Basic Mathematical Programming
Theory
Giorgio Giorgi
Department of Economics and Management, University of Pavia, Pavia, Italy
Bienvenido Jimenez
Department of Applied Mathematics, National University of Distance Education, Madrid, Spain
Vicente Novo
Department of Applied Mathematics, National University of Distance Education, Madrid, Spain
ISSN 0884-8289
e-ISSN 2214-7934
International Series in Operations Research & Management Science
ISBN 978-3-031-30323-4 e-ISBN 978-3-031-30324-1 https://doi.org/10.1007/978-3-031-30324-1
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
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For Elena, Elisa, Marcello and Lucia
Giorgio Giorgi
For my wife Elena and my children Roberto, Cristina and Elena
Bienvenido Jiménez
For my grandchildren Leo, Vega, Eva and Sara Vicente Novo
Preface
It is quite super luous to stress that mathematical optimization is a cornerstone of not only several scienti ic subjects, such as Economic Analysis, Operations Research, Management Sciences, Engineering, Chemistry, Physics, Statistics, Computer Science, but also Biology and other Social Sciences. The subject of (static) optimization, called also mathematical programming, is, besides its various applications, one of the most important and wide branches of modern mathematics, with deep and beautiful theoretical results. The present book is concerned with the main theoretical questions on optimality and duality theory of mathematical programming problems in the Euclidean space . Also in order of limiting the dimension of the book, we have decided to devote our efforts only to expose the basic mathematical tools and results concerning the most treated mathematical programming problems formulated in a inite-dimensional setting. We wish to make clear that it would be desirable that the student or the researcher interested in these questions would complete his knowledge by learning also the basic questions on the various algorithmic methods and on the most important particular applications of mathematical programming problems.
It is however our opinion that the student or researcher interested in learning the fundamental facts of mathematical programming, before being acquainted with the various algorithms and numerical questions, must possess the basic mathematical notions and the basic theoretical background used in analyzing optimization problems.
As it happens for any book, also the present one re lects the tastes of its authors. We have been compelled to make choices, as some subjects of Mathematical Programming have grown at an exponential rate (e.g., Linear Programming) and it would be impossible to cover in a single volume every aspect of mathematical programming theory. Consequently, we have left out certain important subjects, such as, for example, Complementarity Theory, Discrete Optimization, Stochastic Programming, Semi-in inite Programming, Bilevel Programming, and Fractional Programming. Also, set-valued optimization problems are not treated in the present book. However, the last chapter is concerned
with an introduction to vector optimization problems, but only in inite dimensions (as done for the scalar case), that is the multiobjective case.
The text assumes no previous experience in optimization theory and the treatment of the various topics is largely self-contained. The prerequisites are the basic tools of differential calculus for functions of several variables, the basic notions of topology in and of Linear Algebra. Some of these concepts are recalled in Chap. 1. We do not claim any originality on the structure of this book; however, we hope that the gradual and self-contained treatment of the subjects will reveal to be a “friendly” approach to the non-experienced reader. We have accompanied several theoretical results with examples and we have often made reference to papers and books, in order to give further information to the curious and willing reader. The list of works in the bibliographical references is quite long.
The book addresses not only to both undergraduate and postgraduate students interested in mathematical programming problems but also to those professionals who use in their jobs optimization methods and wish to learn more theoretical aspects of these questions. We hope that it can be useful in courses of optimization, operations research, economic analysis, and in other courses where optimization theory and methods are investigated. The book is organized into 11 chapters.
Chapter 1 includes background material on classical analysis and linear algebra, together with some basic de initions and properties concerning optimization problems. We present also the main types of optimization problems that will be studied in the next chapters.
Chapter 2 is concerned with the fundamental facts of convex analysis, a subject which is at the heart of optimization theory. We give particular attention to the theorems of the alternative for linear systems. One of the most used and known results of this type, from which all other theorems of the alternative for linear systems can be obtained, is the famous Theorem (or Lemma) of Farkas (or of FarkasMinkowski). However, some excellent books on optimization theory present an unsatisfactory proof of this theorem, as it is given for granted that a polyhedral cone is a closed set. But this is, in a certain sense, just equivalent to the statement of Farkas’ theorem. In the same
chapter, we give also the main de initions and properties of some local cone approximations used in optimization theory.
Chapter 3 provides an overview of convex functions and generalized convex functions, with emphasis on their most important properties regarding optimality conditions. Generalized convexity and generalized monotonicity of functions have attracted several researchers, both in mathematics and in other professional disciplines: the Working Group of Generalized Convexity (WGGC) is a section of the Mathematical Programming Society (since 2010 named Mathematical Optimization Society). The chapter concludes with some theorems of the alternative for nonlinear systems.
Chapter 4 treats optimality conditions for an unconstrained optimization problem, denoted by (P1) for a constrained optimization problem with a set constraint (or abstract constraint), denoted by (P2), and for a “classical” constrained optimization problem, denoted by (P3); in other words, (P3) is an optimization problem with only equality constraints. These problems have been called “classical” constrained optimization problems, as they were studied for the irst time by J. L. Lagrange in the second half of the 18th century.
Chapter 5 treats a “modern” constrained optimization problem, i.e., a constrained minimization problem with inequality constraints, problem denoted by (P4). The basic necessary and suf icient irst-order and second-order optimality conditions are given (Fritz John conditions and Karush-Kuhn-Tucker conditions). A special attention is devoted to the question of constraint quali ications: various constraint quali ications are presented, together with their inclusion properties. Other formulations of (P4) are taken into consideration and several simple numerical examples are discussed.
Chapter 6 treats constrained optimization problems with mixed constraints, i.e., with both equality and inequality constraints, problems denoted by (P5). We have preferred to treat separately these types of problems, as the related optimality conditions have some particular features. Also for these problems, irst-order conditions and secondorder conditions are discussed with a certain wideness and the question of constraint quali ications is extensively treated, with various constraint quali ications proposed and analyzed. The chapter concludes
with some insights on a constrained optimization problem with both equality and inequality constraints and with a set constraint, problem denoted by (P6), and on asymptotic optimality conditions for (P5).
Chapter 7 is concerned with sensitivity analysis for problem (P5) a subject usually not treated in textbooks on optimization theory, but important for its economic and inancial applications and also for computational questions. We follow mainly the classical approach of Fiacco (1983).
Chapter 8 is concerned with optimality conditions for a convex programming problem of the type (P4) expressed in terms of saddle points conditions, and with Lagrangian duality. From a historical point of view, optimality conditions via saddle points are important, as one of the aims of the pioneering paper of Kuhn and Tucker (1951) was to extend to the nonlinear (convex) case the duality and saddle points results previously obtained for the linear case (Linear Programming). As for what concerns duality theory, we have chosen to give some basic insights on the so-called “Lagrangian duality”, because this approach is equivalent (at least for convex programming) to Fenchel’s approach which uses conjugate functions. We brie ly treat also the general approach to duality which makes reference to the “minimax theory”.
Chapter 9 is concerned with Linear Programming and Quadratic Programming. Linear programming problems are treated with reference to their fundamental formal properties, therefore with no mention to the famous “simplex method” of G. B. Dantzig, nor to other more modern algorithmic questions. For these last aspects, we have given in the bibliographical references some suggestions. The same holds true for quadratic programming problems, which are important for some applications in Finance, Econometrics and Statistics. A special attention has been devoted to duality theory, both for linear programming problems and for quadratic programming problems.
Chapter 10 is an introduction to nonsmooth optimization. The term “nonsmooth analysis” is due to the Canadian mathematician F. H. Clarke; the related theory begins with the necessity to treat, within convex optimization problems, non-necessarily differentiable functions. The basic contributions to this case are the works of Fenchel (1953), Moreau (1963), and the celebrated book Convex Analysis by Rockafellar (1970). Subsequently, Clarke (1983) extended the domain of
nonsmooth analysis from convex to locally Lipschitz functions. The irst and second sections of Chap. 10 are, respectively, concerned with nonsmooth convex optimization problems and with nonsmooth locally Lipschitz optimization problems. The third and last section is concerned with an axiomatic approach to nonsmooth analysis and nonsmooth optimization, due to Elster and Thierfelder (1985, 1988a, b, 1989).
The inal Chap. 11 is an introduction to vector optimization problems, but only in inite dimensions, that is the multiobjective case. Necessary and suf icient optimality conditions are given, together with the so-called “weighted sum method”.
Acknowledgements This work, for the second and third authors, was partially supported by Ministerio de Ciencia e Innovacion and Agencia Estatal de Investigacion (Spain) under the project with reference PID2020-112491GB-I00/AEI/10.13039/501100011033.
Giorgio Giorgi
Bienvenido
Jiménez
Vicente Novo
Pavia, Italy
Madrid, Spain
December 2022
Contents
1 Basic Notions and De initions
1.1 Introduction
1.2 Basic Notions of Analysis and Linear Algebra
1.3 Basic De initions and Properties of Optimization Problems
References
2 Elements of Convex Analysis. Linear Theorems of the Alternative. Tangent Cones
2.1 Elements of Convex Analysis
2.2 Theorems of the Alternative for Linear Systems
2.3 Tangent Cones
References
3 Convex Functions and Generalized Convex Functions
3.1 Convex Functions
3.2 Generalized Convex Functions
3.3 Optimality Properties of Convex and Generalized Convex Functions. Nonlinear Theorems of the Alternative
References
4 Unconstrained Optimization Problems. Set-Constrained Optimization Problems. Classical Constrained Optimization Problems
4.1 Unconstrained Optimization Problems
4.2 Set-Constrained Optimization Problems
4.3 Optimization Problems with Equality Constraints (“Classical Constrained Optimization Problems”)
References
5 Constrained Optimization Problems with Inequality Constraints
5.1 First-Order Conditions
5.2 Constraint Quali ications
5.3 Second-Order Conditions
5.4 Other Formulations of the Problem. Some Examples
References
6 Constrained Optimization Problems with Mixed Constraints
6.1 First-Order Conditions
6.2 Constraint Quali ications
6.3 Second-Order Conditions
6.4 Problems with a Set Constraint. Asymptotic Optimality Conditions
References
7 Sensitivity Analysis
7.1 General Results
7.2 Sensitivity Results for Right-Hand Side Perturbations
References
8 Convex Optimization: Saddle Points Characterization and Introduction to Duality
8.1 Convex Optimization: Saddle Points Characterization
8.2 Introduction to Duality
References
9 Linear Programming and Quadratic Programming
9.1 Linear Programming
9.2 Duality for Linear Programming
9.3 Quadratic Programming
References
10 Introduction to Nonsmooth Optimization Problems
10.1 The Convex Case
10.2 The Lipschitz Case
10.3 The Axiomatic Approach of K.-H. Elster and J. Thierfelder to Nonsmooth Optimization
References
11 Introduction to Multiobjective Optimization
11.1 Optimality Notions
11.2 The Weighted Sum Method and Relations with Proper Ef iciency
11.3 Optimality Conditions
References
Index
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Giorgi et al., Basic Mathematical Programming Theory, International Series in Operations Research & Management Science 344 https://doi org/10 1007/978-3-031-30324-1 1
1. Basic Notions and De initions
Giorgio Giorgi1 , Bienvenido Jimenez2 and Vicente Novo2
(1) (2)
Department of Economics and Management, University of Pavia, Pavia, Italy
Department of Applied Mathematics, National University of Distance Education, Madrid, Spain
Giorgio Giorgi (Corresponding author)
Email: giorgio.giorgi@unipv.it
Bienvenido Jiménez
Email: bjimenez@ind.uned.es
Vicente Novo
Email: vnovo@ind.uned.es
1.1 Introduction
It is well-known that the central problem of mathematical programming is that of minimizing or maximizing a given numerical function of several variables, where the variables are free to move over the whole domain of the function or (more usually) are constrained by a system of constraints.
Mathematical programming, called also nonlinear programming, can be viewed as that ield of optimization theory which treats static and inite-dimensional optimization problems. It seems that the term “mathematical programming” was irst introduced by the American economist Robert Dorfman in 1949, as a generalization of the term “linear programming”, introduced by the American mathematician
George B. Dantzig a couple of years before. The term “nonlinear programming” appears for the irst time in 1951 in the title of the famous pioneering paper of Kuhn and Tucker [1]. Mathematical programming problems are at the heart of many theoretic and applied sciences: indeed, these problems occur in different contexts, such as for example, economic analysis, operations research, management sciences, games theory, statistics, physics, engineerings, etc.
Mathematical programming problems have attracted a wide interest also because of their applications in many practical questions arising in industry, commerce, government and military questions.
In the present text-book we shall be concerned with the basic aspects of optimality conditions and duality conditions related to a mathematical programming problem. We have preferred to point out the basic ideas on the mathematical structure of these problems, rather than on the various algorithmic techniques, available in many excellent books treating numerical optimization.
1.2 Basic Notions of Analysis and Linear Algebra
In this section we recall some basic notions of Analysis and Linear Algebra, which will be useful to understand several notions developed in the next chapters. In order to deepen the concepts expounded here, the reader is referred to the texts quoted in the Bibliographical References.
General concepts of basic topology on R
Given a vector (point) in
we denote by its Euclidean norm:
A neighborhood or also of is given by an open ball of a given radius
When it is useful to specify the radius we shall write etc.
In any case we have
A point is an interior point of a set if and there exists a neighborhood contained in S.
The set is open if all its points are interior points. The set is closed if its complementary set is open (equivalently: if it contains all its boundary points or if it contains all its accumulation points). The boundary of S, denoted by or is the set of points x for which contains a point in S and a point not in S, for all
The interior of a set is given by all its interior points, i.e., if we denote the interior of S by
The closure of S, denoted by or also by is given by the intersection of all closed sets which contain S. It holds
In other words, consists of S and all boundary points of S, i.e. .
The algebraic sum of two sets of is de ined as
The multiplication of a set by a real number or real scalar is de ined as
We have the following calculus rules:
Let be given a set and an element we shall write , instead of Moreover, we shall write to denote the operation
Differentiability
Let be given a function . Then f is said to be (Frechet) differentiable at if there exists a vector such that for all
The Landau symbol (“small o”) o(t) means that
The vector is the gradient of f at its elements are given by the n partial derivatives of f, evaluated at We observe that if, in the above de inition, with then
We accept the following de inition of twice differentiability. A function , differentiable at is said to be twice differentiable at if each component of is differentiable at Then, there exists a square matrix, denoted or also of order n, such that
for all
The matrix is the Hessian matrix of f evaluated at its elements are given by the second-order partial derivatives of f, evaluated at
The function f is differentiable on the open set , respectively, twice differentiable on the open set if it is differentiable, respectively, twice differentiable, at every point If the irstorder partial derivatives of f are also continuous, we say that f is continuously differentiable or also that f belongs to the -class and shall write If then f is differentiable (the vice-versa does not hold).
If all second-order partial derivatives of f are also continuous, we say that f is twice-continuously differentiable and shall write If then f is twice differentiable (the vice-versa does not hold).
Obviously, twice continuous differentiability is more easy to check than twice differentiability, however several results on optimization theory hold under the assumption of twice differentiability. If a function f is twice differentiable (and, even more, if f is twice-continuously differentiable), then its Hessian matrix is symmetric (Schwarz theorem), i. e.
for all
A vector function is differentiable at if each component , is differentiable at In this case we can build the matrix of order (n, p), called the Jacobian matrix of h at We draw the reader’s attention on the fact that many authors de ine the Jacobian matrix as a matrix whose rows are the gradients of the functions
Now, let be given and both differentiable, and consider the composite function de ined by means of the composition operation
This function will be differentiable and it holds (“chain rule”):
We recall Taylor’s formula for twice-continuously differentiable functions.
(a) Let be a -function on the open set and let be Then, for any we have for (Taylor’s formula with remainder in Peano’s form).
(b) Let be a -function on the open set and let be Then, for any we have where with (Taylor’s formula with remainder in Lagrange’s form). Putting Taylor’s formula with remainder in Peano’s form becomes and putting with and
Systems of linear equations
Let be given a vector and a scalar . The set
is a hyperplane in The vector a is called the normal to the hyperplane Note that a hyperplane is a level set of a nonidentically zero linear function.
Let be given a (real) matrix A of order (m, n), with non-zero row vectors, and a vector The solutions of the linear system are given by the intersection of m hyperplanes. If the set of solutions is not empty, it will be a linear manifold (a linear subspace if ), with dimension where denotes the rank of matrix A.
Let the row vectors of A be linearly independent, i.e. matrix A has full rank. It is then possible (maybe by reordering the rows of A) to decompose A into the form with square non-singular matrix of order m. Correspondingly, vector will be partitioned as where is called vector of basic variables and is the vector of nonbasic variables.
Then, the original system has now the form from which
Systems of nonlinear equations and Implicit Function Theorem
In the general case, the vector equation is not formed by linear equations. Nevertheless, if the Jacobian matrix of h has full rank at a point which solves the equation it is possible to get some important solvability results. These results are contained in the following classical version of the Implicit Function Theorem. In essence, this theorem gives suf icient conditions for the existence of a function
that locally expresses a set of basic variables in terms of non basic variables, i.e. a nonlinear version of what previously seen for the linear case.
Implicit Function Theorem
Let ( ) be on the open set X and let be a solution of Let be of full rank, i.e. decomposed as (after a possible reordering of the variables)
where is a non-singular square matrix of order m. Then, there exists a neighborhood U of where the system de ines “implicitly” a system of functions of the type More precisely:
(a) There exists a unique function de ined in a neighborhood U of of class in this neighborhood.
(b) It holds for all in the said neighborhood U.
(d) It holds
(c) for all in the said neighborhood U.
There are also weaker versions of this theorem; in particular, it is possible to consider differentiable functions, instead of functions. See, e.g., Halkin [2].
Quadratic forms
Let be given a (real) symmetric matrix A of order n. The related quadratic form is de ined as
The above quadratic form (and its related matrix A) is classi ied as follows.
(a) Q is positive (negative) de inite if ( ), for all
(b) Q is positive (negative) semide inite if ( ), for all
(c) Q is inde inite if there exist two vectors such that and
Obviously the quadratic form (i.e. the symmetric matrix A which characterized the said quadratic form) is positive (negative) de inite (semide inite) if and only if is negative (positive) de inite (semide inite).
We have the following two criteria which establish the sign of a quadratic form.
Theorem 1.1 The quadratic form (the real symmetric matrix A) is:
(a) Positive (negative) de inite if and only if all eigenvalues of A (which are real, being A symmetric) are positive (negative).
(b) Positive (negative) semide inite if and only if all eigenvalues of A are nonnegative (nonpositive) and at least one eigenvalue of A is zero.
(c) Inde inite if and only if A has at least one positive eigenvalue and least one negative eigenvalue.
In order to introduce the next criterion (“Sylvester criterion”), we recall that, given a matrix A, of order (m, n), we have the following de initions.
The minors of order k of A, are all those determinants formed by k rows and k columns of A. It is shown that we
have
minors of order k. We recall that
Now, let A be a square matrix of order n.
The principal minors of A of order k, are all those determinants formed by k rows of A and the corresponding k columns. It is shown that we have
principal minors of order k.
The leading principal minors or North-West principal minors of A of order k, are all those determinants formed by the irst k rows and the irst k columns of A. Obviously, we have n leading principal minors:
Theorem 1.2 (Sylvester criterion for quadratic forms) Let be given the quadratic form Then Q(x) (i.e. the symmetric matrix A) is:
(i) Positive de inite if and only if all leading principal minors of A are positive.
(ii) Negative de inite if and only if all leading principal minors of A alternate in sign, beginning with
(iii) Positive semide inite if and only if all principal minors of A are nonnegative and
(iv) Negative semide inite if and only if all principal minors of A of odd order are nonpositive, of even order are nonnegative, and
(v) Inde inite on the residual cases.
Note that if a symmetric matrix A is positive (negative) semide inite and then A is positive (negative) de inite, since A cannot have in this case, a zero eigenvalue.
In many applications of real quadratic forms we are interested in determining the sign of a quadratic form Q(x) when must belong to some subset of In particular, an important case is when x must belong to the set of nonzero solutions of a homogeneous linear system of the form where B is a matrix of order (m, n), with and Let us consider the following “bordered matrix”, square of order
Theorem 1.3 Let be and, without loss of generality, suppose that the irst m columns of B are linearly independent. Then is positive de inite on the set of nonzero solutions of if and only if the leading principal minors of M, of order have the sign of : where is the h-th leading principal minor of M. is negative de inite on the same set if and only if the sign of alternate, for beginning with the sign of
See, e.g., Debreu [3].
Corollary 1.4 If i.e. we have one constraint of the type with the previous conditions become:
(a) Q(x) is positive de inite on the set of nonzero solutions of if and only if
(b) Q(x) is negative de inite on the same set of (a) if and only if
We point out that the assumption in Theorem 1.3 stating that the irst m columns of B give the rank of B (if necessary, renumber the variables) is essential to obtain necessary and suf icient conditions for checking the sign of a constrained quadratic form. In absence of the said assumption, we have only suf icient conditions. Consider, e.g., , , which is positive de inite on (i.e. here ). Yet we have
1.3 Basic De initions and Properties of Optimization Problems
Any non-empty set S of real numbers that is bounded above has a least upper bound , i.e. is an upper bound for S and for every upper bound b of S; is called the supremum of S and we write If then is the maximum of S and we write
Any non-empty set S of real numbers that is bounded below has a greatest lower bound i.e. is a lower bound for S and for every lower bound a of S; is called the in imum of S and we write If then is the minimum of S and we write
If is not bounded above, we write and if S is not bounded below, we write One usually de ines and
Obviously, with and , we have that the in imum value of f is the minimum value of f is the supremum value of f is the maximum value of f is
In the following we will consider a minimization problem or a maximization problem of the form where is a real-valued function, called the objective function, and is a subset of the domain of f, The case is not excluded. The set S is called also the feasible set or feasible region or opportunity set. When S is an open set (e.g. when
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Suckow, G. A., 296, 297, 302
Surface-soils, 55–60
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Sycidium, 173
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Trizygia, 411, 412
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CAMBRIDGE: PRINTED BY J AND C F CLAY, AT THE UNIVERSITY PRESS
FOOTNOTES:
[1] Ward (84).
[2] Göppert (36)
[3] Parkinson (11), vol. .
[4] For an account of the early views on fossils, v. Lyell (67), Vol. . Vide also Leonardo da Vinci (83).
[5] Woodward, J. (1695), Preface.
[6] Woodward, J. (1728), p. 59.
[7] Mendes da Costa (1758), p 232
[8] Scheuchzer (1723), p. 7, Pl. . fig. 1.
[9] Parsons (1757), p. 402.
[10] Plot (1705), p 125, Pl fig 2
[11] Ibid. Pl. . fig. 2.
[12] Lhwyd (1760)
[13] Artis (25).
[14] Steinhauer (18).
[15] Schlotheim (04)
[16] Sternberg (20).
[17] Brongniart (28) (282) (49)
[18] Brongniart (39).
[19] Heer (55) (68) (76).
[20] Lesquereux (66) (70) (80) etc
[21] Zigno (56).
[22] Massalongo (51)
[23] Saporta (72) (73).
[24] Ettingshausen (79). Also numerous papers on fossil plants from Austria and other countries.
[25] Sprengel (28).
[26] Witham (33)
[27] ibid., p. 3.
[28] Witham (33), p 5
[29] Nicol (34) See note by Prof Jameson on p 157 of the paper quoted, to the effect that he has long known of this method of preparing sections.
[30] Limpricht (90) in Rabenhorst, vol. . p. 73.
[31] Knowlton (89)
[32] Corda (45).
[33] Binney (68), Introductory remarks
[34] Williamson (71), etc.
[35] Solms-Laubach (95), p. 442.
[36] Lindley and Hutton (31)
[37] Woodward (1729), Pt. ii. p. 106.
[38] Humboldt (48), vol. . p. 274.
[39] Vide Hooker, J D (81), for references to other writers on this subject; also Darwin (82), ch
[40] Darwin (87), vol. . p. 247.
[41] Brongniart (49), p 94
[42] Grand’Eury (77), Potonié (96), Kidston (94), &c.
[43] Ward (92), Knowlton (94), Grand’Eury (90), p. 155.
[44] Old Persian writer, quoted by E G Browne in A Year among the Persians, p 220, London, 1893
[45] W. R. W. Stephens, Life of Freeman, p. 132, London, 1895.
[46] Rothpletz (94)
[47] Heim (78).
[48] Geikie (93), p. 706.
[49] Hughes (79), p 248
[50] Whitney and Wadsworth (84).
[51] Murchison (72), p 5
[52] Kayser and Lake (95), p 88
[53] Kidston (94).
[54] Geikie (93), p 825
[55] Woodward, H. B. (87), p. 197.
[56] Hinde and Fox (95), p. 662.
[57] Kidston (94)
[58] Vide Zeiller (92) for a list of species of Coal-Measure plants found in the pieces of shale included in the core brought up by the borer.
[59] Jukes-Browne (86), p 252
[60] Kayser and Lake (95), p. 196.
[61] Neumayr (83).
[62] Woodward, H B (87), p 255
[63] Kayser and Lake (95), p. 326.
[64] Huxley (93), p 27
[65] Discussed at greater length in vol. II.
[66] Woodward, H. B. (95), Figs. 124 and 133 from photographs by Mr Strahan.
[67] Buckland (37) Pl LVII
[68] Young, Glen, and Kidston (88).
[69] Gardner (87), p. 279.
[70] Treub (88)
[71] Morton (91), p. 228.
[72] Lyell (45), vol I p 180
[73] Mantell (44), vol. I. p. 168.
[74] Forbes, H. O. (85), p. 254.
[75] Hooker, J D (91), p 477
[76] Hooker, J. D. (91), p. 1. There are several good specimens of the black pyritised nipadite fruits in the British Museum and other collections.
[77] Rodway (95), p 106
[78] Bates (63), p. 139.
[79] Bates (63), p. 239.
[80] Lyell (67) vol II p 361
[81] Lyell (67) vol. I. p. 445.
[82] Darwin (90) p 443
[83] Challenger (85), Narrative, vol I Pt ii p 679
[84] Bates (63) p. 389.
[85] Challenger (85), Narrative, vol I p 459
[86] Zeiller (82) and Renault (95).
[87] Heer (76).
[88] Schimper and Mougeot (44)
[89] Sharpe, S. (68) p. 563.
[90] There are still more perfect casts from Sézanne in Prof MunierChalmas’ Geological collection in the Sorbonne The best examples have not yet been figured
[91] Saporta (68).
[92] Darwin (90) p. 432.
[93] For figures of fossil plants in amber, vide Göppert and Berendt (45), Conwentz (90), Conwentz (96) &c
[94] Thomas (48).
[95] Adamson (88)
[96] Williamson (87) Pl. . p. 45. A very fine specimen, similar to that in the Manchester Museum, has recently been added to the School of Mines Museum in Berlin; Potonié (90).
[97] The British Museum collection contains a specimen of Stigmaria preserved in the same manner as the example shown in fig 12
[98] Lyell (45) vol. . p. 60.
[99] Lyell (45) vol. . p. 147.
[100] Warming (96) p 170
[101] Bornemann (56), Schenk (67), Zeiller (82).
[102] Solms-Laubach (952)
[103] Nathorst (86) p. 9. See also Delgado (86).
[104] Parkinson (11) vol. . p. 431.
[105] The British Museum collection contains many good examples of the Solenhofen plants
[106] There is a splendid silicified tree stem from Tasmania of Tertiary age several feet in height in the National Museum.
[107] Darwin (90) p 317
[108] Holmes (80) p. 126, fig. 1.
[109] Marsh (71).
[110] Conwentz (78)
[111] A large piece from one of these South African trees is in the Fossilplant Gallery of the British Museum.
[112] Barton (1751) p 58
[113] Gardner (84) p. 314.
[114] Stokes (40) p 207
[115] Witham (81), Christison (76).
[116] Cole (94), figs 1 and 3
[117] Harker (95) p 233, fig 56
[118] I am indebted to Dr Renault of Paris for showing to me several preparations illustrating this method of petrifaction.
[119] Cash and Hick (78)
[120] Stur (85).
[121] Thiselton-Dyer (72) Pl
[122] Carruthers (70).
[123] Schultze (55).
[124] I am indebted to Prof Lebour of the Durham College of Science for the loan of this letter
[125] Seward (97).
[126] Williamson and Scott (96) Pl. . fig. 16.
[127] Bryce (72) p 126, fig 23
[128] An erroneous interpretation of the Arran stems is given in Lyell’s Elements of Geology: Lyell (78) p. 547.
[129] Guillemard (86) p 322
[130] Heer (55).
[131] Göppert (36), etc.
[132] Hirschwald (73)
[133] Kuntze (80) p. 8.
[134] Schweinfurth (82)
[135] Solms-Laubach (91), p. 29.
[136] Göppert (57). Some of the large silicified trees mentioned by Göppert may be seen in the Breslau Botanic gardens.
[137] An example referred to by Carruthers (71) p 444
[138] Williamson (71) p. 507.
[139] Dealt with more fully in vol
[140] Bentham (70).
[141] See also Bunbury (83) p. 309.
[142] Seward (96) p 208
[143] Darwin (90) p. 416.
[144] Solms-Laubach (91) p 9
[145] Balfour (72) p. 5.
[146] Grand’Eury (77) Pt i , p 3
[147] 1 Renault and Zeiller (88) Pl fig 1
[148] Williamson (73) p. 393, Pl. . Described in detail in vol. . See also Solms-Laubach (91) p. 7, fig. 1.
[149] A good example is figured by Newberry (88) Pl as a decorticated coniferous stem of Triassic age
[150] Potonié (87).
[151] Lindley and Hutton (31) vol p 4 See also Schenk (88) p 202
[152] Saporta (79) (77). Eopteris is included among the ferns in Schimper and Schenk’s volume of Zittel’s Handbuch der Palaeontologie (p. 115), and in some other modern works.
[153] Reinsch (81).
[154] Williamson has drawn attention to the occurrence of such borings and coprolites in Coal-Measure plant tissues. E.g. Williamson (80) Pl. 20, figs. 65 and 66.
[155] Renault (96) p. 437.
[156] Slide No 1923 in the Williamson collection
[157] Crépin (81).
[158] Rules for Zoological Nomenclature, drawn up by the late H. E. Strickland, M A , F R S , London, 1878
[159] Lhwyd (1699).
[160] Knowlton (96) p. 82.
[161] Thiselton-Dyer (95) p 846
[162] Saporta (75) p. 193.
[163] Seward (95) p. 173.
[164] Ward (96) p 874
[165] Challenger (85) p. 934.
[166] Ehrenberg (36) p 117, Pl I figs 1 and 4, and Ehrenberg (54) Pl fig vii
[167] Schütt (96) p. 22.
[168] Bütschli (83–87) p. 1028.
[169] Challenger Reports (85) p 939
[170] Challenger Reports (91) p. 257.
[171] Hensen (92), Schütt (93) p 44
[172] Murray, G., and Blackman, V. H. (97).
[173] Dixon and Joly (97)
[174] Sorby (79) p 78
[175] Rothpletz (96), p. 909, Pl. . fig. 4.
[176] Challenger (85) passim Schütt (93)
[177] Phillips W. (93).
[178] Kippis (78) p. 115.
[179] Darwin (90) p 13
[180] Rothpletz (92).
[181] Walther (88)
[182] Cohn (62).
[183] Murray, G. (952).
[184] Thiselton-Dyer (91) p 226
[185] Nicholson and Etheridge (80) p. 28, Pl. . fig. 24.
[186] Wethered (93) p 237
[187] For figures of the sheaths of Cyanophyceous algae, see Murray (952), Pl. . fig. 5. Gomont (88) and (92); etc.
[188] Brown (94) p 203
[189] For references to the papers of Wethered and others, see Seward (94), p. 24.
[190] E. G. Bornemann (87), Pl. .
[191] Moseley, H N (75), p 321
[192] Weed (87–88), vide also Tilden (97).
[193] Bornet and Flahault (892) Pl
[194] Batters (92).
[195] Quekett (54), fig. 78.
[196] Kölliker (59) and (592); good figures in the latter paper
[197] Rose (55), Pl. .
[198] For other references vide Bornet and Flahault (892)
[199] Duncan (76) and (762).
[200] Similar borings are figured by Kölliker (592), Pl. . 14, in a scale of Beryx ornatus from the Chalk
[201] Bornet and Flahault (892).
[202] E. G. Wedl (59). Good figures are given in this paper.
[203] Bornemann (86), p 126, Pls and
[204] Renault (961) p. 446.
[205] Treub (88)
[206] Williamson (88).
[207] Heer (55) vol. . p. 21, Pl. . fig. 2.
[208] de Bary (87) p 9 A good account of the Schizomycetes has lately been written by Migula in Engler and Prantl’s Pflanzenfamilien, Leipzig, 1896.
[209] 1 µ = 0·001 millimetres.
[210] James (982), translation of a paper by M Ferry in the Revue Mycologique, 1893
[211] Zeiller (82).
[212] Renault (951), (961) p 478, (962) p 106 (Several figures of the cuticles are given in these publications )
[213] Renault (961) p. 492.
[214] Renault (952), (961), (962)
[215] Renault and Bertrand (94). See also Renault (952) p. 3, (961) p. 449, Pl. . (962) p. 94, and (963) p. 280, fig. 3.
[216] Renault (952) p. 17, fig. 9, (961) p. 460, fig. 102, and (963) p. 292, fig. 10.
[217] Renault (963) p. 297, fig. 14.
[218] Van Tieghem (77)
[219] Van Tieghem (79).
[220] Vogelsang (74) Vide also Rutley (92)
[221] I am indebted to Prof. Kanthack for calling my attention to an interesting account of Bacilli in small stones found in gall-bladders; a manner of occurrence comparable to that of the fossil forms in petrified tissues. Vide Naunyn (96) p. 51.
[222] Renault (963) p 277
[223] Hooker, J. D. (44) p. 457. Pls. . . and . D.
[224] An American writer has recently discussed the literature and history of Fucoides; he gives a list of 85 species. It is very doubtful if such work as this is worth the labour (James [93] )
[225] Wille (97) p 136, also Murray, G (95) p 121
[226] Linnarsson (69) Pl. . fig. 3. There are many good specimens of this fossil in the Geological Survey Museum, Stockholm.
[227] Nathorst (812), and (96)
[228] Nathorst (81) p. 14.
[229] Mantell (33) p 166 Vide also Morris (54) p 6
[230] Bateson (88).
[231] Nathorst (81), (86) &c. Dawson (88) p. 26 et seq. Dawson (90) Delgado (86) Williamson (85) Hughes (84) Zeiller (84) Saporta (81) (82) (84) (86) Fuchs (95) Rothpletz (96).
[232] Kinahan (58)
[233] Sollas (86).
[234] Barrois (88) References to other records of this genus may be found in Barrois’ paper
[235] Zeiller (84). Phymatoderma is probably a horny sponge (vide p. 154).
[236] Newberry (88) p 82, Pl There are some large specimens of this supposed alga in the National Museum, Washington; they are undoubtedly of the nature of rill-marks
[237] Vide Williamson (85).
[238] Dawson (90) p. 615.
[239] Salter (78) p 99
[240] Lapworth (81) p. 176, Pl. . fig. 7.
[241] Rothpletz (96)
[242] A term applied to a certain facies of Eocene and Oligocene rocks in Central Europe.
[243] Hall (47) Pl. . 1 and 2, p. 261.
[244] Nathorst (83) p 453
[245] Seward (942) p. 4.
[246] Kidston (83) Pl fig 2 Specimens of this form may be seen in the British Museum collection
[247] Cf. Matthew, G. F. (89). Hall called attention in 1852 to the prevalent habit of describing ‘algae’ from the older strata, without any evidence for a
vegetable origin. (Hall [52] p. 18.)
[248] Credner (87) p 431
[249] Saporta (84) p. 45, Pl. .
[250] Solms-Laubach (91) p 51
[251] A monograph on the Diatomaceae has recently been written by Schütt for Engler and Prantl’s systematic work. See also Murray, G. (97) and Pfitzer (71).
[252] Darwin (90) p. 5.
[253] Weed (87)
[254] Wilson (87).
[255] Ehrenberg (54)
[256] Noll (95) p. 248.
[257] Hooker, J. D. (44) vol. . p. 503.
[258] Murray, J and Renard (91) p 208
[259] Nansen, Daily Chronicle, Nov. 2, 1896.
[260] Schütt (93) p. 10.
[261] Ehrenberg (36) p 77
[262] Cayeux (92), (97).
[263] Shrubsole and Kitton (81)
[264] I am indebted to Mr Murton Holmes for specimens of these London Clay Diatoms.
[265] Rothpletz (96) p. 910, fig. 3, Pl. . fig. 203.
[266] Schütt (96) p 62
[267] Castracane (76).
[268] Heer (76) p 66, Pl and (53) p 117, Pl
[269] Stefani (82) p. 103.
[270] The Chlorophyceae have recently been exhaustively dealt with by Wille (97) in Engler and Prantl’s Pflanzenfamilien.
[271] Vide p 142
[272] Murray G. (95) p. 123.
[273] Göppert (60) p 439 Pl fig 8
[274] Zeiller (84).
[275] Murray G. (92) p. 11; also (95) p. 127.
[276] Damon (88) Pl fig 12
[277] Vide also Rothpletz (96) p. 894.
[278] Ellis (1755) Pl α p 86
[279] Fuchs (96) Pl. . fig. 3.
[280] Murray G (95) Pl figs 1 and 2
[281] Rothpletz (90), and (91) Pls and
[282] Bornemann (87) p. 17, Pl. . pp. 1–4.
[283] Lamouroux (21) Pl fig 5, p 23
[284] Munier-Chalmas (79).
[285] For references to genera of calcareous algae previously referred to Foraminifera, vide Sherborn (93)
[286] Lamarck (16) p. 193.
[287] Defrance (26) Pl. . fig. 2, and Pl. . fig. 6.
[288] Carpenter (62) p 179, Pl figs 9 and 10
[289] Michelin (40–47) Pl. . fig. 24.
[290] Munier-Chalmas (79).
[291] Lamouroux (21) Pl fig 6, p 23
[292] Harvey (58) Vol. I. Pl. . fig. 3.
[293] Fuchs (94)
[294] Lamouroux gives a figure of Acetabularia, and includes this genus with several other algae in the animal kingdom (Lamouroux [21] p. 19), Pl. .).
[295] Solms-Laubach (953)
[296] D’Archiac (43) p. 386, Pl. . fig. 8.
[297] Solms-Laubach loc. cit. p. 33, Pl. .
[298] D’Archiac (43) p 386, Pl fig 8
[299] Michelin (40) p. 176, Pl. . fig. 14.
[300] Carpenter (62) p. 137, Pl. . figs. 27–82.
[301] Solms-Laubach loc cit p 32
[302] Ibid. p. 34, Pl. . fig. 13.
[303] Andrussow (87)
[304] Solms-Laubach (953) p. 11.
[305] Carpenter (62) Pl. . fig. 32.
[306] Reuss (61) p 8, figs 5–8
[307] Ellis (1755) Pl. . C.
[308] Solms-Laubach (91) p 38 gives a detailed description with two figures of a recent species of Cymopolia
[309] Murray G. (95).
[310] Wille (97)
[311] Munier-Chalmas (77).
[312] Cramer (87) (90).
[313] Solms-Laubach (91) (93) (953)
[314] Church (95).
[315] Gümbel (71)
[316] Benecke (76).
[317] Defrance (26) p. 453.
[318] Munier-Chalmas (77) p 815
[319] Munier-Chalmas ibid.
[320] Defrance (26) Pl fig 1
[321] Stolley (93).
[322] Deecke (83).
[323] Benecke (76) Pl
[324] Solms-Laubach (91) p. 42.
[325] Rothpletz (94) p. 24.
[326] Lamarck (16) p 188
[327] Carpenter (62) Pl. .
[328] Gümbel (71) Vide also Solms-Laubach (91) p 39
[329] Stolley (93).
[330] Schlüter (79).
[331] Saporta (91) Pl &c
[332] Solms-Laubach (93), Pl. . figs. 1, 8.
[333] Cramer (90)
[334] Rothpletz (922) p. 235.
[335] Steinmann (80).
[336] Solms-Laubach (91), p 40 fig 3 Vide also Deecke (83) Pl fig 12
[337] Brongniart (28) p. 211.
[338] Bornemann (91) p. 485. Pls. 42 and 43.
[339] Seward (952) p 367
[340] Report of the Trial (62).
[341] Bertrand and Renault (92) p 29
[342] Bertrand (93), Bertrand and Renault (92) (94), Bertrand (96), Renault (96). Additional references may be found in these memoirs.
[343] Batters (92). Vide also Schmitz (97) p. 315.
[344] Hauck (85) in Rabenhorst’s Kryptogamen Flora, vol
[345] Kent (93) p. 140.
[346] Agassiz (88) vol p 82
[347] Walther (85).
[348] Ibid. (88) p. 478.
[349] Philippi (37) p 387
[350] Rosanoff(66) Pl. . fig. 10.
[351] Rothpletz (91) Pl fig 4
[352] Früh (90) fig. 12.
[353] Kjellman (83).
[354] Holmes and Batters (90) p 102
[355] Hauck (85). Rosanoff (66). Rosenvinge (93) p. 779. Kjellman (83) p. 88. Solms-Laubach (81).
[356] Unger (58).
[357] A microscopic section of the Vienna Leithakalk is figured in Nicholson and Lydekker’s Manual of Palæontology (89) vol. . p. 1497.
[358] Gümbel (71) Pl. . fig. 7, p. 41.
[359] Hauck (85) p 272
[360] Rothpletz (91) Pl. . fig. 4.
[361] Vide Walther (88) p. 499; also Jukes-Browne and Harrison (91) passim I am indebted to Mr G F Franks, who has studied the Barbadian reefs, for the opportunity of examining sections of West-Indian coral-rock
[362] Brown A. (94).
[363] Brown A. (94) p. 147.
[364] ibid p 200
[365] e.g. Saporta (82) p. 12.
[366] Turner (11) vol p 51
[367] Rothpletz (96).
[368] Penhallow (96) p 45
[369] Dawson (59).
[370] Vide ‘Academy’ 1870, p 16
[371] Carruthers (72)
[372] Penhallow (89).
[373] Barber (92)
[374] Solms-Laubach (952).
[375] Penhallow (96)
[376] loc. cit. p. 83.
[377] Dawson (59), also (71) p. 17.
[378] Penhallow (89) and (96) p 46
[379] Barber (92) p. 336.
[380] Penhallow (89) and (93).
[381] Dawson (81) p 302
[382] Barber (92).
[383] Hicks (81) p 490
[384] Lake (95) p. 22.
[385] Solms-Laubach (952).
[386] A similar method of fossilisation has been noted by Rothpletz in the case of the Lower Devonian alga Hostinella. [Rothpletz (96) p. 896.]
[387] Penhallow (96) p. 47.
[388] Carruthers (72) p 162 regards this species as identical with N Logani
[389] Seward (953).
[390] Strickland and Hooker (53)
[391] Hicks (81) p. 484.
[392] Dawson (82) p. 104.
[393] Barber (89) and (90)
[394] Murray G. (953).
[395] Hooker J D (89)
[396] loc. cit.
[397] Solms-Laubach (952) p 81
[398] Seward (942) p. 4.
[399] An excellent monograph on the Mycetozoa has lately been issued by the Trustees of the British Museum under the authorship of Mr A. Lister (94). Vide also Schröter (89) in Engler and Prantl’s Natürlichen Pflanzenfamilien.
[400] Renault (96) p. 422, figs. 75 and 76.
[401] Schröter (89) p 32, fig 18 B
[402] Cash and Hick (782) Pl. . fig. 3.
[403] Göppert and Menge (83) Pl fig 106
[404] Harper (95).
[405] e.g. Ludwig (57) Pl. . fig. 1.
[406] Potonié (93) p 27, Pl fig 8
[407] References are given by Potonié to illustrations by Zeiller (922) Pl. . fig. 6, Grand’ Eury (77) Pl. . fig. 7, and others in which possible fungi are represented.
[408] Engelhardt (87)
[409] For figures of the Coccineae, see Comstock (88), Maskell (87), Judeich and Nitsche (95) &c.
[410] Massalongo (59) Pl fig 1, p 87
[411] Meschinelli (92).
[412] James, J F (932)
[413] Lesquereux (87).
[414] Herzer (93).
[415] Conwentz (90) Pl fig 5
[416] Feilden, H. W. (96); Seward (962) p. 62, appendix to Feilden’s paper. I am indebted to Dr Bonney for an opportunity of examining the plant remains from the Feilden collection.
[417] Hartig (78)
[418] Conwentz (80) Pl. . fig. 17.
[419] Carruthers (70) Pl. . fig. 3.
[420] Renault (96) p 427, fig 80, d
[421] ibid. p. 427, fig. 80, a–c.
[422] p 127
[423] Etheridge (92) Pl. .
[424] Hartig (78) and (94), Göppert and Menge (83).
[425] Renault (96) p 425, fig 78
[426] Fischer in Rabenhorst, vol. i. (92) p. 144.
[427] Carruthers (76) p 22, fig 1
[428] Smith, W O (77) p 499
[429] Williamson (81) Pl. . p. 301.
[430] Renault (96) p 439, figs 88 and 89
[431] Cash and Hick (782).
[432] Cash and Hick, Pl fig 3
[433] Felix (94) p. 276, Pl. . fig. 1.
[434] ibid. p. 274, Pl. . figs. 5 and 6.
[435] Williamson (78) and (80)
[436] Conwentz (90) p. 119, Pl. . pp. 2, 3, Pl. . fig. 8.
[437] Migula (90) in Rabenhorst’s Kryptogamen Flora, vol. .
[438] Vaillant (1719) p 17
[439] Migula (90) p. 53.
[440] Knowlton (892)
[441] Meek (73) p. 219.
[442] Saporta (73) p. 214, Pl. . figs. 8–11.
[443] Seward (942) p. 13, fig. 1.
[444] Woodward, H. B. (95) pp, 234, 261, etc.
[445] Forbes, E (56) p 160, Pl
[446] Vide p. 69, fig. 10.
[447] Lyell (29)
[448] Skertchly (77) p 60
[449] Schiffner and Müller in Engler and Prantl (95), Campbell (95), Dixon and Jameson (96) are among the best of modern writers on the Bryophyta.
[450] Hooker, J D (91) p 513
[451] Schiffner (95) p. 140.
[452] Hooker, W J (20) Pl
[453] Bennett and Brown (38), Pl. .
[454] Bennett and Brown (38) p. 35.
