On Subadditive Duality for Conic Mixed-integer Programs

Abstract

In this paper, we show that the subadditive dual of a feasible conic mixed-integer program (MIP) is a strong dual whenever it is feasible. Moreover, we show that this dual feasibility condition is equivalent to feasibility of the conic dual of the continuous relaxation of the conic MIP. In addition, we prove that all known conditions and other 'natural' conditions for strong duality, such as strict mixed-integer feasibility, boundedness of the feasible set or essentially strict feasibility imply that the subadditive dual is feasible. As an intermediate result, we extend the so-called 'finiteness property' from full-dimensional convex sets to intersections of full-dimensional convex sets and Dirichlet convex sets. * burakkocuk@sabanciuniv.edu, Industrial Engineering Program, Sabancı University, Istanbul, Turkey 34956. † diego.moran@uai.cl, School of Business, Universidad Adolfo Ibáñez, Santiago, Chile 7941169.solution. Both of these properties are crucial in the development of effective optimization algorithms.It is well-known that linear programming (LP) and conic programming (CP) problems and their respective duals satisfy strong duality under mild conditions (such as boundedness and feasibility or strict feasibility) [6]. The case of mixed-integer linear programs (MILP)is more involved and requires the definition of a functional dual problem, the so-called subadditive dual, which is a strong dual when the data defining the primal problem is rational [15,21]. Recently, this latter duality result was extended to conic MIPs in [20] under a mixedinteger strict feasibility requirement, similar to the one needed in the continuous conic case. We also note that other types of duals have been studied in the case of general mixed-integer nonlinear programming (MINLP) problems. For instance, the Karush-Kuhn-Tucker (KKT) optimality conditions are generalized for MINLPs in [3], Lagrangian-based methods are used in [10,14], and other geometric [4] or algebraic [27] approaches are utilized to obtain strong duality results in particular cases.Conic MIPs: Conic MIP problems generalize MILPs and have significantly more expressive power in terms of modeling. To name just a few application areas, conic MIPs are used in options pricing [22], power distribution systems [17], Euclidean k-center problems [8] and engineering design [12]. We note here that all the conic MIPs used in these applications include binary variables, and that this feature, rather than being the exception, is a general rule when modeling real life problems.In spite of the growing interest in conic MIP applications, conic MIP solvers are not as mature as their MILP counterparts. Although the subadditive dual for linear/conic MIPs do not yield straightforward solution procedures, any dual feasible solution generates a valid inequality for the primal problem. Moreover, if strong duality holds, all cutting planes are equal to or dominated by a cutting plane obtained from such a solution [30,20]. We know that these valid inequalities are extremely useful for MILPs (...

Taming Binarized Neural Networks and Mixed-Integer Programs

“…Property (B) of Definition 8 together with Thm. 5 of (Kocuk and Morán 2019)), we thus obtain o-minimality.…”

“…The feasible set is a product of {0, 1} Lnm and the set S. For any value in {0, 1} Lnm , we obtain a finite value within S. The feasible set is then compact. Theorem 5 of (Kocuk and Morán 2019) then tells us that condition (B) of Definition 8 is satisfied. The objective function is a finite sum of loss functions for the original BNN, and as such it has a finite number of non-differentiable points, satisfying condition (C).…”

“…Subadditive dual In the context of MIPs, the notion of duality is much more involved than in convex optimization (Güzelsoy, Ralphs, and Cochran 2010). Only recently (Kocuk and Morán 2019;Morán R, Dey, and Vielma 2012), it is emerging that subadditive duals (Jeroslow 1978;Johnson 1980Johnson , 1974Johnson , 1979Guzelsoy and Ralphs 2007;Wolsey and Nemhauser 1999) can be used to establish strong duality for MIPs. To introduce the subadditive dual, we use the modern language of (Morán R, Dey, and Vielma 2012; Kocuk and Morán 2019):…”

“…Theorem 2 (Thm. 3 of (Kocuk and Morán 2019)) If the primal conic MIP (6) and the subadditive dual (7) are both feasible, then (7) is a strong dual of (6). Furthermore, if the primal problem is feasible, then the subadditive dual is feasible if and only if the conic dual of the continuous relaxation of (6) is feasible.…”