lecture. - CASTLE Lab - Princeton University

Motivation
Introduction and Notation
Perturbation analysis in optimization deals with the sensitivity of optimal values
and optimal solution sets to perturbations of the the objective function and
feasible set. Perturbation analysis provides a theoretical foundation that helps
analyzing algorithms for solving approximately stochastic optimization problems.
Notations
Let us write minCf ∈ R = R {−∞} {+∞} for the value of the minimum of
a function f : X → R over the closed set C ⊂ X. The set X is a metric
space,usually R n endowed with the Euclidean norm. We assume that C is in the
domain of f. In minimization problem, domf = {x ∈ X : f(x) < ∞}.
The optimal solution set is the set argmin Cf = {x ∈ C : f(x) = minCf}. For
ɛ > 0, the ɛ-optimal solution set is the set
argmin C,ɛ f = {x ∈ C : f(x) ≤ minCf +ɛ}. If argmin C f is a singleton {s}, we
write s = argmin Cf. For an illustration, see Figure 1 one the next slide.
Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 3 / 26