Stuart M. Harwood

A reliable simulator for dynamic flux balance analysis

Dynamic flux balance analysis (DFBA) provides a platform for detailed design, control and optimization of biochemical process technologies. It is a promising modeling framework that combines genome‐scale metabolic network analysis with dynamic simulation of the extracellular environment. Dynamic flux balance analysis assumes that the intracellular species concentrations are in equilibrium with the extracellular environment. The resulting underdetermined stoichiometric model is solved under the assumption of a biochemical objective such as growth rate maximization. The model of the metabolism is coupled with the dynamic mass balance equations of the extracellular environment via expressions for the rates of substrate uptake and product excretion, which imposes additional constraints on the linear program (LP) defined by growth rate maximization of the metabolism. The linear program is embedded into the dynamic model of the bioreactor, and together with the additional constraints this provides an accurate model of the substrate consumption, product secretion, and biomass production during operation. A DFBA model consists of a system of ordinary differential equations for which the evaluation of the right‐hand side requires not only function evaluations, but also the solution of one or more linear programs. The numerical tool presented here accurately and efficiently simulates large‐scale dynamic flux balance models. The main advantages that this approach has over existing implementation are that the integration scheme has a variable step size, that the linear program only has to be solved when qualitative changes in the optimal flux distribution of the metabolic network occur, and that it can reliably simulate behavior near the boundary of the domain where the model is defined. This is illustrated through large‐scale examples taken from the literature. Biotechnol. Bioeng. 2013; 110: 792–802.

Efficient solution of ordinary differential equations with a parametric lexicographic linear program embedded

This work analyzes the initial value problem in ordinary differential equations with a parametric lexicographic linear program (LP) embedded. The LP is said to be embedded since the dynamics depend on the solution of the LP, which is in turn parameterized by the dynamic states. This problem formulation finds application in dynamic flux balance analysis, which serves as a modeling framework for industrial fermentation reactions. It is shown that the problem formulation can be intractable numerically, which arises from the fact that the LP induces an effective domain that may not be open. A numerical method is developed which reformulates the system so that it is defined on an open set. The result is a system of semi-explicit index-one differential algebraic equations, which can be solved with efficient and accurate methods. It is shown that this method addresses many of the issues stemming from the original problem’s intractability. The application of the method to examples of industrial fermentation processes demonstrates its effectiveness and efficiency.

Recent Progress Using Matheuristics for Strategic Maritime Inventory Routing

This chapter presents an extensive computational study of simple, but prominent matheuristics (i.e., heuristics that rely on mathematical programming models) to find high quality ship schedules and inventory policies for a class of maritime inventory routing problems. Our computational experiments are performed on a test bed of the publicly available MIRPLib instances. This class of inventory routing problems has few constraints relative to some operational problems, but is complicated by long planning horizons. We compare several variants of rolling horizon heuristics, K-opt heuristics, local branching, solution polishing, and hybrids thereof. Many of these matheuristics substantially outperform the commercial mixed-integer programming solvers CPLEX 12.6.2 and Gurobi 6.5 in their ability to quickly find high quality solutions. New best known incumbents are found for 26 out of 70 yet-to-be-proved-optimal instances and new best known bounds on 56 instances.

Lower level duality and the global solution of generalized semi-infinite programs

The reformulation of generalized semi-infinite programs (GSIP) to simpler problems is considered. These reformulations are achieved under the assumption that a duality property holds for the lower level program (LLP). Lagrangian duality is used in the general case to establish the relationship between the GSIP and a related semi-infinite program (SIP). Practical aspects of this reformulation, including how to bound the duality multipliers, are also considered. This SIP reformulation result is then combined with recent advances for the global, feasible solution of SIP to develop a global, feasible point method for the solution of GSIP. Reformulations to finite nonlinear programs, and the practical aspects of solving these reformulations globally, are also discussed. When the LLP is a linear program or second-order cone program, specific duality results can be used that lead to stronger results. Numerical examples demonstrate that the global solution of GSIP is computationally practical via the solution of these duality-based reformulations.

Bounds on Reachable Sets Using Ordinary Differential Equations with Linear Programs Embedded

Abstract This work considers the computation of time-varying enclosures of the reachable sets of nonlinear control systems via the solution of an initial value problem in ordinary differential equations (ODEs) with linear programs (LPs) embedded. To ensure the numerical tractability of such a formulation, the properties of the ODEs with LPs embedded are discussed including existence and uniqueness of the solutions of the initial value problem in ODEs with LPs embedded. This formulation is then applied to the computation of rigorous componentwise time-varying bounds on the states of a nonlinear control system. The bounding theory used in this work exploits physical information to yield tight bounds on the states; this work develops a new implementation of this theory. Finally, the tightness of the bounds are demonstrated for a model of a reacting chemical system with uncertain rate parameters.

How to solve a design centering problem

This work considers the problem of design centering. Geometrically, this can be thought of as inscribing one shape in another. Theoretical approaches and reformulations from the literature are reviewed; many of these are inspired by the literature on generalized semi-infinite programming, a generalization of design centering. However, the motivation for this work relates more to engineering applications of robust design. Consequently, the focus is on specific forms of design spaces (inscribed shapes) and the case when the constraints of the problem may be implicitly defined, such as by the solution of a system of differential equations. This causes issues for many existing approaches, and so this work proposes two restriction-based approaches for solving robust design problems that are applicable to engineering problems. Another feasible-point method from the literature is investigated as well. The details of the numerical implementations of all these methods are discussed. The discussion of these implementations in the particular setting of robust design in engineering problems is new.