bs for all s = 1, . . . , n;
Definition 1.2.1 A function f : X → R is said to be continuous if at point x0 ∈ X ,
x→x0
lim f (x) = f (x0 ),
or equivalently, for any ϵ > 0, there is a δ > 0 such that for any x ∈ X satisfying |x − x0 | < δ , we have |f (x) − f (x0 )| < ϵ A function f : X → R is said to be continuous on X if f is continuous at every point x ∈ X. The idea of continuity is pretty straightforward: There is no disconnected point if we draw a function as a curve. A function is continuous if “small” changes in x produces “small” changes in f (x). 12
Geometrically the convex set means every point on the line segment joining any two points in the set is also in the set. Theorem 1.2.2 (Separating Hyperplane Theorem) Suppose that A, B ⊂ Rm are convex and A ∩ B = ∅. Then, there is a vector p ∈ Rm with p ̸= 0, and a value c ∈ R such that px c py ∀x ∈ A & y ∈ B.
The so-called upper semi-continuity and lower semi-continuity continuities are weaker than continuity. Even weak conditions on continuity are transfer continuity which characterize many optimization problems and can be found in Tian (1992, 1993, 1994) and Tian and Zhou (1995), and Zhou and Tian (1992). Definition 1.2.2 A function f : X → R is said to be upper semi-continuous if at point x0 ∈ X , we have lim sup f (x)
n ∑ ∂f (x) i=1
Rn → R is homogeneous of
∂xi
xi .
1.2.2
Separating Hyperplane Theorem
A set X ⊂ Rn is said to be compact if it is bounded and closed. A set X is said to be convex if for any two points x, x′ ∈ X , the point tx + (1 − t)x′ ∈ X for all 0 t 1.
Furthermore, suppose that B ⊂ Rm is convex and closed, A ⊂ Rm is convex and compact, and A ∩ B = ∅. Then, there is a vector p ∈ Rm with p ̸= 0, and a value c ∈ R such that px < c < py ∀x ∈ A & y ∈ B.
tf (x) + (1 − t)f (x′ )
The function f is said to be strictly concave on X if f (tx + (1 − t)x′ ) > tf (x) + (1 − t)f (x′ ) 14
for all x ̸= x′ ∈ X an 0 < t < 1. A function f : X → R is said to be (strictly) convex on X if −f is (strictly) concave on X . Remark 1.2.1 A linear function is both concave and convex. The sum of two concave (convex) functions is a concave (convex) function. Remark 1.2.2 When a function f defined on a convex set X has continuous second partial derivatives, it is concave (convex) if and only if the Hessian matrix D2 f (x) is negative (positive) semi-definite on X . It is it is strictly concave (strictly convex) if the Hessian matrix D2 f (x) is negative (positive) definite on X . Remark 1.2.3 The strict concavity of f (x) can be checked by verifying if the leading principal minors of the Hessian must alternate in sign, i.e., f11 f12 f21 f22 f11 f12 f13 f21 f22 f23 f31 f32 f33 and so on, where fij = conditions. In economic theory quasi-concave functions are used frequently, especially for the representation of utility functions. Quasi-concave is somewhat weaker than concavity. Definition 1.2.6 Let X be a convex set. A function f : X → R is said to be quasiconcave on X if the set {x ∈ X : f (x) c}
1.2.3
Concave and Convex Functions
Concave, convex, and quasi-concave functions arise frequently in microeconomics and have strong economic meanings. They also have a special role in optimization problems. Definition 1.2.5 Let X be a convex set. A function f : X → R is said to be concave on X if for any x, x′ ∈ X and any t with 0 f (tx + (1 − t)x′ ) t 1, we have
x→x0
f (x0 ),
or equivalently, for any ϵ > 0, there is a δ > 0 such that for any x ∈ X satisfying |x − x0 | < δ , we have f (x) < f (x0 ) + ϵ. Although all the three definitions on the upper semi-continuity at x0 are equivalent, the second one is easier to be versified. A function f : X → R is said to be upper semi-continuous on X if f is upper semicontinuous at every point x ∈ X . Definition 1.2.3 A function f : X → R is said to be lower semi-continuous on X if −f is upper semi-continuous. It is clear that a function f : X → R is continuous on X if and only if it is both upper and lower semi-continuous, or equivalently, for all x ∈ X , the upper contour set U (x) ≡ {x′ ∈ X : f (x′ ) are closed subsets of X . Let f be a function on Rk with continuous partial derivatives. We define the gradient of f to be the vector ] ∂f (x) ∂f (x) ∂f (x) Df (x) = , ,..., . ∂x1 ∂x2 ∂xk [ f (x)} and the lower contour set L(x) ≡ {x′ ∈ X : f (x′ ) f (x)}