Logic-Based Outer Approximation for the Design of Discrete-Continuous Dynamic Systems with Implicit Discontinuities

We address the optimization of discrete-continuous dynamic optimization problems using a disjunctive multistage modeling framework, with implicit discontinuities, which increases the problem complexity since the number of continuous phases and discrete events is not known a-priori. After setting a fixed alternative sequence of modes, we convert the infinite-dimensional continuous mixed-logic dynamic (MLDO) problem into a finite dimensional discretized GDP problem by orthogonal collocation on finite elements. We use the Logic-based Outer Approximation algorithm to fully exploit the structure of the GDP representation of the problem. This modelling framework is illustrated with an optimization problem with implicit discontinuities (diver problem).The authors acknowledge financial support from «Estancias de movilidad en el extranjero "Jose Castillejo" (JC2011-0054)» of the Spanish “Ministerio de Educación”, and from the Spanish “Ministerio de Ciencia e Innovacion” (CTQ2012-37039-C02-02)

Branch-and-Prune Search Strategies for Numerical Constraint Solving

When solving numerical constraints such as nonlinear equations and inequalities, solvers often exploit pruning techniques, which remove redundant value combinations from the domains of variables, at pruning steps. To find the complete solution set, most of these solvers alternate the pruning steps with branching steps, which split each problem into subproblems. This forms the so-called branch-and-prune framework, well known among the approaches for solving numerical constraints. The basic branch-and-prune search strategy that uses domain bisections in place of the branching steps is called the bisection search. In general, the bisection search works well in case (i) the solutions are isolated, but it can be improved further in case (ii) there are continuums of solutions (this often occurs when inequalities are involved). In this paper, we propose a new branch-and-prune search strategy along with several variants, which not only allow yielding better branching decisions in the latter case, but also work as well as the bisection search does in the former case. These new search algorithms enable us to employ various pruning techniques in the construction of inner and outer approximations of the solution set. Our experiments show that these algorithms speed up the solving process often by one order of magnitude or more when solving problems with continuums of solutions, while keeping the same performance as the bisection search when the solutions are isolated.Comment: 43 pages, 11 figure

Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data

Constraint Programming (CP) has proved an effective paradigm to model and solve difficult combinatorial satisfaction and optimisation problems from disparate domains. Many such problems arising from the commercial world are permeated by data uncertainty. Existing CP approaches that accommodate uncertainty are less suited to uncertainty arising due to incomplete and erroneous data, because they do not build reliable models and solutions guaranteed to address the user's genuine problem as she perceives it. Other fields such as reliable computation offer combinations of models and associated methods to handle these types of uncertain data, but lack an expressive framework characterising the resolution methodology independently of the model. We present a unifying framework that extends the CP formalism in both model and solutions, to tackle ill-defined combinatorial problems with incomplete or erroneous data. The certainty closure framework brings together modelling and solving methodologies from different fields into the CP paradigm to provide reliable and efficient approches for uncertain constraint problems. We demonstrate the applicability of the framework on a case study in network diagnosis. We define resolution forms that give generic templates, and their associated operational semantics, to derive practical solution methods for reliable solutions.Comment: Revised versio