In this work, we address optimization of large-scale, nonlinear, block-structured problems with a significant number of coupling variables. Solving these problems using interior-point methods requires the solution of a linear system that has a block-angular structure at each iteration. Parallel solution is possible using a Schur-complement decomposition. In an explicit Schur-complement decomposition, the computational cost of forming and factorizing the Schur-complement is prohibitive for problems with many coupling variables. In this paper, we show that this bottleneck can be overcome by solving the Schur-complement equations implicitly, using a quasi-Newton preconditioned conjugate gradient method. This new algorithm avoids explicit formation and factorization of the Schur-complement. The computational efficiency of this algorithm is compared with the serial full-space approach, and the serial and parallel explicit Schur-complement approach. These results show that the PCG Schur-complement approach dramatically reduces the computational cost for problems with many coupling variables.

It is common that only a subset of the parameters of models can be accurately estimated. One approach for identifying a subset of parameters for estimation is to perform clustering of the parameters into groups based upon their sensitivity vectors. However, this has the drawback that uncertainty cannot be directly incorporated into the procedure as the sensitivity vectors are based upon the nominal values of the parameters. This article addresses this drawback by presenting a parameter set selection technique that can take uncertainty in the parameter space into account. This is achieved by defining sensitivity cones, where a sensitivity cone includes all sensitivity vectors of a parameter for different values, resulting from the uncertainty, in the parameter space. Parameter clustering can then be performed based upon the angles between the sensitivity cones, instead of the angle between sensitivity vectors. The presented technique is applied to two case studies. © 2013 American Institute of Chemical Engineers AIChE J, 60: 181–192, 2014

This paper presents a decomposition strategy applicable to DAE constrained optimization problems. A common solution method for such problems is to apply a direct transcription method and solve the resulting nonlinear program using an interior-point algorithm. For this approach, the time to solve the linearized KKT system at each iteration typically dominates the total solution time. In our proposed method, we exploit the structure of the KKT system resulting from a direct collocation scheme for approximating the DAE constraints in order to compute the necessary linear algebra operations on multiple processors. This approach is applied to find the optimal control profile of a combined cycle power plant with promising results on both distributed memory and shared memory computing architectures with speedups of over 50 times possible.

This paper investigates optimization of operational strategies of an industrial ethanol fermentation process. One of the challenges associated with this type or process is that most of the measurements are only taken sporadically, thereby complicating process monitoring and optimization. The one exception to this rule involves temperature measurements, which are readily available. However, an existing model of the plant investigated in this paper does not include an energy balance and, accordingly, the temperature measurements cannot be used to estimate model parameters. This paper addresses these deficiencies and proposes modifications to an existing ethanol fermentation model. The proposed changes include the derivation of an energy balance, modification of the reaction kinetics to include additional inhibition terms, and also estimation of model parameters from industrial data. The new model is validated against plant data and then used for optimization of the process operations. It is shown that modifications of the input profiles for the cooling rate and the glucoamylase addition can lead to an approximately 10% increase in ethanol yield. These are promising results, even though these findings will ultimately need to be validated during real plant operations.

Infectious diseases remain a significant health concern around the world. Mathematical modeling of these diseases can help us understand their dynamics and develop more effective control strategies. In this work, we show the capabilities of interior-point methods and nonlinear programming (NLP) formulations to efficiently estimate parameters in multiple discrete-time disease models using measles case count data from three cities. These models include multiplicative measurement noise and incorporate seasonality into multiple model parameters. Our results show that nearly identical patterns are estimated even when assuming seasonality in different model parameters, and that these patterns show strong correlation to school term holidays across very different social settings and holiday schedules. We show that interior-point methods provide a fast and flexible approach to parameterizing models that can be an alternative to more computationally intensive methods.

We consider the problem of estimating transmission parameters in a model of childhood disease transmission. We show the structural equivalence between parameter estimation and two-stage stochastic programming, and use the Progressive Hedging decomposition strategy to efficiently compute parameter values on large-scale, real-world observations.

Mathematical models can enhance our understanding of childhood infectious disease dynamics, but these models depend on appropriate parameter values that are often unknown and must be estimated from disease case data. In this paper, we develop a framework for efficient estimation of childhood infectious disease models with seasonal transmission parameters using continuous differential equations containing model and measurement noise. The problem is formulated using the simultaneous approach where all state variables are discretized, and the discretized differential equations are included as constraints, giving a large-scale algebraic nonlinear programming problem that is solved using a nonlinear primal–dual interior-point solver. The technique is demonstrated using measles case data from three different locations having different school holiday schedules, and our estimates of the seasonality of the transmission parameter show strong correlation to school term holidays. Our approach gives dramatic efficiency gains, showing a 40–400-fold reduction in solution time over other published methods. While our approach has an increased susceptibility to bias over techniques that integrate over the entire unknown state-space, a detailed simulation study shows no evidence of bias. Furthermore, the computational efficiency of our approach allows for investigation of a large model space compared with more computationally intensive approaches.

Many important factors affect the spread of childhood infectious disease. To better understand the fundamental drivers of infectious disease spread, several researchers have estimated seasonal transmission coefficients in discrete-time models. In this paper, we build upon this previous work and also develop a framework for efficient estimation using continuous differential equation models. We introduce nonlinear programming formulations to efficiently estimate model parameters and seasonal transmission profiles from existing case count data for the childhood disease, measles. We compare results from discrete time and continuous time models and address several shortcomings of the discrete-time method, including removing the need for the data reporting interval to match the time between successive cases in the chain of transmission or serial interval of the disease. Using a simultaneous approach for optimization of differential equation systems, we demonstrate that seasonal transmission parameters can be effectively estimated using continuous time models instead of discrete-time models.

While large-scale nonlinear programming (NLP) has seen widespread use within the process industries, the desire to solve larger and more complex problems drives continued improvements in NLP solvers. Because of physical hardware limitations, manufacturers have shifted their focus towards multi-core and other modern parallel computing architectures, and we must focus efforts on the development of parallel computing solutions for large-scale nonlinear programming. In this paper we briefly describe some existing and emerging architectures for parallel computing and demonstrate the potential of some of these architectures for parallel solution of nonlinear programming problems. In particular, we show the scalability of an integrated design and control problem using two techniques within a multi-core architecture.