In the rest of this chapter, we introduce and characterize the problem of minimizing the performance measure by optimal design of the probability vector Π.
Problem 3.6.1 (Optimal Random Switching). Given a set of networks R ⊂ S and cost
vector C := [C1,C2, . . . ,Cm]T 0, find a probability mass function π(Wi) = πi for all
10 20 30 40 50 60 0 1 2 3 Time (seconds)
Mean Machine Time Vs. Network Size
Lower Bound Performance Measure
10 20 30 40 50 60
n - Size of the Network 0
0.2 0.4 0.6
Time (seconds)
Figure 3.7: The mean time of the computations versus number of the nodes n (the lower figure magnifies the details of the upper figure)
performance measure and cost of the operation; i.e. for some augmenting coefficient γ ≥0, solve
minimize
Π ρ(W) +γΠ
TC (3.25)
subject to : Π0 and ΠT1m = 1.
Next, we reshape (3.25) as a tractable convex program; i.e. a Semi-Definite Program (SDP) [72], provided that the output matrix is MN.
Theorem 3.6.2. Optimization Problem (3.6.1) with C=MN is equivalent to
minimize Π, t t (3.26) subject to :Π0, ΠT1m = 1 and IN2 −GW vec(MN) vec(MN)T t−γΠTC 0.
This convex reformulation implies that we can directly look at the convexity of the performance measure in variable Π. Our next result magnifies this point.
0 10 20 30 40 50 60 70 80 90 100
i - The Indices Corresponding to Graphs
0 1 2
πi
Representation of an Optimal Vector of probabilities Π
Figure 3.8: Representation of an optimal vector of probabilities Π for a network with m= 100 underlying graphs.
0 10 20 30 40 50 60 70 80 90 100
i - The Indices Corresponding to Graphs
0 2 4
πi
Representation of an Optimal Vector of probabilities Π
Figure 3.9: Representation of a sparse optimal Π with reweighted `1 penalization, where
more entries are on the horizontal axis.
function of the probability vector Π = [π1, π2, . . . , πm]T.
Although we can directly verify the convexity of this problem, the mere fact that the optimization problem is equivalent to a SDP implies that the objective function is a convex function of the variable Π.
3.6.1 Scenarios for the Design Problem
Assume a hypothetical scenario of coordination where at each sampling time only a subset of communication links exist between the agents. One may think of different advantages for this: (i) the power-consuming continuous communications are no longer needed, (ii) the amount of information to be processed (i.e. computational burden of the operations on an agent) may decrease (iii) the control input magnitudes may shrinks as we consider partial synchronization at each step. The next example considers this scenario.
Example 3.6.4 (Mixing Disconnected Networks). Consider 100 different graphs overN = 24 nodes. Each graph has the same number of randomly chosen links|Ei|= 4. An LTI consen- sus network corresponding to each graph has unbounded measure because each individual graph is disconnected. Using Theorem 3.6.2, we implement (3.26) with γ = 0 in CVX [39] to find an optimal probability vector Π. In Fig. 3.8, we show an optimal Π (in percent) found in 113 seconds.
10-1 100 101 γ - Penalization Coefficient 0.1 1 10 30 %Perf. Loss
Performance Loss Vs. Penalization Coefficient
10-1 100 101 γ - Penalization Coefficient 20 40 60 80 %Density Level
Density Level Vs. Penalization Coefficient
Figure 3.10: The Performance Loss and Density Level tradeoff illustrated by increasing the value of γ (see Example 3.6.6).
Example 3.6.5 (Sparse Selection of Disconnected Networks). The number of nonzeros in the optimal Π in Example 3.6.4 is high. As a remedy, we use Reweighted `1-Penalization [73].
To do so, we have a vector of weights at iteration j, wj ∈ Rm is initially w1 = 1
m. In our problem, we set C = wj. Then, at each iteration we compute the optimal solution Πj = [π1j, πj2, . . . , πjm] of (3.26) with γ = 5 . For the next iteration, we update the weights according to
wj+1(i) = 1
e+πji, for all i∈ {1,2, . . . , m},
withe= 10−6. We display the elements of the optimal value (in percent) in Fig. 3.9 after 5 iterations. Due to this penalization, the density level (i.e, the percent of nonzero elements
of π) dropped from 90% to 26%, and the performance measure grew from around 852 to
902, a loss of around 6%.
An influential parameter in Example 3.6.5 is γ. On one side of the spectrum, γ = 0 implies the lack of requirement on the sparsity of Π. However, as γ increases, the sparsity is more favored, and we expect observing more performance loss. Our final example reflects this tradeoff.
Forγ ∈[0.1,15], we examine the density level of the optimal vector Π and the performance measure at the same time and the performance loss and density level versus γ appear in Fig. 3.10.
3.6.2 Additional Convex Attributes of the Performance Measure
Theorem 3.6.3 asserts thatρ(W) is a convex function of the probability vector Π. Here, we note that it is also a convex function of the stacked matrix of all Wi ∈R. Thus, first we introduce a stacked matrix
Wst :=
W1,W2, . . . ,Wm T
, for all Wi ∈R.
Proposition 3.6.7. Ifλmax(GW)<1, the performance measure (4.18) is a convex function of stacked matrix Wst.
Next, we point out that the performance measure inherits this convexity as a function of the weights of all graphs of the switching network. For graph i with edge set Ei, let Ei := vec(wjk) (for all{j, k} ∈ Ei) be a vector representing the weights of the corresponding graph. Stacking these vectors for allWi∈ Wm, we build a vector of weights
Est :=
ET1, E2T, . . . , EmTT .
Theorem 3.6.8. If for all Wi ∈R, Wi =IN−iLi ∈ S for some graph LaplacianLi and a proper i >0, then the performance measure (4.18) is a convex function ofEst.
The notion of convexity with respect to the weights of the graph is useful for design or redesign purposes, since numerous convex optimization techniques and their operational guarantees could be potentially considered.