66 research outputs found
Introduction to Nonsmooth Analysis and Optimization
This book aims to give an introduction to generalized derivative concepts
useful in deriving necessary optimality conditions and numerical algorithms for
infinite-dimensional nondifferentiable optimization problems that arise in
inverse problems, imaging, and PDE-constrained optimization. They cover convex
subdifferentials, Fenchel duality, monotone operators and resolvents,
Moreau--Yosida regularization as well as Clarke and (briefly) limiting
subdifferentials. Both first-order (proximal point and splitting) methods and
second-order (semismooth Newton) methods are treated. In addition,
differentiation of set-valued mapping is discussed and used for deriving
second-order optimality conditions for as well as Lipschitz stability
properties of minimizers. The required background from functional analysis and
calculus of variations is also briefly summarized.Comment: arXiv admin note: substantial text overlap with arXiv:1708.0418
Selected Applications of Linear Semi-Infinite Systems Theory
In this paper we, firstly, review the main known results on systems of an arbitrary (possibly infinite) number of weak linear inequalities posed in the Euclidean space Rn (i.e., with n unknowns), and, secondly, show the potential power of this theoretical tool by developing in detail two significant applications, one to computational geometry: the Voronoi cells, and the other to mathematical analysis: approximate subdifferentials, recovering known results in both fields and proving new ones. In particular, this paper completes the existing theory of farthest Voronoi cells of infinite sets of sites by appealing to well-known results on linear semi-infinite systems.This research was partially supported by PGC2018-097960-B-C22 of the Ministerio de Ciencia, Innovación y Universidades (MCIU), the Agencia Estatal de Investigación (AEI), and the European Regional Development Fund (ERDF); by CONICET, Argentina, Res D No 4198/17; and by Universidad Nacional de Cuyo, Secretaría de Investigación, Internacionales y Posgrado (SIIP), Res. 3922/19-R, Cod.06/D227, Argentina
On the Phase Space of Fourth-Order Fiber-Orientation Tensors
Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation tensor is obtained via a closure approximation. Unfortunately, such a description limits the predictive capabilities of the modeling process significantly, because the wealth of possible fourth-order fiber-orientation tensors is not exploited by such closures, and the restriction to second-order fiber-orientation tensors implies artifacts. Closures based on the second-order fiber-orientation tensor face a fundamental problem – which fourth-order fiber-orientation tensors can be realized? In the literature, only necessary conditions for a fiber-orientation tensor to be connected to a fiber-orientation distribution are found. In this article, we show that the typically considered necessary conditions, positive semidefiniteness and a trace condition, are also sufficient for being a fourth-order fiber-orientation tensor in the physically relevant case of two and three spatial dimensions. Moreover, we show that these conditions are not sufficient in higher dimensions. The argument is based on convex duality and a celebrated theorem of D. Hilbert (1888) on the decomposability of positive and homogeneous polynomials of degree four. The result has numerous implications for modeling the flow and the resulting microstructures of fiber-reinforced composites, in particular for the effective elastic constants of such materials. Based on our findings, we show how to connect optimization problems on fourth-order fiber-orientation tensors to semi-definite programming. The proposed formulation permits to encode symmetries of the fiber-orientation tensor naturally. As an application, we look at the differences between orthotropic and general, i.e., triclinic, fiber-orientation tensors of fourth order in two and three spatial dimensions, revealing the severe limitations inherent to orthotropic closure approximations
Recent contributions to linear semi-infinite optimization
This paper reviews the state-of-the-art in the theory of deterministic and uncertain linear semi-infinite optimization, presents some numerical approaches to this type of problems, and describes a selection of recent applications in a variety of fields. Extensions to related optimization areas, as convex semi-infinite optimization, linear infinite optimization, and multi-objective linear semi-infinite optimization, are also commented.This work was supported by the MINECO of Spain and ERDF of EU, Grant MTM2014-59179-C2-1-P, and by the Australian Research Council, Project DP160100854
Recent contributions to linear semi-infinite optimization: an update
This paper reviews the state-of-the-art in the theory of deterministic and uncertain linear semi-infinite optimization, presents some numerical approaches to this type of problems, and describes a selection of recent applications in a variety of fields. Extensions to related optimization areas, as convex semi-infinite optimization, linear infinite optimization, and multi-objective linear semi-infinite optimization, are also commented.This is an updated version of the paper “Recent contributions to linear semi-infinite optimization” that appeared in 4OR, 15(3), 221–264 (2017). It was supported by the MINECO of Spain and ERDF of EU, Grant MTM2014-59179-C2-1-P, and by the Australian Research Council, Project DP160100854
Riemannian game dynamics
We study a class of evolutionary game dynamics defined by balancing a gain
determined by the game's payoffs against a cost of motion that captures the
difficulty with which the population moves between states. Costs of motion are
represented by a Riemannian metric, i.e., a state-dependent inner product on
the set of population states. The replicator dynamics and the (Euclidean)
projection dynamics are the archetypal examples of the class we study. Like
these representative dynamics, all Riemannian game dynamics satisfy certain
basic desiderata, including positive correlation and global convergence in
potential games. Moreover, when the underlying Riemannian metric satisfies a
Hessian integrability condition, the resulting dynamics preserve many further
properties of the replicator and projection dynamics. We examine the close
connections between Hessian game dynamics and reinforcement learning in normal
form games, extending and elucidating a well-known link between the replicator
dynamics and exponential reinforcement learning.Comment: 47 pages, 12 figures; added figures and further simplified the
derivation of the dynamic
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