Variational convergence of bivariate functions: lopsided convergence

Optimality Functions and Lopsided Convergence

“…(The stronger notion of epi/hypoconvergence [19,21] appears less suitable as it is directed toward saddle points; see the discussion in [15].) In the process, we extend some of the results in [14,15] …”

“…Proof We follow nearly the same argument as in the proof of [14,Theorem 3.4]. Let x ∈ X be such that h(x) is finite.…”

“…Proof We follow the same arguments as in [14,Proposition 3.2], where X = R n is considered. From Definition 4.1(ii) there exists x ν → x, with x ν ∈ X ν , such that the functions {F ν (x ν , ·)} ν∈N and F(x, ·) satisfy Definition 3.1(i).…”

“…In an application to nonlinear programming, we establish the convergence of a primal interior point method in the absence of constraint qualifications and convexity assumptions. We show that lopsided convergence of bifunctions [13][14][15] is a useful tool for analyzing optimality functions and the associated stationary points. In particular, we prove that lopsided convergence of certain bifunctions, defining optimality functions of approximating problems, to a bifunction associated with an optimality function of the original prob-lem, guarantees the axiomatic requirements on optimality functions.…”

“…Using lopsided convergence, we provide results on existence of stationary points as well as characterizations of stationary points under perturbations and approximations. In the process, we extend the primary definitions and results on lopsided convergence in [14,15] from finite dimensions to metric spaces.…”