Abstract
In Chap. 2, a general model of an utility maximization problem is defined, cf. Definition 2.2.1. The data of the problem is specified in terms of a tuple \((u,h,\mathit{r},\mathcal{X}1)\). In this chapter, different methods to solve a utility maximization problem are presented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Recall from Definition 2.2.1 that a parameter setup x1 is an element of the parameter space \(\mathcal{D}1\). Intuitively, the problem dimension is then the dimension of \(\mathcal{D}1\). However, it may not be possible to assign a dimension to an arbitrary metric space \(\mathcal{D}1\). In the following, it is assumed that the set \(\mathcal{D}1\) is a subset (but not necessarily a subspace) of a finite-dimensional vector space \(\mathcal{V}\), which then immediately defines the problem dimension as the dimension of \(\mathcal{V}\).
- 2.
For an example of a sigmoidal utility function, see Sect. 5.2.
- 3.
In fact, [24] starts with the assumption that
$$\begin{array}{rcl} \mathcal{R}\cap \mathcal{H}\subset \mathbb{R}{++}^{K}.& & \end{array}$$(3.58)It is shown in [24] that if Condition (3.58) holds, there exists a compact set \(\tilde{\mathcal{H}}\subset \mathbb{R}{++}^{K}\) such that
$$\begin{array}{rcl} \mathcal{R}\cap \mathcal{H} = \mathcal{R}\cap \tilde{\mathcal{H}}.& & \\ \end{array}$$In other words, if Condition (3.58) holds, \(\mathcal{H}\) can be replaced by a set that satisfies Condition (3.59). Based on this result, in [24], the polyblock algorithm is derived under the assumption that \(\mathcal{H}\) satisfies Condition (3.59).
- 4.
Applying the notion of “proportional fairness” to nonconvex systems is not straightforward, as indicated in Sect. 5.1.2. See [46] for a detailed discussion.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Brehmer, J. (2012). Solution Methods. 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_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-17438-4_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17437-7
Online ISBN: 978-3-642-17438-4
eBook Packages: EngineeringEngineering (R0)