Duality for mixed-integer convex minimization

Abstract

We extend in two ways the standard Karush-Kuhn-Tucker optimality conditions to problems with a convex objective, convex functional constraints, and the extra requirement that some of the variables must be integral. While the standard Karush-Kuhn-Tucker conditions involve separating hyperplanes, our extension is based on lattice-free polyhedra. Our optimality conditions allow us to define an exact dual of our original mixed-integer convex problem.

Optimality certificates for convex minimization and Helly numbers

“…where bd(C) denotes the boundary of C (to see this, consider the line segment connectings and x 0 and a point at which this line segment intersects bd(C)). Thus, an S-free neighborhood of x 0 can be interpreted as a "dual object" that provides a lower bound of the type (2). As a consequence, the following is true.…”

“…Proposition 1 (Strong duality). If there exists ∈ S and C ⊆ R n that is an S-free neighborhood of x 0 , such that equality holds in (2), thens is an optimal solution to (1).…”

“…Of course, the first question to settle is whether these three families actually enjoy strong duality, i.e., do we have strong duality between (1) and the F max -dual, F ∂ -dual and F ∂,S -dual? It turns out that the main result in [2] shows that for the mixed-integer case, i.e., S = C ∩ (Z n1 × R n2 ) for some convex set C, the F ∂ -dual enjoys strong duality under conditions of the Slater type from continuous optimization. It is not hard to strengthen their result to also show that the F max ∩ F ∂ -dual is a strong dual, under some additional assumptions.…”

Constructing Lattice-Free Gradient Polyhedra in Dimension Two

“…As neither z nor z + u 1 + u 2 can strictly cut both, we may assume that z strictly cuts z + u 1 and z + u 1 + u 2 strictly cuts z + u 2 . ⊓ ⊔ We update U by 'flipping' the columns of U to a new matrix U and preprocessing (z, U ) to satisfy (5). Flip(U ) denotes the matrix U obtained from this flipping, and Flip(U) denotes the unimodular set obtained after preprocessing (z, U ).…”

“…Let U 0 be a unimodular set satisfying (5). For i ∈ Z ≥0 set U i+1 = Flip(U i ) and preprocess U i+1 to satisfy (5). By Theorem 4 and Lemma 4, there exists T 1 ∈ Z ≥1 such that GP(U i ) ∩ U i contains an optimal solution of (CM) for all i ≥ T 1 .…”