Optimal control of switched systems arising in fermentation processes 2014

Springer Optimization and Its Applications 97Optimal Control of Switched Systems Arising in Fermentation Processes Chongyang Liu Zhaohua Gong... 1.2.2 Optimal Switching Control For op

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Springer Optimization and Its Applications 97

Optimal Control

of Switched

Systems Arising

in Fermentation Processes

Chongyang Liu

Zhaohua Gong

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Springer Optimization and Its Applications

J Birge (University of Chicago)

C.A Floudas (Princeton University)

F Giannessi (University of Pisa)

H.D Sherali (Virginia Polytechnic and State University)

T Terlaky (Lehigh University)

Y Ye (Stanford University)

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Aims and Scope

Optimization has been expanding in all directions at an astonishing rate during thelast few decades New algorithmic and theoretical techniques have been developed,the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge

of all aspects of the field has grown even more profound At the same time, one ofthe most striking trends in optimization is the constantly increasing emphasis on theinterdisciplinary nature of the field Optimization has been a basic tool in all areas

of applied mathematics, engineering, medicine, economics, and other sciences

The series Springer Optimization and Its Applications publishes undergraduate

and graduate textbooks, monographs and state-of-the-art expository work thatfocus on algorithms for solving optimization problems and also study applicationsinvolving such problems Some of the topics covered include nonlinear optimization(convex and nonconvex), network flow problems, stochastic optimization, optimalcontrol, discrete optimization, multiobjective programming, description of softwarepackages, approximation techniques and heuristic approaches

More information about this series athttp://www.springer.com/series/7393

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Chongyang Liu • Zhaohua Gong

Optimal Control of Switched Systems Arising in

Fermentation Processes

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Chongyang Liu

Zhaohua Gong

Mathematics and Information Science

Shandong Institute of Business

and Technology

Yantai, Shandong, China

ISSN 1931-6828 ISSN 1931-6836 (electronic)

ISBN 978-3-662-43792-6 ISBN 978-3-662-43793-3 (eBook)

DOI 10.1007/978-3-662-43793-3

Springer Heidelberg New York Dordrecht London

Jointly published with Tsinghua University Press, Beijing

ISBN: 978-7-302-37332-2 Tsinghua University Press, Beijing

Library of Congress Control Number: 2014949499

Mathematics Subject Classification: 49J15, 49J21, 65K10, 49M37, 92C42

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The modern fermentation industry, which is largely a product of the twentiethcentury, is dominated by aerobic/anaerobic cultivations intended to make a range

of high-value products However, since most fermentation processes create verydilute and impure products, there is a great need to increase volumetric productivityand to increase the product concentration As a result, significant work is needed

to optimize the operation and design of bioreactors to make production moreefficient and more economical It is obvious that a model-based efficient approach

is necessary to ensure maximum productivity with the lowest possible cost infermentation processes, without requiring a human operator Nevertheless, themathematical determination of optimal control in a fermentation process can be verydifficult and open-ended due to the presence of nonlinearities in process models,inequality constraints on process variables, and implicit process discontinuities

In this book, we present some mathematical models arising in tion processes They are in the form of nonlinear multistage system, switchedautonomous system, time-dependent switched system, state-dependent switchedsystem, multistage time-delay system, and switched time-delay system On the basis

fermenta-of these dynamical systems, we consider the optimization problems including the

v

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vi Preface

optimal control problems and the optimal parameter selection problems We discusssome important theories, such as existence of optimal controls and optimizationalgorithms for the optimization problems mentioned above

The objective of this book is to present, in a systematic manner, the optimal trols under different mathematical models in fermentation processes By bringingforward fresh novel methods and innovative tools, we are to provide a state-of-the-art and comprehensive systematic treatment of optimal control problems arising

con-in fermentation processes This can not only develop nonlcon-inear dynamical system,optimal control theory, and optimization algorithms but also increase processproductivity of product and serve as a reference for commercial fermentationprocesses

Acknowledgments

For the completion of the book, we are indebted to many distinguished individuals

in our community We would like to thank Prof Enmin Feng and Prof ZhilongXiu, Dalian University of Technology, China, for bringing our attention to this area.Almost all the materials presented in this book are extracted from work done jointlywith them It is our pleasure to express our gratitude to Prof Kok Lay Teo, Dr RyanLoxton, and Dr Qun Lin, Curtin University, Australia, for their valuable commentsduring our visiting at Curtin University from January 2013 to July 2014

We gratefully acknowledge the unreserved support, constructive comments,and fruitful discussions from Dr Lei Wang, Dr Yaqin Sun, and Dr QingruiZhang, Dalian University of Technology, China; Dr Jianxiong Ye, Fujian NormalUniversity, China; Dr Bangyu Shen, Huaiyin Normal University, China; and

Dr Jin’gang Zhai, Ludong University, China

We are also grateful to Prof Yuliang Han and Prof Guang’ai Song, ShandongInstitute of Business and Technology, China, for their kind invitations in publishingthe book

Financial Support

We acknowledge the financial support from the National Natural Science tion of China under Grants 11201267, 11001153, and 11126077, from the ShandongProvince Natural Science Foundation of China under Grant ZR2010AQ016, andfrom Shandong Institute of Business and Technology under Grant Y2012JQ02

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1 Introduction 1

1.1 Switched System 1

1.2 Optimal Control 2

1.2.1 Standard Optimal Control 2

1.2.2 Optimal Switching Control 4

1.3 Fermentation Process 5

1.3.1 Generic Fermentation Process 5

1.3.2 1,3-Propanediol Fermentation 7

1.3.3 Kinetics and Physiological Modeling 8

1.4 Outline of the Book 9

2 Mathematical Preliminaries 13

2.1 Lebesgue Measure and Integration 13

2.2 Normed Spaces 17

2.3 Linear Functionals and Dual Spaces 20

2.4 Bounded Variation 22

3 Constrained Mathematical Programming 25

3.1 Introduction 25

3.2 Gradient-Based Algorithms 26

3.2.1 Optimality Conditions 27

3.2.2 The Quadratic Penalty Method 28

3.2.3 Augmented Lagrangian Method 30

3.2.4 Sequential Quadratic Programming 32

3.3 Evolutionary Algorithms 35

3.3.1 Particle Swarm Optimization 35

3.3.2 Differential Evolution 36

3.3.3 Constraint-Handling Techniques 38

vii

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viii Contents

4 Elements of Optimal Control Theory 41

4.1 Introduction 41

4.2 Dynamical Systems 41

4.2.1 Ordinary Differential System 41

4.2.2 Delay-Differential System 44

4.2.3 Switched System 48

4.3 Optimal Control Problems 49

4.3.1 Standard Optimal Control Problem 49

4.3.2 Optimal Multiprocess Control Problem 50

4.4 Necessary Optimality Conditions 52

4.4.1 Necessary Conditions for Standard Optimal Control Problem 52

4.4.2 Necessary Conditions for Optimal Multiprocesses 54

5 Optimal Control of Nonlinear Multistage Systems 59

5.1 Introduction 59

5.2 Controlled Multistage Systems 60

5.3 Properties of the Controlled Multistage Systems 63

5.4 Optimal Control Models 66

5.5 Computational Approaches 68

5.6 Numerical Results 73

5.7 Conclusion 76

6 Optimal Control of Switched Autonomous Systems 77

6.1 Introduction 77

6.2 Switched Autonomous Systems 78

6.6 Conclusion 86

7 Optimal Control of Time-Dependent Switched Systems 89

7.1 Introduction 89

7.2 Time-Dependent Switched Systems 90

7.3 Constrained Optimal Control Problems 93

7.4.1 Approximate Problem 94

7.4.2 Continuous State Constraints 96

7.4.3 Optimization Algorithms 98

7.6 Conclusion 103

8 Optimal Control of State-Dependent Switched Systems 105

8.1 Introduction 105

8.2 State-Dependent Switched Systems 106

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Contents ix

8.4 Solution Methods for the Inner Optimization Problem 113

8.6 Conclusion 119

9 Optimal Parameter Selection of Multistage Time-Delay Systems 123

9.2 Problem Formulation 124

9.2.1 Multistage Time-Delay Systems 124

9.2.2 Properties of the Multistage Time-Delay Systems 126

9.3 Parametric Sensitivity Analysis 128

9.3.1 Sensitivity Functions 128

9.3.2 Numerical Simulation Results 132

9.4 Optimal Parameter Selection Problems 135

9.4.1 Optimal Parameter Selection Models 135

9.4.2 A Computational Procedure 136

9.4.3 Numerical Results 139

9.5 Conclusion 142

10 Optimal Control of Multistage Time-Delay Systems 143

10.2 Controlled Multistage Time-Delay Systems 144

10.3 Constrained Optimal Control Problems 148

10.6 Conclusion 158

11 Optimal Control of Switched Time-Delay Systems 159

11.2 Switched Time-Delay Systems 160

11.3 Optimal Control Problems 163

11.3.1 Free Time Delayed Optimal Control Problem 163

11.3.2 The Equivalent Optimal Control Problem 164

11.4 Numerical Solution Methods 166

11.4.1 Approximation Problem 166

11.4.2 A Computational Procedure 167

11.6 Conclusion 174

References 177

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Chapter 1

Introduction

By a switched system, we mean a hybrid dynamical system consisting of a family of

continuous-time subsystems and a rule that orchestrates the switching between them[123] Many systems encountered in practice exhibit switching between several sub-systems depending on various environmental factors [63,262,281] Another source

of motivation for studying switched systems comes from the rapidly developingarea of switching control Control techniques based on switching between differentcontrollers have been applied extensively in recent years, where they have beenshown to improve control performance [100,128,181] Switched systems havenumerous applications in the control of mechanical systems, automotive industry,aircraft and air traffic control, switching power converters, and many other fields

The switching rules in switched systems can be classified into state-dependent versus time-dependent switching and autonomous versus controlled switching [59].For a state-dependent switching, we suppose that the continuous state space(e.g.,Rn) is partitioned into a finite or infinite number of operation regions by

means of a family of switching surfaces, or guards In each of these regions, a

continuous-time dynamical system (described by differential equations, with orwithout controls) is given Whenever the system trajectory hits a switching surface,

the continuous state jumps instantaneously to a new value, specified by a reset map.

In contrast, for a time-dependent switching, the continuous-time dynamical system’sswitchings are activated according to time functions, i.e., a switching occurs at acertain time instant These switching instants can be prescribed a priori and fixed ordesigned arbitrarily by engineers On the other hand, by autonomous switching, wemean a situation where we have no direct control over the switching mechanism thattriggers the discrete events This category includes systems with state-dependentswitching in which locations of the switching surfaces are predetermined as well assystems with time-dependent switching in which the rule that defines the switching

C Liu, Z Gong, Optimal Control of Switched Systems Arising

in Fermentation Processes, Springer Optimization and Its Applications 97,

DOI 10.1007/978-3-662-43793-3 1

1

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2 1 Introduction

signal is unknown (or was ignored at the modeling stage) In contrast with theautonomous switching, in many situations the switching is actually imposed by thedesigner in order to achieve a desired behavior of the system In this case, we havedirect control over the switching mechanism (which can be state-dependent or time-dependent) and may adjust it as the system evolves For various reasons, it may

be natural to apply discrete control actions, which leads to systems with controlledswitching

As a special class of hybrid systems, switched systems are inherently nonlinearand non-smooth, and therefore many of the results available from the vast literature

on linear systems and smooth nonlinear systems do not apply Consequently,many basic system theoretic problems like well-posedness, stability, controllability,observability, safety, etc., and many design methods for controllers have to be

reconsidered within the hybrid context A system is said to be well posed if a

solution of the system exists and is unique given an initial condition (and possiblyinput signals) [67] The well-posedness property indicates that the system doesnot exhibit deadlock behavior (no solutions from certain initial conditions) andthat determinism (uniqueness of solutions) is satisfied The basic problems ofstability for switched systems were discussed in [134] Then, various methodshave been developed to analyze stability through various types of Lyapunovfunctions such as common Lyapunov function [59], multiple Lyapunov function[35], surface Lyapunov function [89], etc The other stability results of switchedsystems are presented in [64,106,137,178,277] For the controllability conceptand its historical comments, one may refer to [232] and references therein Thecomplexity of characterizing controllability and stabilizability has been studied

in [33] Controllability problem for piecewise linear systems has been studied;see, for example, [76,86,130,269] A similar story holds for observability anddetectability [11,22,56] For switched systems, a wide body of literature exists onthe development of stabilizing controllers [178,261] and model predictive control[25,176,182] In this book, we shall focus on the optimal control of switchedsystems arising in fermentation processes

Optimal control problem is to determine the control policy that will extremize

(maximize or minimize) a specific performance criterion, subject to the constraintsimposed by the physical nature of the problem Over the years, optimal controltheory has been applied to a diverse collection of problems [38,114,205]

1.2.1 Standard Optimal Control

Optimal control theory is an outcome of the calculus of variations, with a historystretching back over 300 years [216] In 1638, G Galileo posed two shape problems:

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1.2 Optimal Control 3

the shape of a heavy chain suspended between two points (the catenary) and theshape of a wire such that a bead sliding along it under gravity traverses the distancebetween its endpoints in minimum time (the brachistochrone) Later, L Eulerformulated the problem in general terms as one of finding the curve x.t / over theinterval a6 t 6 b, with given values x.a/, x.b/, which minimizes

a

for some given function L.t; x; Px/, where Px WD dx=dt, and he gave a necessary

condition of optimality for the curve x./

d

where the suffix x or Px implies the partial derivative with respect to x or Px In

a letter to Euler in 1755, J.L Lagrange described an analytical approach, based

on perturbations or “variations” of the optimal curve and using his “undeterminedmultipliers,” which led directly to Euler’s necessary condition, now known asthe “Euler-Lagrange equation.” Euler enthusiastically adopted this approach andrenamed the subject “the calculus of variations.”

However, modern optimal control theory was established in the late 1950s

since R Bellman introduced dynamic programming to solve discrete-time optimal

control problems [21], L.S Pontryagin developed minimum principle [202], andR.E Kalman provided linear quadratic regulator and linear quadratic Gaussiantheory to design optimal feedback controls [113] Subsequently, the existence of theoptimal control for optimal control problems was widely investigated [41,42,78,

214,233] The optimality conditions were also discussed in [51,69,154,155,201].Some optimization problems involve optimal control problems, which areconsiderably complex and involve a dynamic system There are very few real-world optimal control problems that lend themselves to analytical solutions As aresult, using numerical algorithms to solve the optimal control problems becomes

a common approach that has attracted attention of many researchers, engineers,and managers The numerical solution of the optimal control problems can becategorized into two different approaches: (1) the direct and (2) the indirect method[236] Direct methods are based on discretization of state and/or control variablesover time and then solving the resulting problem using a nonlinear programmingsolver Based on the discretization of the state and/or control, direct methods can

be categorized into three different approaches The first approach is based on stateand control variable parameterization [73,74,77,212,250] The second approach iscontrol parameterization [101,127,139,156,211,240] The third approach is based

on state parameterization only [107,230] Indirect method solves the optimal controlproblem by deriving the necessary conditions based on Pontryagin’s minimumprinciple The first step of this method is to formulate an appropriate two-pointboundary value problem (TPBVP), and the second step is to solve the TPBVPnumerically [37,116,177,190]

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4 1 Introduction

For a dynamic system in the optimal control problem, a system which is governed

by a set of ordinary differential equations is called lumped parameter system In

contrast, if a system is governed by a set of partial differential equations, then the

system is called a distributed parameter system In this book, we shall only deal

with optimal control problems involving lumped parameter systems For the optimalcontrol of distributed parameter systems, we refer the interested reader to [1,39,58,

79,142,244] for details

1.2.2 Optimal Switching Control

For optimal control problem of switched systems, the added flexibility of being able

to switch between subsystems greatly increases the complexity of searching for anoptimal control In the most general case, determining an optimal control strategyfor a switched system involves determining an optimal continuous input functionand an optimal switching sequence

The problem of determining optimal control laws for switched systems has beenwidely investigated in the last years, both from theoretical and from computationalpoints of view [274] The available theoretical results usually extend the classicalminimum principle or the dynamic programming approach to switched systems.For continuous-time hybrid systems, general necessary conditions for the existence

of optimal control laws were discussed in [36] by using dynamic programming.Necessary and/or sufficient optimality conditions for a trajectory of a hybridsystem with a fixed sequence of finite length were derived using the minimumprinciple in [71,199,223,238] The existence of optimal control for switchedsystems was investigated [68,221,279] The computational results take advantage

of efficient nonlinear optimization techniques and high-speed computers to developefficient numerical methods for the optimal control of switched systems Theproblem of optimal control of switched autonomous systems was studied for aquadratic cost functional on an infinite horizon and a fixed number of switches in[87,220] Gradient-based algorithms for solving the switching instants in switchedautonomous systems were developed in [72,160,273] A two-stage optimizationmethodology was proposed for optimal control of switched systems with controlinput [272,275] Based on a parameterization of the switching instants, an optimalcontrol approach was developed in [133,158] Essentially different from the resultsmentioned above, the switched system was embedded into a larger family ofnonlinear systems that can be handled directly by classical control theory [26–28]

By adopting such problem transformation, there is no need to make any assumptionsabout the number of switches nor about the mode sequence at the beginning

of the optimization The possible numerical nonlinear programming techniqueunder this framework was explored in [259] It showed that sequential quadraticprogramming can be utilized to reduce the computational complexity introduced

by mixed integer programming The effectiveness of the proposed approach was

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demonstrated through several examples Recently, the problem of computing theschedule of modes in switched systems was investigated in [9,40,138,224,258].The vast majority of optimization techniques for switched systems, includingthose mentioned above, are restricted to switched systems without time delays.However, time delays are common in practical engineering systems [208] Indeed,switched systems with time delays have various applications in areas such as powersystems [175] and network control systems [121] The presence of delays in aswitched system complicates the search for an optimal control policy Necessaryconditions for determining optimal switching times and/or optimal impulse mag-nitudes for such systems were derived in [66,248,249] via classical variationaltechniques Based on a parameterization scheme in which the switching instantsare expressed in terms of the subsystem durations, an effective optimal controlalgorithm for switched autonomous systems with single time delay was presented

in [268]

Fermentation is a very ancient practice indeed, dating back several millennia Morerecently, fermentation processes have been developed for the manufacture of a vastrange of materials from chemically simple feedstocks right up to highly complexprotein structures

1.3.1 Generic Fermentation Process

The origins of fermentation are lost in ancient history, perhaps even in prehistory.However, “fermentation” has many different and distinct meanings for differinggroups of individuals In the present context, we intend it to mean the use of selectedstrains of microorganisms and plant or animal cells for the manufacture of someuseful products or to gain insights into the physiology of these cell types [170]

By contrast, the modern fermentation industry, which is largely a product of thetwentieth century, is dominated by aerobic/anaerobic cultivations intended to make

a range of high-value products

There are three main modes of fermentation technique: batch, continuous,and fed-batch A batch fermentation process is characterized by no addition toand withdrawal from the culture of biomass, fresh nutrient medium, and culturebroth (with the exception of gas phase) In a continuous fermentation process, anopen system is set up Nutrient solution is added to the bioreactor continuously,and an equivalent amount of converted nutrient solution with microorganisms issimultaneously taken out of the system In a fed-batch fermentation, substrate isadded according to a predetermined feeding profile as the fermentation progresses

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6 1 Introduction

A fed-batch operation may be followed by a terminal batch operation, with culturevolume being equal to maximum permissible volume, to utilize the nutrientsremaining in the culture at the end of fed-batch operation A fed-batch operation isusually preceded by a batch operation A typical run involving fed-batch operationtherefore very often consists of the fed-batch operation sandwiched between twobatch operations This entire sequence (batch!fed-batch!batch) may be repeatedmany times leading to serial (or repeated) fed-batch operation

Although fermentation operations are abundant and important in industriesand academia which touch many human lives, high costs associated with manyfermentation processes have become the bottleneck for further development andapplication of the products Developing an economically and environmentally soundoptimal cultivation method becomes the primary objective of fermentation processresearch nowadays The goal is to control the process at its optimal state and toreach its maximum productivity with minimum development and production cost;

in the meantime, the product quality should be maintained A fermentation processmay not be operated optimally for various reasons For instance, an inappropriatenutrient feeding policy will result in a low production yield, even though the level

of feeding rate is very high An optimally controlled fermentation process offers therealization of high standards of product purity, operational safety, environmentalregulations, and reduction in costs [246] Nevertheless, different combinations andsequence of process conditions and medium components are needs to be biologi-cally investigated to determine the growth condition that produces the biomass withthe physiological state best constituted for product formation [195] Moreover, themathematical determination of optimal control in a fermentation process can be verydifficult and open-ended due to frequent presence of nonlinearity in process models,inequality constraints on process variables, and implicit process discontinuities [17].This presence gives rise to a multimodal and noncontinuous relation between aperformance index and a control function

Optimal control of fermentation processes has been a topic of research formany years Considerable emphasis has been placed on the control of fed-batchfermenters because of their prevalence in industry [111,129] From a processoperation point of view, most of studies are to calculate an optimal feed-rateprofile that will optimize a given objective function For the fed-batch processincluding one single operation, optimal control problem [34,125,231] and optimaladaptive control problem [18,108,247] have been discussed Some useful tools such

as Green function [193], the calculus of variations [135,136,179,191], iterativedynamic programming [163], evolutionary algorithm [50,210,213], and geneticalgorithm [217] are used to determine this profile in fed-batch processes Forthe serial fed-batch operations, parameter optimization problem [252,253,278]and optimal impulsive control problem [84,85,254] have been reported For thecontinuous process, time optimal control problem [60], maximum harvest problem[61,62], optimal operation problem [239], and parameter optimization problem[226,227] have been discussed For the batch process, dynamic optimizationproblem [3,245,255,256], optimal operation problem [34], and robust optimalcontrol problem [183] have been investigated

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In this book, we focus on optimal control of fed-batch process including a serial

of operations This process is more complex and the abovementioned theories andmethods are not applicable for this problem Thus, new theory and computationmethods are needed for the optimal control problems in this book

1.3.2 1,3-Propanediol Fermentation

Biodiesel (green diesel) fuels already constitute an alternative type of fuel forvarious types of diesel engines and heating systems [102] Due to the increasingcost of conventional fuels, the application of biofuels in a large commercial scale

is strongly recommended by various authorities, and this fact could likely result

in the generation of tremendous quantities of glycerol in the near future [283].Furthermore, besides biodiesel production units, concentrated glycerol-containingwaters are also produced as the main by-product from fat saponification andalcoholic beverage fabrication units [16,197] For all of these reasons, glyceroloverproduction and disposal is very likely to cause severe environmental problems

in the near future Therefore, conversion of glycerol to various value products by the means of chemical and/or fermentation technology currentlyattracts much interest [31] The most obvious target of biotechnological glycerolvalorization is referred to its biotransformation into 1,3-propanediol (1,3-PD) Thisproduct is a substance of importance for the textile industry, due to its application

higher-added-as monomer for the synthesis of aliphatic polyesters [131] Plastics based on thismonomer exhibit good product properties [264] Additionally, a recent development

of a new polyester (polypropylene terephalate), presenting unique properties forthe fiber industry, necessitated the drastic increase in the production of 1,3-PD[131] Moreover, 1,3-PD can present various interesting applications in the chemicalindustry [31,283]

1,3-PD is one of the oldest known fermentation products It was reliably fied as early as in 1881 [83], in a glycerol fermentation mixed culture containing

identi-Clostridium pasteurianum as an active organism The majority of commercial

syntheses of 1,3-PD are from acrolein by Degussa (now owned by DuPont) andfrom ethylene oxide by Shell Problems in these conventional processes are the highpressure applied in the hydroformylation and hydrogenation steps along with hightemperature, use of expensive catalyst, and release of toxic intermediates Consid-ering the yield, product recovery, and environmental protection, much attention hasbeen paid to its microbial production [49,65,185,276] The principal way of thebiotechnological conversion of raw materials to 1,3-PD is referred to transformation

of glycerol into 1,3-PD conducted by a number of microorganisms The most

extensively studied microorganisms belong to the species Citrobacter freundii, Klebsiella pneumoniae (K pneumoniae), Klebsiella oxytoca, Enterobacter agglom- erans, Clostridium butyricum and Clostridium acetobutylicum [196] Among these

organisms, K pneumoniae is considered as one of the best “natural producers”

and is paid more attention because of its appreciable substrate tolerance, yield, and

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8 1 Introduction

productivity [173] The enzymes and pathways involved in glycerol dissimilation to

1,3-PD production by K pneumoniae have been elucidated in [82] Regarding thefermentation, batch fermentation [16], continuous fermentation [173], and fed-batchfermentation [48,287] have been performed Substrate and product inhibitions are

the main limiting factors for the microbial production of 1,3-PD by K pneumoniae.

In order to investigate the possibility of maximization of 1,3-PD production,

genetically modified strains of the wild strain K pneumoniae have been created

[184,286]

1.3.3 Kinetics and Physiological Modeling

The optimization and control of bioprocesses often requires the establishment of

a mathematical model that describes the metabolic activities of microorganisms,especially with respect to the responses of cells to a change in the physiologicalenvironment Rate equations for microbial growth, substrate uptake, and productformation that describe the kinetics of a process are the basis for mathematicalmodeling The rate equations used for microbial growth can be generally clas-sified into two categories, i.e., unstructured models and structured models Theformer treat a culture as a lumped quantity of biomass and does not considerintracellular components; the latter consider the heterogeneity of a culture andthe intracellular components [13] Despite impressive progress made recently indeveloping structured models for microbial growth [188], the unstructured models

or semimechanistic models are still the most popular ones used in practice.The unstructured models include the most fundamental observations concerningmicrobial growth and are simple and easy to use, particularly for process controlpurposes

The fermentation of glycerol by K pneumoniae is a complex bioprocess, since

microbial growth is subjected to multiple inhibitions of substrate and products, e.g.,glycerol, 1,3-PD, ethanol, and acetate The following kinetic model was proposed

to describe microbial growth inhibited by several inhibitors [285]:

substrate-sufficient culture could be expressed as follows:

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S and mS are the maximum growth yield and maintenance requirement

of substrate under substrate-limited conditions, respectively; qm

S is the maximumincrement of substrate consumption rate under substrate-sufficient conditions; KS

is a saturation constant; mP i and Ym

P i are formation rate constants; qm

P i is themaximum increase or decrease of product formation rate due to substrate excess;and KPi is a saturation constant An improved model was proposed to describesubstrate consumption and product formation in a large range of feed glycerolconcentrations in medium [271] The main improvement is using the followingexpression to formulate the specific formation rate of ethanol qEtOH:

of both extracellular substances and intracellular substances was proposed in [237]

The book is organized in eleven chapters Except for Chap.1that briefly introducesthe switched system, optimal control and fermentation process, and their literaturereviews Besides this short introduction, there are ten major chapters, which arebriefly summarized as follows

For the convenience of the reader, some mathematical preliminaries aboutmeasure theory and functional analysis are stated without proofs in Chap.2.Engineers and applied scientists should be able to follow the mathematical proofs

in the subsequent chapters with the aid of Chap.2

In Chap.3, we review some results in constrained mathematical programming.This is important because after control parameterization, an optimal control problem

is reduced to an optimal parameter selection problem, which is essentially amathematical programming problem Chapter4presents a crash course in optimalcontrol theory for those readers who are not familiar with the subject

From Chap.5onward, we focus our attention on the optimal control of switchedsystems arising in fermentation processes We start from optimal control of anonlinear multistage system, which is a degenerate switched system since switching

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10 1 Introduction

law is decided a priori, in fed-batch fermentation process in Chap.5 Compared withexisting systems, the proposed system is much closer to the actual fermentationprocess The optimal control model involving the nonlinear multistage system andsubject to continuous state inequality constraint has been developed The existence

of optimal control is established by the theory of bounded variation A globaloptimization algorithm based on the control parameterization concept and theimproved particle swarm optimization algorithm is constructed to solve the optimalcontrol problem Numerical results show that the concentration of target productconcentration at the terminal time is increased considerably compared with theexperimental results

In Chap.6, we propose a switched autonomous system with variable switchinginstants to model the constantly fed-batch process Taking the switching instants

as the control function, we formulate an optimal control problem to optimizethe fermentation process By introducing a time-scaling transform, the optimalcontrol problem is transcribed into an equivalent one with parameters and fixedswitching instants A computational approach to seek the optimal switching instants

is developed This method is based on the constraint transcription technique and thesmoothing approximation method

In Chap.7, a time-dependent switched system, in which the feeding rate isthe control function and the switching instants are the optimization variables,

is proposed to formulate the fed-batch fermentation process We then present aconstrained optimal control problem involving the time-dependent switched system

To seek the optimal control and the optimal switching instants, we use the controlparameterization enhancing transform together with the constraint transcriptiontechnique to convert the constrained optimal control problem into a sequence ofmathematical programming problems An improved particle swarm optimization issubsequently constructed to solve the resultant mathematical programming prob-lem Numerical results show that the target product concentration at the terminaltime can be increased compared with previous results

In Chap.8, considering the hybrid nature in fed-batch fermentation process,

we propose a state-based switched system to model the fermentation process

A constrained optimal switching control model is then presented Because thenumber of the switchings is not known a priori, we reformulate the above optimalcontrol problem as a two-level optimization problem An optimization algorithm isdeveloped to seek the optimal solution on the basis of a heuristic approach and thecontrol parameterization method

In Chap.9, considering the microbial metabolism mechanism, i.e., the production

of new biomass is delayed by the amount of time it takes to metabolize the nutrients,

in fed-batch fermentation process, we propose a multistage time-delay system toformulate the process In view of the effect of time delay and the high number

of kinetic parameters in the system, the parametric sensitivity analysis is used todetermine the key parameters An optimal parameter selection model is presented,and a global optimization method is developed to seek the optimal key parameters.Numerical results show that the multistage time-delay system can describe the fed-batch fermentation process reasonably

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In Chap.10, taking the mass of target product per unit time as the performanceindex, we formulate a constrained optimal control model with free terminal time

to optimize the production process Using a time-scale transformation, the optimalcontrol problem is equivalently transcribed into the one with fixed terminal time

A computational approach is then developed to seek the optimal control and theoptimal terminal time This method is based on the control parameterization inconjunction with an improved differential evolution algorithm Numerical resultsshow that the mass of target product per unit time is increased considerably and theduration of the fermentation is shorted greatly compared with previous results

In Chap.11, taking the switching instants and the terminal time as the controlvariables, a free terminal time-delayed optimal control problem is proposed Using

a time-scaling transformation and parameterizing the switching instants into newparameters, an equivalently optimal control problem is presented A numericalsolution method is developed to seek the optimal control strategy by the smoothingapproximation method and the gradient of the cost functional together with that ofthe constraints Numerical results show that the mass of target product per unit time

at the terminal time is increased considerably

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Chapter 2

Mathematical Preliminaries

For the convenience of the reader, some basic results in measure theory andfunctional analysis are presented without proofs in this chapter The reader canturn to [57,81,99,215,260] for proofs of those theorems and for more detailedinformation

For compactness of notation, we will refer to rectangular parallelepipeds in Rn

whose sides are parallel to the coordinate axes simply as “boxes.”

Definition 2.1 (a) A box inRnis a set of the form

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(e) If E Rnand > 0, then there exists an open set U E such that .U /6

Definition 2.2 A set E Rnis Lebesgue measurable, or simply measurable, if

If E is Lebesgue measurable, then its Lebesgue measure is its exterior Lebesgue

measure and is denoted by .E/ D .E/:

The following result summarizes some of the properties of Lebesgue measurable

Property 2.2 Let E and Ekbe measurable subsets ofRn

(a) If E1, E2,: : : are disjoint measurable subsets ofRn, then

The following concept is often used in the sequel

Definition 2.3 A property that holds except possibly on a set of measure zero is

said to hold almost everywhere, abbreviated a.e.

The essential supremum of a function is an example of a quantity that is defined

in terms of a property that holds almost everywhere

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2.1 Lebesgue Measure and Integration 15

Definition 2.4 The essential supremum of a function f W E !R is

Now, we define the class of measurable functions on subsets ofRn

Definition 2.5 Fix a measurable set E  Rn, and let f W E ! R be given

Then f is a Lebesgue measurable function, or simply a measurable function, if

In particular, every continuous function f WRn! R is measurable However, a

measurable function need not be continuous

Measurability is preserved under most of the usual operations, including tion, multiplication, and limits

addi-Property 2.3 Let E Rnbe measurable

(a) If f W E !R is measurable and g D f a.e., then g is measurable

(b) If f; g W E ! R are measurable, then so are f C ˛g.˛ 2 R/, f g, f =g

(c) If fnW E ! R are measurable for n 2 N, then so are inf

n fn, sup

n

fn, lim inf

n!1 fn,and lim sup

n!1

fn

The following theorem says that pointwise convergence of measurable functions

is uniform convergence on “most” of the set

Theorem 2.1 (Egoroff) LetE Rnbe measurable with.E/ < 1 If fn; f W

every > 0, there exists a measurable set E  E such that .E/ < and fn

converges uniformly to f on E E, i.e.,

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16 2 Mathematical Preliminaries

where N > 0, ak 2 R, Ek is a measurable subset of E and Ek W E ! R is the

indicator function on Ekdefined by

Ek.x/D

(1; if x 2 Ek;

set EkD fx 2 Ej '.x/ D akg, then ' has the form given in Eq (2.9) and the sets

Definition 2.7 If ' is a nonnegative simple function on E with standard

represen-tation, then the Lebesgue integral of ' over E is

If we have functions ffng that are not monotone increasing, then we may not be

able to interchange a limit with an integral The following result states that as long

as ffng are all nonnegative, we do at least have an inequality

Theorem 2.3 (Fatou’s Lemma) Ifffng is a sequence of measurable, nonnegative

functions on a measurable setE Rn, then

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2.2 Normed Spaces 17

The following dominated convergence theorem is one of the most importantconvergence theorems for integrals

Theorem 2.4 (Lebesgue Dominated Convergence Theorem) Assumeffng is a

sequence of Lebesgue measurable functions on a measurable setE Rn such that

(a) f x/ D lim

(b) there exists an integrable function g W E ! R such that

Definition 2.9 A vector space X is called a normed linear space if for each x 2 X ,

there is a (finite) real number kxk, called the norm of x, such that

(a) kxk> 0; for all x 2 X; and kxk D 0 if and only if x D 0;

(b) kcxk D jcj kxk; for all x 2 X and scalar c 2R; and

(c) kx C yk6 kxk C kyk; for all x; y 2 X:

Given a normed space X , it is usually clear from context what norm we mean touse on X Therefore, we usually just write k k to denote the norm on X However,when there is a possibility of confusion, we may write k kX to specify that thisnorm is the norm on X

Definition 2.10 Let X be a normed linear space.

(a) A sequence of vectors fxng in X converges to x 2 X if lim

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Every convergent sequence in a normed space is a Cauchy sequence However,the converse is not true in general

Definition 2.11 A normed space X is complete if it is the case that every Cauchy

sequence in X is a convergent sequence A complete normed linear space is called

a Banach space.

The simplest example of a Banach space is the scalar fieldR, where the norm on

n-tuples of scalars, where n is a positive integer There are many choices of norms for

Rn Writing a generic vector v 2Rnas v WD v1; v2; : : : ; vn/>, each of the followingdefines a norm onRn, andRnis complete with respect to each of these norms:

In fact, there can be many norms on a given Banach space

Definition 2.12 Suppose that X is a normed linear space with respect to a norm

k k and also with respect to another norm jjj jjj These norms are equivalent if

there exist constants C1; C2> 0 such that

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2.2 Normed Spaces 19(i) lpspace

kfxkgklp WD

8ˆ

<

ˆ:

8ˆˆ

In optimal control theory, we shall be concerned with the space C.I;Rn/ of all

continuous functions from I R to Rn The space C.I;Rn/ is a vector space and

becomes a Banach space when it is equipped with the sup norm defined by

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A set A C.I;Rn/ is said to be equicontinuous if for any > 0, there exists a

ı > 0 such that for all f 2 A,

whenever t0; t2 I are such that jt0 tj < ı

Let I WD Œa; b R and f WD f1; f2; : : : ; fn/>2 C.I; Rn/ The function f is

said to be absolutely continuous on I if for any given > 0, there exists a ı > 0

The class of all such absolutely continuous functions is denoted by AC.I;Rn/

Clearly, a Lipschitz continuous function on I is absolutely continuous

Theorem 2.6 Iff 2 L1.I;Rn/ and g is defined by

Definition 2.13 Let X be a normed linear space A map f W X !R is called a

bounded linear functional if

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2.3 Linear Functionals and Dual Spaces 21

Definition 2.14 Given a normed linear space X , the space of all bounded linear

functionals on X with norm

kf k WD sup

x2X nf0g

jf x/j

is the dual space of X and is denoted by

SinceR is complete, the dual space Xof a normed space X is complete, even if

Definition 2.15 A Banach space X is called reflexive if X D X

Note that Xis a Banach space, so X must be a Banach space if we are to beable to identify X with X

Theorem 2.7 A Banach space X is reflexive if and only if Xis reflexive.

Theorem 2.8 Suppose that the Banach space X is not reflexive Then the

inclu-sionsX X X and X X are all strict.

The following theorem gives the duals of some of the classical Banach spaces

Theorem 2.9 For anyp2 Œ1; C1/,

Furthermore, for eachp2 1; C1/, lpandLp.I;Rn/ are all reflexive.

It should be noted that l1/¤ l1and fL1.I;Rn/g¤ L1.I;Rn/

Finally, we give some types of convergence Part (a) of the following definitionrecalls the usual notion of convergence as given in Definition2.10, and parts (b) and(c) introduce some new types of convergence

Definition 2.16 Let X be a Banach space.

(a) A sequence fxng of elements of X converges (strongly) to x 2 X if

lim

We denote this convergence by lim xnD x

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22 2 Mathematical Preliminaries(b) A sequence fxng of elements of X converges weakly to x 2 X if

Theorem 2.10 (Banach–Saks–Mazur) Let X be a normed space andfxng be a

sequence in X converging weakly to x Then there exists a sequence of finite convex

combinations offxng that converges strongly to x.

By a partition of the interval I WD Œa; b R, we mean a finite set of points ti 2 I ,

iD 0; 1; : : : ; m, such that

A function h defined on I is said to be of bounded variation if there is a constant

K > 0 such that for any partition of I ,

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2.4 Bounded Variation 23

where the supremum is taken with respect to all partitions of I The total variation

of a constant function is zero and the total variation of a monotonic function isthe absolute value of the difference between the function values at the endpoints

a and b

The space BV I / is defined as the space of all functions of bounded variation on

I together with the norm defined by

Suppose h 2 BV I / Then, h is differentiable a.e on I If h W I !R is absolutely

continuous, then it is of bounded variation

Theorem 2.11 Ifh2 BV I /, then h is absolutely continuous if and only if

Theorem 2.12 Ifh 2 BV I; Rn/, then h.t C 0/ WD lim

right at t , exists if a 6 t < b; and h.t 0/ WD lim

s"th.s/, the limit from the left at t ,

exists if a < t 6 b.

In order that h approaches a limit in Rn as s approaches t from the right(respectively, from the left), the following condition is necessary and sufficient: Foreach > 0, there corresponds a ı > 0 such that kh. / h.t /k < if s < < t C ı(respectively, t ı < < s)

Theorem 2.13 If h 2 BV I; Rn/, the set of points of discontinuity of h is

countable.

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Let E be a family of functions in BV I;Rn/ It is said to be equibounded with

equibounded total variation if there exist constants K1 > 0; K2 > 0 such that

Theorem 2.14 (Helly) Let E be a family of functions in BV I;Rn/ which is

equibounded with equibounded total variation Then, any sequence˚

hi

of elements

toward a functionh02 BV I; Rn/ with

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Chapter 3

Constrained Mathematical Programming

The optimal control problem can be reduced to optimal parameter selectionproblems by approximating the control functions with an appropriate series ofspline functions Although the constraint on the dynamical system still exists, theproblem may, after the parameterization, be viewed as an implicit mathematicalprogramming problem The solution to the optimal control problem may thus

be obtained through solving a sequence of resulting mathematical programmingproblems, although the computational procedure is much more involved Thus,understanding of the fundamental concepts, theories, and methods of mathematicalprogramming is obviously important

To begin, we note that the notation used in this chapter is applicable only to thischapter For example, subsequent appearance of x is not to be confused with thestate vector in other chapters

As opposed to optimal control problems, mathematical programming problems arestatic in nature The general constrained mathematical programming problem isdescribed by

min

where f and the functions ci are all smooth, real-valued functions on a subset of

RnandE and I are two finite sets of indices f is the objective function, while ci,

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26 3 Constrained Mathematical Programming

define the feasible set ˝ to be the set of points x that satisfy the constraints, that is,

Definition 3.1 A vector xis a local solution of the problem (3.1)–(3.3) if x2 ˝

and there is a neighborhoodN of xsuch that f x/> f x/ for x 2N \ ˝.

Definition 3.2 A vector x is a strict local solution of the problem (3.1)–(3.3)

Definition 3.3 A vector xis a global solution of the problem (3.1)–(3.3) if x2

The following definition is an important terminology in the constrained cal programming

mathemati-Definition 3.4 The active set A.x/ at any feasible x consists of the equality

constraint indices fromE together with the indices of the inequality constraints i

for which ci.x/D 0, that is,

At a feasible point x, the inequality constraint i 2I is said to be active if ci.x/D 0

and inactive if the strict inequality ci.x/ > 0 is satisfied

Optimization techniques, or algorithms, are used to find the solution to theproblem specified in (3.1)–(3.3) Note that, for many problems, more than one opti-mum may exist There are many options for classifying the available optimizationtechniques A short overview of the available algorithms, using a broad classification

as either gradient-based or evolutionary algorithms, is presented

We shall first summarize the main optimality conditions without proofs Then,

we introduce three gradient-based algorithms, i.e., the quadratic penalty method,augmented Lagrangian method, and sequential quadratic programming (SQP)

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Definition 3.5 The point x is said to be a regular point of the constraints (3.2)–(3.3) if xsatisfies all the constraints and if the gradients of the equality and activeinequality constraints

xDx

Note that condition (3.7) is known as a constraint qualification.

As a preliminary to stating the necessary conditions for x to be a local

minimizer, we define the Lagrangian function for the problem (3.1)–(3.3)

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The conditions (3.10) are often known as the Karush–Kuhn–Tucker conditions,

or KKT conditions for short Note that condition (3.10d) implies that if the i thinequality constraint is inactive, then i D 0, and conversely, if i > 0, then the i th

inequality constraint must be active

We turn now to the linearized feasible direction set, which we define as follows

Definition 3.6 Given a feasible point x and the active constraint setA.x/, the set

of linearized feasible directionsF.x/ is

F.x/ Dndj d>rci.x/D 0; for all i 2 E; d>rci.x/> 0; for all i 2 A.x/ \ Io:

(3.11)The next theorem gives a necessary condition involving the second derivatives

Theorem 3.2 Suppose thatxis a local solution of (3.1)–(3.3) and also a regular point for the constraints Let be the Lagrange multiplier vector for which the KKT conditions (3.10) are satisfied Then

(3.13)The second-order sufficient condition stated in the next theorem looks very muchlike the necessary condition just discussed, but it differs in that the constraintqualification is not required and the inequality in (3.12) is replaced by a strictinequality

Theorem 3.3 Suppose that for some feasible pointx2 Rn, there is a Lagrange multiplier vectorsuch that the KKT conditions (3.10) are satisfied Suppose also that

d>r2

xxL

Thenxis a strict local solution for (3.1)–(3.3).

3.2.2 The Quadratic Penalty Method

The penalty methods for constrained mathematical programming replace the nal problem by a sequence of subproblems in which the constraints are represented

origi-by terms added to the objective The simplest penalty function of this type is

the quadratic penalty function, in which the penalty terms are the squares of the

constraint violations

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where > 0 is the penalty parameter By driving to 1, we penalize the constraint

violations with increasing severity It makes good intuitive sense to consider asequence of values fkg with k ! 1 as k ! 1 and to seek the approximate

minimizer xkof Q.xI k/ for each k

For the general constrained mathematical programming problem (3.1)–(3.3), wecan define the quadratic penalty function as

that Q is no longer twice continuously differentiable

We describe some convergence properties of the quadratic penalty method inthe following two theorems We restrict our attention to the equality-constrainedproblem (3.15), for which the quadratic penalty function is defined by (3.16).For the first result we assume that the penalty function Q.xI k/ has a (finite)

minimizer for each value of k

Theorem 3.4 Suppose that each xk is the exact global minimizer of Q.xI k/

defined by (3.16) and that k ! 1 Then every limit point x of the sequence

fxkg is a global solution of the problem (3.15).

Since this result requires us to find the global minimizer for each subproblem, thisdesirable property of convergence to the global solution of (3.15) cannot be attained

in general The next result concerns convergence properties of the sequence fxkg

when we allow inexact (but increasingly accurate) minimizations of Q.I k/ We

make the assumption that the stop test krxQ.xI k/k 6 kis satisfied for all k

Theorem 3.5 Suppose that the tolerances and penalty parameters satisfyk! 0

stationary point of the functionkc.x/k2 On the other hand, if a limit pointxis feasible and also a regular point for the constraints, thenxis a KKT point for the

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problem (3.15) For such points, we have for any infinite subsequence K such that

3.2.3 Augmented Lagrangian Method

The augmented Lagrangian method or multipliers penalty method by introducing

explicit Lagrange multiplier estimates into the objective can preserve the ness and reduce the possibility of ill conditioning

smooth-We describe this approach first in the context of the equality-constrained lem (3.15) The augmented Lagrangian function LA.x; I / for this formulation

where i are the Lagrange multipliers and > 0 is the penalty parameter.

The following result shows that when we have knowledge of the exact Lagrangemultiplier vector , the solution xof (3.15) is a strict minimizer of LA.x; I /

for all sufficiently large

Theorem 3.6 Letxbe a local solution of (3.15) and also a regular point for the constraints If the second-order sufficient conditions specified in Theorem3.3are satisfied forD , then there is a threshold value N such that for all > N, x

is a strict local minimizer ofLA.x; I /

Now, consider the inequality-constrained problem

min

s.t ci.x/> 0; i 2 I: (3.20b)

It is possible to convert (3.21) into an equality-constrained problem by

introduc-ing a vector of additional variables z WD

z1; z2; : : : ; z jIj > This problem is givenby

min

s.t ci.x/ z2D 0; i2 I: (3.21b)

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: (3.22)

In applying the augmented Lagrangian method only involving inequality straints, we must minimize the augmented Lagrangian (3.22) with respect to x; z/

con-for various values of and An important point here is that minimization of

this, note that

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