A representation of generalized convex polyhedra and applications (original) (raw)

On Some Generalized Polyhedral Convex Constructions

Numerical Functional Analysis and Optimization

Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal cone, conjugate function, subdifferential, are studied thoroughly in this paper. Among other things, we show how a generalized polyhedral convex set can be characterized via the finiteness of the number of its faces. In addition, it is proved that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function. The obtained results can be applied to scalar optimization problems described by generalized polyhedral convex sets and generalized polyhedral convex functions.

Generalized polyhedral convex optimization problems

Journal of Global Optimization, 2019

Generalized polyhedral convex optimization problems in locally convex Hausdorff topological vector spaces are studied systematically in this paper. We establish solution existence theorems, necessary and sufficient optimality conditions, weak and strong duality theorems. In particular, we show that the dual problem has the same structure as the primal problem, and the strong duality relation holds under three different sets of conditions.

A ``hidden'' characterization of polyhedral convex sets

Studia Mathematica, 2011

We prove that a closed convex subset C of a complete linear metric space X is polyhedral in its closed linear hull if and only if no infinite subset A ⊂ X\C can be hidden behind C in the sense [x, y] ∩ C = ∅ for any distinct points x, y ∈ A.

Unique representation in convex sets by extraction of marked components

Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, 1983

After introducing the basic concepts of extraction and marking for convex sets, the following marked representation theorem is established: Let C be a lineally closed convex set without lines, the face lattice of which satisfies some descending chain condition, and let p be some marking on C. Then every point of C can be represented in unique way as a convex (nonnegative) linear combination of points (directions) of C which are p-independent, and this representation can be determined by an algorithm of successive extractions. In particular, if C is a finite dimensional closed convex set without lines and p marks extreme points (directions) only, then the marked representation theorem contains some well-known results of convex analysis as special cases, and it yields in the case where C is a polyhedral triangulation which extends available results on polytopes to the unbounded case. The triangulation of unbounded polyhedra then is applied to a certain class of parametric linear programs.

A New Encoding of Not Necessarily Closed Convex Polyhedra

Many applications of static analysis and verification compute on some abstract domain based on convex polyhedra. Traditionally, most of these applications are restricted to convex polyhedra that are topologically closed. When adopting the Double Description (DD) method [8], a closed convex polyhedron can be specified in two ways, using a constraint system or a generator system: the constraint system contains a finite set of linear non-strict inequality constraints; the generator system contains two finite sets of vectors, collectively called generators, which are rays and points of the polyhedron.

Representability of convex sets by analytical linear inequality systems* 1

Linear algebra and its applications, 2004

The solution sets of analytical linear inequality systems posed in the Euclidean space form a transition class between the polyhedral convex sets and the closed convex sets, which are representable by means of linear continuous systems. The constraint systems of many semiinfinite programming problems are analytical, and their feasible sets retain geometric properties of the polyhedral sets which are useful in the numerical treatment of such kind of optimization problems. The Euclidean closed n-dimensional balls admit analytical representation if and only if n < 3. This paper solves, in a negative way, the analytical representation problem for a wide class of n-dimensional convex sets, with n 3, which includes quasi-polyhedral sets and smooth convex bodies.

Representability of convex sets by analytical linear inequality systems

Linear Algebra and its Applications, 2004

The solution sets of analytical linear inequality systems posed in the Euclidean space form a transition class between the polyhedral convex sets and the closed convex sets, which are representable by means of linear continuous systems. The constraint systems of many semiinfinite programming problems are analytical, and their feasible sets retain geometric properties of the polyhedral sets which are useful in the numerical treatment of such kind of optimization problems. The Euclidean closed n-dimensional balls admit analytical representation if and only if n < 3. This paper solves, in a negative way, the analytical representation problem for a wide class of n-dimensional convex sets, with n 3, which includes quasi-polyhedral sets and smooth convex bodies.

Associated polyhedra and dual linear programs

Beitrage Zur Algebra Und Geometrie, 2010

The duality theorem of linear programming is set in a very general context, which is then mediated through the context of associated polyhedra; these latter are related by the representation theory of polyhedra. A feature of this approach is that it is made evident that two complementarity conditions are involved in the theorem.