# Mixed-Integer Nonlinear Optimization

## Final Report Summary - MINO (Mixed-Integer Nonlinear Optimization)

Complex decision-making in enterprises should involve mathematical optimization methods, because a “best choice” has to be made out of a huge number of feasible options. A mathematical description of such decision processes typically involves both “continuous” and “discrete” decisions. If the latter are present, the customary modeling approach is to use integer variables, which are also used to represent all possible nonlinearities, so that the remaining part of the model is linear. This leads to Mixed-Integer Linear Optimization (MILO) problems, which can be handled nowadays by many software packages, but are often very difficult to solve.
The difficulty of MILO problems is often due to the fact that objective functions or constraints, which are structurally nonlinear (e.g. quadratic), are linearized by introducing new integer variables. In many cases, it was observed that this is not the best way to proceed, as facing the nonlinearity directly without the new variables leads to much better mathematical formulations, which, in turn, leads to better results in case those formulations can be solved. However, the algorithmic technology for the resulting Mixed-Integer Nonlinear Optimization (MINO) problems is still at its early stage.
The present situation is that enterprises facing a MINO problem generally give up due to the lack of efficient solvers (in the form of software packages), or try to convert it to a MILO one that is often too hard to be solved in practice. On the other hand, in the academia there is now an increasing expertise in MINO, which is, however, hardly exported outside due to the lack of interaction with the industrial world.
It has been the purpose of this project to help satisfy the increasing demand for highly qualified researchers receiving, at the same time, a state-of-the-art scientific training from the academia and hands-on experience with real-world applications from the industry.
The researchers formed within this project, once recruited by an enterprise at the end of their training, have the potential to apply all the available knowledge to optimize complex decision-making in the real world.

The objectives of the project have been achieved with a tight combination between extremely highly qualified scientific training provided by the twelve partner institutions in Academia and exposure to applied problems provided by the industrial partners IBM, ORTEC and MAIOR.

The successful scientific training, which led already three out of eleven of our Early Stage Researchers to finish their PhD studies, has been organized around the following activities:

.. PhD courses in the partner academic institutions;
.. Mentoring of leading MINO scientists in the form of PhD supervision;
.. Soft-skill activities including organization of events, like workshop and sessions, writing of scientific proposals, presentations/organization for dissemination workshops, tutoring for undergraduate and master students;
.. PhD summer schools specifically tailored for the needs of the training in MINO problems.

Most of above scientific training activities have been attended, of course, by the three Experienced Researchers hired for eighteen months by the industrial partners. The mix of enthusiasm and experience between the two groups of fellows has been especially useful to leverage expectations and improving the smoothness of the training process.
As anticipated, the intense scientific program resulted in a significant number of scientific publications in top journals in the field of Mathematical Optimization and in three PhD dissertations already defended successfully at the University of Bologna, Italy, University of Tilburg, The Netherlands and ETH Zürich, Switzerland:

.. A. Baggio, “Towards Optimal Approximations for Firefighting and Related Problems, PhD Thesis, ETHZ, 2016.
.. M. Seminaroti, “Combinatorial Algorithms for the Seriation Problem”, PhD Thesis, Tilburg University, 2016.
.. S. Wiese, “On the Interplay of Mixed Integer Linear, Mixed Integer Nonlinear and Constraint Programming, PhD Thesis, University of Bologna, 2016.

(The PDFs of the dissertations are attached to the Periodic Report, file “PhD_Theses.zip”.)

The industrial training has been centered around the six months secondment that each of the Early Stage Researchers has spent working at one of the industrial partner facility. The three industrial partners have provided to the consortium quite different exposures to real-world areas. On the one hand, IBM, and especially the connection with the team developing the MILO solver CPLEX, gave the consortium a strong exposure to the challenges of developing commercial mathematical software. On the other hand, ORTEC and MAIOR gave the consortium a very detailed “picture” of the implications of using Mathematical Optimization in consulting, and with a rather different twist concerning the different areas of application, mainly energy for ORTEC, i.e. large, more global companies, and transportation for MAIOR, i.e. smaller, more local enterprises.

The dissemination activity performed by the entire consortium has been very intense. The strong relationship with the COST Action TD1207 “Mathematical Optimization in the Decision Support Systems for Efficient and Robust Energy Networks” has led to the organization of joint events both in terms of PhD schools and conferences, so as to enlarge the outreached scientific community to engineers, economists and industrial practitioners, especially in the energy sector. In addition, a Mathematical Challenge has been organized for the solution of a practical water treatment problem proposed by Shell. The participation to the Challenge both in terms of internal contribution, namely the Early Stage Researchers in secondment at ORTEC, the industrial partner in charge of the challenge, and external players, has been satisfactory, with software companies and academic institutions involved. Finally, a significant series of dissemination talks has been presented by the academic members of the consortium, mainly locally to their institutions.

As a matter of fact, one of the Experienced Researchers has been hired as a full-time scientist by the industrial partner he had worked on and one of the Early Stage Researchers that already graduated is now working as full-time scientists by a spin-off consulting company of the University of Bologna, specialized in Mathematical Optimization.