Convex Optimization Chapter 6

Questions and Answers

What is the primary focus of the notes provided?

Convex Optimization

Which of the following is a type of approximation mentioned?

Norm approximation

Least-squares approximation

Regularized approximation

All of the above (correct)

Penalty function approximation involves minimizing the sum of a convex penalty function and the residuals.

True

What does the variable 'x⋆' represent in norm approximation?

The optimal design that best approximates the desired result b Signup and view all the answers

What equation is satisfied by the solution of the least-squares approximation?

AT Ax = AT b Signup and view all the answers

Which penalty function leads to linear growth for large values of 'u'?

Huber Signup and view all the answers

The term '_____ approximation' involves minimizing the sum of absolute residuals.

sum of absolute residuals Signup and view all the answers

What type of equations must be solved for least-squares approximation?

Normal equations Signup and view all the answers

Study Notes

Approximation and Fitting

Norm approximation aims to minimize the difference between a linear transformation of a variable and a target vector, represented as ( \min kAx - bk ).
Solution interpretation for ( x^* = \arg\min_x kAx - bk ):
- Geometric: ( Ax^* ) is the point in the range of ( A ) closest to ( b ).
- Estimation: In a linear measurement model ( y = Ax + v ), ( y ) represents observed measurements with ( v ) as measurement error; ( x^* ) provides the best estimate of ( x ).
- Optimal Design: Design variables ( x ) yield outputs ( Ax ); ( x^* ) represents the best design to approximate the desired output ( b ).

Examples of Norm Approximation

Least-Squares Approximation ( (k · k_2) ):
- Satisfies normal equations ( A^T A x = A^T b ).
- If ( A ) has full rank ( n ), solution is ( x^* = (A^T A)^{-1} A^T b ).
Chebyshev Approximation ( (k · k_\infty) ):
- Solved through linear programming to minimize ( t ) subject to ( -t \leq Ax - b \leq t ).
Sum of Absolute Residuals Approximation ( (k · k_1) ):
- Also solvable via linear programming, minimizing ( 1^T y ) with constraints ( -y \leq Ax - b \leq y ).

Penalty Function Approximation

Formulated as ( \min \phi(r_1) + · · · + \phi(r_m) ) with ( r = Ax - b ), where ( \phi ) is a convex penalty function.

Types of Penalty Functions

Quadratic Penalty: ( \phi(u) = u^2 )
Deadzone-Linear Penalty: ( \phi(u) = \max{0, |u| - a} ) creates a linear growth after a certain threshold.
Log-Barrier Penalty: ( \phi(u) = -a^2 \log(1 - (u/a)^2) ) is applicable within limits and infinite otherwise.

Characteristics of Penalty Functions

The shape of the penalty function significantly influences the distribution of residuals encountered in approximations.

Huber Penalty Function

Defined with parameter ( M ):
- ( \phi_{hub}(u) = \begin{cases} u^2 & |u| \leq M \ M(2|u| - M) & |u| > M \end{cases} )
Offers linear growth for large ( |u| ), making the approximation robust to outliers compared to simple quadratic penalties.

Description

This quiz covers key concepts from Chapter 6 of 'Convex Optimization' by Stephen Boyd and Lieven Vandenberghe, focusing on approximation and fitting techniques. Topics include norm approximation, least-norm problems, and regularized approximation. Test your understanding of these essential optimization strategies.