An interior-point method for efficient solution of block-structured NLP problems using an implicit Schur-complement decomposition

Kang, Jia; Cao, Yankai; Word, Daniel P.; Laird, Carl D.

doi:10.1016/j.compchemeng.2014.09.013

Cited by 46 publications

(32 citation statements)

References 18 publications

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“…if the original KKT system and each K s block satisfies the inertia condition for descent [11,19]. This property enables the use of a PCG procedure to solve the Schur system [11], leading to the implicit Schur complement method. This approach avoids both the explicit formation and factorization of the dense Schur complement matrix.…”

Section: Efficient Parallel Schur Complement Methods For Stochastic Prmentioning

confidence: 99%

“…There is, to the best knowledge of the author, no efficient modeling language supporting parallel evaluations of functions and gradients for general NLP problems. However, for structured problems such as stochastic programs, Kang et al [11] and Zavala et al [10] build a single AMPL [14] model instance for each scenario and evaluate all these instances in parallel. Several packages (e.g., PySP [15], StochJuMP [16]) have also been developed to support the parallel evaluation of functions and gradients for structured NLP problems.…”

Section: Introductionmentioning

confidence: 99%

“…Kang et al [11] uses a preconditioned conjugate gradient (PCG) procedure to solve the Schur system with an automatic L-BFGS preconditioner. This approach avoids both forming and factorizing the Schur complement explicitly.…”

Section: Introductionmentioning

confidence: 99%

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Parallel Solution of Robust Nonlinear Model Predictive Control Problems in Batch Crystallization

et al. 2016

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Abstract:Representing the uncertainties with a set of scenarios, the optimization problem resulting from a robust nonlinear model predictive control (NMPC) strategy at each sampling instance can be viewed as a large-scale stochastic program. This paper solves these optimization problems using the parallel Schur complement method developed to solve stochastic programs on distributed and shared memory machines. The control strategy is illustrated with a case study of a multidimensional unseeded batch crystallization process. For this application, a robust NMPC based on min-max optimization guarantees satisfaction of all state and input constraints for a set of uncertainty realizations, and also provides better robust performance compared with open-loop optimal control, nominal NMPC, and robust NMPC minimizing the expected performance at each sampling instance. The performance of robust NMPC can be improved by generating optimization scenarios using Bayesian inference. With the efficient parallel solver, the solution time of one optimization problem is reduced from 6.7 min to 0.5 min, allowing for real-time application.

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Section: Efficient Parallel Schur Complement Methods For Stochastic Prmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Parallel Solution of Robust Nonlinear Model Predictive Control Problems in Batch Crystallization

et al. 2016

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“…1 If the solution vector (q 0 , q S ) does not exactly solve (15), it will induce a residual vector that we define as T r :…”

Section: Clustering-based Preconditionermentioning

confidence: 99%

Clustering-based preconditioning for stochastic programs

2015

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We present a clustering-based preconditioning strategy for KKT systems arising in stochastic programming within an interior-point framework. The key idea is to perform adaptive clustering of scenarios (inside-the-solver) based on their influence on the problem at hand. This approach thus contrasts with existing (outside-the-solver) approaches that cluster scenarios based on problem data alone. We derive spectral and error properties for the preconditioner and demonstrate that scenario compression rates of up to 94 % can be obtained, leading to dramatic computational savings. In addition, we demonstrate that the proposed preconditioner can avoid scalability issues of Schur decomposition in problems with large first-stage dimensionality.

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“…More recently, Zavala et al [19] have demonstrated a parallel primal-dual interior-point approach to tackle discretized multi-period dynamic optimization formulations. This has ultimately led to general interior-point approaches to handle discretized nominal dynamic optimization formulations [20] and structured NLP formulations [21]. All of these previously noted studies have been on structured NLP techniques involving explicit objective and constraint functionals; however, our particular interest in the present paper is on solution techniques involving implicit or embedded functionals, which require a secondary solution algorithm for evaluation within the NLP constraints.…”

Section: Problem Statementmentioning

confidence: 99%

Multi-Period Dynamic Optimization for Large-Scale Differential-Algebraic Process Models under Uncertainty

Washington

Swartz

2015

Processes

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Abstract:A technique for optimizing large-scale differential-algebraic process models under uncertainty using a parallel embedded model approach is developed in this article. A combined multi-period multiple-shooting discretization scheme is proposed, which creates a significant number of independent numerical integration tasks for each shooting interval over all scenario/period realizations. Each independent integration task is able to be solved in parallel as part of the function evaluations within a gradient-based non-linear programming solver. The focus of this paper is on demonstrating potential computation performance improvement when the embedded differential-algebraic equation model solution of the multi-period discretization is implemented in parallel. We assess our parallel dynamic optimization approach on two case studies; the first is a benchmark literature problem, while the second is a large-scale air separation problem that considers a robust set-point transition under parametric uncertainty. Results indicate that focusing on the speed-up of the embedded model evaluation can significantly decrease the overall computation time; however, as the multi-period formulation grows with increased realizations, the computational burden quickly shifts to the internal computation performed within the non-linear programming algorithm. This highlights the need for further decomposition, structure exploitation and parallelization within the non-linear programming algorithm and is the subject for further investigation.

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An interior-point method for efficient solution of block-structured NLP problems using an implicit Schur-complement decomposition

Cited by 46 publications

References 18 publications

Parallel Solution of Robust Nonlinear Model Predictive Control Problems in Batch Crystallization

Parallel Solution of Robust Nonlinear Model Predictive Control Problems in Batch Crystallization

Clustering-based preconditioning for stochastic programs

Multi-Period Dynamic Optimization for Large-Scale Differential-Algebraic Process Models under Uncertainty

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