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

The front matter includes the title page, series page, copyright page, dedication, TOC, and preface

Most things can be improved, so engineers and scientists optimize. While designing systems and products requires a deep understanding of influences that achieve desirable performance, the need for an efficient and systematic decision-making approach drives the need for optimization strategies. This introductory chapter provides the motivation for this topic as well as a description of applications in chemical engineering. Optimization applications can be found in almost all areas of engineering. Typical problems in chemical engineering arise in process design, process control, model development, process identification, and real-time optimization. The chapter provides an overall description of optimization problem classes with a focus on problems with continuous variables. It then describes where these problems arise in chemical engineering, along with illustrative examples. This introduction sets the stage for the development of optimization methods in the subsequent chapters.

1.1 Scope of Optimization Problems

From a practical standpoint, we define the optimization task as follows: given a system or process, find the best solution to this process within constraints. This task requires the following elements:

• An objective function is needed that provides a scalar quantitative performance measure that needs to be minimized or maximized. This can be the system's cost, yield, profit, etc.

• A predictive model is required that describes the behavior of the system. For the optimization problem this translates into a set of equations and inequalities that we term constraints. These constraints comprise a feasible region that defines limits of performance for the system.

• Variables that appear in the predictive model must be adjusted to satisfy the constraints. This can usually be accomplished with multiple instances of variable values, leading to a feasible region that is determined by a subspace of these variables. In many engineering problems, this subspace can be characterized by a set of decision variables that can be interpreted as degrees of freedom in the process.

Unconstrained optimization is auseful tool for many engineering applications, particularly in the estimation of parameters and states from data, to assess trade-offs for economic analysis and to analyze a variety of chemical engineering systems. Moreover, an understanding of unconstrained optimization concepts is essential for optimization of constrained problems. These underlying concepts form the basis for many of the developments in later chapters. This chapter describes the formulation of unconstrained optimization problems, discusses fundamentals including continuity, differentiability, and convexity, and develops first and second order conditions for local optimality. The chapter includes several examples and motivates the need for iterative algorithms in the following chapter.

2.1 Introduction

We begin by considering a scalar, real objective function in n variables, i.e., ƒ (x) : ℝⁿ → ℝ. The unconstrained optimization problem can be written as min x ∈ ℝn ƒ (x) . 2.1 Note that this problem is equivalent to max x ∈ ℝn −ƒ (x) 2.2 and that the objective function values are related by − max x (−ƒ (x) ) = min x ƒ (x) . 2.3

To motivate the formulation and solution of unconstrained optimization problems, we consider the following data fitting example problem.

Example 2.1 Consider the reaction of chemical species A, B, and C in an isothermal batch reactor according to the following first order reactions: A → k1 B and A → k2 C.

Newton-type methods are presented and analyzed for the solution of unconstrained optimization problems. In addition to covering the basic derivation and local convergence properties, both line search and trust region methods are described as globalization strategies, and key convergence properties are presented. The chapter also describes quasi-Newton methods and focuses on derivation of symmetric rank one (SR1) and Broyden—Fletcher—Goldfarb—Shanno (BFGS) methods, using simple variational approaches. The chapter includes a small example that illustrates the characteristics of these methods.

3.1 Introduction

Chapter 2 concluded with the derivation of Newton's method for the unconstrained optimization problem min x ∈ ℝn ƒ (x) . 3.1 For unconstrained optimization, Newton's method forms the basis for the most efficient algorithms. Derived from Taylor's theorem, this method is distinguished by its fast performance. As seen in Theorem 2.20, this method has a quadratic convergence rate that can lead, in practice, to inexpensive solutions of optimization problems. Moreover, extensions to constrained optimization rely heavily on this method; this is especially true in chemical process engineering applications. As a result, concepts of Newton's method form the core of all of the algorithms discussed in this book.

On the other hand, given a solution to (3.1) that satisfies first and second order sufficient conditions, the basic Newton method in Algorithm 2.1 may still have difficulties and may be unsuccessful. Newton's method can fail on problem (3.1) for the following reasons:

1. The objective function is not smooth. Here, first and second derivatives are needed to evaluate the Newton step, and Lipschitz continuity of the second derivatives is needed to keep them bounded.

2. The Newton step does not generate a descent direction. This is associated with Hessian matrices that are not positive definite.

This chapter develops the underlying properties for constrained optimization. It describes concepts of feasible regions and reviews convexity conditions to allow visualization of constrained solutions. Karush—Kuhn—Tucker (KKT) conditions are then presented from two viewpoints. First, an intuitive, geometric interpretation is given to aid understanding of the conditions. Once presented, the KKT conditions are analyzed more rigorously. Several examples are provided to illustrate these conditions along with the role of multipliers and constraint qualifications. Finally, two special cases of nonlinear programming, linear and quadratic, are briefly described, and a case study on portfolio optimization is presented to illustrate their characteristics.

4.1 Introduction

The previous two chapters dealt with unconstrained optimization, where the solution was not restricted in ℝⁿ. This chapter now considers the influence of a constrained feasible region in the characterization of optimization problems. For continuous variable optimization, we consider the general NLP problem given by minƒ (x) s.t.g (x)≤0, h (x)=0, 4.1 and we assume that the functions ƒ(x) : ℝⁿ → ℝ, h(x) : ℝⁿ → ℝ^m, and g(x) : ℝⁿ → ℝ^r have continuous first and second derivatives. For ease of notation we will refer to the feasible region as ℱ = {x|g(x) ≤ 0,h(x) = 0}. As with unconstrained problems, we first characterize the solutions to (4.1) with the following definition.

Definition 4.1 Constrained Optimal Solutions.

• A point x* is a global minimizer if ƒ(x*) ≤ ƒ(x) for all x ∈ ℱ.

• A point x* is a local minimizer if ƒ(x*) ≤ ƒ(x) for all x ∈ N ( x* )∩ℱ, where we define N ( x* )= ‖x− x* ‖ <ϵ, ϵ > 0.

This chapter extends the Newton-based algorithms in Chapter 3 to deal with equality constrained optimization. It generalizes the concepts of Newton iterations and associated globalization strategies to the optimality conditions for this constrained optimization problem. As with unconstrained optimization we focus on two important aspects: solving for the Newton step and ensuring convergence from poor starting points. For the first aspect, we focus on properties of the KKT system and introduce both full- and reduced-space approaches to deal with equality constraint satisfaction and constrained minimization of the objective function. For the second aspect, the important properties of penalty-based merit functions and filter methods will be explored, both for line search and trust region methods. Several examples are provided to illustrate the concepts developed in this chapter and to set the stage for the nonlinear programming codes discussed in Chapter 6.

5.1 Introduction to Equality Constrained Optimization

The previous chapter deals with optimality conditions for the nonlinear program (4.1). This chapter considers solution strategies for a particular case of this problem. Here we consider the equality constrained NLP problem of (5.1) given by minƒ (x) s.t.h (x)=0, 5.1 and we assume that the functions ƒ(x) : ℝⁿ → ℝ and h(x) : ℝⁿ → ℝ^m have continuous first and second derivatives. First, we note that if the constraints h(x) = 0 are linear, then (5.1) is a convex problem if and only if ƒ(x) is convex. On the other hand, as discussed in Chapter 4, nonlinear equality constraints imply nonconvex problems even if ƒ(x) is convex. As a result, the presence of nonlinear equalities leads to violation of convexity properties and will not guarantee that local solutions to (5.1) are global minima. Therefore, unless additional information is known about (5.1), we will be content in this chapter to determine only local minima.

This chapter deals with the solution of nonlinear programs with both equality and inequality constraints. It follows directly from the previous chapter and builds directly on these concepts and algorithms for equality constraints only. The key concept of this chapter is the extension for the treatment of inequality constraints. For this, we develop three approaches: sequential quadratic programming (SQP)-type methods, interior point methods, and nested projection methods. Both global and local convergence properties of these methods are presented and discussed. With these fundamental concepts we then describe and classify many of the nonlinear programming codes that are currently available. Features of these codes will be discussed along with their performance characteristics. Several examples are provided and a performance study is presented to illustrate the concepts developed in this chapter.

6.1 Constrained NLP Formulations

We now return to the general NLP problem (4.1) given by minƒ (x) s.t.g (x)≤0, h (x)=0, 6.1 where we assume that the functions ƒ (x), h(x), and g(x) have continuous first and second derivatives. To derive algorithms for nonlinear programming and to exploit particular problem structures, it is often convenient to consider related, but equivalent NLP formulations. For instance, by introducing nonnegative slack variables s ≥ 0, we can consider the equivalent bound constrained problem: min ƒ (x) s.t.g (x)+s=0,s≥0, h (x)=0, 6.2 where the nonlinearities are now shifted away from the inequalities.

The chapter deals with the formulation and solution of chemical process optimization problems based on models that describe steady state behavior. Many of these models can be described by algebraic equations, and here we consider models only in this class. In process engineering such models are applied for the optimal design and analysis of chemical processes, optimal process operation, and optimal planning of inventories and products. Strategies for assembling and incorporating these models into an optimization problem are reviewed using popular modular and equation-oriented solution strategies. Optimization strategies are then described for these models and gradient-based NLP methods are explored along with the calculation of accurate derivatives. Moreover, some guidelines are presented for the formulation of equation-oriented optimization models. In addition, four case studies are presented for optimization in chemical process design and in process operations.

7.1 Introduction

The previous six chapters focused on properties of nonlinear programs and algorithms for their efficient solution. Moreover, in the previous chapter, we described a number of NLP solvers and evaluated their performance on a library of test problems. Knowledge of these algorithmic properties and the characteristics of the methods is essential to the solution of process optimization models.

As shown in Figure 7.1, chemical process models arise from quantitative knowledge of process behavior based on conservation laws (for mass, energy, and momentum) and constitutive equations that describe phase and chemical equilibrium, transport processes, and reaction kinetics, i.e., the “state of nature.” These are coupled with restrictions based on process and product specifications, as well as an objective that is often driven by an economic criterion. Finally, a slate of available decisions, including equipment parameters and operating conditions need to be selected. These items translate to a process model with objective and constraint functions that make up the NLP. Care must be taken that the resulting NLP problem represents the real-world problem accurately and also consists of objective and constraint functions that are well defined and sufficiently smooth.

This chapter provides the necessary background to develop and solve dynamic optimization problems that arise in chemical processes. Such problems arise in a wide variety of areas. Off-line applications range from batch process operation, transition optimization for different product grades, analysis of transients and upsets, and parameter estimation. Online problems include formulations for model predictive control, online process identification, and state estimation. The chapter describes a general multiperiod problem formulation that applies to all of these applications. It also discusses a hierarchy of optimality conditions for these formulations, along with variational strategies to solve them. Finally, the chapter introduces dynamic optimization methods based on NLP which will be covered in subsequent chapters.

8.1 Introduction

With growing application and acceptance of large-scale dynamic simulation in process engineering, recent advances have also been made in the optimization of these dynamic systems. Application domains for dynamic optimization cover a wide range of process tasks and include

• design of distributed systems in chemical engineering, including reactors and packed column separators;

• off-line and online problems in process control, particularly for multivariable systems that are nonlinear and output constrained; these are particularly important for nonlinear model predictive control and real-time optimization of dynamic systems;

• trajectory optimization in chemical processes for transitions between operating conditions and to handle load changes;

• optimum batch process operating profiles, particularly for reactors and separators;

• parameter estimation and inverse problems that arise in state estimation for process control as well as in model building applications.

This chapter develops dynamic optimization strategies that couple solvers for differential algebraic equations (DAEs) with NLP algorithms developed in the previous chapters. First, to provide a better understanding of DAE solvers, widely used Runge—Kutta and linear multistep methods are described, and a summary of their properties is given. A framework is then described for sequential dynamic optimization approaches that rely on DAE solvers to provide function information to the NLP solver. In addition, gradient information is provided by direct and adjoint sensitivity methods. Both methods are derived and described along with examples. Unfortunately, the sequential approach can fail on unstable and ill-conditioned systems. To deal with this case, we extend the sequential approach through the use of multiple shooting and concepts of boundary value solvers. Finally, a process case study is presented that illustrates both sequential and multiple shooting approaches.

9.1 Introduction

Chapter 8 provided an overview of the features of dynamic optimization problems and their optimality conditions. From that chapter, it is clear that the extension to larger dynamic optimization problems requires efficient and reliable numerical algorithms. This chapter addresses this question by developing a popular approach for the optimization of dynamic process systems. Powerful DAE solvers for large-scale initial value problems have led to widely used simulation environments for dynamic nonlinear processes. The ability to develop dynamic process simulation models naturally leads to their extension for optimization studies. Moreover, with the development of reliable and efficient NLP solvers, discussed in Chapter 6, an optimization capability can be implemented for dynamic systems along the lines of modular optimization modes discussed in Chapter 7. With robust simulation models, the implementation of NLP codes can be done in a reasonably straightforward way.

Following on embedded methods for dynamic optimization, this chapter considers “all-at-once” or direct transcription methods that allow a simultaneous approach for this optimization problem. In particular, we consider formulations based on orthogonal collocation methods. These methods can also be represented as a special class of implicit Runge—Kutta (IRK) methods, and concepts and properties of IRK methods apply directly. Large-scale optimization formulations are then presented with the aim of maintaining accurate state and control profiles and locating potential break points. This approach is applied to consider a number of difficult problem classes including unstable systems, path constraints, and singular controls. Moreover, a number of real-world examples are featured that demonstrate the characteristics and advantages of this approach. These include batch crystallization processes, grade transitions in polymer processes, and large-scale parameter estimation for complex industrial reactors.

10.1 Introduction

This chapter evolves from sequential and multiple shooting approaches for dynamic optimization by considering a large nonlinear programming (NLP) formulation without an embedded DAE solver. Instead, we consider the multiperiod dynamic optimization problem (8.5) where the periods themselves are represented by finite elements in time, with piecewise polynomial representations of the state and controls in each element. This approach leads to a discretization that is equivalent to the Runge—Kutta methods described in the previous chapter. Such an approach leads to a fully open formulation, represented in Figure 7.1, and has a number of pros and cons. The large-scale NLP formulation allows a great deal of sparsity and structure, along with flexible decomposition strategies to solve this problem efficiently. Moreover, convergence difficulties in the embedded DAE solver are avoided, and sensitivity calculations from the solver are replaced by direct gradient and Hessian evaluations within the NLP formulation. On the other hand, efficient large-scale NLP solvers are required for efficient solutions, and accurate state and control profiles require careful formulation of the nonlinear program.

To address these problems, we rely on efficient large-scale NLP algorithms developed in Chapter 6.

Steady state and dynamic process models frequently deal with switches and other nonsmooth decisions that can be represented through complementarity constraints. If these formulations can be applied at the NLP level, they provide an “all-at-once” strategy that can be addressed with state-of-the-art NLP solvers. On the other hand, these mathematical programs with complementarity constraints (MPCCs) have not been widely studied in process engineering and must be handled with care. These formulations are nonconvex and violate constraint qualifications, including LICQ and MFCQ, which are required for well-posed, nonlinear programs. This chapter deals with properties of MPCCs including concepts of stationarity and linear independence that are essential for well-defined NLP formulations. NLP-based solution strategies for MPCCs are then reviewed along with examples of complementarity drawn from steady state chemical engineering applications. In addition, we extend these MPCC formulations for the optimization of a class of hybrid dynamic models, where the differential states remain continuous over time. These involve differential inclusions of the Filippov type, and a formulation is developed that preserves the piece wise smoothness properties of the dynamic system. Results on several process examples drawn from distillation optimization, process control, and hybrid systems are used to illustrate MPCC formulations and demonstrate the proposed optimization approach.

11.1 Introduction

Optimization problems were introduced in Chapter 1 with discrete and continuous decision variables. This mixed integer nonlinear programming formulation (1.1) leads to the most general description of process optimization problems. To solve these problems, NLP formulations are usually solved at a lower level for fixed values of the discrete variables. Based on information from the NLP solution, a search of the discrete variables is then conducted at a higher level.

While the scope of this book is restricted to nonlinear programs with smooth objective and constraint functions, the ability to deal with (some) discrete decisions within an “all-at-once” NLP formulation can have an advantage over nested MINLP strategies. This motivates the modeling of these decisions with complementarity constraints.

The back matter includes Bibliography and Index