Outer approximation for global optimization of mixed-integer quadratic bilevel problems

Abstract

Bilevel optimization problems have received a lot of attention in the last years and decades. Besides numerous theoretical developments there also evolved novel solution algorithms for mixed-integer linear bilevel problems and the most recent algorithms use branch-and-cut techniques from mixed-integer programming that are especially tailored for the bilevel context. In this paper, we consider MIQP-QP bilevel problems, i.e., models with a mixed-integer convex-quadratic upper level and a continuous convex-quadratic lower level. This setting allows for a strong-duality-based transformation of the lower level which yields, in general, an equivalent nonconvex single-level reformulation of the original bilevel problem. Under reasonable assumptions, we can derive both a multi- and a single-tree outer-approximation-based cutting-plane algorithm. We show finite termination and correctness of both methods and present extensive numerical results that illustrate the applicability of the approaches. It turns out that the proposed methods are capable of solving bilevel instances with several thousand variables and constraints and significantly outperform classical solution approaches.

Drone-Delivery Network for Opioid Overdose -- Nonlinear Integer Queueing-Optimization Models and Methods

“…While instances of the special variant of model R-MILP when the maximal number M of drones at each DB is fixed and equal to 1 (i.e., each open DB is an M/G/1 system) can be solved, preliminary experiments reveal that state-of-the-art solvers are however unable to solve, within 1 hour, the root node of the continuous relaxation of the moderate-sized R-MILP problem instances for M ≥ 2. To overcome this issue, we derive an MILP relaxation for problem R − MILP using lazy constraints (see, e.g., [32,35]) and embed it in an outer approximation algorithm. The motivation is to alleviate the issue caused by the significantly lifted decision and constraint space due to the linearization method.…”

SOCP-Based Disjunctive Cuts for a Class of Integer Nonlinear Bilevel Programs

“…When it comes to solution approaches, a distinction between problems with convex and non-convex follower problem can be made. For BPs with a convex follower problem, single-level reformulation techniques based on, e.g., Karush-Kuhn-Tucker optimality conditions or strong duality (see, e.g., [12,30,32]) can be used. For MIBLPs with integrality restrictions on (some of) the follower variables, state-of-the-art methods are usually based on B&C (see, e.g., [18,19,49]).…”

“…A solution algorithm for mixed-IBNPs proposed in [40] by Lozano and Smith approximates the value function by dynamically inserting additional variables and big-M type constraints. Recently, Kleinert et al [30] considered BPs with a mixed-integer convex-quadratic leader and a continuous convex-quadratic follower problem. The method is based on outer approximation after the problem is reformulated into a singlelevel one using strong duality and convexification.…”

On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level

“…In the last years, algorithmic research on bilevel optimization focused on more and more complicated lower-level problems such as mixed-integer linear models (Fischetti et al 2017(Fischetti et al , 2018, nonlinear but still convex models (Kleinert, Grimm, et al 2021), or problems in the lower level with uncertain data (Buchheim and Henke 2022;Burtscheidt and Claus 2020). When it comes to the situation of a lowerlevel problem with continuous nonlinearities there is not too much literature-in particular in comparison to the case in which the lower-level problem is convex; see, e.g., Kleniati and Adjiman (2011, 2014a,b, 2015, Mitsos (2010), Mitsos et al (2008), , Paulavičius, Gao, et al (2020), and Paulavičius et al (2016).…”