CSE 203 B Convex Optimization Lecture 3 Convex

CSE 203 B Convex Optimization: Lecture 3: Convex Function CK Cheng Dept. of Computer Science and Engineering University of California, San Diego 1

Outlines 1. Definitions: Convexity, Examples & Views 2. Conditions of Optimality 1. First Order Condition 2. Second Order Condition 3. Operations that Preserve the Convexity 1. Pointwise Maximum 2. Partial Minimization 4. Conjugate Function 5. Log-Concave, Log-Convex Functions 2

Outlines 1. Definitions 1. Convex Function vs Convex Set 2. Examples 1. 2. 3. 4. 5. Norm Entropy Affine Determinant Maximum 3. Views of Functions and Related Hyperplanes 3

1. Definitions: Convex Function vs Convex Set 4

1. Convex Function Definitions: Examples 5

1. Convex Function Definitions: Examples 6

1. Convex Function Definitions: Examples 7

triangle inequality scalability 8

1. 3 Views of Functions and Related Hyperplanes 9

2. Conditions of Optimality: First Order Condition 10

2. Conditions: Second Order Condition 11

2. Conditions: Second Order Condition 12

2. Conditions: Examples 13

2. Conditions: Examples 14

3. Operations that preserve convexity 15

3. Operations that preserve convexity 16

3. Operations that preserve convexity: Dual norm 17

3. Operations that preserve convexity: max function 18

3. Operations that preserve convexity: minimization 19

3. Operations that preserve convexity 20

3. Operations that preserve convexity 21

3. Operations that preserve convexity 22

4. Conjugate Functions 23

4. Conjugate Functions 24

4. Conjugate Functions 25

4. Conjugate Functions 26

4. Conjugate Functions: Examples (single variable) 27

4. Conjugate Functions: Examples (single variable) 28

4. Conjugate Functions 29

4. Conjugate Functions 30

4. Conjugate Functions 31

4. Conjugate Functions 32

4. Conjugate Functions 33

5. Log-Concave, Log-Convex Functions 34

5. Log-Concave, Log-Convex Functions 35

5. Log-Concave, Log-Convex Functions 36

Outlines 1. Definitions: Convexity, Examples & Views 2. Conditions of Optimality 1. First Order Condition 2. Second Order Condition 3. Operations that Preserve the Convexity 1. Pointwise Maximum 2. Partial Minimization 4. Conjugate Function 5. Log-Concave, Log-Convex Functions 37