Minimizing Convex Functions with Bounded Perturbations

When a convex function f : D → R is disturbed by some nonlinear bounded perturbation p : D → R, the arising function f̃ = f + p is no more convex and its local minimizers are no more global minimizers. In order to get some similar properties for f̃ , we use a convexity modulus of f named h1 and its generalized inverse function h−1 1 , and show that f̃ is outer γ-convex for any γ ≥ γ∗ := h−1 1 ( 2 supx∈D |p(x)| ) . As consequence, each γ∗-minimizer x∗ ∈ D defined by f̃(x∗) = infx∈B̄(x∗,γ∗)∩D f̃(x) is a global minimizer, i.e., f̃(x∗) = infx∈D f̃(x), and each local γ∗-infimizer x∗ defined by lim infx∈D, x→x∗ f̃(x) = infx∈B(x∗,γ∗+2)∩D f̃(x) (for some 2 > 0) is a global infimizer, i.e., lim infx∈D, x→x∗ f̃(x) = infx∈D f̃(x). Moreover, the diameter of the set of global infimizers (included global minimizers) of f̃ is not greater than γ∗, and the distance between any global minimizer or infimizer of f̃ and any global minimizer or infimizer of f cannot exceed γ∗.