Part 1.Convex Set and Convex Function 。Definition ·Ball and Ellipsoid Convex Hull and Projection Convex/Concave Function Zeroth,First and Second-order Condition Advanced Optimization(Fall 2023) Lecture 2.Convex Optimization Basics 6
Advanced Optimization (Fall 2023) Lecture 2. Convex Optimization Basics 6 Part 1. Convex Set and Convex Function • Definition • Ball and Ellipsoid • Convex Hull and Projection • Convex/Concave Function • Zeroth, First and Second-order Condition
Convex Set Definition 1(Convex Set).A set Y is convex if for any x,y e,all the points on the line segment connecting x and y also belong to t,i.e., a∈0,1，ax+(1-a)y∈X. convex sets? X X Advanced Optimization(Fall 2023) Lecture 2.Convex Optimization Basics 7
Advanced Optimization (Fall 2023) Lecture 2. Convex Optimization Basics 7 Convex Set convex sets?
Examples .A line segment is convex. ·Aray,which has the form{xo+fv|θ≥O},where v≠0，is convex. Any subspace is convex. Advanced Optimization(Fall 2023) Lecture 2.Convex Optimization Basics 8
Advanced Optimization (Fall 2023) Lecture 2. Convex Optimization Basics 8 Examples
Convex Set Definition 2(Ball).A(Euclidean)ball (or just ball)in Rd has the form B(xc;r)={xc+ruull2 <1}. Definition 3(Ellipsoids).A ellipsoid in Rd has the form E(xe,A)={xe+Au|Iu‖2≤1}， where A is assumed to be symmetric and positive definite. Au Advanced Optimization(Fall 2023) Lecture 2.Convex Optimization Basics 9
Advanced Optimization (Fall 2023) Lecture 2. Convex Optimization Basics 9 Convex Set
Convex Set Definition 4(Convex Hull).The convex hull of a set t,denoted conv t,is the set of all convex combinations of points in A: conv={0x1+…+0xkx:∈X,0:≥0，i∈[k],01+…+0=1. Examples: Advanced Optimization(Fall 2023) Lecture 2.Convex Optimization Basics 10
Advanced Optimization (Fall 2023) Lecture 2. Convex Optimization Basics 10 Convex Set Examples: