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

We call “on/off” constraint an algebraic constraint that is activated if and only if a corresponding boolean variable equals 1. Our main subject of interest is to derive tight convex formulations of Mixed Integer NonLinear Programs featuring “on/off” constraints. We study the simple set defined by one “on/off” constraint with bounded variables. Using Disjunctive Programming, we introduce convex hull formulations of this set defined in higher dimensional spaces. Because the large number of variables in these formulations appears to be practically disadvantageous, we concentrate our efforts on defining explicit projections into lower dimensional spaces. Based on these results, we present new formulations to a well-known telecommunication problem: routing several commodities subject to multiple delay constraints. Numerical results are presented to assess the efficiency of the new models.