Mixed Integer NonLinear Programs featuring “On/Off” constraints: convex analysis and applications

Abstract

In this paper, we study MINLPs featuring "on/off" constraints. An "on/off" constraint is a constraint f (x) ≤ 0 that is activated whenever a corresponding 0-1 variable is equal to 1. Our main result is an explicit characterization of the convex hull of the feasible region when the MINLP consists of simple bounds on the variables and one "on/off" constraint defined by an isotone function f . When extended to general convex MINLPs, we show that this result yields tight lower bounds compared to classical formulations. This allows us to introduce new models for the delayconstrained routing problem in telecommunications. Numerical results show gains in computing time of up to one order of magnitude compared to state-of-the-art approaches.

Perspective Reformulations of the CTA Problem with L2 Distances

“…The results show that, as it could be expected, (MIQP) attains by far the worst results. Similarly to what has been reported several times [11,12,19,1,15,21], the use of "standard" PR techniques, i.e. (P/C) (and (SOCP), which is always worse) significantly improve on (MIQP) by delivering much better lower bounds, which in turn dramatically reduce the number of required B&C nodes.…”

“…We remark that the Perspective Reformulation approach is much more widely applicable than the simple quadratic case we consider here: it not only applies to the objective function but also to constraints f (z) ≤ 0 that are "activated" if and only if a binary variable y is 1, f can be any closed convex (possibly, SOCP-representable) function, z can be a vector whose feasible region can be any bounded polyhedron; see [7,26,18,11,21] and the recent survey [20]. Thus, some of the ideas developed here could be extendable to more complex situations.…”

Delay-Constrained Shortest Paths: Approximation Algorithms and Second-Order Cone Models

“…Clearly, ( 18)-( 21) is a more promising formulation than the one based on ( 11)-( 15): while it has the same number of integer variables and conic constraints, it has only 2m + 2 continuous variables, i.e., only m + 1 more than the structural ones, and clearly the minimum possible number to express the fractional terms in ( 2)-( 3). Furthermore, the continuous relaxation of this formulation is likely to be significantly stronger, since the "optimal" reformulation of some (small) fragments of the model has been used; this has been already shown to yield significant performance improvements in other applications [13][14][15][16]18], and the next section will show that the same holds here.…”

“…To avoid some of the issues in the previous formulation, we exploit a wellknown reformulation technique known as "Perspective Reformulation", that has been introduced in [13] and used in several applications with success (e.g. [14][15][16][17][18]), although usually in a different form than the one that is presented here. The approach is based on the well-known fact (e.g.…”

Segmentation with area constraints

“…Instead of directly enforcing u ∈ {0, 1} we formulate the optimization problem as a mixed integer nonlinear program (MINLP) (Hijazi et al (2009))…”