Abstract. We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert’s projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and half-spaces over the max-plus semiring. 1.

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Abstract. We consider convex sets and functions over idempotent semifields, like the max-plus semifield. We show that if K is a conditionally complete idempotent semifield, with completion ¯ K, a convex function K n → ¯ K which is lower semi-continuous in the order topology is the upper hull of supporting functions defined as residuated differences of affine functions. This result is proved using a separation theorem for closed convex subsets of K n, which extends earlier results of Zimmermann, Samborski, and Shpiz.

Abstract. If A is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that A has an eigenvector in the positive cone, (R+) n. We associate a directed graph to any homogeneous, monotone function, f:(R+) n → (R+) n, and show that if the graph is strongly connected, then f has a (nonlinear) eigenvector in (R+) n. Several results in the literature emerge as corollaries. Our methods show that the Perron-Frobenius theorem is “really ” about the boundedness of invariant subsets in the Hilbert projective metric. They lead to further existence results and open problems.

Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is non-empty, is isomorphic to a convex inf-subsemilattice of R n, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group

In this paper, we present an algebraic approach to idempotent functional analysis, which is an abstract version of idempotent analysis in the sense of [1–3]. Elements of such an approach were used, for example, in [1, 4]. The basic concepts and results are expressed in purely algebraic terms. We consider idempotent versions of certain basic results of linear functional analysis, including the theorem on the general form of a linear functional and the Hahn–Banach and Riesz–Fischer theorems. 1. Recall that an additive semigroup S with commutative addition ⊕ is called an idempotent semigroup (IS) if the relation x ⊕ x = x is fulfilled for all elements x ∈ S. If S contains a neutral element, this element is denoted by the symbol 0. Any IS is a partially ordered set with respect to the following standard order: x ≼ y if and only if x ⊕ y = y. It is obvious that this order is well defined and x ⊕ y = sup{x, y}. Thus, any IS is an upper semilattice; moreover, the concepts of IS and upper semilattice coincide [5]. An idempotent semigroup S is called a-complete (or algebraically complete) if it is complete as an ordered set, i.e., if any subset X in S has the least upper bound sup(X) denoted by ⊕X and the greatest lower bound inf(X) denoted by ∧X. This semigroup is called b-complete (or boundedly complete), if any bounded above subset X of this semigroup (including the empty subset) has the least upper bound ⊕X (in this case, any nonempty subset Y in S has the greatest lower bound ∧Y and S in a lattice). Note that any a-complete or b-complete IS has the zero element 0 that coincides with ⊕Ø, where Ø is the empty set. Certainly, a-completeness implies the b-completeness. Completion by means of cuts [5] yields an embedding S → ̂ S of an arbitrary IS S into an a-complete IS ̂ S (which is called a normal completion 1 International Sophus Lie Centre, Moscow, Russia,

We consider equations of the form Bf = g, where B is a Galois connection between lattices of functions. This includes the case where B is the Fenchel transform, or more generally a Moreau conjugacy. We characterize the existence and uniqueness of a solution f in terms of generalized subdifferentials, which extends K. Zimmermann’s covering theorem for max-plus linear equations, and give various illustrations.

Abstract. We extend the perturbation theory of Viˇsik, Ljusternik and Lidskiĭ for eigenvalues of matrices, using methods of min-plus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturbation formulæ, extending the classical ones and solving cases which where singular in previous approaches. Our results include general weak majorisation inequalities, relating leading exponents of eigenvalues of perturbed matrices and min-plus analogues of eigenvalues. 1.

The lower and average spectral radii measure the minimal and average growth rates, respectively, of long products of matrices taken from a finite set. The logarithm of the average spectral radius is traditionally called the Lyapunov exponent. When one forms these products in the maxalgebra, we obtain quantities that measure the performance of Discrete Event Systems. We show that approximating the lower and average maxalgebraic spectral radii is NP-hard.

Abstract. Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of ‘Idempotent Mathematics ’ with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation. Solution of an interval system is typically NP-hard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined.