Wednesday, December 6, 2017

Workshop: Mixed-integer conic optimization

We are pleased to announce a MOSEK workshop on Mixed-integer conic optimization taking place on Thursday, January 11th, 2018 at our place in the Symbion research park, Copenhagen.

The workshop is free and open to everyone. There will be coffee, refreshments and time for discussions. Please register through this form to help us with planning.

14:00 - 14:05   Welcome (Erling Andersen)
14:05 - 14:50   Tristan Gally
15:00 - 15:45   Julio C. Góez
16:00 - 16:45   Joachim Dahl
17:30+ optional dinner (Nørrebro Bryghus)

  • Tristan Gally, TU Darmstadt, Applications and Solution Approaches for Mixed-Integer Semidefinite ProgrammingMixed-integer semidefinite programming (MISDP) has received increasing attention in recent years. MISDPs appear in many applications either by adding combinatorial decisions to nonlinear problems with natural SDP-formulations or by reformulating combinatorial optimization problems to incorporate stronger SDP-relaxations. While mixed-integer second-order cone programming has been adapted by many commercial solvers, MISDP remains a challenging problem, which so far has mostly been tackled by solution-specific approaches.

    In this talk, we want to present some interesting applications for MISDP from both combinatorial and nonlinear optimization. Afterwards, we will discuss problem-independent solution approaches, mainly concentrating on nonlinear branch-and-bound. Particularly, we will explain the importance of the Slater constraint qualification and its implications for using interior-point methods within a branch-and-bound approach. We will further discuss enhancing techniques like dual fixing and warmstarts and give numerical results comparing the different solution approaches and different implementations.
  • Julio C. Góez, NHH, Disjunctive conic cuts: the good, the bad, and implementation In recent years, the generalization of Balas disjunctive cuts for mixed integer linear optimization problems to mixed integer non-linear optimization problems has received significant attention. Among these studies, mixed integer second order cone optimization (MISOCO) is a special case. For MISOCO one has the disjunctive conic cuts approach. That generalization introduced the concept of disjunctive conic cuts (DCCs) and disjunctive cylindrical cuts (DCyCs). Specifically, it showed that under some mild assumptions the intersection of those DCCs and DCyCs with a closed convex set, given as the intersection of a second order cone and an affine set, is the convex hull of the intersection of the same set with a parallel linear disjunction. The key element in that analysis was the use of pencils of quadrics to find close forms for deriving the DCCs and DCyCs. The first part of this talk will summarize the main results about DCCs and DCyCs including some results about valid conic inequalities for hyperboloids and non-convex quadratic cones when the disjunction is defined by parallel hyperplanes. In the second part, we will discussed some of the limitation of this approach to derive useful valid inequalities in the context of MISOCO. In the last part, we will briefly describe the software libraries that together constitute DisCO, a full-featured solver for MISOCP which we are currently used to explore the potential of DCCs and DCyCs.
  • Joachim Dahl, MOSEK, Mixed-integer conic optimization using MOSEK
    Recently Lubin showed that all the convex instances of the nonlinear mixed-integer bench library MINLPLIB2 can be reformulated as conic optimization problems using 5 different cone types which are the linear, the quadratic, the semidefinite, the exponential and the power cones. The former three cones belong to the class of symmetric cones whereas the latter two belong to the class of nonsymmetric cones. We call modelling with affine expressions and the five previously mentioned cone types extreme desciplined modelling. Based on Lubin et al. and the experience at our company we claim almost all practical convex optimization can be expressed using extreme disciplined modelling so it is a general framework. Now it is much easier to build optimization algorithms and software for extreme disciplined optimization models rather than for general convex (unstructured) convex problems because of limited and explicit structure. This fact is exploited in the software package MOSEK to be discussed. The software package MOSEK has for many years been able to solve conic optimization over the symmetric cones but in the upcoming version 9 MOSEK can also handle two nonsymmetric cones i.e. the exponential and the power cone. In this presentation we will discuss the continuous and mixed-integer conic optimizer in MOSEK. In addition extensive computational results are presented that illustrate the performance of MOSEK on problems including nonsymmetric cones.