Mixed-integer Bilevel Problem Research Articles

Simultaneous design and NMPC control under uncertainty and structural decisions: A discrete‐steepest descent algorithm

AbstractIn this article, we address the integration of design and nonlinear model‐based control under uncertainty and structural decisions for naturally ordered structures. We propose an algorithmic framework to determine the optimal location of process units or streams over an ordered discrete set that operates in closed‐loop with a model‐based controller. The formulation corresponds to a mixed‐integer bilevel problem (MIBLP) that is transformed into a single‐level mixed‐integer nonlinear problem (MINLP) using a KKT transformation strategy. In our methodology, the integer decisions are partitioned into subsets called external variables, such that the MINLP is decomposed into an integer‐based master problem and primal subproblems with fixed discrete variables. The master and primal problems are solved using a Discrete‐Steepest Descent Algorithm (D‐SDA). We illustrate the discrete‐based methodology in a case study for a binary distillation column. The D‐SDA showed an improved performance compared to a benchmark continuous‐based formulation using differentiable distribution functions (DDFs).

On a stochastic bilevel programming problem

AbstractIn this article, a mixed integer bilevel problem having a probabilistic knapsack constraint in the first level is proposed. The problem formulation is mainly motivated by practical pricing and service provision problems as it can be interpreted as a model for the interaction between a service provider and customers. A discrete probability space is assumed which allows a reformulation of the problem as an equivalent deterministic bilevel problem. The problem is further transformed into a linear bilevel problem, which in turn yields a quadratic optimization problem, namely the global linear complementarity problem. Based on this quadratic problem, a procedure to compute upper bounds on the initial problem by using a Lagrangian relaxation and an iterative linear minmax scheme is proposed. Numerical experiments confirm that the scheme practically converges.© 2011 Wiley Periodicals, Inc. NETWORKS, 2012

On a bi-level formulation to protect uncapacitated p-median systems with facility recovery time and frequent disruptions

We consider an uncapacitated p-median system that is subject to external manmade or natural disruptions and formulate the problem of protecting against the worst-case losses when taking into account facility recovery issues. The model is a mixed integer bi-level problem with integer variables controlled by both the upper and lower level. To solve it, we apply two exact decomposition methods: a decomposition algorithm based on a special type of valid inequalities and Benders decomposition coupled with variable reduction and some heuristic rules to speed up the resolution of the master problems. Although we compare the performance of the two decomposition approaches, for brevity, we only show here the Benders decomposition.

Parametric Integer Programming Algorithm for Bilevel Mixed Integer Programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader’s variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case, it yields a “better than fully polynomial time” approximation scheme with running time polynomial in the logarithm of the absolute precision. For the pure integer case where the leader’s variables are integer, and hence optimal solutions are guaranteed to exist, we present an algorithm which runs in polynomial time when the total number of variables is fixed.