Abstract
We study the simultaneous convexification of graphs of bilinear functions gk(x; y) = yAkx over x ∈ Ξ = {x ∈ [0, 1]n |Ex ≥ f} and y ∈ ∆m = {y ∈ Rm+ |1y ≤ 1}. We propose a constructive procedure to obtain a linear description of the convex hull of the resulting set. This procedure can be applied to derive convex and concave envelopes of certain bilinear functions, to study unary expansions of integer variables in mixed integer bilinear sets, and to obtain convex hulls of sets with complementarity constraints. Exploiting the structure of Ξ, the procedure naturally yields stronger linearizations for bilinear terms in a variety of practical settings. In particular, we demonstrate the effectiveness of the approach by strengthening the traditional dual formulation of network interdiction problems and report encouraging preliminary numerical results.
Original language | English (US) |
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Pages (from-to) | 1801-1833 |
Number of pages | 33 |
Journal | SIAM Journal on Optimization |
Volume | 27 |
Issue number | 3 |
DOIs | |
State | Published - 2017 |
Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2017 Society for Industrial and Applied Mathematics
Keywords
- Bilinear functions
- Convex hulls
- Cutting planes
- Envelopes
- Network interdiction