Convex Optimizationselections from Chapter 6

Audrey Hesse

Approximation & Fitting

• Norm Approximation

– Penalty functions

• Least-norm problems

• Regularized Approximation

• Signal Reconstruction

Norm Approximation

• A solution to norm approximation is sometimes an approximate solution

Ax ≈ b

• Residual: vector r = Ax – b

• Residuals: the individual components of the residual associated with x

Minimize: |}||,...,{|max|||| 1 mrrbAx

Minimize the sum of the squares of the residuals 22

2

2

1

2

2 ...|||| mrrrbAx

||...|||||||| 211 mrrrbAx Minimize:

Implicit constraint +1

)(

:subject to

||||minimize

2

2

2

bAxxf

bAx

x

T

4000Rx

Optimal trade-off curve

Knee at