1,051 research outputs found
Nonsmooth Optimization; Proceedings of an IIASA Workshop, March 28 - April 8, 1977
Optimization, a central methodological tool of systems analysis, is used in many of IIASA's research areas, including the Energy Systems and Food and Agriculture Programs. IIASA's activity in the field of optimization is strongly connected with nonsmooth or nondifferentiable extreme problems, which consist of searching for conditional or unconditional minima of functions that, due to their complicated internal structure, have no continuous derivatives. Particularly significant for these kinds of extreme problems in systems analysis is the strong link between nonsmooth or nondifferentiable optimization and the decomposition approach to large-scale programming.
This volume contains the report of the IIASA workshop held from March 28 to April 8, 1977, entitled Nondifferentiable Optimization. However, the title was changed to Nonsmooth Optimization for publication of this volume as we are concerned not only with optimization without derivatives, but also with problems having functions for which gradients exist almost everywhere but are not continous, so that the usual gradient-based methods fail.
Because of the small number of participants and the unusual length of the workshop, a substantial exchange of information was possible. As a result, details of the main developments in nonsmooth optimization are summarized in this volume, which might also be considered a guide for inexperienced users. Eight papers are presented: three on subgradient optimization, four on descent methods, and one on applicability. The report also includes a set of nonsmooth optimization test problems and a comprehensive bibliography
Spectral Optimization Problems
In this survey paper we present a class of shape optimization problems where
the cost function involves the solution of a PDE of elliptic type in the
unknown domain. In particular, we consider cost functions which depend on the
spectrum of an elliptic operator and we focus on the existence of an optimal
domain. The known results are presented as well as a list of still open
problems. Related fields as optimal partition problems, evolution flows,
Cheeger-type problems, are also considered.Comment: 42 pages with 8 figure
Border Basis relaxation for polynomial optimization
A relaxation method based on border basis reduction which improves the
efficiency of Lasserre's approach is proposed to compute the optimum of a
polynomial function on a basic closed semi algebraic set. A new stopping
criterion is given to detect when the relaxation sequence reaches the minimum,
using a sparse flat extension criterion. We also provide a new algorithm to
reconstruct a finite sum of weighted Dirac measures from a truncated sequence
of moments, which can be applied to other sparse reconstruction problems. As an
application, we obtain a new algorithm to compute zero-dimensional minimizer
ideals and the minimizer points or zero-dimensional G-radical ideals.
Experimentations show the impact of this new method on significant benchmarks.Comment: Accepted for publication in Journal of Symbolic Computatio
Numerical approach of collision avoidance and optimal control on robotic manipulators
Collision-free optimal motion and trajectory planning for robotic manipulators are solved by a method of sequential gradient restoration algorithm. Numerical examples of a two degree-of-freedom (DOF) robotic manipulator are demonstrated to show the excellence of the optimization technique and obstacle avoidance scheme. The obstacle is put on the midway, or even further inward on purpose, of the previous no-obstacle optimal trajectory. For the minimum-time purpose, the trajectory grazes by the obstacle and the minimum-time motion successfully avoids the obstacle. The minimum-time is longer for the obstacle avoidance cases than the one without obstacle. The obstacle avoidance scheme can deal with multiple obstacles in any ellipsoid forms by using artificial potential fields as penalty functions via distance functions. The method is promising in solving collision-free optimal control problems for robotics and can be applied to any DOF robotic manipulators with any performance indices and mobile robots as well. Since this method generates optimum solution based on Pontryagin Extremum Principle, rather than based on assumptions, the results provide a benchmark against which any optimization techniques can be measured
Statistical mechanics and dynamics of solvable models with long-range interactions
The two-body potential of systems with long-range interactions decays at
large distances as , with , where is the
space dimension. Examples are: gravitational systems, two-dimensional
hydrodynamics, two-dimensional elasticity, charged and dipolar systems.
Although such systems can be made extensive, they are intrinsically non
additive. Moreover, the space of accessible macroscopic thermodynamic
parameters might be non convex. The violation of these two basic properties is
at the origin of ensemble inequivalence, which implies that specific heat can
be negative in the microcanonical ensemble and temperature jumps can appear at
microcanonical first order phase transitions. The lack of convexity implies
that ergodicity may be generically broken. We present here a comprehensive
review of the recent advances on the statistical mechanics and
out-of-equilibrium dynamics of systems with long-range interactions. The core
of the review consists in the detailed presentation of the concept of ensemble
inequivalence, as exemplified by the exact solution, in the microcanonical and
canonical ensembles, of mean-field type models. Relaxation towards
thermodynamic equilibrium can be extremely slow and quasi-stationary states may
be present. The understanding of such unusual relaxation process is obtained by
the introduction of an appropriate kinetic theory based on the Vlasov equation.Comment: 118 pages, review paper, added references, slight change of conten
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