CERFACS CERFACS Scientific Activity Report Jan. 2010 – Dec. 2011

PARALLEL ALGORITHMS PROJECT
7.16 An active set trust-region method for derivative-free nonlinear
bound-constrained optimization.
S. Gratton : INPT-IRIT, UNIVERSITY OF TOULOUSE AND ENSEEIHT, France ; Ph. L. Toint : FUNDP
UNIVERSITY OF NAMUR, Belgium ; A. Tröltzsch : CERFACS, France
In [ALG59], we consider an implementation of a recursive model-based active-set trust-region method
for solving bound-constrained nonlinear non-convex optimization problems without derivatives using
the technique of self-correcting geometry proposed by K. Scheinberg and Ph. L. Toint [Self-correcting
geometry in model-based algorithms for derivative-free unconstrained optimization. SIAM Journal on
Optimization, 20(6) :3512-3532, 2010]. Considering an active-set method in bound-constrained modelbased
optimization creates the opportunity of saving a substantial amount of function evaluations. It
allows us to maintain much smaller interpolation sets while proceeding optimization in lower-dimensional
subspaces. The resulting algorithm is shown to be numerically competitive.
7.17 A hybrid optimization algorithm for gradient-based and
derivative-free optimization.
S. Gratton : INPT-IRIT, UNIVERSITY OF TOULOUSE AND ENSEEIHT, France ; Ph. L. Toint : FUNDP
UNIVERSITY OF NAMUR, Belgium ; A. Tröltzsch : CERFACS, France
A known drawback of derivative-free optimization (DFO) methods is the difficulty to cope with higher
dimensional problems. When the problem dimension exceeds a few tens of variables, a pure DFO method
becomes rather expensive in terms of number of function evaluations. For this reason, using gradient
information, if accessible, is highly useful in the context of efficient optimization in practice (even if it
is expected to be noisy). This applies especially when working with real-life applications as in aerodynamic
shape optimization. It is well known that when the gradient is known, the L-BFGS method is a very efficient
method for solving bound-constrained optimization problems. Coming back to the derivation of the BFGS
method, it is possible to see it as a way to correct the Hessian information using the so-called secant
equation information. We would like to generate a set of Hessian updates, that would generalize the L-
BFGS approach to situations where the function or the gradient are approximated. We propose a family
of algorithms that will both contain the derivative-free approach and the L-BFGS method, and that would
therefore be able to optimally take into account the error occurring in the cost function or gradient of the
problem.
7.18 How much gradient noise does a gradient-based linesearch
method tolerate ?
S. Gratton : INPT-IRIT, UNIVERSITY OF TOULOUSE AND ENSEEIHT, France ; Ph. L. Toint : FUNDP
UNIVERSITY OF NAMUR, Belgium ; A. Tröltzsch : CERFACS, France
Among numerical methods for smooth unconstrained optimization, gradient-based linesearch methods, like
quasi-Newton methods, may work quite well even in the presence of relatively high amplitude noise in the
gradient of the objective function. We present some properties on the amplitude of this noise which ensure
a descent direction for such a method. Exploiting this bound, we also discuss conditions under which global
convergence can be guaranteed. More details can be found in [ALG59].
CERFACS ACTIVITY REPORT 29