# Practical, Short example of Mixed Integer Conic Program

Mixed Integer Conic Programs is a family of Mixed Integer Programs which are convex in all non integer variables. I am giving presentation on Mixed Integer technology. A large part of the presentation will be about MILP, it's tricks and common problem structures. I will also make people aware of MINLP by giving a practical non-convex MINLP. I aim to do the same with a convex MINLP. Is someone aware of an easy to explain[1] MIP using a Second-order, Rotated second-order, Positive semidefinite or Primal exponential which has a practical application?

[1] It should not involve power networks, trajectory optimisation and ideally fit on a slide. Why the cone is needed should be obvious from the problem description.

Your opening sentence could be more accurately written as Mixed Integer Conic Programs are a family of Mixed Integer Programs whose continuous relaxations are (convex) conic programs.

One easy to understand example is best variable subset selection by Mixed-Integer Least Squares, as presented in BEST SUBSET SELECTION VIA A MODERN OPTIMIZATION LENS, D IMITRIS B ERTSIMAS, ANGELA K ING AND R AHUL M AZUMDER,The Annals of Statistics, 2016, Vol. 44, No. 2, 813–852

• Wireless network design
• Portfolio optimization subject to transaction costs, cardinality constraints, and/or trading size constraints
• Convex piecewise linear regression

The YALMIP wiki has an example of nonconvex regression arising from non-convex penalty function, which is handled as MISOCP.

• Max-Cut (very cleverly, the non-convex rank constraint for the SDP can be relaxed (omitted) with no effect on optimal solution because every integer solution satisfies the omitted rank constraint; therefore the non-convexity has been moved from the rank constraint entirely to the integer (binary) constraints)
• Compressed Sensing
• Truss Topology Design.

Section 9.2 of A Tutorial on Geometric Programming, S. Boyd, S.-J. Kim, L. Vandenberghe, and A. Hassibi has a Mixed-Integer Geometric Programming example for Digital Circuit Gate Scaling in which the scale factors must take integer values. So this would be a Mixed-Integer Exponential Cone problem.

You may also find the video, The conic advantage in MINLP, Henrik A. Friberg: to be of interest.