READING 1 HAN XIAO 1 . Review of Convex Analysis
نویسنده
چکیده
Definition 1.1. Let C be a subset of R. We say C is convex if αx+ (1− α)y ∈ C, ∀x, y ∈ C, ∀α ∈ [0, 1]. Definition 1.2. Let C be a convex subset of R. A function f : C 7→ R is called convex if f(αx+ (1− α)y) ≤ αf(x) + (1− α)f(y), ∀x, y ∈ C, ∀α ∈ [0, 1]. The function f is called concave if −f is convex. The function f is called strictly convex if the above inequality is strict for all x, y ∈ C with x 6= y, and all α ∈ (0, 1). Definition 1.3. Given a convex function f : R 7→ R, we say that a vector d ∈ R is a subgradient of f at a point x ∈ R if f(z) ≥ f(x) + (z − x) d, ∀ z ∈ R. The set of all subgradients of a convex function f at x ∈ R is called the subdifferential of f at x, and is denoted by ∂f(x).
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