Smoothing methods for complementarity problems in process engineering

Gopal, Vipin; Biegler, Lorenz T.

doi:10.1002/aic.690450715

Cited by 58 publications

(49 citation statements)

References 17 publications

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“…Several kinds of probability density functions were proposed by Chen and Harker [21], Pina and Zenios [22], Zang [23], and Gopal and Biegler [24]. In this work, the following density function [21] is adopted:…”

Section: B Approximation Using Parametric Smoothing Functionsmentioning

confidence: 99%

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Three-Dimensional Aircraft Conflict Resolution Based on Smoothing Methods

Chen

Shao

et al. 2016

Journal of Guidance, Control, and Dynamics

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The development of an automated and optimal conflict predicting and resolution strategy is one of the most important components in successfully implementing the Next Generation Air Transportation System concept. The conflict resolution problem, which occurs between multiple aircraft described by a three-degree-of-freedom nonlinear point-mass model, is the focus of this study. A novel method for handling nondifferentiable disjunctive conflict avoidance constraints is developed by using smoothing functions, which are constructed by integrating a probability density function. Thus, the conflict resolution problem is posed as an optimal control problem and discretized into a nonlinear programming problem via the direct transcription approach with a Radau collocation point. The approximation error obtained using the smoothing functions is examined through sensitivity analysis. An estimable linear relationship is established between the approximation error and the smoothing parameter, which indicates that the approximation error is controllable. Numerical results show that the proposed approach efficiently produces conflict-free and feasible maneuvers. The approximation errors are also evaluated and found to be minimal and negligible.

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Section: B Approximation Using Parametric Smoothing Functionsmentioning

confidence: 99%

“…Based on formulations (20)(21)(22)(23)(24)(25), the resulting nonlinear programming problem that corresponds to formulation (18) is given by …”

Section: A Direct Transcriptionmentioning

confidence: 99%

Three-Dimensional Aircraft Conflict Resolution Based on Smoothing Methods

Chen

Shao

et al. 2016

Journal of Guidance, Control, and Dynamics

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“…GDP-based strategies use logical constraints to model these special cases, in order to avoid possible numerical difficulties. Using a continuous strategy, Lang and Biegler (2002) address this potential problem by adding complementary constraints to the original problem, taking advantage of a previously developed smoothing function approach to model the loss of phases during the optimization of a flash unit (Gopal & Biegler, 1999). Their main idea is to rewrite the equilibrium balance equations introducing slack variables that assume different values accordingly to the complementary equations.…”

Section: Selecting the Total Number Of Traysmentioning

confidence: 99%

A robust strategy for optimizing complex distillation columns

Neves

Silva

Oliveira

2005

Computers & Chemical Engineering

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This work introduces a strategy for the optimal design of distillation systems based on continuous optimization. The approach is similar to the one proposed earlier by [Lang, Y. -D., & Biegler, L. T. (2002). A distributed stream method for tray optimization. AIChE Journal, 48, 582], avoiding the need of solving extremely large and non-linear discrete optimization problems. When used with complex distillation units, it can identify interesting design configurations not considered by other continuous formulations, and also relieve some of the numerical difficulties associated with the use of distribution functions for the optimal location of feed and side-streams. The method considers a relaxation of the original problem, where the streams are initially split to several trays in the column, not necessarily adjacent. The optimal location of each stream is converged by constraining the optimization problem, using adjustable parameters that control the minimum amount of aggregation allowed. The methodology is illustrated with the application to several industrial case studies, including sets of distillation columns. Models up to 17,000 variables/equations were solved, revealing large economic benefits in the design of new units and optimization of sets of existing ones.

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“…We begin with the example of complementarity, which in its simplest form, is the relationship between nonnegative variables with the additional constraint that at least one must be zero. In process optimization, complementarity can be used to model (nonsmooth) features such as the disappearance of phases, distillation, flow reversal, safety valve operation, and other discrete events (Gopal and Biegler, 1999;Baumrucker et al, 2008).…”

Section: Extended Mathematical Programsmentioning

confidence: 99%

An extended mathematical programming framework

Ferris

Dirkse²,

Jagla

et al. 2009

Computers & Chemical Engineering

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Extended mathematical programs are collections of functions and variables joined together using specific optimization and complementarity primitives. This paper outlines a mechanism to describe such an extended mathematical program by means of annotating the existing relationships within a model to facilitate higher level structure identification. The structures, which often involve constraints on the solution sets of other models or complementarity relationships, can be exploited by modern large scale mathematical programming algorithms for efficient solution. A specific implementation of this framework is outlined that communicates structure from the GAMS modeling system to appropriate solvers in a computationally beneficial manner.Example applications are taken from chemical engineering.

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Smoothing methods for complementarity problems in process engineering

Cited by 58 publications

References 17 publications

Three-Dimensional Aircraft Conflict Resolution Based on Smoothing Methods

Three-Dimensional Aircraft Conflict Resolution Based on Smoothing Methods

A robust strategy for optimizing complex distillation columns

An extended mathematical programming framework

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