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Learning and propagating Lagrangian variable bounds for mixed-integer nonlinear programming. (English) Zbl 1382.90063
Gomes, Carla (ed.) et al., Integration of AI and OR techniques in constraint programming for combinatorial optimization problems. 10th international conference, CPAIOR 2013, Yorktown Heights, NY, USA, May 18–22, 2013. Proceedings. Berlin: Springer (ISBN 978-3-642-38170-6/pbk). Lecture Notes in Computer Science 7874, 355-361 (2013).
Summary: Optimization-based bound tightening (OBBT) is a domain reduction technique commonly used in nonconvex mixed-integer nonlinear programming that solves a sequence of auxiliary linear programs. Each variable is minimized and maximized to obtain the tightest bounds valid for a global linear relaxation. This paper shows how the dual solutions of the auxiliary linear programs can be used to learn what we call Lagrangian variable bound constraints. These are linear inequalities that explain OBBT’s domain reductions in terms of the bounds on other variables and the objective value of the incumbent solution. Within a spatial branch-and-bound algorithm, they can be learnt a priori (during OBBT at the root node) and propagated within the search tree at very low computational cost. Experiments with an implementation inside the MINLP solver SCIP show that this reduces the number of branch-and-bound nodes and speeds up solution times.
For the entire collection see [Zbl 1263.68017].

MSC:
90C11 Mixed integer programming
68T20 Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.)
90C30 Nonlinear programming
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
90C59 Approximation methods and heuristics in mathematical programming
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