Timeline for Standard solution to semidefinite program

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May 18, 2017 at 19:03	comment	added	Noah Stein		If $b^Ta > 0$ then you can replace $b$ with some $\hat{b} = b - \epsilon a$ for small enough $\epsilon>0$ that $\hat{b}^Ta > 0$. Then use the same argument to construct a psd $\hat{Q} = \frac{\hat{b}\hat{b}^T}{\hat{b}^Ta}$ with $\hat{Q}a = \hat{b}$. Then let $Q = \hat{Q} + \epsilon I$, so $Qa = \hat{b} + \epsilon a = b$.
May 18, 2017 at 18:53	comment	added	Noah Stein		If you unwind the argument I gave, you get that when $b^Ta >0$ then the objective value of zero is achieved at $Q = \frac{bb^T}{b^Ta}$, which is symmetric and positive semidefinite, being a positive scalar multiple of an outer product of a vector with itself.
May 18, 2017 at 17:24	comment	added	user402940		Thanks for the insight. But in case of $b^Ta >0$, what is the expression for symmetric p.s.d matrix $Q$ ? Also, if I strictly want a symmetric positive definite matrix (may be, under newer assumptions like $b^Ta >0$), what should be the matrix $Q$ ?
May 18, 2017 at 17:22	vote	accept	user402940
May 18, 2017 at 15:42	comment	added	Noah Stein		With a bit more effort you can see that $R$ is the interior of $S$ plus the origin, so the infimum is only achieved when $b^T a >0$ or $b=0$.
May 18, 2017 at 15:34	history	answered	Noah Stein	CC BY-SA 3.0