Joseph K. Scott

Generalized McCormick relaxations

Convex and concave relaxations are used extensively in global optimization algorithms. Among the various techniques available for generating relaxations of a given function, McCormick’s relaxations are attractive due to the recursive nature of their definition, which affords wide applicability and easy implementation computationally. Furthermore, these relaxations are typically stronger than those resulting from convexification or linearization procedures. This article leverages the recursive nature of McCormick’s relaxations to define a generalized form which both affords a new framework within which to analyze the properties of McCormick’s relaxations, and extends the applicability of McCormick’s technique to challenging open problems in global optimization. Specifically, relaxations of the parametric solutions of ordinary differential equations are considered in detail, and prospects for relaxations of the parametric solutions of nonlinear algebraic equations are discussed. For the case of ODEs, a complete computational procedure for evaluating convex and concave relaxations of the parametric solutions is described. Through McCormick’s composition rule, these relaxations may be used to construct relaxations for very general optimal control problems.

Input design for guaranteed fault diagnosis using zonotopes

a b s t r a c t An input design method is presented for guaranteeing the diagnosability of faults from the outputs of a system. Faults are modeled by discrete switches between linear models with bounded disturbances and measurement errors. Zonotopes are used to efficiently characterize the set of inputs that are guaranteed to lead to outputs that are consistent with at most one fault scenario. Provided that this set is nonempty, an element is then chosen that is minimally harmful with respect to other control objectives. This approach leads to a nonconvex optimization problem, but is shown to be equivalent to a mixed-integer quadratic program that can be solved efficiently. Methods are given for reducing the complexity of this program, including an observer-based method that drastically reduces the number of binary variables when many sampling times are required for diagnosis.

Bounds on the reachable sets of nonlinear control systems

The computation of rigorous enclosures of the reachable sets of nonlinear control systems is considered, with a focus on applications for which speed is crucial. Low computational costs make interval methods based on differential inequalities an attractive option. Unfortunately, such methods are prone to large overestimation and often produce useless results in practice. From physical considerations, however, it is common that some crude set is known to contain the reachable set. We establish a general bounding result, based on differential inequalities, which enables the effective use of such sets during the bounding procedure. In the case where this set is a convex polyhedron, an efficient implementation using interval computations is developed. Using readily available physical information from practical examples, this method is shown to provide significant advantages over alternative methods in terms of both efficiency and accuracy.

Tight, efficient bounds on the solutions of chemical kinetics models

Abstract An efficient method is presented for computing time-varying, component-wise bounds on the solutions of chemical kinetics models subject to an interval of permissible parameters. The model may be any system of ordinary differential equations whose right-hand side may be written as a stoichiometric matrix premultiplying a vector of potentially nonlinear rate functions, and model parameters may include initial conditions, kinetic rate coefficients, controls, disturbances, etc. The method presented differs from other bounding methods in that it takes advantage of the special structure of chemical kinetics models, including known physical bounds and the presence of affine reaction invariants. The method is demonstrated for several case studies, and it is shown that the resulting bounds are nearly always significantly tighter than those computed by a similar method which does not take reaction invariants into account. Finally, the additional computational cost of taking reaction invariants into consideration is negligible.

Design of active inputs for set-based fault diagnosis

Effective fault diagnosis depends on the detectability of the faults in the measurements, which can be improved by a suitable input signal. This article presents a deterministic method for computing the set of inputs that guarantee fault diagnosis, referred to as separating inputs. The process of interest is described, under nominal and various faulty conditions, by linear discrete-time models subject to bounded process and measurement noise. It is shown that the set of separating inputs can be efficiently computed in terms of the complement of one or several zonotopes, depending on the number of fault models. In practice, it is essential to choose elements from this set that are minimally harmful with respect to other control objectives. It is shown that this can be done efficiently through the solution of a mixed-integer quadratic program. The method is demonstrated for a numerical example.

Improved relaxations for the parametric solutions of ODEs using differential inequalities

A new method is described for computing nonlinear convex and concave relaxations of the solutions of parametric ordinary differential equations (ODEs). Such relaxations enable deterministic global optimization algorithms to be applied to problems with ODEs embedded, which arise in a wide variety of engineering applications. The proposed method computes relaxations as the solutions of an auxiliary system of ODEs, and a method for automatically constructing and numerically solving appropriate auxiliary ODEs is presented. This approach is similar to two existing methods, which are analyzed and shown to have undesirable properties that are avoided by the new method. Two numerical examples demonstrate that these improvements lead to significantly tighter relaxations than previous methods.

Convex and concave relaxations of implicit functions

A deterministic algorithm for solving nonconvex NLPs globally using a reduced-space approach is presented. These problems are encountered when real-world models are involved as nonlinear equality constraints and the decision variables include the state variables of the system. By solving the model equations for the dependent (state) variables as implicit functions of the independent (decision) variables, a significant reduction in dimensionality can be obtained. As a result, the inequality constraints and objective function are implicit functions of the independent variables, which can be estimated via a fixed-point iteration. Relying on the recently developed ideas of generalized McCormick relaxations and McCormick-based relaxations of algorithms and subgradient propagation, the development of McCormick relaxations of implicit functions is presented. Using these ideas, the reduced space, implicit optimization formulation can be relaxed. When applied within a branch-and-bound framework, finite convergence to ε-optimal global solutions is guaranteed.

Constrained zonotopes

This article introduces a new class of sets, called constrained zonotopes, that can be used to enclose sets of interest for estimation and control. The numerical representation of these sets is sufficient to describe arbitrary convex polytopes when the complexity of the representation is not limited. At the same time, this representation permits the computation of exact projections, intersections, and Minkowski sums using very simple identities. Efficient and accurate methods for computing an enclosure of one constrained zonotope by another of lower complexity are provided. The advantages and disadvantages of these sets are discussed in comparison to ellipsoids, parallelotopes, zonotopes, and convex polytopes in halfspace and vertex representations. Moreover, extensive numerical comparisons demonstrate significant advantages over other classes of sets in the context of set-based state estimation and fault detection.

Reverse propagation of McCormick relaxations

Constraint propagation techniques have heavily utilized interval arithmetic while the application of convex and concave relaxations has been mostly restricted to the domain of global optimization. Here, reverse McCormick propagation, a method to construct and improve McCormick relaxations using a directed acyclic graph representation of the constraints, is proposed. In particular, this allows the interpretation of constraints as implicitly defining set-valued mappings between variables, and allows the construction and improvement of relaxations of these mappings. Reverse McCormick propagation yields potentially tighter enclosures of the solutions of constraint satisfaction problems than reverse interval propagation. Ultimately, the relaxations of the objective of a non-convex program can be improved by incorporating information about the constraints.

Convex and Concave Relaxations for the Parametric Solutions of Semi-explicit Index-One Differential-Algebraic Equations

A method is presented for computing convex and concave relaxations of the parametric solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). These relaxations are central to the development of a deterministic global optimization algorithm for problems with DAEs embedded. The proposed method uses relaxations of the DAE equations to derive an auxiliary system of DAEs, the solutions of which are proven to provide the desired relaxations. The entire procedure is fully automatable.