CSE 203 B Convex Optimization Lecture 2 Convex
CSE 203 B Convex Optimization Lecture 2 Convex Set CK Cheng Dept. of Computer Science and Engineering University of California, San Diego 1
Convex Optimization Problem: 2
Convex Optimization Problem: A. Convex Function Definition: 3
Chapter 2 Convex Set 1. Set Convexity and Specification 1. Convexity 2. Implicit vs. Explicit Enumeration 2. Convex Set Terms and Definitions 3. Separating Hyperplanes 4. Dual Cones 4
1. Set Convexity and Specification: Convexity 5
1. Set Convexity and Specification: Convexity 6
1. Set Convexity and Specification: Set Specification via Implicit or Explicit Enumeration Implicit Expression Explicit Enumeration 7
1. Implicit vs Explicit Enumeration of Convex Set Implicit Expression Example: {�� |���� ≤�� } Remark: Simultaneous linear constraints imply AND, Intersection of the constraints 8
1. Implicit vs Explicit Enumeration of Convex Set • 9
1. Specification of Convex Set: Implicit Expression Example: 10
1. Implicit vs Explicit Enumeration of Convex Set Example: 11
1. Specification of Set: Explicit Expression Explicit Enumeration Example: 12
1. Specification of Set: Explicit Expression Implicit and Explicit Conversion 13
1. Implicit vs Explicit Enumeration of Convex Set Remark: Implicit Expression: Constraints of the problem formulation Explicit Enumeration: Formulation of the objective function The interchange may not be trivial. 14
1. Implicit vs Explicit Enumeration of Convex Set Explicit Enumeration 15
1. Implicit vs Explicit Enumeration of Convex Set • We then have 16
2. Convex Set Terms and Definitions 17
2. Sets and Definitions: Hyperplane and Half Spaces 18
2. Sets and Definitions: Hyperplanes 19
2. Sets and Definitions: Hyperplanes 20
2. Sets and Definitions 21
2. Sets and Definitions 22
2. Sets and Definitions 23
2. Sets and Definitions 24
3. Separating Hyperplane 25
3. Separating Hyperplane 26
3. Supporting Hyperplane 27
4. Dual Cones 28
4. Dual Cones 29
4. Dual Cones 30
4. Dual Cones 31
4. Dual Cones 32
- Slides: 32