| dc.description.abstract | This thesis studies the theory and implementation of interior-point methods
for convex optimisation. A number of important problems from mathematics and
engineering can be cast naturally as convex optimisation problems, and a
great many others have useful convex relaxations. Interior-point methods
are among the successful algorithms for solving convex optimisation problems.
One class of interior-point methods, called primal-dual interior-point
methods, have been particularly successful at solving optimisation problems
defined over symmetric cones, which are self-dual cones whose
linear automorphisms act transitively on their interiors.
The main theoretical contribution is the design and analysis of a primal-dual
interior-point method for general convex optimisation that is ``primal-dual
symmetric''---if arithmetic is done exactly, the sequence of iterates
generated is invariant under interchange of primal and dual problems.
The proof of this algorithm's correctness and asymptotic worst-case iteration
complexity hinges on a new analysis of a certain rank-four update formula
akin to the Hessian estimate updates performed by quasi-Newton methods.
This thesis also gives simple, explicit constructions of primal-dual
scalings---linear maps from the dual space to the primal space that map the
dual iterate to the primal iterate and the barrier gradient at the primal
iterate to the barrier gradient at the dual iterate---by averaging the primal
or dual Hessian over a line segment. These scalings are called the primal
and dual integral scalings in this thesis. The primal and dual integral
scalings can inherit certain kinds of good behaviour from the barrier whose
Hessian is averaged. For instance, if the primal barrier Hessian at every
point maps the primal cone into the dual cone, then the primal integral
scaling also maps the primal cone into the dual cone.
This gives the idea that primal-dual interior-point methods based on the
primal integral scaling might be effective on problems in which the primal
barrier is somehow well-behaved, but the dual barrier is not.
One such class of problems is \emph{hyperbolicity cone
optimisation}---minimising a linear function over the intersection of an
affine space with a so-called hyperbolicity cone. Hyperbolicity cones arise
from hyperbolic polynomials, which can be seen as a generalisation of the
determinant polynomial on symmetric matrices. Hyperbolic polynomials
themselves have been of considerable recent interest in mathematics, their
theory playing a role in the resolution of the Kadison-Singer problem.
In the setting of hyperbolicity cone optimisation, the primal barrier's
Hessian satisfies ``the long-step Hessian estimation property'' with which
the primal barrier may be easily estimated everywhere in the interior of the
cone in terms of the primal barrier anywhere else in the interior of the
cone, and the primal barrier Hessian at every point in the interior of the
cone maps the primal cone into the dual cone. In general, however, the dual
barrier satisfies neither of these properties.
This thesis also describes an adaptation of the Mizuno-Todd-Ye method for
linear optimisation to hyperbolicity cone optimisation and its implementation.
This implementation is meant as a window into the algorithm's convergence
behaviour on hyperbolicity cone optimisation problems rather than as a useful
software package for solving hyperbolicity cone optimisation problems that
might arise in practice.
In the final chapter of this thesis is a description of an implementation of
an interior-point method for linear optimisation. This implementation can
efficiently use primal-dual scalings based on rank-four updates to an old
scaling matrix and was meant as a platform to evaluate that technique.
This implementation is modestly slower than CPLEX's barrier optimiser on
problems with no free or double-bounded variables. A computational
comparison between the ``standard'' interior-point algorithm for solving LPs
with one instance of the rank-four update technique is given.
The rank-four update formula mentioned above has an interesting
specialisation to linear optimisation that is also described in this thesis.
A serious effort was made to improve the running time of an interior-point
method for linear optimisation using this technique, but it ultimately failed.
This thesis revisits work from the early 1990s by Rothberg and Gupta on
cache-efficient data structures for Cholesky factorisation. This thesis
proposes a variant of their data structure, showing that, in this variant,
the time needed to perform triangular solves can be reduced substantially
from the time needed by either the usual supernodal or simplicial data
structures.
The linear optimisation problem solver described in this thesis is also used
to study the impact of these different data structures on the overall time
required to solve a linear optimisation problem. | en |
