation. Specifically, this curve cannot be approximated by neighboring admissible curves while obeying the state constraint due to the intricate nature of the dynamics near the boundary of the constraint set. Our example also presents an occupation measure relaxation gap.23 pages, 2 figure
Young measure relaxation gaps for controllable systems with smooth state constraints
No full textIn this article, we tackle the problem of the existence of a gap corresponding to Young measure relaxations for state-constrained optimal control problems. We provide a counterexample proving that a gap may occur in a very regular setting, namely for a smooth controllable system state constrained to the closed unit ball, provided that the Lagrangian density (i.e., the running cost) is non-convex in the control variables. The example is constructed in the setting of sub-Riemannian geometry with the core ingredient being an unusual admissible curve that exhibits a certain form of resistance to state-constrained approximation. Specifically, this curve cannot be approximated by neighboring admissible curves while obeying the state constraint due to the intricate nature of the dynamics near the boundary of the constraint set. Our example also presents an occupation measure relaxation gap.23 pages, 2 figure
Long term dynamics of the subgradient method for Lipschitz path differentiable functions
Get PDFWe consider the long-term dynamics of the vanishing stepsize subgradient method in the case when the objective function is neither smooth nor convex. We assume that this function is locally Lipschitz and path differentiable, i.e., admits a chain rule. Our study departs from other works in the sense that we focus on the behavior of the oscillations, and to do this we use closed measures. We recover known convergence results, establish new ones, and show a local principle of oscillation compensation for the velocities. Roughly speaking, the time average of gradients around one limit point vanishes. This allows us to further analyze the structure of oscillations, and establish their perpendicularity to the general drift
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