Separation and Extension of Cover Inequalities for Conic Quadratic Knapsack Constraints with Generalized Upper Bounds

Abstract

Motivated by addressing probabilistic 0-1 programs we study the conic quadratic knapsack polytope with generalized upper bound (GUB) constraints. In particular, we investigate separating and extending GUB cover inequalities. We show that, unlike in the linear case, determining whether a cover can be extended with a single variable is -hard. We describe and compare a number of exact and heuristic separation and extension algorithms which make use of the structure of the constraints. Computational experiments are performed for comparing the proposed separation and extension algorithms. These experiments show that a judicious application of the extended GUB cover cuts can reduce the solution time of conic quadratic 0-1 programs with GUB constraints substantially.

Technical Note—A Conic Integer Optimization Approach to the Constrained Assortment Problem Under the Mixed Multinomial Logit Model

“…We refer the reader to Lobo et al (1998) and Alizadeh and Goldfarb (2003) for reviews of conic quadratic optimization and its applications. Although there is an extensive body of literature on convex conic quadratic optimization, development of conic optimization with integer variables is quite recent (Ç ezik and Iyengar, 2005;Narayanan, 2007, 2011;Atamtürk et al, 2013). With the growing availability of commercial solvers for these problems (for example, both CPLEX and Gurobi now include solvers for these models), conic quadratic integer models have recently been employed to address problems in portfolio optimization (Vielma et al, 2008), value-at-risk minimization (Atamtürk and Narayanan, 2008), machine scheduling (Aktürk et al, 2010), and supply chain network design (Atamtürk et al, 2012), airline rescheduling with speed control (Aktürk et al, 2014).…”

Optimal response to epidemics and cyber attacks in networks

“…In the following, we empirically compare (3) with two formulations: the first-order (union bound) formulation, with all bilinear terms in (5) satisfying r ijk = 0, and formulation (5) with r ijk = q ij = q ik . To evaluate each, we consider the relative error, defined as u(x (3) ) − u(x (5) ) u(x (3) ) , where x (·) is an optimal solution of the corresponding formulation (·). The computational experiments are run using the state-of-the art CPLEX solver, version 12.3, to compute (the linearized version of) (5) with r ijk = 0 and (5) with r ijk = q ij q ik for all i ∈ V , and {j, k} ∈ T i .…”

“…Such inequalities and their extensions have been used for general mixed‐integer programs (MIPs), and have also been specialized for specific applications; see, for example, , for a recent application to a network optimization problem. Further, cover inequalities have recently been extended for conic‐quadratic nonlinear formulations . However, we note that these recent extensions, as well as standard applications to knapsack constraints, all require that the left‐hand side of the associated constraint is monotone in ; for example, with linear knapsack constraints binary variables with negative coefficients need to be complemented (which in turn, increases the constant in right‐hand side).…”

“…a contradiction to the feasibility of x. Conversely, for an independent set S ⊂V , if x i = 0 for every i ∈V \ S, then x is feasible for (3). Further, the fact that q ij = 0 for {i, j} ∈ E \Ē implies that (16) holds.…”