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I am reading this topic of boyd book from convex optimization, but the following division i-e trade-off least square and l2 norm are difficult to understand for me. If kindly someone can explain equation, that how it has been calculated. Your guidance will be appreciated. Regards,

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Glorfindel
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Sohail Khan
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Solution for the least square regularization

My apologies, please ignore my question. I have solved my problem. I am just posting the answer in case someone needed. In case the question is too irrelevant here, it can be deleted. Regards

Community
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Sohail Khan
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  • While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review – rhermans Jun 04 '18 at 09:29
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The solution to this "multi-objective" task is known to be $A^\dagger \,b$ with the Moore-Penrose pseudoinverse $A^\dagger$. If $A$ is injective, we have $$A^\dagger b = (A^* A)^{-1} A^* b.$$

In case $A$ is surjective, we have $$A^\dagger b = A^* (A\, A^*)^{-1} b$$

In the general case, one may apply Tikhonov regularization:

$$A^\dagger \,b = \lim_{\lambda \searrow 0} (A^*A + \lambda)^{-1} A^\intercal b = \lim_{\lambda \searrow 0} A^* (A\, A^* + \lambda)^{-1} b.$$

See also here.

Henrik Schumacher
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