Norm Typically used vector norms. General p-norm: xlp=(P+…+zal)/ Quadratic norm: lx=VxT Ax,where A is positive semi-definite. Advanced Optimization (Fall 2023) Lecture 1.Mathematical Background 16
Advanced Optimization (Fall 2023) Lecture 1. Mathematical Background 16 Norm • Typically used vector norms
Dual Norm Let‖.∥be a vector norm on Rd.The associated dual norm‖.‖is defined as lyl*=sup{yTx|I川x≤1}. Proposition 1.The dual of p-norm is the norm with=1. e.g.,the dual of (2-norm is still 2-norm,the dual of e-norm is lo-norm Proposition2.H6lder's inequality:(x,y〉≤lx·lyll*. Advanced Optimization (Fall 2023) Lecture 1.Mathematical Background 17
Advanced Optimization (Fall 2023) Lecture 1. Mathematical Background 17 Dual Norm
Norm Relationship Qualitative: Lemma1(Mathematical Equivalence of Norms)).Suppose that‖·a and‖，lb are norms on Rd,there exist positive"constants"a and B,for all xRd,such that axla≤lxl6≤Bllxlla. Notice:constants may depend on dimension! For example:for any x E Ra,the following inequalities hold: ·aIxl1≤xlx≤xl1 ·x‖≤xl2≤Vdxl Advanced Optimization (Fall 2023) Lecture 1.Mathematical Background 18
Advanced Optimization (Fall 2023) Lecture 1. Mathematical Background 18 Norm Relationship Qualitative: Notice: constants may depend on dimension!