Ivan Singer

Topical and sub-topical functions, downward sets and abstract convexity

We study topical and sub-topical functions (i.e., functions which are increasing in the natural partial ordering of ℝn and additively homogeneous, respectively additively sub-homogeneous), and downward sets (i.e., subsets of ℝn which contain, along with each element, all smaller elements), in the framework of abstract convex analysis, with the aid of the additive min-type coupling function . We study primal and dual link between topical functions and closed downward sets, via plus-Minkowski gauges and ϕ-support functions of sets and level sets and ϕ-support sets of functions. Also, we give characterizations of topical functions in terms of their Fenchel-Moreau conjugates and biconjugates with respect to the above coupling function ϕ

Downward Sets and their separation and approximation properties

We develop a theory of downward subsets of the space ℝI, where I is a finite index set. Downward sets arise as the set of all solutions of a system of inequalities x∈ℝI,ft(x)≤0 (t∈T), where T is an arbitrary index set and each ft (t∈T) is an increasing function defined on ℝI. These sets play an important role in some parts of mathematical economics and game theory. We examine some functions related to a downward set (the distance to this set and the plus-Minkowski gauge of this set, which we introduce here) and study lattices of closed downward sets and of corresponding distance functions. We discuss two kinds of duality for downward sets, based on multiplicative and additive min-type functions, respectively, and corresponding separation properties, and we give some characterizations of best approximations by downward sets. Some links between the multiplicative and additive cases are established.

Extensions of functions of 0-1 variables and applications to combinatorial optimization

For in § 2 we introduce and study “tight extensions” fD : [0, 1]n → R, defined on each n-simplex Di of a triangulation D of [0,1]n, with all vertices in {0,1}n, as the unique affine function which interpolates f at the vertices of Di. In § 3 we study convexity of tight extensions. In §4 we show the existence of polyhedral convex (generally, non-tight) extensions. As applications, in §5 we give some duality theorems for minimization and maximization of submodular functions and in §6 (Appendix) we obtain new insight into the “greedy solutions” of a certain linear maximization problem.

Dualities between complete lattices

We study dualities between two complete lattices Eand Fi.e., mappings △:E→ F satisfying for all {x i } ieI ⊆E and all index sets I including the empty set I = O. We give characterizations and representations of dualities △, and some results on the dual △* F→Eof △ and on the associated hull operator △*△:E→Ein the general case and in various particular eases. Among several applications, we devote special attention to Fenchel-Moreau conjugations.

A general theory of dual optimization problems

We construct a unified theory of dual optimization problems, which encompasses, as particular cases, the known dual problems. For each concept of dual problem, we define first an “unperturbational” version, from which we deduce, via a certain scheme, the corresponding “perturbational” version. We generalize simultaneously the Lagrangian and surrogate cases, using coupling functionals. We study the connections between dual problems and define Lagrangian functionals for them. We study the class of perturbation functionals which can be written as the upper sum of the primal objective functional h and a functional with values not depending On h.

The distance to a polyhedron

Abstract We give some exact formulas and some estimates for the distance to a polyhedron in a normed linear space E . We show that in the case when E = R n , endowed with the l ∞ -norm, these estimates are, in general, better than a recent estimate by Cook, Gerards, Schrijver, and Tardos [2].

Max-plus convex sets and max-plus semispaces. II

We give some complements to Nitica, V. and Singer, I., 2007, Max-plus convex sets and max-plus semispaces, I.Optimization, 56, 171–205. We show that the theories of max-plus convexity in and -convexity in are equivalent, and we deduce some consequences. We show that max-plus convexity in Rn is a multi-order convexity. We give simpler proofs, using only the definition of max-plus segments, of the results of loc. cit. on max-plus semispaces. We show that unless ≤ is a total order onA, the results ofloc. cit. on semispaces cannot be generalized in a natural way to the framework ofAn =(An , ≤, ⊗), whereA:=M∪ {−∞}, withM=(M,≤,⊗) being a lattice ordered group and −∞ a “least element’ adjoined toM.

Duality for optimization and best approximation over finite intersections

Recently Deutsch, Li and Swetits [2] have studied, in Hilbert space, a dual problem (Qm ) to the primal problem (P) of minimization of a special class of convex functions f over the intersection of m closed convex sets, where m is finite. In the first part of this paper we obtain, in a locally convex space, some results on problem (Qm ) and on its relations with the usual Lagrangian dual problem (Q) to (P) (studied in [9]), in the case when (P) has a solution. In the second part we give some applications to duality for the distance to the intersection of m closed convex sets in a normed linear space, in the case when a nearest point exists. Most of our results seem to be new even in the particular cases studied in [9] (the case m = 1), [l] (duality formulas for the distance to the intersection of m closed half-spaces in a normed linear space) and [2].

Uniqueness and strong uniqueness of best approximations by spline subspaces and other subspaces

Abstract A complete characterization is given of those functions in C¦a, b¦ which have a unique best approximation from the subspace of spline functions of degree n with k fixed knots. Also, the relationship between unique and strongly unique best approximations from arbitrary finitedimensional subspaces of C0(T) is investigated.

Topologies on lattice ordered groups, separation from closed downward sets and conjugations of type Lau

We extend the results of Rubinov and Glover on the separation of normal subsets of , and those of Martínez-Legaz, Rubinov and Singer on the separation of downward subsets of , to the unifying framework of subsets of An , where A is a continuous conditionally complete lattice ordered group. We also extend earlier results of the second author on the characterizations of conjugations of type Lau of functions , to the framework of functions , where is the canonical enlargement of A. These extensions are based on the notions and results of the theory of continuous lattices, generalized in an earlier work of the first author and here, to the conditionally complete case, in particular the way below order relation ≪, and the Scott and Lawson topologies. We also introduce the “way above order relation ≫”, and consider the “bi-Scott topology” on A, introduced previously by the first author, which under our continuity assumption on A, turns out to be equal to the order topology; moreover, we show that for this topology, (A, ⊗) is a topological group and (A, ≤) is a topological lattice.