Generic primal-dual solvability in continuous linear semi-infinite programming

Abstract

In this paper we consider the space of all the linear semi-infinite programming (LSIP) problems with a given infinite compact Hausdorff index set, a given number of variables, and continuous coefficients, endowed with the topology of the uniform convergence. These problems are classified as inconsistent, solvable with bounded optimal set, bounded (i.e., finite valued) but either unsolvable or having an unbounded optimal set, and unbounded (i.e., with infinite optimal value), giving rise to the socalled refined primal partition of the space of problems. The mentioned LSIP problems can be also classified with a similar criterion applied to the corresponding Haar's dual problems, which provides the refined dual partition of the space of problems. We characterize the interior of the elements of the refined primal and dual partitions as well as the interior of the intersections of the elements of both partitions (the so-called refined primal-dual partition). These characterizations allow to prove that most (primal or dual) bounded problems have simultaneously primal and dual non-empty bounded optimal set. Consequently, most bounded continuous LSIP problems are primal and dual solvable.

Genericity Results in Linear Conic Programming—A Tour d’Horizon

“…There are many papers dealing with generic properties (in the sense of density and stability) of semi-infinite problems in the form (SIP P ), (SIP D ) (see (2.3), (2.4)). We refer to [23] and [15,16,17,18,27]. In [14,Chapter 5] the interested reader finds an overview of stability and genericity results for linear semi-infinite problems.…”

Solvability And Primal-dual Partitions Of The Space Of Continuous Linear Semi-infinite Optimization Problems

“…The continuity property of π = (a, b, c) ensures nice theoretical properties (e.g., in the duality context) and has computational implications (for instance, continuity guarantees the convergence of LSIP discretization algorithms). In particular, Goberna, Lopez, Todorov, Ochoa and Vera de Serio, among others, have investigated conditions under which the primal-dual pair in LSIP satisfies some of the above mentioned properties (see [7,10,11,15]).…”

“…In [10], Goberna and Todorov divided the set of parameters with bounded primal (dual) problem Π P B (Π D B ) into sets of parameters that have solvable primal (dual), problem with bounded optimal set Π P S (Π D S ) and a set of parameters that have unsolvable primal (dual), problem or unbounded optimal set Π P N (Π D N ). This generates what we call a first general primal partition…”

“…In this paper we present a more natural primal-dual partition in continuous LSIP than the one given in [10]. The new partition is generated by dividing the set of bounded primal (dual) problems Π P B (Π D B ) in two: the set of parameters that have solvable primal (dual) problem Π P s (Π D s ), and the set of parameters that have unsolvable primal (dual) problem Π P n (Π D n ).…”

“…This paper is organized as follows. In Section 2, we introduce the necessary notations, recall the basic results on LSIP which are frequently used throughout this article, and summarize the conditions presented in [10] and [11] that characterize the states of the primal-dual partition and the first general primal-dual partition. Section 2.1 provides new conditions which are either necessary of sufficient guaranteeing that a given parameter belongs to a certain element of the second general primal-dual partition.…”