## Algorithms for on-line vertex enumeration problem

##### Author

Kaya, İrfan Caner

##### Advisor

Ulus, Firdevs

##### Date

2017-09##### Publisher

Bilkent University

##### Language

English

##### Type

Thesis##### Item Usage Stats

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Show full item record##### Abstract

Vertex enumeration problem is to enumerate all vertices of a polyhedron P
which is given by intersection of finitely many halfspaces. It is a basis for many
algorithms designed to solve problems from various application areas and there are
many algorithms to solve these problems in the literature. On the one hand, there
are iterative algorithms which solve the so called
'on-line'
vertex enumeration
problem in each iteration. In other words, in each iteration of these algorithms,
the current polyhedron is intersected with an additional halfspace that defines P.
On the other hand, there are simplex-type algorithms which takes the set off all
halfspaces as its input from the beginning.
One of the usages of the vertex enumeration problem is the
Benson-type
multiobjective
optimization algorithms. The aim of these algorithms is to generate or
approximate the Pareto frontier (the set of nondominated points in the objective
space). In each iteration of the Benson's algorithm, a polyhedron which contains
the Pareto frontier is intersected with an additional halfspace in order tofind a
finer outer approximation. The vertex enumeration problem to be used within
this algorithm has a special structure. Namely, the polyhedron to be generated
is known to be unbounded with a recession cone which is equal to the positive
orthant.
In this thesis, we consider the
double description method
which is a method
to solve an on-line vertex enumeration problem where the starting polyhedron
is bounded. (1) We generate an iterative algorithm to solve the vertex enumeration
problem from the scratch where polyhedron P is allowed to be bounded
or unbounded. (2) Then, we slightly modify this algorithm to be more efficient
while it only works for problems where the recession cone of P is known to be
the positive orthant. (3) Finally, we generate an additional algorithm for these
problems. For this one, we modify the double description method such that it
uses the extreme directions of the recession cone more effectively. We provide an
illustrative example to explain the algorithms in detail.
We implement the algorithms using MATLAB; employ each of them as a function
of a
Benson-type
multiobjective optimization algorithm; and test the performances
of the algorithms for randomly generated linear multiobjective optimization
problems. Accordingly, for two dimensional problems, there is no clear
distinction between the run-time performances of these algorithms. However, as
the dimension of the vertex enumeration problem increases, the last algorithm
(Algorithm 3) gets more efficient than the others.