Lorenz T. Biegler

On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming

Abstract.We present a primal-dual interior-point algorithm with a filter line-search method for nonlinear programming. Local and global convergence properties of this method were analyzed in previous work. Here we provide a comprehensive description of the algorithm, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix. Heuristics are also considered that allow faster performance. This method has been implemented in the IPOPT code, which we demonstrate in a detailed numerical study based on 954 problems from the CUTEr test set. An evaluation is made of several line-search options, and a comparison is provided with two state-of-the-art interior-point codes for nonlinear programming.

An algorithmic framework for convex mixed integer nonlinear programs

This paper is motivated by the fact that mixed integer nonlinear programming is an important and difficult area for which there is a need for developing new methods and software for solving large-scale problems. Moreover, both fundamental building blocks, namely mixed integer linear programming and nonlinear programming, have seen considerable and steady progress in recent years. Wishing to exploit expertise in these areas as well as on previous work in mixed integer nonlinear programming, this work represents the first step in an ongoing and ambitious project within an open-source environment. COIN-OR is our chosen environment for the development of the optimization software. A class of hybrid algorithms, of which branch-and-bound and polyhedral outer approximation are the two extreme cases, are proposed and implemented. Computational results that demonstrate the effectiveness of this framework are reported. Both the library of mixed integer nonlinear problems that exhibit convex continuous relaxations, on which the experiments are carried out, and a version of the software used are publicly available.

Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes

This book addresses modern nonlinear programming (NLP) concepts and algorithms, especially as they apply to challenging applications in chemical process engineering. The author provides a firm grounding in fundamental NLP properties and algorithms, and relates them to real-world problem classes in process optimization, thus making the material understandable and useful to chemical engineers and experts in mathematical optimization. Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes shows readers which NLP methods are best suited for specific applications, how large-scale problems should be formulated and what features of these problems should be emphasized, and how existing NLP methods can be extended to exploit specific structures of large-scale optimization models. Audience: The book is intended for chemical engineers interested in using NLP algorithms for specific applications, experts in mathematical optimization who want to understand process engineering problems and develop better approaches to solving them, and researchers from both fields interested in developing better methods and problem formulations for challenging engineering problems. Contents: Preface; Chapter 1: Introduction to Process Optimization; Chapter 2: Concepts of Unconstrained Optimization; Chapter 3: Newton-Type Methods for Unconstrained Optimization; Chapter 4: Concepts of Constrained Optimization; Chapter 5: Newton Methods for Equality Constrained Optimization; Chapter 6: Numerical Algorithms for Constrained Optimization; Chapter 7: Steady State Process Optimization; Chapter 8: Introduction to Dynamic Process Optimization; Chapter 9: Dynamic Optimization Methods with Embedded DAE Solvers; Chapter 10: Simultaneous Methods for Dynamic Optimization; Chapter 11: Process Optimization with Complementarity Constraints; Bibliography; Index

Retrospective on optimization

In this paper, we provide a general classification of mathematical optimization problems, followed by a matrix of applications that shows the areas in which these problems have been typically applied in process systems engineering. We then provide a review of solution methods of the major types of optimization problems for continuous and discrete variable optimization, particularly nonlinear and mixed-integer nonlinear programming (MINLP). We also review their extensions to dynamic optimization and optimization under uncertainty. While these areas are still subject to significant research efforts, the emphasis in this paper is on major developments that have taken place over the last 25 years.

Advances in simultaneous strategies for dynamic process optimization

Abstract Following on the popularity of dynamic simulation for process systems, dynamic optimization has been identified as an important task for key process applications. In this study, we present an improved algorithm for simultaneous strategies for dynamic optimization. This approach addresses two important issues for dynamic optimization. First, an improved nonlinear programming strategy is developed based on interior point methods. This approach incorporates a novel filter-based line search method as well as preconditioned conjugate gradient method for computing search directions for control variables. This leads to a significant gain in algorithmic performance. On a dynamic optimization case study, we show that nonlinear programs (NLPs) with over 800,000 variables can be solved in less than 67 CPU minutes. Second, we address the problem of moving finite elements through an extension of the interior point strategy. With this strategy we develop a reliable and efficient algorithm to adjust elements to track optimal control profile breakpoints and to ensure accurate state and control profiles. This is demonstrated on a dynamic optimization for two distillation columns. Finally, these algorithmic improvements allow us to consider a broader set of problem formulations that require dynamic optimization methods. These topics and future trends are outlined in the last section.

Line Search Filter Methods for Nonlinear Programming: Motivation and Global Convergence

Line search methods are proposed for nonlinear programming using Fletcher and Leyffers filter method [Math. Program., 91 (2002), pp. 239--269], which replaces the traditional merit function. Their global convergence properties are analyzed. The presented framework is applied to active set sequential quadratic programming (SQP) and barrier interior point algorithms. Under mild assumptions it is shown that every limit point of the sequence of iterates generated by the algorithm is feasible, and that there exists at least one limit point that is a stationary point for the problem under consideration. A new alternative filter approach employing the Lagrangian function instead of the objective function with identical global convergence properties is briefly discussed.

Simultaneous optimization and solution methods for batch reactor control profiles

Abstract Differential-algebraic optimization problems appear frequently in process engineering, especially in process control, reactor design and process identification applications. For fed-batch reactor systems the optimal control problem is especially difficult because of the presence of singular arcs and state variable constraints. For problems of this type we propose a simultaneous optimization and solution strategy based on successive quadratic programming (SQP) and orthogonal collocation on finite elements. In solving the resulting nonlinear programming (NLP) problem, a number of interesting analogs can be drawn to more traditional methods based on variational calculus. First, the collocation method has very desirable stability and accuracy properties. Secondly, it will be shown that NLP optimality conditions have direct parallels to general variational conditions for optimal control. To demonstrate this strategy, we consider the optimization of a fed-batch penicillin reactor using a number of cases. For the simplest case, the results presented here agree well with previously obtained, analytically-based solutions. In addition, accurate results are presented for more difficult cases where no analytic solution is available.

The advanced-step NMPC controller: Optimality, stability and robustness

Widespread application of dynamic optimization with fast optimization solvers leads to increased consideration of first-principles models for nonlinear model predictive control (NMPC). However, significant barriers to this optimization-based control strategy are feedback delays and consequent loss of performance and stability due to on-line computation. To overcome these barriers, recently proposed NMPC controllers based on nonlinear programming (NLP) sensitivity have reduced on-line computational costs and can lead to significantly improved performance. In this study, we extend this concept through a simple reformulation of the NMPC problem and propose the advanced-step NMPC controller. The main result of this extension is that the proposed controller enjoys the same nominal stability properties of the conventional NMPC controller without computational delay. In addition, we establish further robustness properties in a straightforward manner through input-to-state stability concepts. A case study example is presented to demonstrate the concepts.

Large-scale nonlinear programming using IPOPT : An integrating framework for enterprise-wide dynamic optimization

Integration of real-time optimization and control with higher level decision-making (scheduling and planning) is an essential goal for profitable operation in a highly competitive environment. While integrated large-scale optimization models have been formulated for this task, their size and complexity remains a challenge to many available optimization solvers. On the other hand, recent development of powerful, large-scale solvers leads to a reconsideration of these formulations, in particular, through development of efficient large-scale barrier methods for nonlinear programming (NLP). As a result, it is now realistic to solve NLPs on the order of a million variables, for instance, with the IPOPT algorithm. Moreover, the recent NLP sensitivity extension to IPOPT quickly computes approximate solutions of perturbed NLPs. This allows on-line computations to be drastically reduced, even when large nonlinear optimization models are considered. These developments are demonstrated on dynamic real-time optimization strategies that can be used to merge and replace the tasks of (steady-state) real-time optimization and (linear) model predictive control. We consider a recent case study of a low density polyethylene (LDPE) process to illustrate these concepts.

Assessment and Future Directions of Nonlinear Model Predictive Control

Foundations and History of NMPC.- Theoretical Aspects of NMPC.- Numerical Aspects of NMPC.- Robustness, Robust Design, and Uncertainty.- State Estimation and Output Feedback.- Industrial Perspective on NMPC.- NMPC and Process Control.- NMPC for Fast Systems.- Novel Applications of NMPC.- Distributed NMPC, Obstacle Avoidance, and Path Planning.