Theses

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

formulation),

- 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.

https://tel.archives-ouvertes.fr/tel-00006861

Contributor : François Glineur <>

Submitted on : Thursday, September 9, 2004 - 9:54:50 PM

Last modification on : Wednesday, June 19, 2019 - 1:15:50 AM

Long-term archiving on: : Thursday, September 13, 2012 - 11:50:37 AM

Contributor : François Glineur <>

Submitted on : Thursday, September 9, 2004 - 9:54:50 PM

Last modification on : Wednesday, June 19, 2019 - 1:15:50 AM

Long-term archiving on: : Thursday, September 13, 2012 - 11:50:37 AM