Control Techniques for Constrained Discrete Time Systems

Control techniques for constrained discrete time systems

Paolo d’Alessandro, Elena De Santis

April 27, 2004 ii Contents

Introduction vii 0.1 Notations terminology and recalls ...... vii

iii iv CONTENTS Preface

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v vi PREFACE Introduction

Reading the book requires some mathematical background. This is proposed in form functional to the present purposes in [7] and consists in the …rst place of rudiments of topology, linear spaces, Hilbert spaces and convex analysis. The style of exposition therein is concise and therefore for most proofs and details it is useful to make reference to the classical books: [8], [11] and [12]. In addition to that, we need here some of the results contained in this latter book, especially some basic ideas underlying the conical approach and the main results of the dual conical methods for the solution of linear feasibility and optimality problems. Rather then duplicating the presentation in [7], the following section aims at establishing the notations used in the present book as well as providing a sort of check list for the reader of the concept and results that will be used in the sequel. It does not substitute reading the above references, unless one is already familiar with the required background. We do not exclude a few re…nements later on, wherever required.

0.1 Notations terminology and recalls

We shall consider, unless otherwise stated, points and sets in the Euclidean real spaces Rn equipped with the usual product topology, even though many of the involved concept are valid for more general linear spaces. At times we refer to a generic linear space, especially when there is no additional e¤ort to achieve maximum generality. We assume familiarity with the theory of linear bases. n (i) (i) We assume the space equipped with the inner product (x; y) = i=1 x y , which makes it into a Hilbert space. The symbol x(i) denotes the i.th component of the vector x. The inner product induces a norm by: P

x 2 = (x; x) k k Any other Haussdor¤ (distinct points have disjoint neighborhoods so that limits are unique) vector topology is equivalent to the one generated by this norm, because of …nite dimensionality of the space [8]. Again at times we shall refer to a generic Hilbert space. We shall leave to the context however the distinction between di¤erent inner product, and use for all of them the same symbol.

vii viii INTRODUCTION

Given an arbitrary integer m > 0, m vectors x1; ::; xm and m scalars 1; ::; m, m we call linear combination of x1; ::; xm the vector i=1 ixi: This is called a¢ ne m combination if i=1 = 1; conical combination if i 0 i; convex combination if it is both a¢ ne and conical. P 8 P n Next we de…ne set operations. If A and B are non-void subsets of R and if and are scalars, then:

n A + B = z : z R ; z = x + y; x A; y B f 2 2 2 g n A + B = z : z R ; z = x + y; x A; y B f 2 2 2 g Henceforth, in formulas similar to the above, we shall omit to specify which symbols denote scalars. This will be left to the context. Sometimes simpli…ed notations are useful: e.g. for a vector x Rn 2 x + A = x + A f g where x is the singleton of x. Thef lineg segment joining two points x and y of a linear space is the set

z : z = x + y; 0; 0; + = 1 f g and is denoted by [x : y]. The vectors x and y are called extremal points of the segment; it may well be the case that x and y coincide and hence the segment reduces to x . Notice thatf g

[x : y] = z : z = x + (1 )y; 0 1 f g A subset F of Rn is said to be a linear subspace of Rn if it is closed under linear combinations. This is equivalent to say x + y F whenever x; y for any reals and . We can also write this as: + 2 . The set 02 is f g a linear subspace of Rn, often called the trivial subspace. Two linear subspaces F and G of a linear space E are called complementary subspaces if any vector of the space can be uniquely expressed as the sum of a vector of F plus a vector of G. In formulas:

z E !x F !y G; z = x + y 8 2 9 2 9 2 where ! means exists and is unique. It is easy to prove that two linear subspaces are complementary9 if and only if their intersection is 0 and their sum is the whole space. Given a subspace F , all linear subspaces Gf gsuch that F and G are complementary have the same dimension, which is called the codimension of F . A translate x+F of a linear subspace is said to be an a¢ ne subspace. A¢ ne subspaces are closed under a¢ ne combinations. >From the theory of linear topological spaces [8], we know that linear and a¢ ne subspaces of Rn are necessarily closed. A subset of a linear space is said to be a convex cone (for brevity cone) if it is closed under conical combinations. This is equivalent to say x + y 2 0.1. NOTATIONS TERMINOLOGY AND RECALLS ix whenever x; y , 0; 0 and + = 1. We can also write this as: + ,2 ; 0 : + = 1. Finally another equivalence: is convex if 8

x; y ; [x : y] 8 2 The whole space Rn is a set of all the above sorts. In other words it is a linear subspace (of itself), an a¢ ne subspace, a cone and a convex set. It is also both closed and open. The linear, a¢ ne, conical, convex hull (or extension) of a set is, respectively, the minimum linear, a¢ ne, conical, convex set that contains . They are respectively denoted by: ( ); ( ); o( ) and ( ). It is immediate to verifyL thatA theseC extensionsC always exist. In fact, for example, ( ) is readily seen to be the intersection of all linear subspaces that contain L. Such family of sets is non-void because the whole space is one of its members. A similar proposition holds with the obvious changes, for the other hulls, and is proved with exactly the same argument mutatis mutandis. This provides external descriptions of hulls, because they are obtained by intersecting a family of sets. The linear subspace ( ) is the set of all linear combinations built using points of . Similarly forL ( ); o( ) and ( ), substituting, respectively, a¢ ne, conical and convex combinationsA C to linearC combinations. These state- ments yield the internal descriptions of these sets as opposed to the external descriptions depicted above. Of course there is the problem of giving compact, that is minimal in some sense, descriptions as well. For example we know from the theory of linear bases that ( ) is also the linear extension of a maximal linearly independent set containedL in . For a¢ ne sets it is easy to follow a similar paradigm introducing the notion of a¢ ne independence. The same problem is much more substantial for convex sets and cones. Later on, for the …rst case we will recall a major result of topology, namely the Krein-Milman theorem and for the second we shall recall the solution of this problem for the special case of polyhedral cones. Because linear subspaces and a¢ ne subspaces of Rn are always closed, it follows that ( ) and ( ) are also the minimal closed subspace and the minimal closed a¢L ne subspaceA containing . In other words they are also the closed linear hull, denoted by ( ) and the closed a¢ ne hull, denoted by ( ), of . L A The situation is di¤erent for convex sets and cones that are not necessarily closed. Denote with o( ) and ( ) the minimal closed cone and, respectively, the minimal convexC set containingC ; which are respectively the intersection of all closed cones containing and all convex sets containing . In general o( ) o( ) and ( ) ( ). But what is their internal description? AgainC in view C of the factC that the C topology is a linear one a simpli…cation occurs. More precisely it is possible to determine these closed extension by the formulas: o( ) = ( o( )) C C x INTRODUCTION

( ) = ( ( )) C C The proofs readily follow directly from linearity of the underlying topology. Before passing on we recall two more noteworthy computation rules for convex extensions for arbitrary non-void sets A and B and scalars and :

( A + B) = (A) + (B) C C C

(A B) = [x : y]; x (A)& y (B) C [ f[ 2 C 2 C g Linear subspaces are particular examples of cones and a cone may well contain a (non-trivial) linear subspace (all cones contain the subspace 0 ). Thus it is interesting to single out cones that do not contain linear subspaces.f g The key concept to this purpose is that of lineality space of a cone, which is, intuitively, the maximum linear subspace contained in the cone in question Given a cone , the lineality space of , denoted as lin(), is given by lin() = ( ) If the lineality space of is the singleton of the origin then is called a pointed\ cone. We shall see momentarily that if a cone is not pointed, it can be decomposed in the sum of a pointed cone plus its lineality space. A noteworthy example of cone is the recession cone of a convex set . The recession cone of is the set de…ned by: y : x + y ; 0; x . To proceed on, it is now in point tof introduce a2 few elementary8 8 concepts2 g pertaining Hilbert spaces. Recall that two subsets A and B of a Hilbert space H are said to be mutually orthogonal (in symbols A B) if ? (x; y) = 0 x A; y B 8 2 8 2 Given a set A in H, the orthogonal complement of A, to be denoted by A? is given by the set: A? = y :(y; x) = 0 x A f 8 2 g Note that A? is a closed linear subspace (in view of the continuity of inner product). For any sets A and B:

A? = (A)? L A? ? = (A) L A B A? B? )

Thus, if A is a closed linear subspace A? ? = A. Moreover, if A is a closed linear subspace, A and A? are a pair of complementary subspaces ([8] p. 66). Finally, a frequently useful result for computations on orthogonal complements is given by the following formula:

(A B)? = (A + B)? = A? B? [ \ 0.1. NOTATIONS TERMINOLOGY AND RECALLS xi

This formula has an interesting consequence when A and B are linear subspaces of a …nite dimensional space and hence are closed. In fact, taking the orthogonal complement of each member and exchanging A with A? and B with B? it is obtained:

(A B)? = A? + B? \ Let’s now turn to cones in Hilbert spaces. Notice that if the cone is closed then its lineality space is certainly a closed linear subspace. If the lineality space of a cone is closed then it is possible, so to say, to single out the pointed and unpointed part of the cone. In fact the cone can be decomposed in the sum of its lineality space plus a pointed cone. This result is in [12] for the case of a …nite dimensional space, but their proof depend only on the fact that the lineality space be closed. Formally, consider a convex cone C in a Hilbert space and assume that its lineality space be closed. Then the cone can be expressed as:

C = lin(C) + (lin(C)? C) (1) \ and the cone (lin(C)? C) is pointed. \ If E and F are two linear spaces and T is a function on E to F , we say that T is a linear function, if it preserves the algebraic operations, that de…ne the linear structure of the space E. In formulas T ( x + y) = T (x) + bT (y) for all scalars and and for all x and y in E. For linear functions we shall use, in keeping with a practically universal convention, the juxtaposition notation. Thus T (x) will be denoted by T x. Since a linear function preserves linear operations, it is expected to be well- behaved with respect to structures based on such operations. We will encounter a number of examples that con…rm that the expectation is correct. The range of a linear function is a linear subspace of the space in which the function takes values, as it is immediately veri…ed. More in general, the same is true for the image of any linear subspace. Nicely, things go well also in the reverse direction. In fact the inverse image of any linear subspace is a linear subspace. The instance of the trivial subspace 0 is particularly interesting. The linear 1 f g subspace T ( 0 ) is called the null space of the linear function T . A linear f g 1 function is one to one if and only if T ( 0 ) = 0 . This is another example of localization of properties in linear spaces.f Ag onef tog one and onto linear function is a linear isomorphism. The inverse of such a function is a linear isomorphism too. Similarly, images and inverse images under a linear function of a¢ ne subspaces, of convex sets and of convex cones are, respectively, a¢ ne subspaces, convex sets and convex cones. If the spaces E and F are endowed with a vector topology and the linear function T is continuous, we use to say that T is an operator. This terminology is particularly established for Hilbert spaces. xii INTRODUCTION

If the domain space of the linear function is …nitely dimensional (so that the same is true for the range) the function is necessarily continuous. This means that, for the purposes of studying problems de…ned in …nite dimensional spaces, we deal exclusively with linear operators. Notice also that the fact that any …nite dimensional linear topological space is topologically isomorphic to a Euclidean space allows us to represent any linear function with a …nite dimensional domain by means of a matrix. We shall not distinguish, neither terminologically nor symbolically the operator and the corresponding matrix, according to an established use. A linear function is called bounded if it carries each bounded set into a bounded set. In a Hilbert space a linear function is continuous if and only if it is bounded. An interesting topology for the linear space of the operators from a Hilbert space to another Hilbert space is de…ned taking advantage of this boundedness result. It is called the norm topology, and is de…ned by means of the following norm. If T is an operator from a Hilbert space H1 to a Hilbert space H2 then the norm T of T is given by: k k T = sup T x : x 1 k k fk k k k g Beware that there are three di¤erent meaning of the symbol : in this formula. On the lhs we de…ne a norm for the linear space of all lineark continuousk operators on H1 to H2. Inside the parenthesis on the rhs there are the di¤erent norms of H1 and H2. Henceforth, any norm will be denoted with the symbol : , leaving to the context to determine the normed space to which it is referred.k Ak similar convention will hold for inner products, which will always be denoted by (:; :) whatever is the Hilbert space to which they refer. Consider again a linear space E and two complementary subspaces F and G. As a consequence of uniqueness of the expression of any vector as sum of a vector in F and a vector in G, it is well de…ned the function, that associates to each vector of the space E the …rst vector of the decomposition. Such function PFG, is de…ned by: PFG(x) = xF , for all x E. It is a linear function and 2 is called the projection of E onto F along G. Notice that the range of PFG is given by the linear subspace F , whereas the null space of PFG is the linear subspace G. This simple remark implies the interesting consequence that any linear subspace F of E can always be expressed either as the range space of a linear function or as the null space of a linear function. In fact it su¢ ces to consider a subspace G, that forms, together with F , a pair of complementary subspaces. Then F = range (PFG) and G = kernel(PFG). Finally notice that the projection is idempotent, that is, PFGPFG = PFG. As we said above, in a Hilbert space a closed linear subspace and its orthogonal complement are a pair of complementary linear closed subspaces. They are called orthogonally complementary subspaces. Let H be the Hilbert space and F be the closed linear subspace. The projection of H onto F along F ? , 0.1. NOTATIONS TERMINOLOGY AND RECALLS xiii

denoted by PF , is called the orthogonal projection of H onto F and is always continuous Thus it is an operator. Notice that PF = 1. There is a particular instance of orthogonalk subspaces,k that is particularly important and will be used extensively in this book. First some premises. It is an immediate consequence of continuity that the null space of an operator is always closed. Notice that the range space is not closed, in general. Recall that, if T is an operator from a Hilbert space H1 to a Hilbert space H2, the adjoint operator T of T is de…ned by:

(T x; y) = (x; T y); x H1; y H2 8 2 8 2 Orthogonal projection operators are self-adjoint, that is, for any closed subspace F of an Hilbert space, PF = PF . Note also that T = T . In the …nite dimensional case, if and operator is represented by a matrix, then the adjoint operator is represented by the transposed matrix. At this point we can state the following result. Let T be a linear operator on a Hilbert space H1 to a Hilbert space H2, and T its adjoint operator. Then:

(T )? = (T ) R N and interchanging T and T :

(T )? = (T ) R N However taking orthogonal complements it is obtained:

(T )? = (T ) N R

Caution: (T )? = (T ) is, in general, false. That is not so evidently in the …nite dimensionalN case,R in which case we can freely use this formula. A linear function with values in the scalar …eld is called linear functional. It is also natural to identify with a linear functional any linear function whose range space is one dimensional. The null space of a linear functional is a particularly important kind of linear subspace. It has always codimension one and is called a hyperplane. Conversely, in view of a remark we made above for projections, any linear subspace of codimension one, is the null space of a linear functional. We are of course particularly interested to closed hyperplanes. Recall in this respect that a linear functional is continuous if and only if its null space is closed [8]. Thus the class of closed hyperplanes is identi…ed with the class of null spaces of continuous linear functionals. Recall, also, that, if the domain space is …nite dimensional, any linear functional is necessarily continuous. We shall also call hyperplane (leaving to the context the distinction) any a¢ ne subspace, whose parallel linear subspace is an hyperplane. In a Hilbert space H, for any vector x H, the functional obtained taking inner product de…ned by such vector, namely2 the functional (x; :), is a continuous linear functional. Conversely, thanks to the Riesz Theorem, we can uniquely xiv INTRODUCTION represent in this way any continuous linear functional. That is, given any continuous non zero linear functional f on H, there exists a unique non zero vector x in H, such that f = (x; :). In view of the Riesz Theorem, any closed hyperplane can be represented in the form z :(x; z) = b for some vector x and scalar b. The hyperplane is a closed linearf subspace (thatg is, it passes through the origin) if and only if b is zero. We turn now to give a di¤erent interpretation to orthogonal projection, which paves the way to generalize orthogonal projections to closed convex sets. Such generalizations is a key tool in our analysis, and is particularly useful in investigating many geometric as well as mathematical programming issues. For example, it is invoked frequently to develop conical algorithm of linear programming. Consider again the Hilbert space H and a closed linear subspace F . Then for any x the point PF x can be reinterpreted as the unique solution of the optimization problem: min x y : y F fk k 2 g This fact can be easily proved by direct computations on norms. The nice thing of this reinterpretation is that it can be generalized substituting closed convex subsets to closed linear spaces (which are special instances of closed convex subsets). In this way the concept of projection is generalized too. To this e¤ect we can state the following result.

Theorem 1 Let C be a closed convex subset of a Hilbert space H. Then (i) if x is any point of H, there exists an unique vector PC (x) in C such that: x PC (x) = inf x z : z C k k fk k 2 g The map PC is called the projection of H onto C. (ii) z C is the projection of x onto F if and only if 2 (x z; y z) 0; y C 8 2 (iii) For all x and y in H:

PC x PC y x y k k k k

Consequently the map PC is continuous.

This key result is stated e.g. in [8], in a problem section and without proof. Notice that, in view of statement (ii) the projection of any point not in C lies in the boundary of C. Henceforth and with minor exceptions, we con…ne ourselves to the …nite dimensional case. Thus, to …x ideas and unless otherwise stated, our ambient space will be Rn. Not all convex sets have interior. A convex set with non-void interior is called a convex body. However, if, for example, we consider a segment in R2 0.1. NOTATIONS TERMINOLOGY AND RECALLS xv

(which has a void interior), we still intuitively perceive all of it points, but the extremal ones, as internal. This disappearance of the interior is due to the too large topological environment, which has a sort of dimensional mismatch with respect to our convex set. Thus, the apparent contradiction should disappear if we more tightly adapt the environment to the convex set. This tight environment will be, for any convex set, the a¢ ne extension of the convex set, endowed with the relative topology. When we talk of relative topological concepts, without further speci- …cation, we always intend to refer to such relative topology. That this is the right choice is made apparent by the following result: every (non void) convex set has a non void relative interior ([11] page 45). The concepts of relative interior, relative closure and relative boundary, (which is the set di¤erence between relative closure and relative interior) play an important role in this book. We pass to introduce extreme sets and extreme points. Let be a convex set. A convex subset of is called an extreme subset if none of its points are included in an open segment joining two points of , which are not both in . An extreme subset consisting of one point is an extreme point. Rephrasing: a point of a convex set is an extreme point, if it is not a relative internal point of any segment, joining two points of the convex set. Notice that, if we consider a segment joining x and y, then the points x and y, which constitute the relative boundary of this convex set, are also extreme points of the set, and, actually, the only extreme points. The concept of extreme point plays a fundamental role in the generation of convex subset. In fact, given a convex set, the convex extension of any set properly contained in the set of its extreme points is properly contained in the set. In other words, we cannot exclude extreme points from a generating set. However, is the set of extreme points enough to generate the given convex set? In other words, is it exactly the minimal generating set? There is a class of well-behaved convex sets, for which the set of extreme points is a sort of convex base. This assertion is based on the celebrated Krein- Milman Theorem. In [8] there is a very general formulation. Here it su¢ ces to give a statement for Hilbert spaces:

Theorem 2 Any convex and compact subset of a Hilbert space is the closed convex extension of the set of its extreme points.

This theorem solves in Rn the problem of …nding the minimal internal description for the class of closed compact sets. Intuitively we cannot escape closedness, but, how about unboundedness? And how about cones, which are necessarily unbounded? As we shall see later on, at least for certain classes of convex sets, of primary interest for us, the problem is solved adding extreme directions to extreme points. A convex subset of a convex set is called a face of the set if, whenever a segment, contained in the set, has a relative internal point in common with the subset, the extreme points of the segment belong to the convex subset. xvi INTRODUCTION

Notice that the singleton of any extreme point is a face. Moreover, the whole convex set is a face of itself. Thus we stipulate to call a face proper, if it does not coincide with the whole set. Sometimes, we shall leave the speci…cation proper to the context. Any proper face of a convex set is contained in the relative boundary of the convex set ([11], Corollary 18.1.3). A face of a face of a convex set is a face of the convex set. The intersection of any family of faces is a face. The set of all faces of a convex set, regarded as partially ordered by inclusion, has a greatest element (the given convex set) and a least element (the intersection of the family, possibly the void set). These facts have a very important consequence: the family of all faces of a convex set is always a complete lattice ([11] page 164). The greatest lower bound of a set of faces is its intersection. And the least upper bound is the intersection of the family of all faces that contain every face of the set (such a family is not void because the whole convex set is one of its elements). We can retrieve a convex set from its faces. Theorem 18.2 (in the same reference [11]) asserts that the collection of all relative interiors of non-empty faces of a non-void convex set is a partition of the convex set. This result, along with the previous one on proper faces, immediately implies that the relative boundary of a relatively closed convex set is the union of its proper faces. We now turn to normal cones and polarity First of all, we need to de…ne the concept of semispace. Given a linear functional (x; :) and a scalar b, a (closed) semispace is the set z :(x; z) b . The hyperplane z :(x; z) = b is the boundary hyperplane off the semispace. g Note that we couldf have takeng the inequality . However we make the convention of using systematically . Next we introduce the fundamental concept of normal cone at a point of a set. In [12] a very general de…nition is given, considering arbitrary sets and arbitrary points. Since we don’tneed such a generality, we con…nes ourselves to closed convex sets and points in their boundary. In this way we devise a more direct connection with the concept of projection. Thus consider a closed convex set C and a point z in its boundary. Then we shall say that the vector n is normal to C at z if:

(n; y z) 0; y C: 8 2 Notice that, in view of the Theorem 1 above, all points of the ray generated by n and emanating from z, that is, the set given by z + an : a 0 , have a projection on C that coincides with z. f g The de…nition of normal vector induces some interesting properties to the hyperplane w :(n; w) = (n; z) . The point z belongs to the hyperplane. The hyperplane isf the boundary hyperplaneg of the semispace w :(n; w) (n; z) , which contains the set C. Thus, we have de…ned a semispacef that contains ourg convex set and has the boundary hyperplane tangent to it. Such a hyperplane is called support hyperplane to C at z. Support hyperplanes have a very important property, which is not di¢ cult to prove by direct application of the involved de…nitions: the intersection of a 0.1. NOTATIONS TERMINOLOGY AND RECALLS xvii support hyperplane with the convex set is always a face of the set. Next, consider the set of all normal vectors to C at z. This is given by:

n :(n; y z) 0; y C f 8 2 g This is readily seen to be a closed convex cone. It is called the normal cone to C at z and it is denoted by N(C; z). The set z + N(C; z) is precisely the set of all points of the space, whose projection onto C coincides with z. Finally we introduce another of the fundamental tools of our theory. Given an arbitrary subset S in our space, the polar cone of the set, denoted by Sp is given by:

Sp = n :(n; y) 0; y S f 8 2 g Polar cones have particular interest for cones. It is clear from the preceding discussion that the polar cone of a convex cone is nothing but the normal cone at the origin to the given convex cone. Thus the polar cone of a closed convex cone is nothing but the set of all points in the space, whose projection onto the cone itself coincides with the origin. We give two noteworthy examples, which are both veri…ed by straightforward direct computation. If the cone is a linear subspace F then:

p F = F ?

If we consider the non negative orthant P then:

P p = P ”In the theory of cones, polar cones play the role which orthogonal complements play in the theory of subspaces” ([12]). As a matter of facts in this reference it is shown how, by means of a cone and its polar cone, it is possible to decompose the space in a way parallel to the decomposition based on a pair of orthogonal complementary subspaces. The concept of polar cone is the main ingredient of duality theory. In this respect we make often use of another result, parallel to the theory of orthogonal subspaces, and that pertains to computation on polars. The sum of two convex cones, as already stated, is a convex cone. What is the polar cone of the sum? It is easy to prove that the most expected result, in view of the analogy with orthogonality, holds good. In fact if C1 and C2 are two convex cones then: p p p (C1 + C2) = C C 1 \ 2 As may also be expected, for any convex cone C ([11] p.121):

pp C = C

For special cases of well- behaved sum of cones, we can give, as we did for linear subspaces, the computation formula, that goes the other way around. More xviii INTRODUCTION

precisely, if C1 and C2 are closed convex cones and the sum of their polar cones is a closed convex cone then:

p p p (C1 C2) = C + C \ 1 2 The hypothesis we just made might appear a bit restrictive, but it is just what happens for the class of cones we are most interested in, namely polyhedral cones. A polyhedron in Rn is de…ned to be a …nite intersection of closed semispaces. Hence, given a …nite set of vectors n1; ::; nm; and a …nite set of scalars v1; ::; vm, a polyhedron is a set of the form:

x :(x; ni) vi : i = 1; ::; m f g Note that we may consider in this expression both the symbols and = instead of . In fact, in the …rst case, we can invert the inequality multiplying both sides by 1. In the second, we can express the equality as double inequality and then apply the same arti…ce to the part. As we shall see later on, this means that in all the formulations of Linear Programming problems, the constraints de…ne a polyhedral region. Thus there is a very direct connection between Linear Programming and polyhedral theory. Polyhedra have a …nite number of faces and any face of a polyhedron is in turn a polyhedron ([11], pp. 171- 172) We can clearly rewrite the de…nition of polyhedron in matrix form. Indeed, the same set can be de…ned by:

x : Gx v f g where G is a matrix and v is a vector, the entries of the ith row of G are the components of the vector ni and the components of the vector v are the scalars vi. A polyhedron is a cone (thus a polyhedral convex cone) if and only if all the scalars vi are zero. As we shall see, many results valid for polyhedra hold good for polyhedral cones. Polyhedral cones are an essential structure in the context of our approach, applied to an Euclidean setting. In the …rst place because the positive cone of the ordering is polyhedral. Obviously, a …nite intersection of polyhedra is a polyhedron and a …nite intersection of polyhedral cones is a polyhedral cone. Moreover, faces of polyhedra are polyhedra and faces of polyhedral cones are polyhedral cones ( [12] Chapter 2). A …rst example of polyhedral cone is a linear subspace. In fact recall that any linear subspace F can be expressed as null space of some matrix Q. Thus it has the form: F = x : Qx = 0 . Now the desired conclusion follows rewriting this formula as: f g Q F = x : x 0 Q 0.1. NOTATIONS TERMINOLOGY AND RECALLS xix

>From this simple result we can draw an important consequence. Let us apply the decomposition of cones to the case of a polyhedral cone C. Recall that C = lin(C) + (lin(C)? C). Then both cones on the rhs are polyhedral cones. \ Another important example of pointed polyhedral cone is the non-negative orthant in Rn. Notice that, consequently, the intersection of any subspace with the non-negative orthant is a pointed polyhedral cone. The polar of a polyhedral cone is a polyhedral cone. More than that, if the polyhedral cone is given by: x :(x; ni) vi : i = 1; ::; m then: f g p C = Co( ni : i = 1; ::; m ) f g This is a …nitely generated cone and, by the Theorem of Weyl ([12] p. 56), every …nitely generated cone is a polyhedral cone. Images and inverse images of polyhedra under linear transformations are polyhedra ([11] Theorem 19.3). Consequently images and inverse images under linear transformations of polyhedral cones are polyhedral cones. If C1 and C2 are polyhedra then C1 + C2 is also a polyhedron (Corollary 19.3.2 in [11]). Consequently, if C1 and C2 are polyhedral cones then C1 + C2 is a polyhedral cone. At this point, we can say that both the following formulas are true, for any two polyhedral cones C1 and C2:

p p p (C1 + C2) = C1 C2 p p \ p (C1 C2) = C + C \ 1 2

As an example, if C1 is a linear subspace F and C2 is the non negative orthant P , it is obtained:

p (F + P ) = F ? ( P ) p \ (F P ) = F ? + ( P ) \ These two formulas will be exploited frequently in the sequel. It is now in order to get back to the important problem of internal description, in the new context of polyhedral theory. The de…nitions of polyhedron and of polyhedral cones are based on external descriptions. Thus, as usual, we are interested in passing to internal descriptions. More speci…cally, we want to individuate a subset of the polyhedron or the polyhedral cone in question, from whose elements, by means of convex or conical combinations, the whole set is reconstructed. And, possibly, we want this set to be minimal. In this respect, let us …rst consider a special class of polyhedra. We de…ne a convex set to be a polytope if it is …nitely generated, that is, if it is the convex extension of a …nite set. A polytope is always a polyhedron. This is a consequence of theorem 19.1 in [11]. It is easy to prove that any polytope is bounded and hence compact. Things also go the other way around, because any bounded polyhedron is a polytope (Theorem 2.12.2 in [12]). Thus, by the Krein-Milman Theorem, a bounded xx INTRODUCTION polyhedron admits, as minimal generating set, the set of its extreme points, also called vertexes. This solves completely and satisfactorily the problem of giving an internal description, for at least the bounded species of polyhedra. The general case is solved via a decomposition theorem. In fact Theorem 2.12.6 in [12] says that any polyhedron is the sum of a polytope plus a linear subspace plus a pointed polyhedral cone. Actually, we may develop this result in two stages. First we can exploit the result asserting that any polyhedron is the sum of a polytope plus a polyhedral cone ([11] Theorem 19.1). Then we can decompose the polyhedral cone on the base of (1). This result reduces the problem to that of the internal description of a polyhedral cone. To this purpose we can immediately solve the problem for the lineality space. In fact, any linear subspace is trivially the conical extension of the union of a linear base of its own and opposite of the base. Hence it remains to see how the problem is solved for a polyhedral pointed cone. To this purpose, we need to introduce another fundamental concept. A ray (recall that a ray is the set of all non negative multiples of a non zero vector) is said to be an extreme ray for a cone if it is a face of the cone. Extreme rays play in conical theory a role analogous to vertexes for polytope theory. A polyhedral cone has always a …nite number of extreme rays, because, as we stated earlier, any polyhedron has a …nite number of faces. Exploiting the concept of extreme rays, the internal description problem for a pointed polyhedral cone is solved by the following result. A pointed polyhedral cone is the convex extension of the union of its extreme rays. This statement can easily be deduced from Corollary 18.5.2 in [11]. Because it is more natural to use conical extensions, the same result can be easily restated, so to use such extensions. We de…ne set of generators of the pointed polyhedral cone any set, obtained choosing a non zero vector from each extreme ray of the cone. Then the above result can be clearly rephrased, saying that any pointed polyhedral cone is the conical extension of a set of generators of the cone itself. Notice that each generator is unique only up to a positive scalar constant. All the results we have mentioned regarding internal descriptions, are exis- tence results. They say nothing about the problem of determining the internal description. In [7] the case the cone intersection of a linear subspace and the non-negative orthant is extensively studied and solved. Such case is crucial in the dual conical methods for linear programming problems illustrated therein. It may be useful to conclude this section with a few remarks on the structure of the non negative orthant in Rn. The extreme rays of this cone are the coordinate axes, that is, the rays generated by the vectors ei; i = 1; ::; n where ei has all components equal to zero, except the ith, whichf is equal to one.g The lattice of faces has an elegant structure, which is easy to describe ex- plicitly. All the faces contain the origin. Thus, the lower bound of the lattice is the face 0 . Any other face is the convex hull the union of a subset of the set of extremef g rays. Or, if one prefers to speak in terms of generators, it is the conical hull of a subset of the orthonormal base ei; i = 1; ::; n . Such subset, f g 0.1. NOTATIONS TERMINOLOGY AND RECALLS xxi

of the form ei : i I with I 1; ::; n ; completely identify the face. Thus the set I toof completely2 g identify the f face.g Note also that the set I also represent the set of indexes, corresponding to non zero components of an arbitrary vector in the relative interior of the face. It is natural to stipulate that the set I be void in correspondence to the face 0 . The number of elements in the set I gives the ”dimension” of the face f(withg the stipulation that there are zero elements in the void set and that the trivial subspace has dimension zero). It would be easy, in view of this identi…cation of the face with the corresponding index set, to give the count of faces for each dimension and, hence, also the total number of all faces. We omit this for the sake of brevity. It is also possible to express operations on faces in terms of corresponding operations on the associated index sets. This facts con…rm that many questions relative to polyhedra and, in particular, to polyhedral cones have a combinatorial nature. Here is an important example. Consider a set of faces Qj : j = 1; ::; r , each f g corresponding to the index set Ij. Then, for this set, the greatest lower bound face in the lattice is that corresponding to the index set Ij : j = 1; ::; r , and [f g the least upper bound face is that corresponding to the index set Ij : j = 1; ::; r . \f g xxii INTRODUCTION Bibliography

[1] P. d’Alessandro and M. Dalla Mora, ”Systems, memory, causality, evolution and recursive equations”, Computers and Mathematics with Applications, Vol. 10, No. 1; pp:61 69; (1984). [2] P. d’Alessandro, M. Dalla Mora, E. De Santis. Techniques of Linear Pro- gramming Based on the Theory of Convex Cones, Optimization, 20 (1989), 761-777. [3] P. d’Alessandro, E. De Santis. Conical Theory of Feasibility and Optimality for Dynamical Linear Systems, Reseach Report, 40 (1990), Department of Electrical Engineering, University of L’Aquila. [4] P. d’Alessandro, E. De Santis. Conical Approach to Linear Dynamic Opti- mization, in P. Gritzmann, R. Hettich, R. Horst, E. Sachs Eds, Operations Reserach ’91, Physica Verlag Heidelberg (1992). [5] P. d’Alessandro, E. De Santis, Reachability in input constrained discrete time linear systems, Automatica, vol. 28 no 1 (1992), pp 227-230. [6] P. d’Alessandro, E. De Santis, General Closed loop optimal solutions for linear dynamic systems with linear constraints. J. of Mathematical Systems, Estimation and Control, vol. 6, no. 2, 1996. [7] P. d’Alessandro, A conical approach to linear programming, scalar and vector optimization problems, Gordon and Breach Science Publishers, 1997. [8] J.L. Kelley and I. Namioka, Linear Topological Spaces, Springer, New York, 1963. [9] A. Marzollo. Controllability and Optimization, Lectures held at the De- partment of Automation and Information, University of Trieste, Courses and Lectures no. 17, International Center for Mechanical Sciences, Udine (Italy) (1969). [10] A. Propoi. Dynamic Linear Programming, WP-79-38, International Insti- tute for Applied System Analysis, Austria. [11] R.T. Rockafellar, ”Convex Analysis”, Princeton Mathematical Series, No. 28, (1970).

1 2 BIBLIOGRAPHY

[12] J. Stoer, C. Witzgall, Convexity and Optimization in …nite dimensions I, Springer-Verlag Berlin-Heidelberg-New York 1970.