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.

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. Given a n × n matrix B = (bij) with real entries, the optimal assignment problem is to find a permutation σ of {1,..., n} maximising the sum Pn i=1 biσ(i). In discrete optimal control and in the theory of discrete event systems, one often encounters the problem of solving the equation Bf = g for a given vector g, where the same symbol B denotes the corresponding maxplus linear operator, (Bf)i: = max1≤j≤n bij + fj. The matrix B is said to be strongly regular when there exists a vector g such that the equation Bf = g has a unique solution f. A result of Butkovič and Hevery shows that B is strongly regular if and only if the associated optimal assignment problem has a unique solution. We establish here an extension of this result which applies to max-plus linear operators over a countable state space. The proofs use the theory developed in a previous work in which we characterised the unique solvability of equations involving Moreau conjugacies over an infinite state space, in terms of the minimality of certain coverings of the state space by generalised subdifferentials. 1.