49 research outputs found
An adaptive discretization method solving semi-infinite optimization problems with quadratic rate of convergence
Semi-infinite programming can be used to model a large variety of complex
optimization problems. The simple description of such problems comes at a
price: semi-infinite problems are often harder to solve than finite nonlinear
problems. In this paper we combine a classical adaptive discretization method
developed by Blankenship and Falk and techniques regarding a semi-infinite
optimization problem as a bi-level optimization problem. We develop a new
adaptive discretization method which combines the advantages of both techniques
and exhibits a quadratic rate of convergence. We further show that a limit of
the iterates is a stationary point, if the iterates are stationary points of
the approximate problems
A generalized projection-based scheme for solving convex constrained optimization problems
In this paper we present a new algorithmic realization of a projection-based
scheme for general convex constrained optimization problem. The general idea is
to transform the original optimization problem to a sequence of feasibility
problems by iteratively constraining the objective function from above until
the feasibility problem is inconsistent. For each of the feasibility problems
one may apply any of the existing projection methods for solving it. In
particular, the scheme allows the use of subgradient projections and does not
require exact projections onto the constraints sets as in existing similar
methods.
We also apply the newly introduced concept of superiorization to optimization
formulation and compare its performance to our scheme. We provide some
numerical results for convex quadratic test problems as well as for real-life
optimization problems coming from medical treatment planning.Comment: Accepted to publication in Computational Optimization and
Application
An improved asymptotic analysis of the expected number of pivot steps required by the simplex algorithm
Let be i.i .d. vectors uniform on the unit sphere in , and let := {} be the random polyhedron generated by. Furthermore, for linearly independent vectors , in , let be the number of shadow vertices of in ). The paper provides an asymptotic expansion of the expectation value for fixed and . The first terms of the expansion are given explicitly. Our investigation of is closely connected to Borgwardt's probabilistic analysis of the shadow vertex algorithm - a parametric variant of the simplex algorithm. We obtain an improved asymptotic upper bound for the number of pivot steps required by the shadow vertex algorithm for uniformly on the sphere distributed data
A Simple Integral Representation for the Second Moments of Additive Random Variables on Stochastic Polyhedra
Let , be an i.i.d. sequence taking values in , whose convex hull is interpreted as a stochastic polyhedron . For a special class of random variables, which decompose additively relative to their boundary simplices, eg. the volume of , simple integral representations of its first two moments are given in case of rotationally symmetric distributions in order to facilitate estimations of variances or to quantify large deviations from the mean
A comparison method for expectations of a class of continuous polytope functionals
Let be independent random points in spherically symmetrically but not necessarily identically distributed. Let be the random polytope generated as the convex hull of and for any -dimensional subspace let be the volume of with respect to the -dimensional Lebesgue measure . Furthermore, let (t):= \)(\(\Vert a_i \|_2\leq t),
, be the radial distribution function of . We prove that the expectation
functional ( := ) is strictly decreasing in
each argument, i.e. if , , but , we show ) > ). The proof is clone in the more general framework
of continuous and - additive polytope functionals
On the Variance of Additive Random Variables on Stochastic Polyhedra
Let be an i.i.d. sequence taking values in . Whose convex hull is interpreted as a stochastic polyhedron . For a special class of random variables which decompose additively relative to their boundary simplices, eg. the volume of , integral representations of their first two moments are given which lead to asymptotic estimations of variances for special "additive variables" known from stochastic approximation theory in case of rotationally symmetric distributions
On the Approximation of a Ball by Random Polytopes
Let ( be a sequence of identically and independently distributed random vectors drawn from the -dimensional unit ball and let := convhull ) be the random polytope generated by . Furthermore, let : = (Vol \ ) be the deviation of the polytope's volume from the volume of the ball. For uniformly distributed and , we prove that tbe limiting distribution of for satisfies a 0-1-law. Especially, we provide precise information about the asymptotic behaviour of the variance of ). We deliver analogous results for spherically symmetric distributions in with regularly varying tail
On the expected number of shadow vertices of the convex hull of random points
Let be independent random points in that are independent and identically distributed spherically symmetrical in . Moreover, let be the random polytope generated as the convex hull of and let be an arbitrary -dimensional
subspace of with . Let be the orthogonal projection image of in . We call those vertices of , whose projection images in are vertices of as well shadow vertices of with respect to the subspace . We derive a distribution independent sharp upper bound for the expected number of shadow vertices of in