MINLPs (Mixed Integer Nonlinear Programs) appear in many different applications in engineering design, computational chemistry, computational biology, communication, finance and other areas. Only few codes for solving nonconvex MINLP are currently available. In particular, there is a lack of MINLP methods for solving large-scale structured MINLPs arising in real-world applications.
In constrast, there exist powerful codes for solving large-scale MILPs (Mixed Integer Linear Programs). These codes are based on a branch-and-cut framework combined with methods from constraint programming. Main difficulties in generalizing MILP-techniques to MINLP are: (i) the LP relaxation has to be replaced by a different relaxation, which is often not tight enough or expensive to generate, (ii) the computation of local solutions can be expensive and (iii) it can be difficult to derive efficient cuts.
As a result, only medium-size MINLPs can usually be solved by branch-and-cut. In practice large problems are often solved either by MILP approximation or by meta-heuristics combined with local search methods.
My research on new MINLP solution techniques includes the following topics: