Tight convex and concave relaxations are of high importance in the field of deterministic global optimization. We present a heuristic to tighten relaxations obtained by the McCormick technique. We use the McCormick subgradient propagation (Mitsos et al., SIAM J. Optim., 2009) to construct simple affine under- and overestimators of each factor of the original factorable … Read more
We describe BASBL, our implementation of the deterministic global optimization algorithm Branch-and-Sandwich for nonconvex/nonlinear bilevel problems, within the open-source MINOTAUR framework. The solver incorporates the original Branch-and-Sandwich algorithm and modifications proposed in the first part of this work. We also introduce BASBLib, an extensive online library of bilevel benchmark problems collected from the literature and … Read more
In this paper, we consider the global solution of bilevel programs involving nonconvex functions. We present algorithmic improvements and extensions to the recently proposed deterministic Branch-and-Sandwich algorithm (Kleniati and Adjiman, J. Glob. Opt. 60, 425–458, 2014), based on the theoretical results and heuristics. Choices in the way each step of the Branch-and-Sandwich algorithm is tackled, … Read more
In this paper we propose a new approach for finding global solutions of mixed-integer nonlinear optimization problems with ordinary differential equation constraints on networks. Instead of using a first discretize then optimize approach, we combine spatial and variable branching with appropriate discretizations of the differential equations to derive relaxations of the original problem. To construct … Read more
We present a flexible framework for general mixed-integer nonlinear programming (MINLP), called Minotaur, that enables both algorithm exploration and structure exploitation without compromising computational efficiency. This paper documents the concepts and classes in our framework and shows that our implementations of standard MINLP techniques are efficient compared with other state-of-the-art solvers. We then describe structure-exploiting … Read more
In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially on Positivstellens\”atze from the late 20th century (e.g., due to Stengle, Putinar, or Schm\”udgen) that certify positivity of a polynomial on an … Read more
In this paper, we consider a fundamental problem in modern digital communications known as multi-input multi-output (MIMO) detection, which can be formulated as a complex quadratic programming problem subject to unit-modulus and discrete argument constraints. Various semidefinite relaxation (SDR) based algorithms have been proposed to solve the problem in the literature. In this paper, we … Read more
There is an undeniable global shortage of skillful nurses. This is a problem of high priority, which is correlated to workforce management issues. These issues can be palliated by increasing nurses’ satisfaction based on flexible rosters using automated nurse rostering. This paper in concerned with nurse rostering based on constraint programming by satisfying global constraints, … Read more
We propose a global optimization algorithm for stochastic nonlinear programs that uses a specialized spatial branch and bound (BB) strategy to exploit the nearly decomposable structure of the problem. In particular, at each node in the BB scheme, a lower bound is constructed by relaxing the so-called non-anticipativity constraints and an upper bound is constructed … Read more
We derive a closed form description of the convex hull of mixed-integer bilinear covering set with bounds on the integer variables. This convex hull description is determined by considering some orthogonal disjunctive sets defined in a certain way. This description does not introduce any new variables, but consists of exponentially many inequalities. An extended formulation … Read more