Regularity Conditions via Quasi-Relative Interior in Convex Programming

Abstract

We give some new regularity conditions for Fenchel duality in separated locally convex vector spaces, written in terms of the notion of quasi interior and quasi-relative interior, respectively. We provide also an example of a convex optimization problem for which the classical generalized interior-point conditions given so far in the literature cannot be applied, while the one given by us is applicable. By using a technique developed by Magnanti, we derive some duality results for the optimization problem with cone constraints and its Lagrange dual problem, and we show that a duality result recently given in the literature for this pair of problems has self-contradictory assumptions.

Constraint Qualifications for Extended Farkas's Lemmas and Lagrangian Dualities in Convex Infinite Programming

“…In particular, even in the special case when (1.4) is satisfied, our results provide improved versions of [17,Theorems 2 and 5] and that of [23,Theorem 4.1]. Moreover, applications to the problem (1.3) are given: we not only extend and improve some recent known results in [3,5,9,18,29,31] but also provide new results as detailed in Section 6.…”

“…Boţ et al [3] established the strong Lagrange duality between problem (P f (S)) and (D f (S)) under the following interiority condition: 0 ∈ qi[(g(C) + S) − (g(C) + S)], 0 ∈ qri(g(C) + S) and (0, 0) / ∈ qri co(ε v(P f (S)) ∪ {(0, 0)}), (6.20) where ε v(P f (S)) = {(f (x) + α − v(P f (S)), g(x) + y) : x ∈ C, α ≥ 0, y ∈ S} ⊆ R × Y.…”

“…Another important and classical example of (1.1) is the following so-called conic programming problem, which recently received much attention (cf. [2,3,5,6,9,18,28,29,30,31,32]), Minimize f (x), s. t.…”

“…For example, f, f t are continuous in [8] and lsc in [16,17,23], f, g are continuous in [9,18,28,29,30,32], f is lsc, g is continuous in [31] and g is S-epi-closed in [5,6]. Very recently, an optimality condition for (1.1) was established in [41] for the case when f t are not necessary lsc, and a Lagrangian duality result via the interiority technique for (1.3) was established in [3] without any continuity assumption on f and g. Indeed, in the mathematical programming, many of the problems naturally involve non-convex and non-continuous functions. For example, in the DC infinite programming (see for example [7,19,20,21,22] and references therein), the objective functions and constraint functions are, in general, assumed to be DC functions (such a function is, by definition, a difference of two convex functions and so can be neither convex nor lsc).…”

“…Recall (cf. [3]) that the quasi-relative interior and the quasi interior of Z are defined respectively by qri Z := {x ∈ Z : cl cone(Z − x) is linear} and qi Z := {x ∈ Z : cl cone(Z − x) = X}.…”