25. Appendix

In this appendix we wish to briefly summarize a few essential definitions and facts which might prove useful when reading this book. We would like, however, to warn the reader that the theory behind many of the models and all of the algorithms needed to solve them requires advanced mathematical skills and cannot be summarized in an appendix like this one. We refer the interested reader to the huge literature available on all of the topics we covered in this volume, from optimization theory, to the simplex method to duality theory or network flows, and so on…

Here we simply would like to summarize a few basic definitions and concepts which are found somewhere in the volume and maybe more efficiently found in a single place like this appendix. We list them here, without any specific logic.

25.1. Basic optimization notions

Let \(n\) be an integer and \(S \subseteq \R^n\) be a subset of the Euclidean \(n\)-dimensional space. Let \(f: S \rightarrow \R\) be a real-valued function. A generic optimization problem is defined as

\begin{align*} \min_{\var{x} \in S} f(\var{x}) \end{align*}

where \(\var{x}\) is the array of unknown variables, \(\set{S}\) the feasible set, \(f\) the objective function. A similar definition can also be given for a maximization problem. Any vector \(\bar{x}\) such that \(\bar{x} \in S\) is called a feasible solution. Any feasible solution \(x^\star\) such that

\begin{align*} f(x^\star) & \leq f(x) & \forall\, x &\in S \end{align*}

is called an optimal solution or a global optimum solution. The value of the objective function at an optimal solution \(f^\star = f(x^\star)\) is called the optimal value, or \(optimum\) of the problem. Sometimes, when it is felt that no confusion should arise, we use the term optimum also to denote the optimal solution, not just its value.

An optimization problem need not possess optimal solutions. In particular, if \(\set{S} = \emptyset\) then the problem is called infeasible.

If on the contrary \(\set{S} \ne \emptyset\) it might happen that

\begin{align*} \forall\,x \in S & \exists\, y \in S: \\ f(y) & < f(x) \end{align*}

In this case the optimization problem is unbounded and no optimal solution can exist.

Let \(\bar{x} \in S\) be a feasible solution. Let \(B(x,\varepsilon) := \{y \in \R^n: \|x-y\| \leq \varepsilon\}\) be an euclidean ball centered at \(x\) with radius \(\varepsilon\), If \(\exists\,\varepsilon > 0\) such that

\begin{align*} f(\bar{x}) & \leq f(y) & \forall\, y \in \set{S} \bigcap B(\bar{x},\varepsilon) \end{align*}

then \(\bar{x}\) is called a local optimum or local minimum. Of course an optimum is always a local optimum, but, in general, the opposite is not true.

An extremely important class of optimization problems is that of convex optimization problems.

A set \(S \subseteq \R^n\) is called a convex set if

\begin{align*} \lambda x + (1-\lambda) y & \in S & \forall\, x,y \in S, \lambda \in [0,1] \end{align*}

Geometrically this means that a convex set is characterized by the fact that for any pair of points in the set, also the segment with those points as extremes entirely belongs to the set.

A function \(f\) defined on a convex set \(S\) is called a convex function if

\begin{align*} f(\lambda x + (1-\lambda)y) & \leq \lambda f(x) + (1-\lambda) f(y) & \forall\, x,y \in S, \lambda \in [0,1] \end{align*}

Also in this case a geometrical interpretation is possible: a convex function is such that for any pair of points a linear interpolation of the function through the two points overestimates the function along the segment with those points as extremes.

A convex optimization problem is defined as a problem in which the feasible set is convex and the objective function is convex over a set that contains the feasible set. A concave function is a function \(f\) whose opposite \(-f\) is convex. By extension, the problem of maximizing a concave function, always on a convex feasible set, is called a convex optimization problem.

Convex optimization problems enjoy many interesting properties, one of the most notable of which is that if a feasible point is a local optimum, than it is also global.

A linear function is an additive and homogeneous function, i.e.,

\begin{align*} f(x+y) & = f(x) + f(y) & \forall\, x,y \\ f(\alpha x) & = \alpha f(x) & \forall\, \alpha \in \R, \forall\,x \end{align*}

It can be shown that in Euclidean spaces a function is linear if and only if there exists a vector \(v \in \R^n, v = [v_1, \ldots, v_n]^T\) such that

\begin{align*} f(x) & = \sum_{j=1}^n v_j x_j \\ & = v^T x \end{align*}

An affine function is defined as a linear function plus a constant: \(v^Tx + u\).

A fundamental convex optimization problem is the linear optimization one:

\begin{align*} \min_\var{x} \param{c}^T \var{x} & \\ \param{A}_1 \var{x} & = \param{b}_1 \\ \param{A}_2 \var{x} & \leq \param{b}_2 \\ \param{A}_3 \var{x} & \geq \param{b}_3 \end{align*}

where vectors and matrix have dimensions which are compatible with the matrix products as required. In the above problem a linear objective function is to be minimized subject to a set of linear equations and inequalities. Quite often, inequalities include sign constraints (all variables non negative) or box constraints (all variables constrained to be in a specific interval). It can be easily shown that this problem is indeed a convex optimization problem, called generic linear optimization problem.

A special case, to which any generic linear optimization problem can be reduced, is the so-called standard linear optimization problem:

\begin{align*} \min_\var{x} \param{c}^T \var{x} & \\ \param{A} \var{x} & = \param{b} \\ \var{x} & \geq 0 \end{align*}

If the feasible set is non empty and the set of linear equations is reduced by eliminating redundant equations, then matrix \(\param{A}\) is reduced to its rank and it can be shown that we can always assume that the number \(m\) of its rows can always be assumed to be equal to the rank and is never larger than the number of variables. In this situation, a square sub-matrix composed of all of the rows and an equal number of columns can always be chosen so that the resulting square matrix \(\param{A}_B\) is non singular. Any such matrix is called a basis for the problem and by solving the sub-problem \(\param{A}_B \var{x}_B = \param{b}\) a solution is always found. Here by \(\var{x}_B\) we denote the subset of variables whose indices are the same as those of the columns of the basis matrix. The \(n\)-dimensional array obtained completing this unique solution with zeros is called a basic solution. If it is composed of non negative elements, it is called a basic feasible solution. A fundamental property in linear optimization says that if a standard linear optimization problem is reduced to its row rank than the problem admits a feasible solution if and only if it admits a basic feasible solution. Moreover, it admits an optimal solution if and only if it admits an optimal solution which is also basic. The simplex method, one of the most popular algorithms to solve linear optimization problems, is strongly associated to this property and, in fact, limits the search for optimal solutions to basic ones.

A fundamental property of linear optimization, as well as of more general optimization problems, is that of duality; a general introduction to the topic is far beyond our aims in this appendix: the interested reader might consult any optimization book, where the topic of Lagrange duality is usually dealt with. Limiting ourselves to the easy case of linear optimization, let us associate to a linear optimization problem (which for the sake of simplicity we assume to be in standard form) another optimization problem, called the dual:

\begin{align*} \max_{\mathvar{\lambda}} \mathvar{\lambda}^T \param{b} & \\ \mathvar{\lambda}^T \param{A} & \leq \param{c}^T \end{align*}

It can be proven that:

  • given a pair of feasible solutions, \(\bar{x}\) for the original problem and \(\bar{\lambda}\) for the dual, it holds that \(\bar{\lambda}^T \param{b} \leq \param{c}^T \bar{x}\). This is called the weak duality theorem and states that in the two problems the objective function of one of the two always limits the other one

  • if, as before, two feasible solutions exist and, furthermore, the above inequality holds as an equality (i.e., the objective functions in the two problems are equal) than each of the two solutions is optimal for the respective optimization problem

  • if one of the two problems admits an optimal solution, the other admits an optimal solution too and the objective functions of the two optimal solutions are identical (strong duality theorem)

  • if one of the two problems in unbounded, the other one is infeasible

  • there exists a one-to-one correspondence between variables in the standard problem and inequalities in the dual. Given two feasible solutions, one for each problem, they are both optimal if and only if for any variable in the problem either the variable is null or the dual constraint associated to this variable holds as an equality. In formulae:

    \begin{align*} \bar{x}_j (\bar{\lambda}^T \param{A}_j - \param{c}_j) & = 0 & \forall\, j \end{align*}

    This property is known as complementarity.

25.2. Graphs: basic definitions

A graph is a pair \(\langle V, E \rangle\), where \(V\) is a finite set called set of nodes or vertices. \(E \subseteq V \times V\) is a set of pair of nodes. Each element of \(E\) is called an edge of the graph and is an (unordered) subset of the set of nodes with cardinality 2.

An edge in a graph is said to be incident to the two vertices which define it; these vertices are called extremes of the edge. A sequence of edges \(e_1, e_2, \ldots, e_k\) is called a walk if there exists a sequence of nodes in the graph \(v_1, v_2, \ldots, v_k, v_{k+1}\) such that

\begin{align*} e_1 & = \{v_1,v_2\} \\ e_2 & = \{v_2, v_3\}\\ \dots \\ e_k & = \{v_k,v_{k+1}\} \end{align*}

If the last node \(v_{k+1}\) coincides with the first one \(v_1\) then the walk is called a chain.

A graph is connected if there exists at least a walk connecting every pair of nodes. A graph is called acyclic if it is not possible to find, in the graph, chains in which no edge is repeated.

A connected and acyclic graph is called a tree. A tree enjoys several important properties. A graph is a tree if and only if any one of the following holds:

  • it is connected and acyclic (this is the definition)

  • in the finite case, it is connected and \(|E| = |V|-1\)

  • in the finite case, it is acyclic and \(|E| = |V|-1\)

  • the graph is connected, but if any edge is removed it gets disconnected

  • the graph is acyclic, but if we add any edge one and only one chain is generated

  • every pair of vertices is connected by a unique path (with no repeated edges)

In most application we are more interested in directed graphs or di-graphs. These are graphs for which edges, which now are called arcs, are an ordered subset of cardinality two of the set of nodes. This means, e.g., that in a di-graph arc \((i,j)\) is different from arc \((j,i)\).

All the definitions above, including that of a tree, can be given for directed arcs too, by simply neglecting the orientation of arcs. However in di-graphs we often consider paths as oriented paths.

It is possible to associate several matrices to graphs and di-graphs. One of the most useful is the incidence matrix. Given a di-graph, its incidence matrix is a \(|V| \times |E|\) matrix whose elements are defined as

\begin{align*} a_{ij} & = \left\{ \begin{array}{rl} +1 & \textrm{ if } \exists\, v_k \in V: e_j = (v_i,v_k) \\ -1 & \textrm{ if } \exists\, v_k \in V: e_j = (v_k,v_i) \\ 0 & \textrm{ otherwise} \end{array} \right. \end{align*}

In order to be able to define such a matrix the graph needs to be finite and loop-free. A loop is defined as an arc \((v,v)\) from a node to itself.


© Fabio Schoen 2024