Idempotent structures in optimization

Consider the set A = R ∪ {+∞} with the binary operations o1 = max and o2 = + and denote by An the set of vectors v = (v1,...,vn) with entries in A. Let the generalised sum u o1 v of two vectors denote the vector with entries uj o1 vj , and the product a o2 v of an element a ∈ A and a vector v ∈ An denote the vector with the entries a o2 vj . With these operations, the set An provides the simplest example of an idempotent semimodule. The study of idempotent semimodules and their morphisms is the subject of idempotent linear algebra, which has been developing for about 40 years already as a useful tool in a number of problems of discrete optimisation. Idempotent analysis studies infinite dimensional idempotent semimodules and is aimed at the applications to the optimisations problems with general (not necessarily finite) state spaces. We review here the main facts of idempotent analysis and its major areas of applications in optimisation theory, namely in multicriteria optimisation, in turnpike theory and mathematical economics, in the theory of generalised solutions of the Hamilton-Jacobi Bellman (HJB) equation, in the theory of games and controlled Marcov processes, in financial mathematics

TOPOLOGY CONTROL ALGORITHMS FOR RULE-BASED ROUTING

In this dissertation, we introduce a new topology control problem for rule- based link-state routing in autonomous networks. In this context, topology control is a mechanism to reduce the broadcast storm problem associated with link-state broadcasts. We focus on a class of topology control mechanisms called local-pruning mechanisms. Topology control by local pruning is an interesting multi-agent graph optimization problem, where every agent/router/station has access to only its local neighborhood information. Every agent selects a subset of its incident link-state in- formation for broadcast. This constitutes the pruned link-state information (pruned graph) for routing. The objective for every agent is to select a minimal subset of the local link-state information while guaranteeing that the pruned graph preserves desired paths for routing. In topology control for rule-based link-state routing, the pruned link-state information must preserve desired paths that satisfy the rules of routing. The non- triviality in these problems arises from the fact that the pruning agents have access to only their local link-state information. Consequently, rules of routing must have some property, which allows specifying the global properties of the routes from the local properties of the graph. In this dissertation, we illustrate that rules described as algebraic path problem in idempotent semirings have these necessary properties. The primary contribution of this dissertation is identifying a policy for pruning, which depends only on the local neighborhood, but guarantees that required global routing paths are preserved in the pruned graph. We show that for this local policy to ensure loop-free pruning, it is sufficient to have what is called an inflatory arc composition property. To prove the sufficiency, we prove a version of Bellman's optimality principle that extends to path-sets and minimal elements of partially ordered sets. As a motivating example, we present a stable path topology control mecha- nism, which ensures that the stable paths for routing are preserved after pruning. We show, using other examples, that the generic pruning works for many other rules of routing that are suitably described using idempotent semirings

Tropical linear algebra with the Lukasiewicz T-norm

The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics "a+b"=max(a,b) and "ab"=max(0,a+b-1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Lukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Lukasiewicz semiring.Comment: 27 page

Complete solution of a constrained tropical optimization problem with application to location analysis

We present a multidimensional optimization problem that is formulated and solved in the tropical mathematics setting. The problem consists of minimizing a nonlinear objective function defined on vectors over an idempotent semifield by means of a conjugate transposition operator, subject to constraints in the form of linear vector inequalities. A complete direct solution to the problem under fairly general assumptions is given in a compact vector form suitable for both further analysis and practical implementation. We apply the result to solve a multidimensional minimax single facility location problem with Chebyshev distance and with inequality constraints imposed on the feasible location area.Comment: 20 pages, 3 figure

Evolutionary distances in the twilight zone -- a rational kernel approach

Phylogenetic tree reconstruction is traditionally based on multiple sequence alignments (MSAs) and heavily depends on the validity of this information bottleneck. With increasing sequence divergence, the quality of MSAs decays quickly. Alignment-free methods, on the other hand, are based on abstract string comparisons and avoid potential alignment problems. However, in general they are not biologically motivated and ignore our knowledge about the evolution of sequences. Thus, it is still a major open question how to define an evolutionary distance metric between divergent sequences that makes use of indel information and known substitution models without the need for a multiple alignment. Here we propose a new evolutionary distance metric to close this gap. It uses finite-state transducers to create a biologically motivated similarity score which models substitutions and indels, and does not depend on a multiple sequence alignment. The sequence similarity score is defined in analogy to pairwise alignments and additionally has the positive semi-definite property. We describe its derivation and show in simulation studies and real-world examples that it is more accurate in reconstructing phylogenies than competing methods. The result is a new and accurate way of determining evolutionary distances in and beyond the twilight zone of sequence alignments that is suitable for large datasets.Comment: to appear in PLoS ON