Wireless Network Pricing Chapter 4: Social Optimal Pricing Jianwei - PowerPoint PPT Presentation

Wireless Network Pricing Chapter 4: Social Optimal Pricing Jianwei Huang & Lin Gao Network Communications and Economics Lab (NCEL) Information Engineering Department The Chinese University of Hong Kong Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 1 / 67

The Book E-Book freely downloadable from NCEL website: http: //ncel.ie.cuhk.edu.hk/content/wireless-network-pricing Physical book available for purchase from Morgan & Claypool ( http://goo.gl/JFGlai ) and Amazon ( http://goo.gl/JQKaEq ) Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 2 / 67

Chapter 4: Social Optimal Pricing Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 3 / 67

Focus of This Chapter Key Focus: This chapter discusses the issue of social optimal pricing, where one service provider chooses prices to maximize the social welfare. Theoretic Approach: Convex Optimization Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 4 / 67

Convex Optimization Mainly follow the book “Convex Optimization” by Boyd and Vandenberghe. Definition (Convex Optimization) Convex optimization studies the problem of minimizing convex functions (or equivalently, maximizing concave functions) over convex sets. Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 5 / 67

Section 4.1 Theory: Dual-based Optimization Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 6 / 67

Prelims Notations ◮ R n : the set of all real n -vectors ⋆ Each vector in R n is called a point of R n . ⋆ R 2 is the set of all 2-D vectors ⋆ R 1 or R denotes the set of all real 1-vectors or all real numbers. ◮ R m × n : the set of all m × n real matrices ◮ f : R n → R m : a function that maps some real n -vectors (called the domain of function f ) into real m -vectors ⋆ D ( f ): the domain of function f Key Concepts ◮ Convex Set ◮ Convex Function ◮ Convex Optimization Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 7 / 67

Convex Set Definition (Convex Set) A nonempty set X ⊆ R n is convex, if for any x 1 , x 2 ∈ X and any θ ∈ R with 0 ≤ θ ≤ 1, we have: θ x 1 + (1 − θ ) x 2 ∈ X Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 8 / 67

Convex Set Geometrically, a set is convex if every point in the set can be reached by every other point, along an inner straight path between them. Examples of convex and non-convex sets: x 1 x 2 Figure: (i) Convex, (ii) Non-convex, and (iii) Non-convex. Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 9 / 67

Convex Combination Definition (Convex Combination) A convex combination of points x 1 , ..., x k can be expressed as y = θ 1 x 1 + ... + θ k x k , with θ 1 + ... + θ k = 1 and θ i ≥ 0 , i = 1 , ..., k . Lemma (4.2) A nonempty set X is convex, if and only if the convex combination of any points in X also lies in X . Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 10 / 67

Convex Hull The convex hull of a set X , denoted H ( X ), is the smallest convex set that contains X . Definition (Convex Hull) The convex hull H ( X ) of a set X consists of the convex combinations of all points in X , i.e., { θ 1 x 1 + ... + θ k x k | θ 1 + ... + θ k = 1 , θ i ≥ 0 , x i ∈ X , i = 1 , ..., k } . Properties ◮ H ( X ) is always convex; ◮ X ⊆ H ( X ); ◮ X = H ( X ) if X is a convex set; ◮ H ( X ) ⊆ Y where Y is any convex set that contains X . Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 11 / 67

Convex Hull Examples of convex hull ◮ Source sets: ◮ Convex hulls: Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 12 / 67

Operations Preserving Convexity of Sets Intersection: Suppose X 1 , ..., X k are convex sets. Then, the intersection of X 1 , ..., X k X � X 1 ∩ ... ∩ X k is also a convex set. Affine Mapping: Suppose X is a convex set in R n , A ∈ R m × n , and b ∈ R m . Then, the affine mapping of X Y � { A x + b | x ∈ X} is also a convex set. Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 13 / 67

Convex (and Concave) Function Definition (Convex Function) A function f : R n → R is convex, if D ( f ) is a convex set, and 1 for all x , y ∈ D ( f ) and θ ∈ R with 0 ≤ θ ≤ 1, we have: 2 f ( θ x + (1 − θ ) y ) ≤ θ f ( x ) + (1 − θ ) f ( y ) Definition (Concave Function) A function f ( · ) is concave if and only if − f ( · ) is convex. A function f ( · ) can be neither convex nor concave, e.g., f ( x ) = x 3 over D ( f ) = R . Question: What about f ( x ) = x 3 over D ( f ) = [0 , ∞ ), f ( x ) = x 3 over D ( f ) = ( −∞ , 0], and f ( x ) = 2 x over D ( f ) = R ? Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 14 / 67

Convex Function Assume that D ( f ) is convex. Geometrically, a function f ( · ) is convex if the chord from any point ( x , f ( x )) to ( y , f ( y )) lies above the graph of f ( · ). Illustration of Convex Function f ( · ): f ( · ) chord � � y , f ( y ) � � x , f ( x ) x y 0 Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 15 / 67

Strictly Convex (and Concave) Function Definition (Strictly Convex Function) A function f : R n → R is strictly convex, if D ( f ) is a convex set, and 1 for all x , y ∈ D ( f ) and θ ∈ R with 0 < θ < 1, we have: 2 f ( θ x + (1 − θ ) y ) < θ f ( x ) + (1 − θ ) f ( y ) Definition (Strictly Concave Function) A function f ( · ) is strictly concave if and only if − f ( · ) is strictly convex. Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 16 / 67

Generalized Definition of Convex Function Definition (Convex Function) A function f ( · ) is convex, if and only if D ( f ) is convex, and 1 For any x 1 , ..., x k ∈ D ( f ), 2 f ( θ 1 x 1 + ... + θ k x k ) ≤ θ 1 f ( x 1 ) + ... + θ k f ( x k ) , when θ 1 + ... + θ k = 1 and θ i ≥ 0 , i = 1 , ..., k . Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 17 / 67

Examples of Convex Functions Examples of convex functions over D ( f ) = (0 , ∞ ) ◮ 2 x , 3 x , e x , etc. ◮ x 2 , x 4 , x 6 , etc. ◮ − log 2 ( x ) , − ln ( x ), etc. Question: What about ◮ x 2 over D ( f ) = R ? ◮ x 2 over D ( f ) = ( −∞ , 0) ∪ [1 , ∞ )? ◮ − log 2 ( x ) over D ( f ) = R ? Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 18 / 67

First-Order Condition Consider a scalar-valued function f ( x ) : R n → R Definition (First-Order Derivative (Gradient)) The first-order derivative of a scalar-valued function f ( · ) at a point x ∈ D ( f ), denoted by ∇ f ( x ), is an n -dimensional vector with the i -th component given by ∇ f ( x ) i = ∂ f ( x ) , i = 1 , ..., n , ∂ x i x i : the i -th coordinate of the vector x ; ∂ f ( x ) /∂ x i : the partial derivative of f ( x ) with respect to x i . Lemma (First-Order Condition) A differentiable function f ( · ) is convex, if and only if D ( f ) is convex and f ( y ) ≥ f ( x ) + ∇ f ( x ) T ( y − x ) , ∀ x , y ∈ D ( f ) . Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 19 / 67

First-Order Condition Geometrically, the first-order condition means that the line passing through any point ( x , f ( x )) along the gradient direction ∇ f ( x ) lies under the graph of f ( · ). Illustration of First-order Condition: f ( · ) � � y , f ( y ) l ( y ) � � x , f ( x ) x y 0 Example: f ( x ) = ( x − 3) 2 , we have f (8) ≥ f (5) + f ′ (5)(8 − 5) Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 20 / 67

Second-Order Condition Definition (Second-Order Derivative (Hessian Matrix)) The second-order derivative of a scalar-valued function f ( · ) at a point x ∈ D ( f ), denoted by ∇ 2 f ( x ), is an n × n matrix, given by ∇ 2 f ( x ) ij = ∂ 2 f ( x ) , i = 1 , ..., n , j = 1 , ..., n . ∂ x i ∂ x j ∂ 2 f ( x ) ∂ x i ∂ x j : the second partial derivative of f ( x ) with respect to x i and x j . Lemma (Second-Order Condition) A twice differentiable function f ( · ) is convex, if and only if D ( f ) is convex and its Hessian matrix is positive semidefinite, i.e., ∇ 2 f ( x ) � 0 , ∀ x ∈ D ( f ) . Question: what about the special case of f ( x ) : R → R ? Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 4 September 13, 2016 21 / 67