Expanding subspace theorem

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Let LaTeX: S_k be the subspace spanned by vectors LaTeX: d^k, which are Q-conjugate. Let LaTeX: x^0 be any initial point in LaTeX: \mathbb{R}^n, and let LaTeX: x^k be generated by these conjugate directions with optimal line search:

LaTeX: \displaystyle x^{k+1} = x^k + s_k d^k \;\; \mbox{ and } \;\;
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<pre>s_k = \frac{-(Qx^k +c)}{(d^k)^T Q d^k}.

Then, LaTeX: x^k minimizes LaTeX: x^{T} Q x + c^{T} x on the affine set, LaTeX: x^0 + S_k.


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