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Definition: The Basic Conjugate Direction Algorithm for QP
We want to minimize the quadratic function of $n $ variables
$$f(x) = \frac{ 1 }{ 2 } \transpose{ \b{x}} \b{Q} \b{x} - \transpose{ \b{x}} \b{b}$$
where $\b{Q} = \transpose{ \b{Q}} > 0, x \in \R^n $. Note that because $\b{Q} > 0 $, the function $f $ has a global minimizer that can be found by solving $\b{Qx} = \b{b} $.
Given a starting point $\b{x}^{(0)} $, and $\b{Q} $-conjugate directions $\b{d}^{(0)}, \b{d}^{(1)}, …, \b{d}^{(n-1)} $; for $k \geq 0 $,
$$\b{g}^{(k)} = \nabla f(\b{x}^{(k)}) = \b{Q}\b{x}^{(k)} - \b{b} $$ $$\alpha_k = - \frac{ \transpose{ \b{g}^{(k)}} \b{d}^{(k)}}{ \transpose{ \b{d}^{(k)}} \b{Q} \b{d}^{(k)}} $$ $$\b{x}^{(k+1)} = \b{x}^{(k)} + \alpha_k \b{d}^{(k)} $$
Theorem
: Convergence Rate in QP
In quadratic programming, for any starting point $\b{x}^{(0)} $, the basic conjugate direction algorithm converges to the unique minimizer $\b{x}^* $ in $n $ steps; that is $\b{x}^{(n)} = \b{x}^* $.
Lemma
In the conjugate direction algorithm,
$$\transpose{ \b{g}^{(k+1)}} \b{d}^{(i)} = 0 $$
for all $k$, $0 \leq k \leq n- 1$, and $0 \leq i \leq k$.