seminaire: T. Westerlund (LIX, Ecole polytechnique, 20/11/07 at 10:30)
Forum 'Annonces' - Sujet créé le 2007-11-03 par Leo Liberti
Tuesday, 20/11/2007. LIX, Ecole Polytechnique. Salle des Seminaires, 10:30
TRANSFORMATION TECHNIQUES WITH APPLICATIONS IN GLOBAL OPTIMIZATION
Professor Tapio Westerlund
Process and Systems Engineering, Åbo Akademi University
Biskopsgatan 8, FIN-20500 ÅBO, Finland
Email: tapio.westerlund@abo.fi
Some transformation techniques, in global optimization, are discussed
in this presentation. With the given techniques a general class of
non-convex MINLP problems, including signomial functions, can be
solved to global optimality. The transformation techniques are based
on exact power transformations and on the use of approximate inverse
transformations. By the power transformations signomial function can
be transformed to convex form. The feasible region of the original
non-convex MINLP problem can, on the other hand, be overestimated when
approximating the inverse power transformations by piecewise linear
functions. By power transformations and using piecewise linear
approximations of the inverse transformations the original non-convex
MINLP problem can, thus, be transformed to convex form having a
feasible region overestimating that of the original one. The
piecewise linear inverse transformations can efficiently be modelled
using continuous special ordered set formulations. The approximations
are required for all inverse transformations but SOS2 variable set are
required only for original variables involved in the transformations,
reducing the complexity of the convexified model. The optimal
solution of a non-convex MINLP problem is obtained by solving a
sequence of convex MINLP sub-problems, in which new grid points in the
piecewise linear functions are added for subsequent iterations. In
case the convex sub-problems are solved using a cutting plane method,
the transformation techniques can fully be integrated into the cutting
plane framework and the original non-convex MINLP problem be solved as
a sequence of MILP or LP problems only. The principles behind the
transformation techniques are given and some numerical examples are
used to illustrate how the global optimal solution is obtained, with
the given technique.
TRANSFORMATION TECHNIQUES WITH APPLICATIONS IN GLOBAL OPTIMIZATION
Professor Tapio Westerlund
Process and Systems Engineering, Åbo Akademi University
Biskopsgatan 8, FIN-20500 ÅBO, Finland
Email: tapio.westerlund@abo.fi
Some transformation techniques, in global optimization, are discussed
in this presentation. With the given techniques a general class of
non-convex MINLP problems, including signomial functions, can be
solved to global optimality. The transformation techniques are based
on exact power transformations and on the use of approximate inverse
transformations. By the power transformations signomial function can
be transformed to convex form. The feasible region of the original
non-convex MINLP problem can, on the other hand, be overestimated when
approximating the inverse power transformations by piecewise linear
functions. By power transformations and using piecewise linear
approximations of the inverse transformations the original non-convex
MINLP problem can, thus, be transformed to convex form having a
feasible region overestimating that of the original one. The
piecewise linear inverse transformations can efficiently be modelled
using continuous special ordered set formulations. The approximations
are required for all inverse transformations but SOS2 variable set are
required only for original variables involved in the transformations,
reducing the complexity of the convexified model. The optimal
solution of a non-convex MINLP problem is obtained by solving a
sequence of convex MINLP sub-problems, in which new grid points in the
piecewise linear functions are added for subsequent iterations. In
case the convex sub-problems are solved using a cutting plane method,
the transformation techniques can fully be integrated into the cutting
plane framework and the original non-convex MINLP problem be solved as
a sequence of MILP or LP problems only. The principles behind the
transformation techniques are given and some numerical examples are
used to illustrate how the global optimal solution is obtained, with
the given technique.