Behrends, Sönke

English

We consider geometric approaches that assist in the solution process of mixed-integer nonlinear programming (MINLP). Amongst others, we compute half-spaces, seminorm balls and
ellipsoids that contain the relaxed feasible set. We also compute norm balls that contain
all optimal solutions, which is formalized in the concept of norm bounds. We then investigate how integrality arguments can be used to shrink these sets and potentially cut
off continuously feasible points. The norm and seminorm balls as well as the ellipsoids
make the integer part of the problem (given some assumptions) accessible to branch and
bound. For the branch and bound part, we also propose a class of underestimators that
yield tight lower bounds. The approaches always involve the task to optimally choose
a geometric object out of a whole class of similar geometric objects – for example, to
choose an ellipsoid of minimal volume containing the relaxed feasible set out of the collection of all axis-parallel ellipsoids. For polynomial objective and constraint functions,
these auxiliary problems, or approximations thereof, become tractable by using sum of
squares programming and tools from real algebra. These auxiliary problems can then
be implemented and assist in the solution process of a given problem. We also present
results that guarantee that the approximate solutions converge eventually to the optimal
solution of the auxiliary problem. A subset of the approaches has been implemented
and we demonstrate that they work in computer experiments

Georg-August-University Göttingen

Geometric and algebraic approaches to mixed-integer polynomial optimization using sos programming

https://ediss.uni-goettingen.de/bitstream/11858/00-1735-0000-0023-3F9C-9/1/dissertation.pdf

Geometric and algebraic approaches to mixed-integer polynomial optimization using sos programming

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Georg-August-University Göttingen