Paul I. Barton

End-to-End Continuous Manufacturing of Pharmaceuticals: Integrated Synthesis, Purification, and Final Dosage Formation†

A series of tubes: The continuous manufacture of a finished drug product starting from chemical intermediates is reported. The continuous pilot-scale plant used a novel route that incorporated many advantages of continuous-flow processes to produce active pharmaceutical ingredients and the drug product in one integrated system.

Efficient sensitivity analysis of large-scale differential-algebraic systems

Abstract A new algorithm and software for numerical sensitivity analysis of differential-algebraic equations are presented. The staggered corrector method proposed has lower computational complexity than both the staggered direct and the simultaneous corrector methods. The results here are especially significant for, though not limited to, large-scale sparse differential-algebraic systems.

Optimally-reduced kinetic models: reaction elimination in large-scale kinetic mechanisms

Abstract A new optimization-based approach to kinetic model reduction is presented. The reaction-elimination problem is formulated as a linear integer program which can be solved to guaranteed global optimality. This formulation ensures that the solution to the integer program is the smallest possible reduced model consistent with the user-set tolerances. The method is applied to generate optimally-reduced models for isobaric, adiabatic homogeneous combustion. The computational cost and accuracy of the reduced models are compared to those of the full mechanism. Results are shown for GRImech 3.0 and the Lawrence Livermore n-heptane combustion mechanism. The accuracy of the integer programming approach is compared to existing reaction elimination methods. The method is also applied to generate a library of reduced kinetic models for an adaptive chemistry simulation of a 2-D laminar, partially-premixed methane burner flame. Preliminary results are presented comparing the computational cost of the full GRImech 3.0 chemistry to that of the reduced model library.

State event location in differential-algebraic models

An efficient discontinuity handling algorithm for initial value problems in differential-algebraic equations is presented. The algorithm supports flexible representation of state conditions in propositional logic, and guarantees the location of all state events in strict time order. The algorithm consists of two phases:(1) event detection and(2) consistent event location. In the event detection phase, the entire integration step is searched for the state event by solving the interpolation polynomials for the discontinuity functions generated by the BDF method. An efficient hierarchical polynomial root-finding procedure based upon interval arithmetic guarantees detection of the state event even if multiple state condition transitions exist in an integration step, in which case many existing algorithms may fail. As a second phase of the algorithm, a consistent even location calculation is developed that accurately locates the state event detected earlier while completely eliminating incorrect reactivation of the same state event immediately after the consistent initialization calculation that may follow. This numerical phenomenon has not been explained before and is termed discontinuity sticking. Results from various test problems are presented to demonstrate the correctness and efficiency of the algorithm.

Modeling, simulation, sensitivity analysis, and optimization of hybrid systems

Hybrid (discrete/continuous) systems exhibit both discrete state and continuous state dynamics which interact to such a significant extent that they cannot be decoupled and must be analyzed simultaneously. We present an overview of the work that has been done in the modeling, simulation, sensitivity analysis, and optimization of hybrid systems, paying particular attention to the interaction between discrete and continuous dynamics. A concise intuitive framework for hybrid system modeling is presented, together with discussions on robust state event location, transfer functions of the continuous state at discontinuities, parametric sensitivity analysis of hybrid systems, and challenges in optimization.

Parametric sensitivity functions for hybrid discrete/continuous systems

Abstract The general equations for the parametric sensitivity functions of a broad class of hybrid discrete/continuous dynamic systems where the continuous part is described by differential–algebraic equations (DAEs) are presented. For the cases where this continuous part is an ordinary differential equation system (ODEs) or a linear time invariant DAE, sufficient conditions for the existence and uniqueness of the sensitivity functions are derived. Numerical computation of these sensitivity functions has been implemented as a generic functionality in a mathematical modeling environment. Special cases and application examples are used for illustration.

Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs

Abstract.A rigorous decomposition approach to solve separable mixed-integer nonlinear programs where the participating functions are nonconvex is presented. The proposed algorithms consist of solving an alternating sequence of Relaxed Master Problems (mixed-integer linear program) and two nonlinear programming problems (NLPs). A sequence of valid nondecreasing lower bounds and upper bounds is generated by the algorithms which converge in a finite number of iterations. A Primal Bounding Problem is introduced, which is a convex NLP solved at each iteration to derive valid outer approximations of the nonconvex functions in the continuous space. Two decomposition algorithms are presented in this work. On finite termination, the first yields the global solution to the original nonconvex MINLP and the second finds a rigorous bound to the global solution. Convergence and optimality properties, and refinement of the algorithms for efficient implementation are presented. Finally, numerical results are compared with currently available algorithms for example problems, illuminating the potential benefits of the proposed algorithm.

Global Optimization with Nonlinear Ordinary Differential Equations

This paper examines global optimization of an integral objective function subject to nonlinear ordinary differential equations. Theory is developed for deriving a convex relaxation for an integral by utilizing the composition result defined by McCormick (Mathematical Programming 10, 147–175, 1976) in conjunction with a technique for constructing convex and concave relaxations for the solution of a system of nonquasimonotone ordinary differential equations defined by Singer and Barton (SIAM Journal on Scientific Computing, Submitted). A fully automated implementation of the theory is briefly discussed, and several literature case study problems are examined illustrating the utility of the branch-and-bound algorithm based on these relaxations.

Mixed-integer dynamic optimization I: problem formulation

A rigorous decomposition approach is presented for mixed-integer dynamic optimization problems. The approach combines dynamic optimization with insight based targeting techniques to decompose the optimization into subproblems providing rigorous upper and lower bounds on the objective. This approach has the potential to eliminate total enumeration of the discrete space, assures termination in a finite number of iterations and yields a rigorous bound on the distance between the solution found and the global solution.

Global solution of bilevel programs with a nonconvex inner program

A bounding algorithm for the global solution of nonlinear bilevel programs involving nonconvex functions in both the inner and outer programs is presented. The algorithm is rigorous and terminates finitely to a point that satisfies ε-optimality in the inner and outer programs. For the lower bounding problem, a relaxed program, containing the constraints of the inner and outer programs augmented by a parametric upper bound to the parametric optimal solution function of the inner program, is solved to global optimality. The optional upper bounding problem is based on probing the solution obtained by the lower bounding procedure. For the case that the inner program satisfies a constraint qualification, an algorithmic heuristic for tighter lower bounds is presented based on the KKT necessary conditions of the inner program. The algorithm is extended to include branching, which is not required for convergence but has potential advantages. Two branching heuristics are described and analyzed. Convergence proofs are provided and numerical results for original test problems and for literature examples are presented.