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Is there some off-the-shelf C/C++ library to solve the following problem:

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    This is not a linear regression problem. This is a ridge regression problem. If you change your question to 'How to solve the ridge regression problem?' you will likely get many good answers on cross validated. – user189035 Mar 4 '15 at 8:17
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I'm not sure about the particular formula you've referred to, but you have an extremely wide variety of libraries to choose from for regression, optimization and other numerical solutions: http://en.wikipedia.org/wiki/List_of_numerical_libraries (see C, C++ and multi-language sections). In addition, take a look at IMHO very interesting Accord.NET framework, which offers this functionality and much more. Hope this helps.

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