Stability and Error Analysis for Optimization and Generalized Equations

Abstract

Stability and error analysis remain challenging for problems that lack regularity properties near solutions, are subject to large perturbations, and might be infinite dimensional. We consider nonconvex optimization and generalized equations defined on metric spaces and develop bounds on solution errors using the truncated Hausdorff distance applied to graphs and epigraphs of the underlying set-valued mappings and functions. In the process, we extend the calculus of such distances to cover compositions and other constructions that arise in nonconvex problems. The results are applied to constrained problems with feasible sets that might have empty interiors, solution of KKT systems, and optimality conditions for difference-of-convex functions and composite functions.

Set-Convergence and Its Application: A Tutorial

“…The theorem holds for nearly arbitrary functions as long as minimizers are attained with finite minimum values and ρ is large enough. The bounds are sharp as discussed in [26,28].…”

“…For any norm • on R n , a main choice for norm on R n+1 is max x −x , |α −ᾱ| for (x, α), (x,ᾱ) ∈ R n × R, (6.2) which typically results in the simplest formulae. In the context of minimization, we then obtain the following bounds; see [28,Proposition 2.2].…”

“…Computation of the truncated Hausdorff distance is supported by the following condition that originated with Kenmochi [16] and fully realized by Attouch and Wets [7]. For the present form, see [28,Proposition 4.1]. We denote lower level-sets of f : R n → R by {f ≤ δ} = {x ∈ R n | f (x) ≤ δ} for any δ ∈ R.…”

“…For other rules to compute the truncated Hausdorff distance, we refer to [7,9,25,26,28]. We next turn to solution errors for generalized equations.…”

“…The "output space" R m × R m × R n is assigned the norm max{ u 2 , v 2 , w 2 } for (u, v, w) ∈ R m × R m × R n . Then, by [28,Theorem 5.3], for ρ ∈ R + , dl ρ (gph S, gph T ) ≤ sup where • F is the Frobenius norm. In calculating distances between gph ∂ϕ and gph ∂ψ, we adopt the norm max{ z 2 , y 2 } for (z, y) ∈ R m × R m .…”