Topics in Convex Optimization: Interior-Point Methods, Conic Duality and Approximations

Abstract : Optimization is a scientific discipline that lies at the boundary
between pure and applied mathematics. Indeed, while on the one hand
some of its developments involve rather theoretical concepts, its
most successful algorithms are on the other hand heavily used by
numerous companies to solve scheduling and design problems on a
daily basis.

Our research started with the study of the conic formulation for
convex optimization problems. This approach was already studied in
the seventies but has recently gained a lot of interest due to
development of a new class of algorithms called interior-point
methods. This setting is able to exploit the two most important
characteristics of convexity:

- a very rich duality theory (existence of a dual problem that is
strongly related to the primal problem, with a very symmetric
- the ability to solve these problems efficiently,
both from the theoretical (polynomial algorithmic complexity) and
practical (implementations allowing the resolution of large-scale
problems) points of view.

Most of the research in this area involved so-called self-dual
cones, where the dual problem has exactly the same structure as the
primal: the most famous classes of convex optimization problems
(linear optimization, convex quadratic optimization and semidefinite
optimization) belong to this category. We brought some contributions
in this field:
- a survey of interior-point methods for linear optimization, with
an emphasis on the fundamental principles that lie behind the design
of these algorithms,
- a computational study of a method of linear approximation of convex
quadratic optimization (more precisely, the second-order cone that
can be used in the formulation of quadratic problems is replaced by a
polyhedral approximation whose accuracy can be guaranteed a priori),
- an application of semidefinite optimization to classification,
whose principle consists in separating different classes of patterns
using ellipsoids defined in the feature space (this approach was
successfully applied to the prediction of student grades).

However, our research focussed on a much less studied category of
convex problems which does not rely on self-dual cones, i.e.
structured problems whose dual is formulated very differently from
the primal. We studied in particular
- geometric optimization, developed in the late sixties, which
possesses numerous application in the field of engineering
(entropy optimization, used in information theory, also belongs to
this class of problems)
- l_p-norm optimization, a generalization of linear and convex
quadratic optimization, which allows the formulation of constraints
built around expressions of the form |ax+b|^p (where p is a fixed
exponent strictly greater than 1).

For each of these classes of problems, we introduced a new type of
convex cone that made their formulation as standard conic problems
possible. This allowed us to derive very simplified proofs of the
classical duality results pertaining to these problems, notably weak
duality (a mere consequence of convexity) and the absence of a
duality gap (strong duality property without any constraint
qualification, which does not hold in the general convex case). We
also uncovered a very surprising result that stipulates that
geometric optimization can be viewed as a limit case of l_p-norm
optimization. Encouraged by the similarities we observed, we
developed a general framework that encompasses these two classes of
problems and unifies all the previously obtained conic formulations.

We also brought our attention to the design of interior-point
methods to solve these problems. The theory of polynomial algorithms
for convex optimization developed by Nesterov and Nemirovski asserts
that the main ingredient for these methods is a computable
self-concordant barrier function for the corresponding cones. We
were able to define such a barrier function in the case of
l_p-norm optimization (whose parameter, which is the main
determining factor in the algorithmic complexity of the method, is
proportional to the number of variables in the formulation and
independent from p) as well as in the case of the general
framework mentioned above.

Finally, we contributed a survey of the self-concordancy property,
improving some useful results about the value of the complexity
parameter for certain categories of barrier functions and providing
some insight on the reason why the most commonly adopted definition
for self-concordant functions is the best possible.
Document type :
Mathématiques [math]. Polytechnic College of Mons, 2001. Français
Contributor : François Glineur <>
Submitted on : Thursday, September 9, 2004 - 9:54:50 PM
Last modification on : Wednesday, June 1, 2016 - 11:27:43 PM
Document(s) archivé(s) le : Thursday, September 13, 2012 - 11:50:37 AM


  • HAL Id : tel-00006861, version 1



François Glineur. Topics in Convex Optimization: Interior-Point Methods, Conic Duality and Approximations. Mathématiques [math]. Polytechnic College of Mons, 2001. Français. <tel-00006861>



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