Abstract
The functionality provided by a wireless communication system can usually be partitioned into submodules, the so-called layers [21]. With each layer, a set of variable parameters can be associated. For example, common physical layer parameters are power allocation, allocation of time slots, beamforming vectors, choice of modulation and coding schemes, etc. Possible parameters at upper layers are scheduling decision, choice of a route, and, at the application layer, source coding rate. Parameters are variable in the sense that for each parameter there exists a set of possible values to choose from. Moreover, each parameters impacts overall system performance in a certain way. Generally speaking, then, a parameter optimization corresponds to optimizing system performance with respect to all variable parameters over the set of all possible parameter setups.
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- 1.
\(\mathcal{L}\) is a finite subset of \(\mathbb{R}2\). Accordingly, any function \(f : \mathcal{L}\rightarrow \mathbb{R}\) is continuous. This result follows by considering the subspace topology in \(\mathcal{L}\)—intersecting the open sets in \(\mathbb{R}2\) with \(\mathcal{L}\) yields the open sets in \(\mathcal{L}\). Obviously, any subset of \(\mathcal{L}\) is an open set (the discrete topology). As a result, the preimage of any open set \(I \subseteq \mathbb{R}\) under f is open, showing that f is continuous, cf. Proposition A.2.1.
- 2.
The function u is defined using the set of extended real numbers. A definition of the set of extended real numbers is provided in Appendix A.6.
- 3.
Note that concavity of u and h implies that their domains are convex sets, cf. Definition A.5.2. Moreover, concavity of h implies that the set
$$\begin{array}{rcl} \left \{\mathit{s} \in {\mathcal{D}}_{h} : h(\mathit{s}) \geq 0\right \}& & \\ \end{array}$$is convex.
- 4.
A prerequisite for defining concavity of r and convexity of \(\mathcal{X}1\) is that the domain of r is a subset of a real or complex vector space, cf. Definitions A.5.1 and A.5.2. Moreover, concavity of r implies that the domain of r is convex.
- 5.
For a definition of comprehensiveness, see Appendix A.4.
- 6.
A definition of the term topological manifold is provided in Appendix A.8.
- 7.
A homeomorphism is a continuous function that has a continuous inverse function, cf. Definition A.2.2 in Appendix A.2.
- 8.
As a simple example, any subset of ℝ + + K has no single-link points.
- 9.
The statement follows by letting x1′ = (e 1, x1, …, x1). Then r ′(x1′) = r(x1). Thus, if r is nonconcave, there exist \(\mathit{y},\mathit{z} \in \mathcal{X}1\) such that r ′ violates the concavity condition on the line segment \(\left \{\alpha ({\mathit{e}}_{1},\mathit{y}) + (1 - \alpha )({\mathit{e}}_{1},\mathit{z}),\alpha \in \left [0, 1\right ]\right \} \subset {\mathcal{X}}^{{\prime}}\).
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© 2012 Springer-Verlag Berlin Heidelberg
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Brehmer, J. (2012). General Problem Setup. In: Utility Maximization in Nonconvex Wireless Systems. Foundations in Signal Processing, Communications and Networking, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17438-4_2
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DOI: https://doi.org/10.1007/978-3-642-17438-4_2
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