Abstract
Having established the duality relationship between the MIMO BC and the MIMO MAC in Chaps. 3 and 4, we can handle optimizations originally arising in the BC in its dual MIMO MAC where they have more favorable properties like being convex, for example, as in the weighted sum rate maximization problem. Due to the limited resource power, we have to deal with constrained optimizations. In particular, an upper bound on the dissipated sum power is imposed for the transceiver design in the following two chapters. An attractive iterative scheme that is targeted at solving constrained optimization problems is the gradient-projection algorithm which can be regarded as an extension of the steepest ascent method derived by Cauchy to the case of a constrained optimization.
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- 1.
The unconstrained infimum of \(f\) is assumed not to be attained in \(\mathcal C \) and every unconstrained gradient step is assumed to leave \(\mathcal C \) to let this statement hold.
- 2.
For a convenient notation, it is necessary to define the (Wirtinger) derivative with respect to a block-diagonal matrix as a block-diagonal matrix itself, whose main diagonal blocks correspond to the (Wirtinger) derivative of the function with respect to the main diagonal blocks of the argument.
- 3.
Clearly, the point that has to be projected has to lie outside \(\mathcal C \). Otherwise, the projection is super fluous.
- 4.
The supremum of \(f\) is assumed to lie outside \(\mathcal C \).
- 5.
Clearly, the under-estimated slope must remain positive.
- 6.
Choosing \(m^{(n)}\in \mathbb Z \) works fine if the exact slope (squared Frobenius norm of the tangent cone projection) as in (5.28) and (5.31) is used. However, if the under-estimated slope from (5.30) and (5.32) shall be used, this estimation should not depend on \(m\). As a consequence, \(\beta ^m\) should be replaced by \(\beta ^0=1\) in the right hand side of (5.30) and (5.32). Otherwise, a smallest integer \(m\) might not exist since the slope estimate would go to zero for \(m\rightarrow -\infty \).
- 7.
The lower bound \(1<s\) ensures that \(z^{(1)}<-0.5\) in the first step and that the sign of \(z^{(n)}\) is alternating in \(n\). Due to the symmetry, \(1<s\) leads to \(|z^{(n)}|>0.5\) for all \(n\). The upper bound enforces an increase of the utility. Due to the symmetry and concavity of \(f\), an increase of the utility corresponds to the necessity that \(|z^{(n+1)}|<|z^{(n)}|\) which leads to the upper bound for \(s\).
- 8.
According to Footnote 7, the sign of \(z^{(n)}\) is alternating and \(|z^{(n)}|>0.5 \ \forall n\). Hence, the limit in (5.34) can be obtained by solving \(-z=z+s\cdot \frac{\partial f(z)}{\partial z}\) for \(z\).
- 9.
An exception to this is when the update direction is orthogonal to the precoder.
- 10.
The block-diagonal structure constraint for \(\varvec{E}\) is automatically fulfilled which can be seen from the solution (5.47).
- 11.
The factor \(\frac{1}{2}\) in front of the distance \(\Vert \varvec{E}\Vert _\mathrm{F }^2\) in (5.50) does not change the minimizer \(\check{\varvec{E}}\). Its only purpose is to get rid of the factor \(\frac{1}{2}\) in front of the Lagrangian multiplier \(\nu \) when solving for \(\check{\varvec{E}}\) after setting the Wirtinger derivative with respect to \(\varvec{E}^\mathrm{T }\) to zero. This is due to the fact that \(\frac{\partial _\mathrm{w }}{\partial _\mathrm{w }\varvec{E}^\mathrm{T }}\Vert \varvec{E}\Vert _\mathrm{F }^2=2\varvec{E}\) for \(\varvec{E}=\varvec{E}^\mathrm{H }\) with a pre-factor of \(2\), see (A28) in Appendix A12.
- 12.
To illustrate this circumstance, assume that \(\varvec{Q}=\left[\begin{array}{cc} 1&0\\ 0&0\end{array}\right]\) and \(\varvec{V}=\left[\begin{array}{cc} v_1&v_2\\ v_2&v_3\end{array}\right]\in \mathbb R ^{2\times 2}\). To let \(\varvec{V}\) be a feasible direction, \(\varvec{Q}+\alpha \varvec{V}\succcurlyeq \varvec{0}\) must hold for sufficiently small \(\alpha >0\). For a real-valued, symmetric \(2\times 2\) matrix, positive semi-definiteness means that both the trace and the determinant are nonnegative. This implies first, that \(1+\alpha (v_1+v_3)\ge 0\), which is always fulfilled for sufficiently small \(\alpha \), and second, that \((1+\alpha v_1)\alpha v_3-\alpha ^2v_2^2=\alpha v_3 + \alpha ^2(v_1v_3-v_2^2)\ge 0\). The first condition does not entail any limitations on \(v_1\) and \(v_3\). For \(v_3>0\), the second condition is always satisfied for any \(v_1\) and \(v_2\). However, to let \(\varvec{V}\) be a feasible direction, \(v_3=0\) implies \(v_2=0\). The (open) set of feasible directions is thus given by \(\mathcal F _{\mathcal C }(\varvec{Q})=\{\varvec{V}|v_3>0\}\cup \{\varvec{V}|v_2=v_3=0\}\). If we neglect the quadratic term in the condition \(\alpha v_3 + \alpha ^2(v_1v_3-v_2^2)\ge 0\) and thus replace the second condition by \(\alpha v_3\ge 0\), we end up with \(v_3\ge 0\), which defines the tangent cone to \(\mathcal T _{\mathcal C }(\varvec{Q})=\{\varvec{V}|v_3\ge 0\}={\mathrm{cl }}(\mathcal F _{\mathcal C }(\varvec{Q}))\) which clearly coincides with the closure of the set of feasible directions. This example explains the fact that a first-order analysis of the eigenvalues is sufficient to determine the closure of the set of feasible directions.
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Hunger, R. (2013). Matrix-Based Gradient-Projection Algorithm. In: Analysis and Transceiver Design for the MIMO Broadcast Channel. Foundations in Signal Processing, Communications and Networking, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31692-0_5
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DOI: https://doi.org/10.1007/978-3-642-31692-0_5
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