Current research includes:

Bounding strategies for global and robust dynamic optimisation

Dynamic optimisation problems encountered in various engineering and scientific fields frequently exhibit multiple suboptimal solutions. A prototypical example is in the fields of model identification, where failure to determine the best possible fit can lead to false conclusions regarding the validity of a candidate model. Other applications are in the field of robust and scenario-integration optimisation.

In this project, we are investigating deterministic global optimisation methods, such as spatial branch-and-bound, which have the appealing property that a global optimum can be located, at an arbitrary precision, in a finite number of iterations. Pivotal to the success of these procedures is the ability to compute tight estimators for the solutions of the underlying differential equations (ODEs).

Among the approaches currently available for parametric ODEs, the method of differential inequalities proceeds by constructing an auxiliary system of ODEs, the solutions of which provide the desired estimators, either as interval bounds or as convex/concave relaxations. We have developed an approach that combines differential inequalities with Taylor models, namely estimators which consist of a multivariate polynomial part and an interval remainder term.

The advantage of propagating Taylor model estimators, as opposed to interval bounds, is that the former enjoy a high-order of convergence, which makes their use particularly attractive in the context of global dynamic optimisation. An illustration of the resulting bounds for Taylor models of various orders is shown in the figure opposite for the case study of a Lotka-Volterra (predator-prey) system.

Another class of approaches builds upon interval methods for ODEs to determine a rigorous enclosure of the ODE solution set. Traditional interval methods discretize the integration domain into a finite number of steps, and each step proceeds in two phases.

In the first phase, an a priori enclosure of the solutions over the current integration step is computed; in the second phase, a tightened enclosure at the end of the current step is then computed, with special care taken to mitigate the wrapping effect.

An extension of this approach was proposed recently, whereby Taylor models are used instead of simple natural interval extensions in order to reduce the dependency problem. In this project, we went one step further and computed convex/concave relaxations based on a new type of Taylor models, called McCormick-Taylor models.

For a majority of problems, the methods based on Taylor models and McCormick-Taylor models have proved to be more efficient for global dynamic optimisation than the ones based on interval analysis and McCormick relaxations, as a result of their higher order of convergence.

Dynamic optimisation problems with up to 10-12 decision variables could be solved to global optimality with such bounding techniques. All these methods are automated in the software library MC++ that is currently developed in the group of Dr. Chachuat.

Global optimisation of transient processes

Dynamic optimisation problems encountered in various engineering and scientific fields frequently exhibit multiple suboptimal solutions. A prototypical example is in the fields of model identification, where failure to determine the best possible fit can lead to false conclusions regarding the validity of a candidate model.

Other applications are in the field of robust and scenario-integration optimisation. Here, we are investigating deterministic global optimisation methods, such as spatial branch-and-bound, which have the appealing property that a global optimum can be located, at an arbitrary precision, in a finite number of iterations.

Pivotal to the success of these procedures is the ability to compute tight bounds or, better, convex/concave relaxations for the solutions of the underlying differential equations. Our current focus is on discretize-then-relax approaches that build upon verified ODE methods to compute the bounds, and their extension to handle more general DAE systems.