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A question on vector norm error analysis

Exists solution for this optimization problem?norm over differentiable functions computable from derivatives onlyConcave approximation and errorCalculating the global optimumMinimum value of $a_{k+1}left(2^{a_k}-1right)$, for $a_0,dots,a_ninmathbb{N}^ast$Taking a magnitude of a vector consisting of other vectorsAre three optimization equations are equal?Minimize $ {L}_{p} $ Norm Regularized with a Linear Term (Conjugate Function of the Norm Function)Is it possible to find an upper bound on the moment generating function of $sqrt{|X|}$, where $Xsim mathcal{N}(0,1)$?Show That Minimizer of Rayleigh Quotient Is the Smallest Eigenvalue - $ min_{x : left| x right| = 1} {x}^{*} {A}^{*} A x = {lambda}_{1}^{2} $

1

$begingroup$

${x^*} in {R^n}$ is the optimal solution of an optimization problem and leads to the minimum objective function ${left| {A{x^*} - b} right|^2}$, where $A in {R^{m times n}}$ and $b in {R^m}$. But I can only obtain an approximately optimal solution denoted as ${x^*} + Delta x$ which leads to the objective function ${left| {Aleft( {{x^*} + Delta x} right) - b} right|^2}$. Now I can already bound the difference of the two objective functions by ${Delta lambda }$:

${left| {Aleft( {{x^*} + Delta x} right) - b} right|^2} - {left| {A{x^*} - b} right|^2} le theta left| {Delta lambda } right|$, where $theta $ is a constant.

My question is can I bound $left| {Delta x} right|$ or $frac{{{{left| {Delta x} right|}^2}}}{{{{left| {{x^*}} right|}^2}}}$ by some function of ${Delta lambda }$, $A$ and $b$, i.e.,
${left| {Delta x} right|^2} le fleft( {Delta lambda ,A,b} right)$ or $frac{{{{left| {Delta x} right|}^2}}}{{{{left| {{x^*}} right|}^2}}} le fleft( {Delta lambda ,A,b} right)$.

Or anyone can recommend some relevant materials to me?

Thanks!

linear-algebra functions inequality

share|cite|improve this question

edited Mar 19 at 2:04

Guangyang_ZJU

asked Mar 18 at 12:50

Guangyang_ZJUGuangyang_ZJU

63

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Or can anyone recommend some relevant materials to me? Thanks~
  $endgroup$
  – Guangyang_ZJU
  Mar 19 at 2:01
  

  
  
  
  

add a comment | 

1

$begingroup$

${x^*} in {R^n}$ is the optimal solution of an optimization problem and leads to the minimum objective function ${left| {A{x^*} - b} right|^2}$, where $A in {R^{m times n}}$ and $b in {R^m}$. But I can only obtain an approximately optimal solution denoted as ${x^*} + Delta x$ which leads to the objective function ${left| {Aleft( {{x^*} + Delta x} right) - b} right|^2}$. Now I can already bound the difference of the two objective functions by ${Delta lambda }$:

${left| {Aleft( {{x^*} + Delta x} right) - b} right|^2} - {left| {A{x^*} - b} right|^2} le theta left| {Delta lambda } right|$, where $theta $ is a constant.

My question is can I bound $left| {Delta x} right|$ or $frac{{{{left| {Delta x} right|}^2}}}{{{{left| {{x^*}} right|}^2}}}$ by some function of ${Delta lambda }$, $A$ and $b$, i.e.,
${left| {Delta x} right|^2} le fleft( {Delta lambda ,A,b} right)$ or $frac{{{{left| {Delta x} right|}^2}}}{{{{left| {{x^*}} right|}^2}}} le fleft( {Delta lambda ,A,b} right)$.

Or anyone can recommend some relevant materials to me?

Thanks!

linear-algebra functions inequality

share|cite|improve this question

edited Mar 19 at 2:04

Guangyang_ZJU

asked Mar 18 at 12:50

Guangyang_ZJUGuangyang_ZJU

63

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Or can anyone recommend some relevant materials to me? Thanks~
  $endgroup$
  – Guangyang_ZJU
  Mar 19 at 2:01
  

  
  
  
  

add a comment | 

$begingroup$

${x^*} in {R^n}$ is the optimal solution of an optimization problem and leads to the minimum objective function ${left| {A{x^*} - b} right|^2}$, where $A in {R^{m times n}}$ and $b in {R^m}$. But I can only obtain an approximately optimal solution denoted as ${x^*} + Delta x$ which leads to the objective function ${left| {Aleft( {{x^*} + Delta x} right) - b} right|^2}$. Now I can already bound the difference of the two objective functions by ${Delta lambda }$:

${left| {Aleft( {{x^*} + Delta x} right) - b} right|^2} - {left| {A{x^*} - b} right|^2} le theta left| {Delta lambda } right|$, where $theta $ is a constant.

My question is can I bound $left| {Delta x} right|$ or $frac{{{{left| {Delta x} right|}^2}}}{{{{left| {{x^*}} right|}^2}}}$ by some function of ${Delta lambda }$, $A$ and $b$, i.e.,
${left| {Delta x} right|^2} le fleft( {Delta lambda ,A,b} right)$ or $frac{{{{left| {Delta x} right|}^2}}}{{{{left| {{x^*}} right|}^2}}} le fleft( {Delta lambda ,A,b} right)$.

Or anyone can recommend some relevant materials to me?

Thanks!

linear-algebra functions inequality

share|cite|improve this question

edited Mar 19 at 2:04

Guangyang_ZJU

asked Mar 18 at 12:50

Guangyang_ZJUGuangyang_ZJU

63

$endgroup$

share|cite|improve this question

edited Mar 19 at 2:04

Guangyang_ZJU

asked Mar 18 at 12:50

Guangyang_ZJUGuangyang_ZJU

63

share|cite|improve this question

edited Mar 19 at 2:04

Guangyang_ZJU

asked Mar 18 at 12:50

Guangyang_ZJUGuangyang_ZJU

63

asked Mar 18 at 12:50

Guangyang_ZJUGuangyang_ZJU

63

asked Mar 18 at 12:50

Guangyang_ZJUGuangyang_ZJU

63