The original contributions of this work are:
The other ideas used inside CONDOR are mostly coming from recent
work of M.J.D.Powell [Pow00].
- The free, stand-alone implementation in C++ of the
whole algorithm. Most optimizer are expensive piece of software.
Usually ``free'' optimizers are using expensive, external,
unavailable libraries (such libraries are usually the Harwell
Libraries, the NAG tools, the NPSOL non-linear solver,...). In
CONDOR, there is no call to expensive, external libraries.
- The algorithm used for the parallelization of the
- The algorithm used for the constrained case.
- The assembly of different well-known algorithms in one code
(In particular, the code of the Chapter 4 is a
completely new implementation of the Moré and Sorensen
algorithm which has never been used in conjunction with
quadratical model build by interpolation).
- The bibliographical work needed to:
- Understand and implement all the parts of the algorithm.
- Acquire a huge amount of background knowledge in constrained
and unconstrained continuous optimization. This is necessary to be
able to review all the available techniques and choose the best
promising one. However, I will not present here a full review of
state-of-the-art techniques (for a starting point, see the
excellent book ``Trust-region Methods'' by Andrew R. Conn,
Nicholas I.M.Gould, and Philippe L.Toint [CGT00a]). A very short
review for the constrained case is given in Chapter
8. In particular, this Section contains:
- A short, concise and original introduction to Primal-Dual
methods for non-linear optimization. The link between Primal-Dual
methods and Barrier methods is motivated and intuitively
- A short, complete, intuitive and original description and
justification of the Second Order Step (SOC) used inside SQP
- The redaction of the complete description of the CONDOR
algorithm (``COnstrained, Non-linear, Direct, parallel
Optimization using trust Region method for high-computing load
function''). This description is intended for graduate school
level and requires no a priori knowledge in the field of
continuous optimization. Most optimization books are very formal
and give no insight to the reader. This thesis is written in a
informal way, trying to give an intuition to the reader of what is
going on, rather than giving pure formal demonstration which are
usually hiding the intuition behind many formalism. The thesis is
a complete start from scratch (or nearly) and is thus a good
introduction to the optimization theory for the beginner.
- The comparison of the new algorithm CONDOR with other famous
algorithms: CFSQP, DFO, UOBYQA, PDS,etc. The experimental results
of this comparison shows very good performance of CONDOR compared
to the other algorithms. On nearly all the unconstrained problems,
CONDOR finds the optimal point of the objective function in
substantially fewer evaluation of the objective function than its
competitor, especially when more than one CPU is used. In the
constrained case, the performances of CONDOR are comparable to the
best algorithms. Preliminary results indicates that on box and
linear constraints only, CONDOR also outperforms its competitors.
However, more numerical results are needed to assert definitively
this last statement.
- The application of a gradient-based optimizer on a objective
function based on CFD code is unusual. This approach is usually
rejected and is considered by many researcher as a ``dead-end''.
The validation of the usefulness of the gradient-based approach is
a primordial result.
- The algorithm to solve the secondary Trust-Region subproblem
described in Section 5 is slightly different that
the algorithm proposed by Powell. Numerical results exhibit better
performances for the CONDOR algorithm.
Frank Vanden Berghen