Joint Power Allocation and Reflecting-Element Activation for Energy Efficiency Maximization in IRS-Aided Communications Under CSI Uncertainty

This article has been accepted for publication in IEEE Communications Letters, December 2025. ©2025 IEEE. 1 Joint Power Allocation and Reflecting-Element Activation for Energy Efficiency Maximization in IRS-Aided Communications Under CSI Uncertainty Christos N. Efrem and Ioannis Krikidis, Fellow, IEEE Abstract—We study the joint power allocation and reflectingelement (RE) activation to maximize the energy efficiency (EE) in communication systems assisted by an intelligent reflecting surface (IRS), taking into account imperfections in channel state information (CSI). The robust optimization problem is mixedinteger, i.e., the optimization variables are continuous (transmit power) and discrete (binary states of REs). In order to solve this challenging problem we develop two algorithms. The first one is an alternating optimization (AO) method that attains a suboptimal solution with low complexity, based on the Lambert Wfunction and a dynamic programming (DP) algorithm. The second one is a branch-and-bound (B&B) method that uses AO as its subroutine and is formally guaranteed to achieve a globally optimal solution. Both algorithms do not require any external optimization solver for their implementation. Furthermore, numerical results show that the proposed algorithms outperform the baseline schemes, AO achieves near-optimal performance in most cases, and B&B has low computational complexity on average. Index Terms—Mixed-integer optimization, dynamic programming, alternating optimization, branch-and-bound method. I. INTRODUCTION Intelligent reflecting surfaces (IRSs) can dynamically reconfigure the phase of the incident electromagnetic waves by using adjustable electronic circuits [1]. The optimization of EE in IRS-assisted wireless networks has been an important area of research [2]. Furthermore, there has been increased interest in IRS beamforming to maximize the signal-to-noise ratio (SNR) [3] and the sum rate [4]. Moreover, the joint optimization of IRS location and number of REs has been examined in [5] from the outage-probability perspective. The optimal number of REs that maximizes the spectral efficiency and EE, considering channel estimation and feedback, has been studied in [6]. In addition, the authors in [7] have investigated a lowcost strategy for only tuning the on/off states of REs, while keeping their phase shifts fixed. A comparison between IRS and decode-and-forward relaying has been presented in [8]. This letter extends our previous work [9] on the robust optimization of EE under CSI uncertainty. In particular, the focus now is on the joint optimization of the REs’ on/off states and the transmit power. The considered problem becomes more complicated due to the mixed nature of optimization variables: the transmit power is continuous, while the states of REs are discrete. Firstly, we design an AO method based on the Lambert Wfunction and the DP algorithm given in [9]. Also, we develop a B&B method that is based on the AO algorithm. Interestingly, the convergence of B&B to an optimal solution is guaranteed provided that the AO initialization point is appropriately chosen. In this way, we can evaluate the AO performance by comparing with the global optimum of B&B. This work was supported by the Cyprus Research and Innovation Foundation under Grant DUAL USE/0922/0031 (RISE), and by the European Union’s Horizon Europe programme through the European Research Council (ERC) under Grant Agreement No. 101112697 (WAVE). The authors are with the Department of Electrical and Computer Engineering, University of Cyprus, 1678 Nicosia, Cyprus (e-mail: {efrem.christos, krikidis}@ucy.ac.cy). The rest of this letter is structured as follows. Section II presents the system model and formulates the robust optimization problem. Next, Sections III and IV develop and analyze the AO and B&B algorithms, respectively. Finally, Section V gives numerical results, while Section VI concludes the letter. The mathematical notation is the same as [9, Section I-D]. We write x:=y, or y=:x, whenever xis by definition equal to y. In addition, for every integer k,Wk(z)denotes the kth complex branch of the Lambert Wfunction. II. SYSTEM MODEL AND PROBLEM FORMULATION We consider a single-antenna transmitter (Tx) that communicates with a single-antenna receiver (Rx) via a passive IRS with NREs, denoted by N:={1, . . . , N}. Let h0∈Cbe the Tx-Rx direct channel, and hn∈Cbe the cascaded channel corresponding to the nth RE whose phase shift is denoted by ϕn∈[0,2π)[9, Section II-A].1For convenience, we define h= [h0, . . . , hN]⊤∈CN+1 and express each channel in polar coordinates as hn=αnejθn, where αn=|hn|≥0 and θn= Arg(hn)∈[0,2π), for all n∈ N0:={0, . . . , N}. Regarding the CSI, we model the actual channel has the sum of two terms: the estimated channel b h= [b h0,...,b hN]⊤∈ CN+1 (after quantization), and the unknown estimation error e h= [e h0,...,e hN]⊤∈CN+1, i.e., h=b h+e h. The elements of b hand e hcan also be expressed in polar coordinates: b hn=bαnej b θnand e hn=eαnej e θn. Furthermore, we adopt a deterministic CSI-error model: ∥e h∥2≤ξ, where ξ∈[0,bαmin]is the CSI-uncertainty radius and bαmin := minn∈N0{bαn}; see [9, Section II-B] for more details.2Because the estimation error e his unknown, the IRS phase shifts are selected using only the estimated channel b h so that the ideal SNR (with h=b h) is maximized, i.e., ϕn= (b θ0−b θn) mod 2π, for all n∈ N [9, Eq. (6)]. In addition, every RE is either on/activated (operating in reflection mode) or off/deactivated (operating in absorption mode) [4], [7]. For this reason, we use a binary vector x= [x1, . . . , xN]⊤∈ {0,1}N, where xn= 1 if and only if the nth RE is on. Given the CSI uncertainty, the worst-case SNR is given by [9, Theorem 3] γw(p, x;ξ) = p σ2(f(x)−g(x;ξ))2,(1) where p≥0is the transmit power, σ2>0is the noise power, f(x) = bα0+Pn∈N xnbαn, and g(x;ξ)=ξp1 + Pn∈N xn. Note that f(x)≥g(x;ξ), for all x∈ {0,1}N. 1The channel coefficients are fixed in a given time frame (flat fading). Also, the number of bits used for controlling the IRS phase shifts is assumed to be sufficiently large, so the phase shifts are (approximately) continuous. 2In practice, given the channel-estimation and quantization procedures, we can determine the parameters ξand qmin (the minimum quantization level for channel magnitudes, which is independent of N). If we ensure that ξ≤qmin, then we have ξ≤bαmin since qmin ≤bαmin. This condition may not apply in cases with a very large number of REs due to increased ξ. Nevertheless, a moderate number of REs is usually preferable, since more REs (despite the SNR improvement) result in higher energy consumption and channelestimation overhead. This article has been accepted for publication in IEEE Communications Letters, December 2025. ©2025 IEEE. 2 Afterwards, the worst-case energy efficiency is defined by EEw(p, x;ξ) = log2(1+γw(p, x;ξ)) Ptot(p, x),(2) where Ptot(p, x)is the total power consumption. In particular, Ptot(p, x) = η−1p+ (Pon −Poff)Pn∈N xn+Pfix, with Pfix =Pstatic +NPoff, where η∈(0,1] is the power amplifier’s efficiency, and Pstatic >0accounts for the dissipated power in the remaining signal-processing blocks at the transmitter and receiver. In addition, Pon and Poff (with Pon ≥Poff >0) represent the power consumption of each activated and deactivated RE, respectively [9]. Note that Pfix is defined differently from [9, Eq. (30)]: Pold fix =η−1p+Pstatic. Now, we formulate a robust optimization problem as follows EE∗ w:=max p,x EEw(p, x;ξ)(3a) s.t. γw(p, x;ξ)≥γmin,(3b) 0≤p≤pmax,x∈ {0,1}N,(3c) where γmin >0is the minimum required SNR, and pmax >0 is the maximum transmit power. Specifically, we want to maximize the worst-case EE by jointly adjusting the transmit power and tuning the on/off states of REs, while satisfying a minimum-SNR requirement and a maximum-power constraint. Problem (3) is a challenging mixed-integer optimization problem, i.e., with continuous and discrete variables.3 For convenience in the design of B&B algorithm (see Section IV), we define the following problem EE∗∗ w(pℓ, pu):=max p,x EEw(p, x;ξ)(4a) s.t. γw(p, x;ξ)≥γmin,(4b) pℓ≤p≤pu,x∈ {0,1}N,(4c) which is a generalization of problem (3), obtained by setting pℓ= 0 and pu=pmax. In general 0≤pℓ≤pu≤pmax, so every feasible solution to problem (4) is also feasible for problem (3). The following proposition gives a necessary and sufficient condition for feasibility. Proposition 1. Problem (4) is feasible if and only if γw(pu,1N;ξ)≥γmin. Proof: It is sufficient to show that ∂γw(p, x;ξ)/∂p ≥0 and ∂γw(p, x;ξ)/∂xn≥0,∀n∈ N . From (1) we deduce that ∂γw(p, x;ξ)/∂p = (f(x)−g(x;ξ))2/σ2≥0. In addition, the monotonicity of γw(p, x;ξ)with respect to each xnfollows from [9, Proposition 5]. III. ALTERNATING OPTIMIZATION ALGORITHM In this section, we will design an AO algorithm in order to achieve a suboptimal solution to problem (4); the original problem (3) is just a special case.4 3In [9] we assume fixed transmit power, thus having only discrete variables (the REs’ on/off states) without the maximum-power constraint. 4More specifically, we start with a feasible solution, and then we alternately optimize a subset of variables with the remaining variables being fixed. A. Transmit Power Allocation with Fixed REs’ On/Off States Given the vector x∈ {0,1}N, problem (4) reduces to max pEEw(p, x;ξ)(5a) s.t. γw(p, x;ξ)≥γmin, pℓ≤p≤pu.(5b) Note that (f(x)−g(x;ξ))2>0, because γmin >0 =⇒ γw(p, x;ξ)>0. For convenience, let us define the functions u(x;ξ) = (f(x)−g(x;ξ))2/σ2,(6) v(x) = (Pon −Poff)X n∈N xn+Pfix.(7) Observe that u(x;ξ), v(x)>0. Problem (5) is equivalent to max pEEw(p, x;ξ)(8a) s.t. p′ ℓ≤p≤pu,(8b) where p′ ℓ= max γmin u(x;ξ), pℓ. Proposition 2. The optimal solution to problem (5)/(8) is popt = min (max (p′ ℓ,epopt), pu),(9) where epopt = (eW0((u(x;ξ)v(x)η−1)e−1)+1 −1)/u(x;ξ). Proof: The worst-case EE in (2) can be expressed as EEw(p, x;ξ) = log2(1+u(x;ξ)p) η−1p+v(x). By differentiating with respect to p, we obtain ∂ ∂p EEw(p, x;ξ) = s(p,x;ξ) log(2)(1+u(x;ξ)p)(η−1p+v(x))2,(10) where s(p, x;ξ)=u(x;ξ)η−1p+v(x)− η−1(1+u(x;ξ)p) log (1 + u(x;ξ)p). To find the optimal (unconstrained) transmit power p≥0, we should solve the equation ∂ ∂p EEw(p, x;ξ)=0which yields s(p, x;ξ)=0, i.e., 1+u(x;ξ)p elog 1+u(x;ξ)p e=u(x;ξ)v(x)η−1 e. By applying the transformation y= log 1+u(x;ξ)p e⇐⇒ p= (ey+1 −1)/u(x;ξ), the above equation becomes yey= (u(x;ξ)v(x)η−1) e−1. Lemma 1 ([10]).The equation yey=δ, where δis a real number, has the following real solution(s) y=W0(δ)or W−1(δ),if −e−1≤δ < 0, W0(δ),if δ≥0.(11) If δ < −e−1, then the equation has no real solution. There is a unique real solution y=W0(−e−1)=W−1(−e−1)=−1 when δ=−e−1, whereas there are exactly two (distinct) real solutions when −e−1<δ<0(because W0(δ)>−1> W−1(δ)in this region). Also, W0(δ)≥0when δ≥0. In our case, u(x;ξ)p≥0 =⇒y≥ −1, so there is a unique solution given by y=W0(u(x;ξ)v(x)η−1) e−1>−1. Hence, the optimal (unconstrained) transmit power, epopt >0, is expressed as shown in Proposition 2. In particular, it holds that ∂2 ∂p2EEw(epopt,x;ξ)<0, because ∂ ∂p s(epopt,x;ξ) = −η−1u(x;ξ) log (1 + u(x;ξ)epopt)<0. Therefore, epopt is a This article has been accepted for publication in IEEE Communications Letters, December 2025. ©2025 IEEE. 3 local maximum that is also the unique global maximum of EEw(p, x;ξ)for p≥0, since EEw(0,x;ξ)=0and limp→∞ EEw(p, x;ξ)=0. Finally, the (unique) optimal solution to the constrained problem (8) can be easily computed by (9), because EEw(p, x;ξ)is increasing for p∈[0,epopt)and decreasing for p∈(epopt,∞). B. Reflecting-Element Activation with Fixed Transmit Power Given the transmit power p∈[pℓ, pu], problem (4) becomes max x EEw(p, x;ξ)(12a) s.t. γw(p, x;ξ)≥γmin,x∈ {0,1}N.(12b) Problem (12) is a discrete (binary) optimization problem that can be globally solved using the dynamic programming algorithm given in [9, Algorithm 1], which has polynomial complexity O(Nlog N); please refer to [9, Theorem 4]. C. Algorithm Analysis The AO procedure is presented in Algorithm 1. First, the algorithm decides the feasibility of problem (4) (step 1) and initializes some parameters (step 2). Subsequently, it computes two solutions in an alternating manner by first optimizing with respect to: i) the transmit power p(steps 3–7), and ii) the binary vector x(steps 8–13). Finally, it returns the best of them (steps 14–15).5The following theorem demonstrates the convergence and complexity of the AO algorithm. Algorithm 1 Alternating Optimization (AO) for problem (4) 1: if γw(pu,1N;ξ)< γmin then return ‘Infeasible’ end if 2: Choose a feasible solution (p(0) opt ,x(0) opt )to problem (4), and a convergence tolerance ϵ>0.EE(0) w←EEw(p(0) opt ,x(0) opt ;ξ),i←0 3: repeat 4: i←i+ 1,p(i) opt ←popt according to (9) with x=x(i−1) opt 5: Solve problem (12) with p=p(i) opt , using [9, Algorithm 1], and let x(i) opt be its globally optimal solution. EE(i) w←EEw(p(i) opt ,x(i) opt ;ξ) 6: until |EE(i) w−EE(i−1) w|< ϵ 7: (p′ opt,x′ opt)←(p(i) opt ,x(i) opt ),EE′ w←EE(i) w,i←0 8: repeat 9: i←i+ 1. Solve problem (12) with p=p(i−1) opt , using [9, Algorithm 1], and let x(i) opt be its globally optimal solution. 10: p(i) opt ←popt according to (9) with x=x(i) opt 11: EE(i) w←EEw(p(i) opt ,x(i) opt ;ξ) 12: until |EE(i) w−EE(i−1) w|< ϵ 13: (p′′ opt,x′′ opt)←(p(i) opt ,x(i) opt ),EE′′ w←EE(i) w 14: if EE′ w≥EE′′ wthen return (p′ opt,x′ opt)and EE′ w 15: else return (p′′ opt,x′′ opt)and EE′′ wend if Theorem 1. Algorithm 1 returns either ‘Infeasible’ if problem (4) is not feasible, or a feasible solution otherwise. In the latter case, the algorithm produces a nondecreasing sequence of objective values in each loop, i.e., EE(i) w≥EE(i−1) wfor all i≥1, and terminates in finitely many iterations. Moreover, its complexity is O(INlog N), where Iis the total number of iterations and Nis the number of reflecting elements. 5In addition to the ϵ-convergence criterion, we can also use a (predetermined) maximum number of iterations for robustness against numerical errors. Proof: It can be easily seen that Algorithm 1 correctly returns ‘Infeasible’ due to Proposition 1. Moreover, we can show by induction on ithat (p(i) opt,x(i) opt)is a feasible solution to problem (4), for all iterations i≥0. Also, we have EE(i−1) w:= EEw(p(i−1) opt ,x(i−1) opt ;ξ)≤EEw(p(j) opt ,x(k) opt ;ξ) ≤EEw(p(i) opt,x(i) opt;ξ) =:EE(i) w,∀i≥1,(13) where (j, k) = (i, i −1) and (j, k) = (i−1, i) in the first and second loop, respectively. As a result, the sequence {EE(i) w}i≥0is nondecreasing and, since it is upper bounded (i.e., EEw(p, x;ξ)≤ P−1 fix log2(1+γw(pmax,1N;ξ)) <∞), it converges to a finite value. Therefore, limi→∞(EE(i) w−EE(i−1) w) = 0, which by the limit definition means that: for every ϵ > 0, there exists an integer msuch that if i≥mthen |EE(i) w−EE(i−1) w|< ϵ. Thus, Algorithm 1 terminates in a finite number of iterations. Finally, the complexity of Algorithm 1 is O(INlog N), because each iteration requires O(Nlog N)time, mainly due to [9, Algorithm 1] in steps 5 and 9. IV. BRANCH-AND-BOUND METHOD A powerful approach for global optimization is the B&B technique. The main idea is to develop a B&B method based on the AO algorithm to achieve fast convergence to an optimal solution. In a nutshell, B&B generates multiple subproblems by recursively splitting the feasible set and using bounds on their optimum values. In our case, we have the original problem (3) and subproblems in the form of (4). A generated subproblem is called active if it has not been examined yet. Moreover, Qis the first-in-first-out list (i.e., a queue) that contains the active subproblems. For each subproblem (4) we just store the interval [pℓ, pu]in Q. Initially the list contains the original problem (3), i.e., Q={[0, pmax]}, whereas Q=∅in the last iteration. We also denote its cardinality by Q=|Q|, and the maximum Qover all iterations by Qmax (which is proportional to the space complexity of B&B). In addition, (ep, e x)is the best feasible solution found so far, and f EEwis the current best energy efficiency, i.e., f EEw= EEw(ep, e x;ξ). Note that B&B produces a nondecreasing sequence of f EEw values over its iterations. Finally, EEw= EEw(pℓ, pu)and EEw= EEw(pℓ, pu)are upper and lower bounds (U/LB) on the optimum value of the subproblem, respectively, i.e., EEw(pℓ, pu)≤EE∗∗ w(pℓ, pu)≤EEw(pℓ, pu). In particular, the UB is chosen as follows EEw(pℓ, pu):=max x log2(1+γw(pu,x;ξ)) Ptot(pℓ,x)(14a) s.t. γw(pu,x;ξ)≥γmin,(14b) x∈ {0,1}N.(14c) It holds that EE∗∗ w(pℓ, pu)≤EEw(pℓ, pu), because the objective function of (4) is upper bounded by the objective function of (14), and the feasible set of (4) is a subset of the feasible set of (14). This problem can be (globally) solved using again [9, Algorithm 1] with Pold fix =η−1pℓ+Pstatic and γ=pu/σ2. Regarding the LB, EEw(pℓ, pu)is computed by Algorithm 1 provided that subproblem (4) is feasible. In particular, Algo- This article has been accepted for publication in IEEE Communications Letters, December 2025. ©2025 IEEE. 4 rithm 1 is initialized with p(0) opt =pu; this is very important for the B&B convergence (see the proof of Theorem 2). The proposed B&B is given in Algorithm 2.6Specifically, it performs the following basic operations: 1) subproblem selection from the list Q, 2) infeasibility: remove a subproblem when it is not feasible, 3) bounding: compute upper and lower bounds on the optimum value of the subproblem, 4) update of f EEwand (ep, e x), 5) pruning/fathoming: remove a subproblem when EEw≤f EEw+ε(the subproblem does not contain any better solution with respect to the given tolerance ε > 0), or when EEw= EEw(we have already found an optimal solution to the subproblem), and 6) branching: split the feasible set into two subsets using the standard bisection technique, i.e., [pℓ, pu]=[pℓ, pm]∪[pm, pu], where pm=1 2(pℓ+pu). Algorithm 2 Branch-and-Bound (B&B) for problem (3) 1: Choose an optimal-solution accuracy ε>0. 2: Q←{[0, pmax]},f EEw← −∞,i←0 3: while Q =∅do 4: i←i+ 1 5: ▷Subproblem selection 6: Let [pℓ, pu]be the first subproblem in the front of the list Q. 7: Q ← Q \ {[pℓ, pu]} 8: ▷Pruning due to infeasibility 9: if γw(pu,1N;ξ)< γmin then continue end if 10: ▷Bounding 11: Compute the bound EEwby solving problem (14), using [9, Algorithm 1] with Pold fix =η−1pℓ+Pstatic and γ=pu/σ2. 12: Compute a feasible solution (p, x)to subproblem (4), using Algorithm 1 with (p(0) opt ,x(0) opt )=(pu,1N), and let EEw= EEw(p, x;ξ)be its objective value. 13: ▷Updating f EEwand (ep, e x) 14: if EEw> f EEwthen f EEw←EEw,(ep, e x)←(p, x)end if 15: ▷Pruning/Fathoming 16: if (EEw≤ f EEw+ε)∨(EEw= EEw)then continue end if 17: ▷Branching and adding the new subproblems at the back of Q 18: pm←1 2(pℓ+pu),Q ← Q ∪ {[pℓ, pm],[pm, pu]} 19: end while 20: if f EEw=−∞ then return ‘Infeasible’ 21: else return (ep, e x)and f EEwend if In general, the B&B method has exponential complexity in the worst case, but much lower (possibly polynomial) complexity on average. Since the computation of an exact optimal solution is generally impossible due to finite precision, we need a definition of approximately-optimal solutions. Definition 1 (ε-optimal solution).Consider the following optimization problem: φ∗:= max {φ(y):y∈ D}, where φ(y) is a continuous function and D ⊂ Rnis a compact set. Given an ε > 0, we say that e y∈ D is a (globally) ε-optimal solution to the aforementioned problem if φ∗−ε≤φ(e y) (≤φ∗). Now, we can prove the correctness of the B&B algorithm. Theorem 2. If the original problem (3) is not feasible, then Algorithm 2 returns ‘Infeasible’ after a single iteration. Otherwise, it terminates within a finite number of iterations and (ep, e x)is an ε-optimal solution to problem (3) with EE∗ w−ε≤f EEw= EEw(ep, e x;ξ)≤EE∗ w. Proof: See the Appendix. 6With ‘continue’ the program skips any remaining statements in the whileloop for the current iteration, and continues from the next iteration. Fig. 1: Worst-case energy efficiency versus the number of reflecting elements, with CSI-uncertainty radius ξ=τbαmin. V. NUMERICAL RESULTS The simulation parameters are similar to [9, Section VI] with the following differences: transmitter, receiver and IRS locations (0,0,0),(80,0,0) and (40,10,5), respectively, channel Rician factors κu=κv= 6 dB, ξ=τbαmin, where τ∈[0,1],γmin =χγw(pmax,1N;bαmin), where χ∈[0,1] (the feasibility of the original problem (3) is ensured, because γw(pmax,1N;ξ)≥γw(pmax,1N;bαmin)≥γmin), N= 50, pmax = 27 dBm, σ2=−85 dBm, Poff = 0.4mW, χ= 0.4. In addition, we select (p(0) opt ,x(0) opt )=(pu,1N)and ϵ=ε= 10−3. For comparison purposes, we consider three baseline schemes: 1) Only Reflecting-Element Optimization (OREO) with maximum transmit power, using [9, Algorithm 1] with p=pmax,2)Only Power Allocation (OPA) with all reflecting elements being activated, according to (9) with (pℓ, pu,x) = (0, pmax,1N), and 3) Maximum Power and All-ReflectingElements Activation (MPAREA), i.e., p=pmax and x=1N. Fig. 1 shows the worst-case EE against the number of REs. Firstly, we can observe that severe CSI uncertainty results in EE reduction for all schemes. Secondly, AO achieves nearly the same performance with B&B, i.e., it is very close to the global optimum. Moreover, AO outperforms all benchmarks, with MPAREA having the worst performance. Thirdly, OPA is better than OREO for small N, whereas OREO shows higher performance than OPA for large N. Finally, it is interesting to note that the average running time was of the order of 10−3s for B&B, 10−4sfor AO, and 10−5sfor all the benchmarks. AO required only 4–6 iterations (2–3 iterations for each repeatuntil loop) to converge on average. Also, the average number of B&B iterations was 143–584, and the average maximum number of active subproblems in the list Q(Qmax) was 32– 164 (i.e., low memory requirement or space complexity). Furthermore, the evolution of the B&B algorithm for a problem instance is illustrated in Fig. 2. In particular, the total number of B&B iterations is 159, while Qmax = 33. We can also observe that: a) Q= 1 at the beginning, whereas Q= 0 at the end, and b) f EEwis nondecreasing with iterations. Finally, we investigate the impact of the minimum SNR on the EE. Based on Fig. 3, the worst-case EE is a nonincreasing function of χfor all algorithms, since the increase of χresults in a more restricted feasible set. Although AO outperforms the baseline schemes and remains close to the global optimum in This article has been accepted for publication in IEEE Communications Letters, December 2025. ©2025 IEEE. 5 Fig. 2: B&B evolution for a particular problem instance, with CSI-uncertainty radius ξ=τbαmin and τ= 0.7. Fig. 3: Worst-case energy efficiency versus the minimum-SNR control parameter, with γmin =χγw(pmax,1N;bαmin). the majority of cases, there is a noticeable gap between AO and B&B in some cases (e.g., for χ= 0.5–0.8). VI. CONCLUSION In this letter, we dealt with the robust optimization of EE in IRS-aided communication systems under CSI uncertainty. In particular, we designed two algorithms (AO and B&B) to jointly optimize the transmit power and the on/off states of REs. Numerical simulations verified the theoretical results and showed the effectiveness of the proposed algorithms. APPENDIX PROOF OF THEOREM 2 The first part of the theorem is easy to prove by observing that the original problem will be pruned due to infeasibility in the first iteration, thus f EEwwill remain equal to −∞. For the second part about the B&B convergence to an ε-optimal solution in finite time, it is sufficient to show that lim ∆p→0EEw(pℓ, pu)−EEw(pℓ, pu)= 0,(15) where ∆p=pu−pℓis the length of the interval [pℓ, pu].7 For convenience, using pℓ=pu−∆pand assuming (without loss of generality) that puis fixed, but arbitrary, we define F(∆p;x) = log2(1+γw(pu,x;ξ)) Ptot(pu−∆p, x).(16) 7In Algorithm 2, immediately after step 14 we have f EEw≥EEw, thus EEw− f EEw≤EEw−EEw. Therefore, condition (15) implies that there is a finite number of iterations after which EEw− f EEw≤ε. Eventually all subproblems will be pruned, resulting in Q=∅and EE∗ w≤ f EEw+ε. The first-order derivative of F(∆p;x)with respect to ∆pis ∂F(∆p;x) ∂(∆p)=η−1log2(1+γw(pu,x;ξ)) P2 tot(pu−∆p, x).(17) Observe that the derivative is continuous for ∆p≤pu, and can be bounded by a positive constant M, i.e., |∂F(∆p;x)/∂(∆p)| ≤ M, given by M=η−1P−2 fix log2(1+γw(pmax,1N;ξ)) <∞.(18) Next, according to Taylor’s theorem, we can express F(∆p;x)around ∆p= 0 as follows F(∆p;x) = F(0; x)+R(∆p;x),(19) where R(∆p;x) = R∆p 0 ∂F (t;x) ∂t dtis the remainder term, which can be bounded by |R(∆p;x)|≤M|∆p|,∀x∈ {0,1}N.(20) Now, we define the set X={x∈ {0,1}N:γw(pu,x;ξ)≥ γmin}and the optimization problem EEw(pu):= max x∈X EEw(pu,x;ξ) = max x∈X F(0; x).(21) In step 12 Algorithm 1 is initialized with p(0) opt =pu, hence EEw(pℓ, pu)≥EEw(pu).(22) By combining (14) and (19)–(22), we can write EEw(pℓ, pu) = max x∈X F(∆p;x) ≤max x∈X F(0; x) + max x∈X R(∆p;x) ≤EEw(pℓ, pu)+M|∆p|. (23) Therefore, we have 0≤EEw(pℓ, pu)−EEw(pℓ, pu)≤M|∆p|. Finally, by taking the limit as ∆p→0, we obtain (15). 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