2.5 Convergence Analysis
2.5.2 Sub-optimality in wireless control system
Because the proposed Newton method indeed solves (2.15) to within a statistical approxima- tion ˆV, it is important to consider the effect of such an approximation on the original WCP in (WCPk). In this section we provide a sequence of results that characterize the accuracy of the solutions generated by the Newton update in (2.24) in the original primal control problem in (WCPk). Firstly, recall the constraints in (WCPk) reflect both a power budget limited by pmax and that the auxiliary variable yi should not exceed the expected packet
success function q(·). In solving the dual problem approximately, we may then also violate these constraints by a small margin. We can specifically characterize such a constraint violation, as well as address the suboptimality in terms of the primal objective. Both these results together can then be combined to demonstrate the stability of the switched system WCP introduced in Example 1. To do so, we first introduce an assumption regarding the feasibility and boundedness of the dual loss solutions L∗k and the optimal dual pointµ∗k. AS5. For all epochs k, the problem in (WCPk) under distribution Hk is strictly feasible.
There also exists constants K and Kˆ such that the optimal dual objective value L∗k is bounded as L∗k≤ K and optimal dual variable bounded as kµ∗kk ≤Kˆ.
From strict feasibility of the primal problem in (WCPk), we also obtain a finite upper bound on the value of the dual function. This can be used with the suboptimality result in Theorem 1 to bound the norm of the dual variables µk generated from the Newton update in (2.24). This is presented in the following corollary.
Corollary 1. The norm of the dual variablesµkgenerated by the update in (2.24)is bounded
as kµkk ≤p(2/α) + ˆK.
Proof: From strong convexity we have thatkµk−µ˜∗kk2 ≤(2/αVˆ)( ˜Rk(µk)−R˜∗
k). Using the reverse triangle inequality with 2.35 and Assumption 5, we obtain the intended result.
Observe that the boundedness of the solutions to the regularized dual function in Assumption 5 in effect states that, for all distributions Hk, the empirical, or sampled,
versions of the constrained problem in (WCPk) will be strictly feasible. From here, we can establish through duality a bound on each constraint violation that may occur from solving the dual problem to its statistical accuracy. This result is stated in the following proposition. Proposition 2. Consider µk to be a Vˆ-optimal minimizer of R˜k, i.e. R˜k(µk)−R˜∗k ≤Vˆ.
Further consider p(h,µk) and y(µk) to be the Lagrangian maximizers over dual parameter µk. If Assumptions 1 and 5 hold, then the norm of the constraint violations in (WCPk) can
each be upper bounded as
m X i=1 Ehi k(p i(hi k,µ))−pmax ≤ q 2∆( ˜VN+CVˆ), (2.36) ky(µk)−Ehk{q(hk,p(hk,µk))}k ≤ q 2∆( ˜VN+CVˆ), (2.37)
where C := 1 +ρ+βκand κ such that 1Tlog(µk)≤κ.
Proof: See Appendix.
In Proposition 2, we establish a bound that is proportional to ˆV on the violation of the constraints in (WCPk). There are two points to be stressed here. First, is that this constraint violation can indeed be made small by controlling the target accuracy ˆV. Additionally, we point out that the violation of the budget constraint can be controlled by adding a slack term to the maximum power as ˆpmax=pmax−2∆CVˆ. In this way, any such violation will
still be within the true intended budgetpmax.
We proceed by establishing suboptimality of the generated variables y(µk) in terms of control performance. Recall the final result in Theorem 1 that establishes at each epoch
k, the current dual function value ˜Rk(µk) will be within accuracy ˆV of the optimal value ˜
Rk( ˜µ∗k) (after satisfying the necessary conditions). To establish that the control systems induced by such dual parametersµk remain stable, we first connect the accuracy of the dual function value to the accuracy of associated primal variablesp(h,µk) andy(µk) with respect to their optimal values p∗k(h) := p(h,µ∗k) andyk∗ :=y( ˜µ∗k). This bound is established in the following theorem.
Theorem 2. Consider µk to be a Vˆ-optimal minimizer of R˜k, i.e. R˜k(µk)−R˜∗k ≤ Vˆ.
Further consider p(h,µk) and y(µk) to be the Lagrangian maximizers over dual parameter µk. Under Assumptions 1-5 the primal objective function sub-optimality J(y(µk))−J(y∗k)
can be upper bounded as
J(y(µk))−J(yk∗)≤(1 +C)∆ 1 α + 2 ˆV( p 2/α+ ˆK) . (2.38)
In Theorem 2, we derive a bound on the suboptimality of the primal objective function
J(y) that is proportional also to the statistical accuracy ˆV plus a constant. Recall that this function is, in general, a measure of the control performance of the system. Thus, solving the dual problem approximately indeed can be translated into approximate accuracy in terms of our original utility metric with respect to the control system. In many problems, the performanceJ(y) will also effectively establish a stability margin for control systems that have unstable regions of operation. To demonstrate the effect of using the proposed Newton’s method over a non-stationary wireless channel, we return to the switched dynamical system in Example 1.