We propose an algorithm for solving bilevel problems with mixed-integer convex-quadratic upper level as well as convex-quadratic and continuous lower level. The method is based on a classic branch-and-bound procedure, where branching is performed on the integer constraints and on the complementarity constraints resulting from the KKT reformulation of the lower-level problem. However, instead of … Read more
Bilevel problems are used to model the interaction between two decision makers in which the lower-level problem, the so-called follower’s problem, appears as a constraint in the upper-level problem of the so-called leader. One issue in many practical situations is that the follower’s problem is not explicitly known by the leader. For such bilevel problems … Read more
Bilevel optimization is a very active field of applied mathematics. The main reason is that bilevel optimization problems can serve as a powerful tool for modeling hierarchical decision making processes. This ability, however, also makes the resulting problems challenging to solve—both in theory and practice. Fortunately, there have been significant algorithmic advances in the field … Read more
Bilevel optimization has been successfully applied to many important machine learning problems. Algorithms for solving bilevel optimization have been studied under various settings. In this paper, we study the nonconvex-strongly-convex bilevel optimization under a decentralized setting. We design decentralized algorithms for both deterministic and stochastic bilevel optimization problems. Moreover, we analyze the convergence rates of … Read more
In this paper we present BilevelJuMP, a new Julia package to support bilevel optimization within the JuMP framework. The package is a Julia library that enables the user to describe both upper and lower-level optimization problems using the JuMP algebraic syntax. Due to the generality and flexibility our library inherits from JuMP’s syntax, our package … Read more
Bilevel optimization is one of the fundamental problems in machine learning and optimization. Recent theoretical developments in bilevel optimization focus on finding the first-order stationary points for nonconvex-strongly-convex cases. In this paper, we analyze algorithms that can escape saddle points in nonconvex-strongly-convex bilevel optimization. Specifically, we show that the perturbed approximate implicit differentiation (AID) with … Read more
It is well known that bilevel optimization problems are hard to solve both in theory and practice. In this paper, we highlight a further computational difficulty when it comes to solving bilevel problems with continuous but nonconvex lower levels. Even if the lower-level problem is solved to ɛ-feasibility regarding its nonlinear constraints for an arbitrarily … Read more
Developing solution methods for mixed-integer bilevel problems is known to be a challenging task – even if all parameters of the problem are exactly known. Many real-world applications of bilevel optimization, however, involve data uncertainty due to some kind of bounded rationality. We study mixed-integer min-max problems with a follower who faces uncertainties regarding the … Read more
Two-level stochastic optimization formulations have become instrumental in a number ofmachine learning contexts such as continual learning, neural architecture search, adversariallearning, and hyperparameter tuning. Practical stochastic bilevel optimization problemsbecome challenging in optimization or learning scenarios where the number of variables ishigh or there are constraints. In this paper, we introduce a bilevel stochastic gradient method … Read more
In this article we address the problem of finding lower bounds for small Ramsey numbers $R(m,n)$ using circulant graphs. Our constructive approach is based on finding feasible colorings of circulant graphs using Integer Programming (IP) techniques. First we show how to model the problem as a Stackelberg game and, using the tools of bilevel optimization, … Read more