Section: Scientific Foundations
Non-linear integer programming
Many engineering, management, and scientific applications involve not only discrete decisions, but also nonlinear relationships that significantly affect the feasibility and optimality of solutions. Mixed-integer nonlinear programming (MINLP) problems combine the difficulties of MIP with the challenges of handling nonlinear functions. MINLP is one of the most flexible modeling paradigms available. Hence, an expanding body of researchers and practitioners, including engineers, operations managers, economists, statisticians, computer scientists, and mathematical programmers are now interested in solving large-scale MINLPs (for recent examples of studies based on MINLP models, see, e.g.,  ,  ,  ,  ,  ,  ).
The wealth of applications that can be accurately modeled by using MINLP is not yet matched by the capability of available optimization solvers. Both MIP and nonlinear programming (NLP) have witnessed tremendous progress over the past 15 years. Some of the factors that have gone into the development of effective MIP algorithms and powerful academic, open source, and commercial MIP solvers are described above; similarly, new paradigms and a better theoretical understanding have created faster and more reliable NLP solvers that work well even under adverse conditions such as failures of constraint qualifications (more details on recent improvements in NLP solvers, as well as on challenges that remain, can be found in  ,  ,  ,  ,  , among others.)
The time is right to synthesize these advances and inspire new ideas in order to transform MINLP into an area in which researchers and practitioners can access robust tools and methods capable of solving a wide range of important, commonly occurring decision support problems. While there remains enormous room for progress, initial efforts towards the development of such algorithmic tools are already under way. Our team is involved in the enhancement of existing ideas, and the development of new ones, towards these ends (see, e.g.,  , ).