On Solving Mixed-Integer Constraint Satisfaction Problems with Unbounded Variables

Many mixed-integer constraint satisfaction problems and global optimization problems contain some variables with unbounded domains. Their solution by branch and bound methods to global optimality poses special challenges as the search region is infinitely extended. Many usually strong bounding methods lose their efficiency or fail altogether when infinite domains are involved. Most implemented branch and bound solvers add artificial bounds to make the problem bounded, or require the user to add these. However, if these bounds are too small, they may exclude a solution, while when they are too large, the search in the resulting huge but bounded region may be very inefficient. Moreover, if the global solver must provide a rigorous guarantee (as for the use in computer-assisted proofs), such artificial bounds are not permitted without justification by proof. We developed methods based on compactification and projective geometry as well as asymptotic analysis to cope with the unboundedness in a rigorous manner. Based on projective geometry we implemented two different versions of the basic idea, namely (i) projective constraint propagation, and (ii) projective transformation of the variables, in the rigorous global solvers COCONUT and GloptLab. Numerical tests demonstrate the capability of the new technique, combined with standard pruning methods, to rigorously solve unbounded global problems. In addition, we present a generalization of projective transformation based on asymptotic analysis. Compactification and projective transformation, as well as asymptotic analysis, are fruitless in discrete situations but they can very well be applied to compute bounded relaxations, and we will present methods for doing that in an efficient manner.