Convergence in distribution of random closed sets and applications in stability theory of stochastic optimisation [Elektronische Ressource] / vorgelegt von Oliver Gersch

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• Mathematics

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• Publié par: technische_universitat_ilmenau
• Publié le: 01 janvier 2007
• Nombre de lectures: 20
• Langue: English

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Convergence in Distribution of
Random Closed Sets and Applications
in Stability Theory of Stochastic
Optimisation
Dissertation zur Erlangung des akademischen Grades
Doctor rerum naturalium (Dr. rer. nat.)
Der Fakult¨at fur¨ Mathematik und Naturwissenschaften
der Technischen Universi¨at Ilmenau
vorgelegt von
Dipl. Math. Oliver Gersch
Gutachter:
Prof. Dr. rer. nat. Silvia Vogel, TU Ilmenau
Doc. RNDr. Petr Lachout, Charles University in Prague
Prof. Dr. rer.nat Eckhard Liebscher, HS Merseburg
Tag der Einreichung: 18.04.2006
Tag der wissenschaftlichen Aussprache: 26.01.2007
urn:nbn:de:gbv:ilm1-2007000011Contents
Introduction iii
1. Convergence in Distribution of Random Closed Sets 1
1.1. Set Convergence and Associated Topologies . . . . . . . . . . . . . . . . 1
1.2. Convergence in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3. Random Lower Semicontinuous Functions . . . . . . . . . . . . . . . . . 18
2. Sufficient Conditions 21
2.1. Convergence in Distribution from Convergence in Probability . . . . . . . 22
2.2. Finite Dimensional Convergence . . . . . . . . . . . . . . . . . . . . . . . 34
2.3. Skorohod Convergence in D[0,∞) . . . . . . . . . . . . . . . . . . . . . . 63
2.4. Dependence on Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.5. Pointwise Convergence of Convex Functions . . . . . . . . . . . . . . . . 76
2.6. Convergence in Distribution in Product Spaces . . . . . . . . . . . . . . . 83
3. Stochastic Optimisation 97
3.1. Restriction Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.2. Minimum and Argmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.3. Argmin Continuous Mapping Theorems . . . . . . . . . . . . . . . . . . . 118
3.4. Approximate Confidence Regions . . . . . . . . . . . . . . . . . . . . . . 123
4. Multiobjective Optimisation 125
A. Appendix 139
B. Notations 146
C. Kurzzusammenfassung in deutscher Sprache 147
Bibliography 149
iiIntroduction
In this work we investigate one-sided convergence in distribution of random closed sets.
Set convergence methods, most notably the concept of epi-convergence, have proved to
be useful tools in the investigation of parametric optimisation problems (cf. [3], [33]).
They are often favoured above uniform convergence methods, as they allow for lower
semicontinuous objective functions instead of continuous functions. One-sided conver-
gence, in the sense of inner/outer set convergence, is needed for example in stability
theory of parametric optimisation problems. Frequently the original problem is approx-
imated by a sequence of (easier to solve or numerically obtained) surrogate problems.
Since in general the solutions of the surrogate problems only approximate a subset of
the solutions for the original problem, inner convergence is a fruitful concept. Stochastic
optimisation problems bear a close resemblance to parametric optimisation problems
and can for example have their origin in the estimation of the parameters or in simula-
tions. To derive stochastic versions of stability results, the set valued methods have to
be combined with probability theory (see [24] and for convergence in distribution [34]).
Convergence in distribution methods are especially useful when the original problem is
also random. They are asked for to obtain information about the distribution of opti-
mal values and solutions for the original problem. In [27], [43] and [22] one-sided set
convergence in distribution (predominantly inner convergence) has been investigated.
In this work our approach and contributions are organised as follows.
The first chapter contains a collection of properties of the topologies which describe
one-sided set convergence and allow a sound topological foundation for the definition
of one-sided convergence in distribution. The noted differences to the case of conver-
gence in distribution in metric spaces hint on the expected difficulties and the need for
workarounds. The basic notations for the different types of convergence are given.
The conditions in the definition of convergence in distribution are hard to verify for a
given sequence of random closed sets or stochastic processes. We thus accompany them
with useful sufficient conditions for one-sided convergence in distribution in the second
iiiIntroduction
chapter. Mostly we derive convergence in distribution in our set valued framework from
other types of convergence (in distribution). These criteria can serve as a bridge be-
tween classical convergence for stochastic processes for important classes (e.g. D[0,∞))
and set convergence in distribution. A central result are new convergence criteria for
the epigraphs of random lower semicontinuous functions, which are in line with the fre-
quently used finite dimensional approach to convergence in distribution for stochastic
processes. They use the idea of stochastic equi lower semicontinuity. As a combination
of the two one-sided criteria we obtain a corrected version of the convergence criterion
for the full epi-convergence in distribution in [19]. Our sufficient conditions can help to
make recent applications of epi-convergence to statistics (cf. [38],[23],[29]) accessible for
stochastic processes. The second chapter is closed with a partial result for the conver-
gence in distribution of vectors of random closed sets, which is important for stochastic
optimisation problems with random restriction sets.
Starting with the third chapter we show, how one-sided set convergence in distribution
can be applied in stability theory of stochastic optimisation with random constraints.
We show generalisations to the case of ε- resp. ε -optimality for known results andn
provide the complementary outer convergence part to [43]. One-sided convergence in
distribution does in general not yield the distribution of minima and argmins of the
approximated problem. Instead we obtain one-sided bounds, which can for example be
used to find approximate confidence regions for the argmin sets. It is shown, how results
from [12] about argmax distributions in the non unique case can be obtained with the
set convergence in distribution approach. The most important technique used in this
chapter is to transfer results from parametric optimisation (found in [2]) to the setting
of convergence in distribution with the help of the Continuous Mapping Theorem and
its semicontinuous versions.
In the fourth and final chapter we derive ‘in distribution’ stability results for stochastic
multiobjective optimisation problems. In these problems in addition to the solutions,
theoptimalvaluesareusuallyset-valuedandarethustractablebysetconvergencemeth-
ods. As in the third chapter we are able to transfer deterministic results (here [35] was a
valueable source) to the case of convergence in distribution and to provideε -optimalityn
extensions.
I with to thank Professor Silvia Vogel for her advice and encouragement, Dr. Eckhard
Liebscher for many discussions on the topic and Dr. Petr Lachout for his hospitality
during my stays in Prague.
iv1. Convergence in Distribution of
Random Closed Sets
Random sets occur in a variety of situations, for example as solutions of random equali-
tiesandinequalitiesandassetsofoptimalpointsinstochasticoptimisationproblems. It
is known (see [3],[33]) that set convergence of closed sets is useful for stability theory of
parametric optimisation problems and that it is topologically very accessible. Random
closedsets([24],[37])areofspecialinterest,forexampleasepigraphsoflowersemicontin-
uous objective functions and as restriction sets in stochastic optimisation problems.. In
this chapter we deal with one-sided convergence in distribution for sequences of random
closedsetsanditstopologicalfoundations. Firstweconsiderinner-/outerconvergenceof
closed sets in the deterministic case and investigate properties of the related topologies.
Throughout this text we denote the space of all closed subsets of a given first countable
topological space X byF(X).
1.1. Set Convergence and Associated Topologies
There are several concepts for convergence of sequences (F ) ⊂F(X). For applicationsn n
instochasticoptimisationithasprovedtobeusefultochoosetheconceptofKuratowski–
Painlev´e convergence, which describes convergence of sequences of closed sets with the
help of sequences of points and their limits and accumulation points.
Definition 1.1 Let(X,τ)beafirstcountabletopologicalspace. Forasequence(F ) ⊂n n
F(X)theKuratowski–Painlev´elimitsuperiorandtheKuratowski–Painlev´elimitinferior
are given by

K–limsupF := x :∃(x ) , x ∈F , x →xn n n n nk k k k k
n→∞
and
K–liminfF :={x :∃(x ), x ∈F , for n≥n , x →x}.n n n n 0 n
n→∞
11. Convergence in Distribution of Random Closed Sets
Note that K−liminfF ⊂ K−limsupF . It is well known, that K−limsupF andn n n
n→∞ n→∞ n→∞
K−liminfF are again closed subsets of X, if X is first countable. In the following wen
n→∞
will write limsup and liminf instead of K−limsup and K−liminf, as it will always
be clear from the context, whether the limit inferior/superior is to be understood in the
classical calculus sense, in the set–algebraic way or in the Kuratowski–Painlev´e sense.
Definition 1.2 Let (X,τ) be a topological space. Let (F ) ⊂F(X), let F ⊂F(X).n n
F is said to inner-converge to