07-02-Convex-Optimization.pdf

# 07-02-Convex-Optimization.pdf - Convex Optimization Convex...

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Convex Optimization

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Convex Optimization: Definition I Convex Optimization Problem: min x f ( x ) s.t. x ∈ F
Convex Optimization: Definition I Convex Optimization Problem: min x f ( x ) s.t. x ∈ F I A special class of optimization problem

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Convex Optimization: Definition I Convex Optimization Problem: min x f ( x ) s.t. x ∈ F I A special class of optimization problem I An optimization problem whose optimization objective f is a convex function and feasible region F is a convex set .
Convex Optimization: Definition I Convex Optimization Problem: min x f ( x ) s.t. x ∈ F I A special class of optimization problem I An optimization problem whose optimization objective f is a convex function and feasible region F is a convex set .

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Convex Optimization: Definition I Convex Optimization Problem: min x f ( x ) s.t. x ∈ F I A special class of optimization problem I An optimization problem whose optimization objective f is a convex function and feasible region F is a convex set .
Convex combination I A point between two points I Given x , y R n , a convex combination of them is any point of the form z = θ x + (1 - θ ) y where θ [0 , 1] . I When θ [0 , 1] , z is called a strict convex combination of x , y .

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Convex Sets I Conceptually : Any convex combination of two points in the set is also in the set
Convex Sets I Conceptually : Any convex combination of two points in the set is also in the set I Mathematically : A set F is convex if x, y ∈ F , θ [0 , 1] , z = θx + (1 - θ ) y ∈ F

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Quiz: Convex Set I Which of the following sets are convex? 1. F = R n 2. F = 3. F = { x 0 } , x 0 R n 4. F = F 1 T F 2 , where F 1 and F 2 are convex sets. 5. F = F 1 S F 2 , where F 1 and F 2 are convex sets.
Convex Function I Value in the middle point is lower than average value I Let F be a convex set. A function f : F → R is convex in F if x, y ∈ F , θ [0 , 1] , f ( θ x + (1 - θ ) y ) θf ( x ) + (1 - θ ) f ( y ) I If F = R n , we simply say f is convex.

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How to determine if a functions is convex? I Prove by definition
How to determine if a functions is convex? I Prove by definition I Use properties

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How to determine if a functions is convex? I Prove by definition I Use properties I Sum of convex functions is convex If f ( x ) = X i w i f i ( x ) , w i 0 , f i ( x ) convex, then f ( x ) is convex.
How to determine if a functions is convex? I Prove by definition I Use properties I Sum of convex functions is convex If f ( x ) = X i w i f i ( x ) , w i 0 , f i ( x ) convex, then f ( x ) is convex. I Convexity is preserved under a linear transformation If f ( x ) = g ( A x + b ) , if g ( x ) is convex, then f ( x ) is convex.

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How to determine if a functions is convex? I Prove by definition I Use properties I Sum of convex functions is convex If f ( x ) = X i w i f i ( x ) , w i 0 , f i ( x ) convex, then f ( x ) is convex. I Convexity is preserved under a linear transformation If f ( x ) = g ( A x + b ) , if g ( x ) is convex, then f ( x ) is convex.
• Winter '15
• Derivative, Convex function, Convex Optimization, Objective f

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