# Parallel algorithms for nonlinear programming and applications in pharmaceutical manufacturing

#### Abstract

Effective manufacturing of pharmaceuticals presents a number of challenging optimization problems due to complex distributed, time-independent models and the need to handle uncertainty. These challenges are multiplied when real-time solutions are required. The demand for fast solution of nonlinear optimization problems, coupled with the emergence of new concurrent computing architectures, drives the need for parallel algorithms to solve challenging NLP problems. The goal of this work is the development of parallel algorithms for nonlinear programming problems on different computing architectures, and the application of large-scale nonlinear programming on challenging problems in pharmaceutical manufacturing. The focus of this dissertation is our completed work on an augmented Lagrangian algorithm for parallel solution of general NLP problems on graphics processing units and a clustering-based preconditioning strategy for stochastic programs within an interior-point framework on distributed memory machines. Our augmented Lagrangian interior-point approach for general NLP problems is iterative at three levels. The first level replaces the original problem by a sequence of bound-constrained optimization problems. Each of these bound-constrained problems is solved using a nonlinear interior-point method. Inside the interior-point method, the barrier subproblems are solved using a variation of Newton's method, where the linear system is solved using a preconditioned conjugate gradient (PCG) method. The primary advantage of this algorithm is that it allows use of the PCG method, which can be implemented efficiently on a GPU in parallel. This algorithm shows an order of magnitude speedup on certain problems. We also present a clustering-based preconditioning strategy for stochastic programs. The key idea is to perform adaptive clustering of scenarios inside-the-solver based on their influence on the problem. We derive spectral and error properties for the preconditioner and demonstrate that scenario compression rates of up to 94% can be obtained, leading to drastic computational savings. A speed up factor of 42 is obtained with our parallel implementation on an stochastic market-clearing problem for the entire Illinois power grid system. In addition, we discuss two applications of nonlinear programming in pharmaceutical manufacturing. The first application is to analyze the effect of different policies to deal with drug shortages and the role of emergency supply in pharmaceutical manufacturing. Simulation results indicate that the availability of an emergency production facility can significantly reduce expected government spending even if the unit price of emergency supply is high and the capacity is limited. In addition, we discuss an important application of nonlinear programming in control of pharmaceutical manufacturing processes. First, we focus on the development of real-time feasible multi-objective optimization based NMPC-MHE formulations for batch crystallization processes to control the crystal size and shape distribution. At each sampling instance, based on a nonlinear DAE model, an estimation problem estimates unknown states and parameters and an optimal control problem determines the optimal input profiles. Both DAE-constrained optimization problems are solved by discretizing the system using Radau collocation and optimizing the resulting algebraic nonlinear problem using IPOPT. NMPC-MHE is shown to provide better setpoint tracking than the open-loop optimal control strategy in terms of setpoint change, system noise, and model/plant mismatch. Second, to deal with the parameter uncertainties in the crystallization model, we also develop a real-time feasible robust NMPC formulation. The size of optimization problems arising from the robust NMPC becomes too large to be solved by a serial solver. Therefore, we use a parallel algorithm to ensure real-time feasibility.

#### Degree

Ph.D.

#### Advisors

Laird, Purdue University.

#### Subject Area

Chemical engineering

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