A Storm of Feasibility Pumps for Nonconvex MINLP

One of the foremost difficulties in solving Mixed Integer Nonlinear Programs, either with exact or heuristic methods, is to find a feasible point. We address this issue with a new feasibility pump algorithm tailored for nonconvex Mixed Integer Nonlinear Programs. Feasibility pumps are algorithms that iterate between solving a continuous relaxation and a mixed-integer relaxation of the original problems. Such approaches currently exist in the literature for Mixed-Integer Linear Programs and convex Mixed-Integer Nonlinear Programs: both cases exhibit the distinctive property that the continuous relaxation can be solved in polynomial time. In nonconvex Mixed Integer Nonlinear Programming such a property does not hold, and therefore special care has to be exercised in order to allow feasibility pumps algorithms to rely only on local optima of the continuous relaxation. Based on a new, high level view of feasibility pumps algorithms as a special case of the well-known successive projection method, we show that many possible different variants of the approach can be developed, depending on how several different (orthogonal) implementation choices are taken. A remarkable twist of feasibility pumps algorithms is that, unlike most previous successive projection methods from the literature, projection is "naturally" taken in two different norms in the two different subproblems. To cope with this issue while retaining the local convergence properties of standard successive projection methods we propose the introduction of appropriate norm constraints in the subproblems; these actually seem to significantly improve the practical performances of the approach. We present extensive computational results on the MINLPLib, showing the effectiveness and efficiency of our algorithm

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques