 contains the optimal Lagrange variables. Then, we reduce the volume of the ellipsoid using gradient descent method as the following. (7.12a) Mk(i+1)=Mk(i)+kLk-Rt) (7.12b) (7.12c) (7.12d) where , AMk, Ank, and Am are the respective step size, i is the iteration index. Notice that all the Lagrange variables must be non-negative. If a negative value is obtained, the Lagrange variable will be set to 0 instead. 7.3.3 Update Auxiliary Variables Lastly, the auxiliary variables NK and Bk are updated in the following way. Notice that in Lemma 7.1, the optimal solution Pk, tx, and fk should also satisfy the following system conditions: (7.13) (7.14) Similarly, according to [92], we define functions for notational brevity. Specifically, let = {1,2, ,KK. The optimal solution for Nk 7.4 Numerical Results and Bk can be obtained by solving T(Ak, Bk) = [T1, T2, T2K] = 0. We can apply iterative method to update the auxiliary variables. Specifically, (7.15) Bk(i+1)=(1-(i))Bk(i)+ (7.16) where O(i) is the largest 0 that satisfies - 201) q is the Jacobian matrix of T, l E {1,2, (0,1), and ZE (0,1). Note that when O(i) = 1, it becomes the standard Newton method. To summarize, we list the detailed algorithm in Algorithm 7.1. Algorithm 7.1 Computation Efficiency Maximization Algorithm 1: Initialization: the algorithm accuracy indicator t1 and t2, set i = 0, Ak(i) and k(i) 2: while|T(AkBk)ll>tdo Initialization:ak(j),Mk(j),nkj), and m(j), and let j = 0 while lak(j+1)-ak(j)l>t2do Calculate p* and f* based on Proposition 7.1. Solve for problem P5, obtain the timing variable tk. Update Lagrange variables based on gradient descent method in (7.12). Let tjjj1 end while Let i = i + 1, update auxiliary variables Ak(i+1) and Bk(i+1) from (7.15) and (7.16). 11: end while 12: Output the optimal computation efficiency. Notice that in the inner loop, the stop criterion can also be the convergence of other Lagrange multipliers or the condition that their combined value is less than a threshold. 7.3.4 Complexity Analysis Since the algorithm involves t