Raymond Cuninghame-Green

Minimax algebra and applications

Publisher Summary This chapter discusses the applications of minimax algebra. The chapter discusses a discrete-event system (DES), in which the individual components move from event to event rather than varying continuously through time. A characteristic of many such DESs, which is focussed in the chapter, is that any given component must wait before proceeding to its next event until certain others have completed their current events. The chapter presents a hypothetical DES, containing four machines called the “model system.” Max algebra depends on a crucial change of notation. In place of the operator max, use of the symbol ⊕, reminiscent of an addition symbol instead of + use ⊗, reminiscent of a multiplication symbol. An AND-gate is a device used in signal processing. It is characterized by a single output and a number of inputs, each of which may be active or quiescent. A delay is characterized by a single input, a number of outputs, and a given fixed time-interval. The chapter discusses several processes of Max algebra, scheduling and approximation, Chebyshev approximation, and others.

The equation A ⊗ x = B ⊗ y over (max, +)

For the two-sided homogeneous linear equation system A ⊗ x = B ⊗ y over (max, +), with no infinite rows or columns in A or B, an algorithm is presented which converges to a finite solution from any finite starting point whenever a finite solution exists. If the finite elements of A, B are all integers, convergence is in a finite number of steps, for which a precise bound can be calculated if moreover one of A, B has only finite elements. The algorithm is thus pseudopolynomial in complexity.

Minimal (max,+) Realization of Convex Sequences

We show that the minimal dimension of a linear realization over the (max,+) semiring of a convex sequence is equal to the minimal size of a decomposition of the sequence as a supremum of discrete affine maps. The minimal-dimensional realization of any convex realizable sequence can thus be found in linear time. The result is based on a bound in terms of minors of the Hankel matrix.

An algebra for piecewise-linear minimax problems

Abstract A piecewise-linear function whose definition involves the operator max and min may be reformulated as a ‘sum-of-partial-fractions’ by use of an algebraic structure J and so may be ‘rationalized’ to become a ‘quotient-of-polynomials’ in the notation of J We show that these ‘partial fractions’ and ‘polynomials’ have algebraic properties closely analogous to those of their counterparts in traditional elementary algebra: in particular an analogue of the fundamental theorem of algebra holds. These formal properties lead to straightforward procedures for finding maxima and minima of such functions.

Reducible Spectral Theory with Applications to the Robustness of Matrices in Max-Algebra

Let

The characteristic maxpolynomial of a matrix

a\oplus b=\max(a,b)

Projections in minimax algebra

and

The absolute centre of a graph

a\otimes b=a+b

An O( n 2 ) algorithm for the maximum cycle mean of an n x n bivalent matrix

for

Nearest-neighbour rules for emergency services

a,b\in\overline{\mathbb{R}}:=\mathbb{R}\cup\{-\infty\}