23 research outputs found
The Radius of Metric Subregularity
Get PDFThere is a basic paradigm, called here the radius of well-posedness, which
quantifies the "distance" from a given well-posed problem to the set of
ill-posed problems of the same kind. In variational analysis, well-posedness is
often understood as a regularity property, which is usually employed to measure
the effect of perturbations and approximations of a problem on its solutions.
In this paper we focus on evaluating the radius of the property of metric
subregularity which, in contrast to its siblings, metric regularity, strong
regularity and strong subregularity, exhibits a more complicated behavior under
various perturbations. We consider three kinds of perturbations: by Lipschitz
continuous functions, by semismooth functions, and by smooth functions,
obtaining different expressions/bounds for the radius of subregularity, which
involve generalized derivatives of set-valued mappings. We also obtain
different expressions when using either Frobenius or Euclidean norm to measure
the radius. As an application, we evaluate the radius of subregularity of a
general constraint system. Examples illustrate the theoretical findings.Comment: 20 page
Robust Linear Optimization with Recourse: Solution Methods and Other Properties.
Full text linkThe unifying theme of this dissertation is robust optimization; the study of solving
certain types of convex robust optimization problems and the study of bounds
on the distance to ill-posedness for certain types of robust optimization problems.
Robust optimization has recently emerged as a new modeling paradigm designed
to address data uncertainty in mathematical programming problems by finding an
optimal solution for the worst-case instances of unknown, but bounded, parameters.
Parameters in practical problems are not known exactly for many reasons: measurement
errors, round-off computational errors, even forecasting errors, which created
a need for a robust approach. The advantages of robust optimization are two-fold:
guaranteed feasible solutions against the considered data instances and not requiring
the exact knowledge of the underlying probability distribution, which are limitations
of chance-constraint and stochastic programming. Adjustable robust optimization,
an extension of robust optimization, aims to solve mathematical programming problems where the data is uncertain and sets of decisions can be made at different points in time, thus producing solutions that are less conservative in nature than those produced by robust optimization.
This dissertation has two main contributions: presenting a cutting-plane method
for solving convex adjustable robust optimization problems and providing preliminary
results for determining the relationship between the conditioning of a robust
linear program under structured transformations and the conditioning of the equivalent
second-order cone program under structured perturbations. The proposed algorithm
is based on Kelley's method and is discussed in two contexts: a general convex
optimization problem and a robust linear optimization problem with recourse under
right-hand side uncertainty. The proposed algorithm is then tested on two different
robust linear optimization problems with recourse: a newsvendor problem with
simple recourse and a production planning problem with general recourse, both under
right-hand side uncertainty. Computational results and analyses are provided.
Lastly, we provide bounds on the distance to infeasibility for a second-order cone program
that is equivalent to a robust counterpart under ellipsoidal uncertainty in terms
of quantities involving the data defining the ellipsoid in the robust counterpart.Ph.D.Industrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/64714/1/tlterry_1.pd
Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)
Get PDFThe Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..
