Sequential Competitive Facility Location: Exact and Approximate Algorithms

We study a competitive facility location problem (CFLP), where two firms sequentially open new facilities within their budgets, in order to maximize their market shares of demand that follows a probabilistic choice model. This process is a Stackelberg game and admits a bilevel mixed-integer nonlinear program (MINLP) formulation. We derive an equivalent, single-level MINLP reformulation … Read more

Learning Symbolic Expressions: Mixed-Integer Formulations, Cuts, and Heuristics

In this paper we consider the problem of learning a regression function without assuming its functional form. This problem is referred to as symbolic regression. An expression tree is typically used to represent a solution function, which is determined by assigning operators and operands to the nodes. The symbolic regression problem can be formulated as … Read more

An Exact Projection-Based Algorithm for Bilevel Mixed-Integer Problems with Nonlinearities

We propose an exact global solution method for bilevel mixed-integer optimization problems with lower-level integer variables and including nonlinear terms such as, \eg, products of upper-level and lower-level variables. Problems of this type are extremely challenging as a single-level reformulation suitable for off-the-shelf solvers is not available in general. In order to solve these problems … Read more

Solving AC Optimal Power Flow with Discrete Decisions to Global Optimality

We present a solution framework for general alternating current optimal power flow (AC OPF) problems that include discrete decisions. The latter occur, for instance, in the context of the curtailment of renewables or the switching of power generation units and transmission lines. Our approach delivers globally optimal solutions and is provably convergent. We model AC … Read more

Global Optimization for the Multilevel European Gas Market System with Nonlinear Flow Models on Trees

The European gas market is implemented as an entry-exit system, which aims to decouple transport and trading of gas. It has been modeled in the literature as a multilevel problem, which contains a nonlinear flow model of gas physics. Besides the multilevel structure and the nonlinear flow model, the computation of so-called technical capacities is … Read more

On Refinement Strategies for Solving MINLPs by Piecewise Linear Relaxations: A Generalized Red Refinement

We investigate the generalized red refinement for n-dimensional simplices that dates back to Freudenthal in a mixed-integer nonlinear program (MINLP) context. We show that the red refinement meets sufficient convergence conditions for a known MINLP solution framework that is essentially based on solving piecewise linear relaxations. In addition, we prove that applying this refinement procedure … Read more

The confined primal integral

It is a challenging task to fairly compare local solvers and heuristics against each other and against global solvers. How does one weigh a faster termination time against a better quality of the found solution? In this paper, we introduce the confined primal integral, a new performance measure that rewards a balance of speed and … Read more

Improved optimization models for potential-driven network flow problems via ASTS orientations

The class of potential-driven network flow problems provides important models for a range of infrastructure networks that lead to hard-to-solve MINLPs in real-world applications. On large-scale meshed networks the relaxations usually employed are rather weak due to cycles in the network. To address this situation, we introduce the concept of ASTS orientations, a generalization of … Read more

Improving relaxations for potential-driven network flow problems via acyclic flow orientations

The class of potential-driven network flow problems provides important models for a range of infrastructure networks. For real-world applications, they need to be combined with integer models for switching certain network elements, giving rise to hard-to-solve MINLPs. We observe that on large-scale real-world meshed networks the usually employed relaxations are rather weak due to cycles … Read more

Γ-counterparts for robust nonlinear combinatorial and discrete optimization

Γ-uncertainties have been introduced for adjusting the degree of conservatism ofrobust counterparts of (discrete) linear optimization problems under interval uncertainty. Thisarticle’s contribution is a generalization of this approach to (mixed-integer) nonlinear optimizationproblems. We focus on the cases in which the uncertainty is linear but also derive formulationsfor the general case. We present cases where the … Read more