Cognitive radar management Alexander Charlish 1 and Folker Hoffmann 1
5.2 Effective QoS-based resources management
5.2.2.2 Optimality conditions
Conditions for the optimal solution of the resource management problem can be derived by introducing a change of variable, to give an equivalent problem to the resource allocation problem in (5.7). Alternatively, the KKT conditions can be applied directly to the problem in (5.13).
Change of variable
The control parameter selection problem can be related to the resource allocation problem in (5.7) by introducing a change of variable. Consider a one-to-one function fdefined on just a subset of all of the possible control parameter selections^^ ^:
f: ^^7!R (5.16)
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Resource − utility space plot
Resource
Utility
Figure 5.5 Visualization of control parameters in resource utility space. Each line illustrates the variation in one control parameter dimension while the other control parameter dimension is kept static
A subset of control parameters must be taken in order for the functionfto be one- to-one. A control parameter selection from the subset ^^ is denoted as
^
u¼f^u1;^u2;. . .;^uKg 2^^.
As the tasks are assumed to be independent, the functionfcan be broken down into a one-to-one function for each of the tasks:
fð^uÞ ¼XK
k¼1
fkð^ukÞ (5.17)
where fk : ^^k7!Rk is a mapping from the control parameters for task Tk into resource space.
Then, the problem of finding the optimal parameter selections^ut from the sub- set of control parameters can also be formulated as a constrained optimization problem that is equivalent to the problem formulation for the resource allocation problem given in (5.7):
maximize:
^
u uðfð^uÞÞ ¼XK
k¼1
wkukðqkðfkð^ukÞ;ekÞÞ (5.18)
subject to: gðfð^uÞÞ 0 (5.19)
where: gðfð^uÞÞ ¼ XK
k¼1
fkð^ukÞ
!
^r (5.20)
This constrained optimization problem is equivalent to the problem for resource allocation in (5.7); however,rhas been substituted withfð^uÞ. As these problems are equivalent, exactly the same optimality conditions apply to this problem.
Therefore, the optimal parameter selection from^umust be primal and dual feasible, there must be no resource left or no further benefit of further resource allocation and the gradient in resource utility space at the parameter selections for all tasks must be equal.
Although this optimization problem is equivalent to the resource allocation problem (Equation (5.7)), it is not equivalent to the resource management problem (Equation (5.13)) as it selects control parameters from the reduced subset of control parameter selections ^^ and not the complete set of possible control parameter selections^. However, reducing the problem to the set of parameter selections^^ can be a very useful step as long as the optimal parameter selections from ^are contained in ^^. A logical choice for the functionfis the mapping of the control parameters that lie on the Pareto frontier in Figure 5.5 into resource, as this subset ensures that utility is maximized for all resource. However, the subset^^ must be chosen such that the objective function remains convex. Consequently, the concave majorant is typically taken, which is discussed further in Section 5.2.3.
Application of KKT conditions
Instead of applying a change of variable, the KKT conditions can be directly applied to (5.13), if it is assumed that the objective function ^uðuÞ is a concave
differentiable function and the resource function ^gðuÞ is a convex differentiable function, then the following KKT conditions can then be derived:
r^uðuÞ þmr^gðuÞ ¼0 (5.21)
^
gðuÞ 0 (5.22)
m0 (5.23)
m^gðuÞ ¼0 (5.24)
wheremis a KKT multiplier.
As shown in [16], the stationarity condition can be rearranged as follows:
m¼@lwkukð^qkðuk;ekÞÞ
@l^gkðuk;ekÞ 8k2f1;2;. . .;Kg 8l2f1;2;. . .;Mkg
(5.25)
where@l denotes the partial derivative with respect toulk, which occurs as uk is itself a vector given byuk¼ ðu1k;u2k;. . .;uLkÞ.
This set of conditions implies that the stationarity condition is satisfied when the gradients of resource over utility in all dimensions and for all tasks are equal to a common value, which is the KKT multiplierm. The optimal solution is found when the stationarity condition is satisfied (Equation (5.21) and Equation (5.25)) and the solution is primal feasible (Equation (5.22)) and dual feasible (Equation (5.23)) and either all the resource has been allocated or there would be no utility increase from further resource allocation (Equation (5.24)).
Applying the KKT conditions directly yields the optimality conditions;
however, it must be assumed that the utility and resource functions are concave functions of the control parameters. Introducing a change of variable demonstrates that the utility and resource functions do not need to be concave functions of the control parameters, as long as a subset can be found that is concave in resource utility space.
5.2.3 Quality of service algorithms
If the resource, quality and utility functions are closed-form expressions, then the KKT conditions can be solved analytically. However, it is often the case that these models do not have a closed-form, and instead require numerical evaluation. In the previous subsection, it was assumed that the resource, quality and utility functions are defined on a continuous space. However, the control parameters may in fact be discrete, or it may be desirable to discretize the control parameters due to the need to perform the numerical evaluations. Figure 5.6 illustrates a quality function (Figure 5.6(a)) as well as a discretized resource utility space (Figure 5.6(b)) gen- erated from discretized control parameters. It can be seen that even though the quality function is a non-convex function of the control parameters, a concave majorant can still be generated in resource utility space. However, as the concave majorant does not cover the complete Pareto front, a suboptimal solution could
be found. Careful design of the performance model and adequate sampling of the discrete control parameters can ensure that the deviation from the optimal is small.
Once control parameters are selected by the resource management algorithm, these selected control parameters are used to schedule radar dwells for each task.
Therefore, it is assumed that a scheduler at the measurement level below has access to the control parameters at every resource management frame. Schedulers are discussed in detail in Volume 1, Chapter 3. The scheduler may not be able to perfectly resolve radar dwell conflicts and therefore the actual behaviour may deviate from the desired control parameters. However, as the quality of service management ensures that the resource allocated is matched to the resource avail- able, the scheduler is never overloaded and therefore the deviation should be small.
It is also assumed that the mission level above provides the utility function and importance weighting for each task, with respect to the current mission.
This section describes two algorithms for solving the resource management problem: the quality of service resource allocation method (Q-RAM) and the continuous double auction parameter selection (CDAPS) algorithm.