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Performance Limits of Solutions to Network Utility Maximization Problems R. L. G. Cavalcante and S. Stanćzak Fraunhofer Heinrich Hertz Institute and Technical University of Berlin email: {renato.cavalcante,slawomir.stanczak}@hhi.fraunhofer.de 7 Abstract—We study performance limits of solutions to utility the bounds in previous studies [4], the bounds derived here 1 maximization problems (e.g., max-min problems) in wireless are asymptotically tight, and they are valid for a larger class 0 networks as a function of the power budget p¯ available to 2 of utility maximization problems. These bounds, which do transmitters. Special focus is devoted to the utility and the b not depend on any unknown constants, are particularly useful transmit energy efficiency (i.e., utility over transmit power) of e to determine power regions in which wireless networks are F the solution. Briefly, we show tight bounds for the general class 3 of network utility optimization problems that can be solved expected to be noise limited and interference limited. In by computing conditional eigenvalues of standard interference addition, they reveal that the network utility and the energy ] T mappings.Theproposedbounds,whicharebasedontheconcept efficiency scale as Θ(1) and Θ(1/p¯), respectively, as p¯→∞ I of asymptotic functions, are simple to compute, provide us with . (inthisstudy,bigthetaΘisdefinedasinthestandardfamilyof s good estimates of the performance of networks for any value of c Bachmann-Landau notations). The main tools for the analysis p¯ in many real-world applications, and enable us to determine [ points in which networks move from a noise limited regime to shownherearetheresultsin[1]andtheconceptofasymptotic 3 v aninterferencelimitedregime.Furthermore,theyalsoshowthat functions [7], which so far have received limited attention 1 the utility and the transmit energy efficiency scales as Θ(1) and from the wireless community [2], [4], [8]. To the best of our 9 Θ(1/p¯), respectively, as p¯→∞. 4 knowledge,weshowforthefirsttimepropertiesofasymptotic 6 0 I. INTRODUCTION functions associated with standard interference functions that . are not necessarily convex or concave. We illustrate the 1 Recent studies [1]–[4] have shown a strong connection 0 theoretical findings by studying the utility obtained in a dense 7 between the solution to many utility maximization problems 1 wirelessnetworkinastadium,oneoftheuse-casesconsidered : in wireless networks and conditional eigenvalues of nonlinear v for the fifth generation of wireless networks [2]. i mappings [5] (typically, standard interference mappings [6]). X In general, the utility of the solution increases as we increase II. MATHEMATICALPRELIMINARIES r a the power budget p¯ available to transmitters [1], so we can One of the main objectives of this section is to derive unlock fundamental bounds on the network performance by properties of asymptotic functions associated with standard studying the asymptotic behavior of the solutions as p¯→∞, interference functions. This section also clarifies much of the aswehaverecentlyshownintheyetunpublishedstudyin[4]. notation, and it reviews standard results and definitions that However,todate,boundsofthistypearerareintheliterature, are required for the contributions in the next sections. and the existing bounds on the utility are not asymptotically In more detail, by R and R we denote the set of non- + ++ sharp for many well-known utility maximization problems. negative reals and positive reals, respectively. The effective Against this background, in this study we derive upper domain of a function f : RN → R ∪ {∞} is given by bounds for the utility and for the transmit energy efficiency domf := {x ∈ RN | f(x) < ∞}, and f is proper if (i.e., utility over power) achieved by solutions to utility dom f (cid:54)= ∅. We say that f is continuous when restricted to maximization problems for a given power budget p¯. Unlike C ⊂ RN if (∀x ∈ C)(∀(xn)n∈N ⊂ C)limn→∞xn = x ⇒ lim f(x ) = f(x), and we write lim x = x if the domain RN are standard interference functions, but the n→∞ n n→∞ n + and only if lim (cid:107)x −x(cid:107)=0 for an arbitrary norm (cid:107)·(cid:107) converse does not hold in general. Therefore, as it will soon n→∞ n (thechoiceofthenormisarbitrarybecauseoftheequivalence become clear, the bounds derived in the next section are more of norms in finite dimensional spaces). The notions of upper general than those in the recent unpublished work in [4]. One and lower semicontinuity for functions f : RN → R∪{∞} of the key results that we further explore in the next section restricted to sets C ⊂ RN are defined similarly. Given is the following: (x,y)∈RN ×RN, vector inequalities such as x≤y should Fact 1 ([1],[5])Let(cid:107)·(cid:107)beamonotonenormandT :RN → beunderstoodcoordinate-wise.IfC ⊂RN isaconvexset,we + RN be either a standard interference mapping or a concave say that a mapping T :C →RN :x(cid:55)→[t (x),··· ,t (x)] is + 1 N mapping that is positively homogeneous and, for every x ∈ concave if, for every i∈{1,...,N}, the function t :C →R i RN\{0}, there exists m ∈ N for which Tn(x) > 0 for all is concave. A norm (cid:107)·(cid:107) in RN is said to be monotone if + n ≥ m (these mappings are said to be primitive, and Tn (∀x∈RN)(∀y ∈RN) 0≤x≤y ⇒(cid:107)x(cid:107)≤(cid:107)y(cid:107). denotes the n-fold composition of T with itself). Then each of A fundamental mathematical tool used in this study is the the following holds: analyticrepresentationofasymptoticfunctions,whichwestate (i) There exists a unique solution (x(cid:63),λ(cid:63)) ∈ RN ×R as a definition: ++ ++ to the conditional eigenvalue problem Definition 1 ( [7, Theorem 2.5.1] Asymptotic function) The Problem 1 Find (x,λ)∈RN ×R such that T(x)= + + asymptotic function associated with a proper function f : λx and (cid:107)x(cid:107)=1. RN →R∪{∞} is the function given by f :RN → R∪{∞} (ii) The sequence (xn)n∈N ⊂RN+ generated by ∞ (cid:26) (cid:27) 1 x (cid:55)→ inf linm→i∞nf f(hhnnxn) | hn →∞, xn →x , xn+1 =T(cid:48)(xn):= (cid:107)T(xn)(cid:107)T(xn), x1 ∈RN++, (2) (1) converges geometrically to the uniquely existing vector where (xn)n∈N and (hn)n∈N are sequences in RN and R, x(cid:63) ∈Fix(T(cid:48)):={x∈RN | x=T(cid:48)(x)}, which is also respectively. Equivalently, f : RN → R ∪ {∞} : x (cid:55)→ + ∞ thevectorx(cid:63)ofthetuple(x(cid:63),λ(cid:63))thatsolvesProblem1. liminf f(hy)/h. h→∞,y→x Furthermore, the sequence (λn := (cid:107)T(xn)(cid:107))n∈N ⊂ In particular, here we are mostly interested in computing R converges to λ(cid:63). ++ asymptotic functions associated with standard interference We now proceed to the study of asymptotic functions functions, defined as follows: associated with standard interference functions. We start with Definition 2 ([6]Standardinterferencefunctions)Afunction the following simple result. f : RN → R ∪{∞} is said to be a standard interference ++ Lemma 1 Let f : RN → R ∪ {∞} be a standard ++ function if the following properties hold: interference function. Then we have: 1) (Scalability) (∀x∈RN) (∀α>1) αf(x)>f(αx). + (i) (∀x∈RN)(∀α∈ ]0,1[) f(αx)>αf(x) + 2) (Monotonicity) (∀x ∈ RN) (∀x ∈ RN) x ≥ x ⇒ 1 + 2 + 1 2 (ii) (∀x∈RN)(∀α ∈R )(∀α ∈R ) + 1 ++ 2 ++ f(x )≥f(x ). 1 1 1 2 α >α ⇒ f(α x)> f(α x)>0 3) f(x)=∞ if and only if x∈/ RN. 2 1 α1 1 α2 2 + (iii) (∀x ∈ RN) lim f(hx)/h ∈ R (i.e., the limit + h→∞ + Given N standard interference functions t : RN → R ∪ i ++ always exists). {∞}, i = 1,...,N, we call the mapping T : RN → RN : + ++ Proof:(i)Thedesiredresultfollowsfromasimpleimplication x(cid:55)→[t (x),...,t (x)] a standard interference mapping. 1 N of Definition 2.1: As shown in [8, Proposition 1] and the references therein, 1 (cid:16)α (cid:17) (∀x∈RN)(∀α∈ ]0,1[) f(αx)>f x =f(x). functions that are positive and concave when restricted to + α α (ii) Let α > α > 0 and define β := α /α > 1. The Let α ∈ ]0,1[ be arbitrary. From the definition of the 2 1 2 1 proof is now a direct consequence of Definition 2.1: asymptotic function in (1) and the definition of standard 1 β 1 1 interferencefunctions,weknowthat,foreveryx∈RN+,there f(α x)= f(α x)> f(βα x)= f(α x)>0. α1 1 βα1 1 βα1 1 α2 2 exist a sequence (xn)n∈N ⊂ RN+ and an increasing sequence (iii) By (ii), for every x ∈ RN, the function (hn)n∈N ⊂ R++ such that f∞(x) = limn→∞f(hnxn)/hn, + lim x = x, and lim h = ∞. Therefore, as an R → R : h (cid:55)→ f(hx)/h is monotonically n→∞ n n→∞ n ++ ++ implication of lim x =x∈RN, we have αx≤x for decreasing and bounded below by zero, so the limit n→∞ n + n everyn≥LwithL∈Nsufficientlylarge.Lemma1(i)andthe lim f(hx)/h∈R exists as claimed. (cid:3) h→∞ + monotonicitypropertyofstandardinterferencefunctionsyield (∀n ≥ L)(∀x ∈ RN)0 < αf(h x)/h < f(αh x)/h ≤ The next proposition shows important properties of asymp- + n n n n f(h x )/h . Taking the limit as n → ∞ and considering toticfunctionsassociatedwithstandardinterferencefunctions. n n n Lemma 1(iii), we verify that In particular, it establishes the connection between the limit in Lemma 1(iii) and asymptotic functions. This connection is (∀x∈RN) 0≤α lim f(hnx) =α lim f(hx) ≤f (x). of practical significance because the limit in Lemma 1(iii) is + n→∞ hn h→∞ h ∞ (3) ofteneasytocomputeintheapplicationsweconsiderhere.In fact,Proposition1(i)showsthatausefulsimplificationforthe Since α can be made arbitrarily close to one, (3) proves that computation of f∞ when f is convex [7, Corollary 2.5.3] is f∞(x)≥limh→∞f(hx)/h≥0, which completes the proof. also available when f is a standard interference function that (ii) First recall that asymptotic functions associated with is not necessarily convex or concave. proper functions are lower semicontinuous and positive homogeneous [7, Proposition 2.5.1(a)]. Now, assume that f is Proposition 1 The asymptotic function f∞ :RN →R∪{∞} continuous when restricted to RN+, and let (hn)n∈N ⊂ R++ associated with a standard interference function f : RN → be an arbitrary sequence such that limn→∞hn = ∞. Define R++∪{∞} has the following properties: gn : RN+ → R+ : x (cid:55)→ f(hnx)/hn, and note that the function g is continuous for every n ∈ N because f is (i) (∀x∈RN) f (x)=lim f(hx)/h∈R . n + ∞ h→∞ + continuouswhenrestrictedtoRN byassumption.Theproperty (ii) f islowersemicontinuousandpositivelyhomogeneous. + ∞ in (i) and Lemma 1(ii)-(iii) imply that (∀x ∈ RN)f (x) = If f is in addition continuous when restricted to the + ∞ non-negative orthant RN, then f is continuous when infn∈Ngn(x). This shows that, restricted to RN+, f∞ is the + ∞ pointwise infimum of continuous functions, so f is upper restricted to the non-negative orthant RN. ∞ + semicontinuous, in addition to being lower semicontinuous as (iii) (Monotonicity) (∀x ∈ RN) (∀x ∈ RN) x ≥ x ⇒ 1 + 2 + 1 2 already shown. Therefore, f restricted to RN is continuous. f (x )≤f (x ). ∞ + ∞ 2 ∞ 1 (iv) Let x ∈ RN be arbitrary. If f is continuous when (iii) Let (hn)n∈N ⊂ R++ be an arbitrary monotone se- + restricted to RN, then f (x) = lim f(h x )/h quence such that limn→∞hn = ∞. If x1 ≥ x2 ≥ 0, + ∞ n→∞ n n n for all sequences (xn)n∈N ⊂ RN+ and (hn)n∈N ⊂ R++ the monotonicity property of standard interference functions shows that (∀n ∈ N) f(h x )/h ≤ f(h x )/h . Now let such that lim x =x and lim h =∞. n 2 n n 1 n n→∞ n n→∞ n (v) If f is also concave when restricted to RN, then f is n→∞ and use the property in (i) to obtain the desired result + ∞ concave when restricted to RN. f∞(x2)≤f∞(x1). + (iv) Let the arbitrary sequences (xn)n∈N ⊂ RN+ and Proof: (i) The inequality (∀x ∈ RN+) f∞(x) ≤ (hn)n∈N ⊂R++ satisfy limn→∞xn =x and limn→∞hn = limh→∞f(hx)/h ∈ R+ is immediate from Lemma 1(iii) ∞. Denote by 1∈RN the vector of ones. As a consequence and the definition in (1), so it is sufficient to prove that of lim x =x, we know that n→∞ n (∀x∈RN) f (x)≥lim f(hx)/h to obtain the desired + ∞ h→∞ (∀(cid:15)>0)(∃L∈N)(∀n≥L) x ≤x+(cid:15)1. (4) result. n As a result, for every (cid:15)>0, we have Problem 2 (Canonical form of the network utility maximization problem) (a) 1 (b) 1 f (x) ≤ liminf f(h x ) ≤ limsup f(h x ) ∞ n∈N hn n n n∈N hn n n maximizep,c c (≤c)limsup 1 f(h (x+(cid:15)1))(=d)f (x+(cid:15)1), (5) subject to p∈Fix(cT):=(cid:8)p∈RN+ | p=cT(p)(cid:9) n∈N hn n ∞ (cid:107)p(cid:107)a ≤p¯ where (a) follows from (1), (b) is a basic property of limits, p∈RN,c∈R , + ++ (c) follows from (4) and monotonicity of f, and (d) is a (6) consequence of the property we proved in (i). By assumption, where p¯∈R is a design parameter hereafter called power ++ f restrictedtoRN+ iscontinuous,sof∞restrictedtoRN+ isalso budget, (cid:107)·(cid:107)a is an arbitrary monotone norm, and T :RN+ → continuous as shown in (ii). Therefore, by (5) and continuity RN is an arbitrary standard interference mapping. ++ of f when restricted to RN, we have ∞ + The main objective of this section is to study selected 1 1 f (x)≤liminf f(h x )≤limsup f(h x ) properties of the solution to Problem 2 as a function of the ∞ n∈N hn n n n∈N hn n n power budget p¯. In particular, these properties enable us to ≤ lim f (x+(cid:15)1)=f (x), ∞ ∞ characterizethepowerregionsofanoiselimitedregimeandan (cid:15)→0+ interferencelimitedregimeinwirelessnetworks.Theresultsin which implies that limn∈Nf(hnxn)/hn =f∞(x). the following not only generalize those of [4] to a larger class (v) Let g be the function defined in the proof of (ii). n Note that, if f is concave in RN, then g is also concave ofmappingsT,buttheyalsosharpentheboundsofthatstudy. + n in RN for every n ∈ N. In (ii) we have showed that TheyareobtainedfromadeepconnectionbetweenProblem2 + (∀x ∈ RN+) f∞(x) = infn∈Ngn(x), so, in RN+, f∞ is the and conditional eigenvalues of nonlinear mappings: pointwise infimum of a family of concave functions, and the Fact 2 [1] Consider the assumptions and definitions in proof is complete because concavity is preserved by taking Problem 2, and denote by (p ,c )∈RN ×R a solution the pointwise infimum. (cid:3) p¯ p¯ ++ ++ to this problem for a given power budget p¯. Then: (i) The solution (p ,c ) ∈ RN ×R always exists, and p¯ p¯ ++ ++ We end this section by defining a mapping that plays a it is unique. crucial role in the analysis of network utility maximization (ii) Let (x ,λ ) ∈ RN × R be the solution to the p¯ p¯ ++ ++ problems. following conditional eigenvalue problem: Definition 3 (Asymptotic mappings) Let the function t(i) : Problem 3 Find (x ,λ ) ∈ RN × R such that p¯ p¯ ++ ++ RN → R ∪{∞} be a standard interference function for ++ T(x ) = λ x and (cid:107)x (cid:107) = 1, where (cid:107)·(cid:107) denotes the p¯ p¯ p¯ p¯ each i∈{0,...,N}. Given a standard interference mapping monotone norm (cid:107)x(cid:107):=(cid:107)x(cid:107) /p¯. a T :RN →RN :x(cid:55)→[t(1)(x),··· ,t(N)(x)], the asymptotic + ++ mapping associated with T is given by T∞ : RN+ → RN+ : Then pp¯=xp¯ and cp¯=1/λp¯. x(cid:55)→[t(1)(x),··· ,t(N)(x)], where, for each i∈{1,··· ,N}, (iii) The function R++ → R++ : p¯ (cid:55)→ cp¯ is monotonically ∞ ∞ t(∞i) is the asymptotic function associated with t(i). increasing; i.e., p¯1 >p¯2 >0 implies cp¯1 >cp¯2 >0. (iv) The function R → RN : p¯ (cid:55)→ p is monotonically ++ ++ p¯ III. PROPERTIESOFUTILITYMAXIMIZATIONPROBLEMS increasing in each coordinate; i.e., p¯ > p¯ > 0 implies 1 2 p >p >0. As shown in [1]–[4] and the references therein, a large p¯1 p¯2 class of (weighted max-min) utility maximization problems An important implication of Fact 2(ii) is that the simple in wireless networks are particular instances of the following iterative scheme in (2) is able to solve Problem 2. Fact 2 also canonical optimization problem: motivates the use of the following functions [4]: Definition 4 (Utility, power, and (cid:107)·(cid:107) -energy efficiency func- (ii) Letthescalarλ beasdefinedin(i),andassumethatT b ∞ tions) Denote by (pp¯, cp¯) ∈ RN++ × R++ the solution to iscontinuous.Inaddition,let(p¯n)n∈N ⊂R++ denotean Problem 2 for a given power budget p¯ ∈ R . The utility arbitrary monotonically increasing sequence satisfying ++ and power functions are defined by, respectively, U :R → lim p¯ = ∞, and define x := (1/(cid:107)p (cid:107) )p . ++ n→∞ n n p¯n a p¯n R :p¯(cid:55)→c and P :R →RN :p¯(cid:55)→p . In turn, given If x ∈ RN is an accumulation point of the bounded ++ p¯ ++ ++ p¯ ∞ + a monotone norm (cid:107)·(cid:107)b, the (cid:107)·(cid:107)b-energy efficiency function is sequence (xn)n∈N ⊂ RN++, then the tuple (x∞,λ∞) defined by E : R → R : p¯(cid:55)→ U(p¯)/(cid:107)P(p¯)(cid:107) , and note solves the following conditional eigenvalue problem: ++ ++ b that (∀p¯∈R ) E(p¯)=1/(cid:107)T(p )(cid:107) =c /(cid:107)p (cid:107) . ++ p¯ b p¯ p¯ b Problem 4 Find (x,λ)∈RN×R such that T (x)= + + ∞ By Fact 2(iii)-(iv), the utility function U and each coordi- λx and (cid:107)x(cid:107) =1. a nate of the power function P are monotonically increasing. (iii) Considertheassumptionsandthenotationin(i)and(ii). However, in the next lemma, we show that the utility cannot If Problem 4 has a unique solution denoted by (x(cid:48),λ(cid:48))∈ grow faster than the transmit power. RN ×R ,thenlim x =x =x(cid:48)andλ(cid:48) =λ . ++ ++ n→∞ n ∞ ∞ Lemma 2 The (cid:107)·(cid:107) -efficiency function E : R → R is b ++ ++ Proof: (i) The limit lim λ =: λ ≥ 0 exists because n→∞ p¯n ∞ non-increasing; i.e., p¯1 >p¯2 >0 implies E(p¯1)≤E(p¯2). the function R → R : p¯ (cid:55)→ λ is monotonically ++ ++ p¯ decreasing (and bounded below by zero) by Fact 2(iii). Proof: The result follows from (ii) First note that T is continuous as a consequence of ∞ (a) (b) p¯1 >p¯2 >0 ⇒ P(p¯1)>P(p¯2)⇒T(P(p¯1))≥T(P(p¯2)) Proposition 1(ii). Since (∀n ∈ N) T(pp¯n) = λp¯npp¯n and (c) (d) (cid:107)p (cid:107) =p¯ >0, we have ⇒(cid:107)T(P(p¯1))(cid:107)b ≥(cid:107)T(P(p¯2))(cid:107)b >0⇔E(p¯1)≤E(p¯2), p¯n a n where (a) is an implication of Fact 2(iv), (b) is a consequence (∀n∈N)λ x = λp¯n p = 1 T (cid:18)(cid:107)pp¯n(cid:107)ap (cid:19) p¯n n (cid:107)p (cid:107) p¯n (cid:107)p (cid:107) (cid:107)p (cid:107) p¯n of the monotonicity of standard interference functions, (c) p¯n a p¯n a p¯n a 1 = T(p¯ x ). (7) results from the monotonicity of the norm (cid:107)·(cid:107)b and positivity p¯ n n n of T, and (d) is obtained from a basic property of E shown Now, if x is an accumulation point of the bounded ∞ in Definition 4. (cid:3) sequence (xn)n∈N ⊂ RN++, then there exists a convergent subsequence(x ) ,K ⊂N,suchthatlim x =x ∈ n n∈K n∈K n ∞ WecanalsoprovethatthefunctionsU,P,andE inDefini- RN. Therefore, from the result in (i) and (7), we have + tion4arecontinuousinR++,butweomittheproofowingto 1 λ x = lim T(p¯ x )=T (x ), thespacelimitation(itissimilartothatshownin[4]forutility ∞ ∞ n∈K p¯n n n ∞ ∞ optimizationproblemswithconcavemappingsT).Wearenow where the last equality follows from Proposition 1(iv) and readytoshowthatspectralpropertiesofasymptoticmappings continuity of T . We now conclude the proof by noticing ∞ areusefultocharacterizetheperformanceofwirelessnetworks that (cid:107)x (cid:107) =lim (cid:107)x (cid:107) =1. ∞ a n∈K n a in a unified way. To this end, we need the following technical (iii) If (x(cid:48),λ(cid:48)) ∈ RN × R is the unique solution to ++ ++ result: Problem 4, then, as an immediate consequence of the result in (ii), the only accumulation point of the bounded sequence Proposition 2 Consider the assumptions and definitions of (xn)n∈N is x(cid:48), which implies that limn→∞xn = x∞ = x(cid:48). Problem 2. For notational convenience, let (pp¯,λp¯) := As a result, T∞(x∞) = λ(cid:48)x∞ ∈ RN++, which also proves (P(p¯),1/U(p¯)) ∈ RN++ ×R+ be the solution to Problem 3 that λ(cid:48) =λ∞ by considering the result in (ii). (cid:3) for a given power budget p¯∈ R (see Fact 1), and denote ++ by T∞ : RN+ → RN+ the asymptotic mapping associated with Theassumptionofuniquenessandpositivityofthesolution the standard interference mapping T :RN+ →RN++. Then: to Problem 4 in Proposition 2(iii) is valid in many utility (i) The limit lim λ =:λ ≥0 exists. maximization problems. In particular, Fact 1 shows sufficient p¯→∞ p¯ ∞ conditions that are easily verifiable, and it also shows a these two last inequalities, we obtain the desired result (∀p¯∈ simple fixed point algorithm able to solve Problem 4. For the R )U(p¯)≤min{p¯/(cid:107)T(0)(cid:107) ,1/λ }. ++ a ∞ large class of utility maximization problems for which this (iii) By (i), Fact 2(ii), and the definition of the energy assumptionisvalid,wecanobtainsimpleperformancebounds efficiency function, we deduce for every p¯>0: that are tight and fast to compute: U(p¯) αU(p¯) αU(p¯) α E(p¯)= ≤ = ≤ . (cid:107)P(p¯))(cid:107) (cid:107)P(p¯))(cid:107) p¯ λ p¯ b a ∞ Proposition 3 Let the assumptions in Proposition 2(iii) be The desired result is now obtained by combining the previous valid,andconsiderthefunctionsinDefinition4.Furthermore, bound with the bound (∀p¯>0) E(p¯)≤1/(cid:107)T(0)(cid:107) , which is b denoteby(x ,λ )∈RN ×R thesolutiontoProblem4. ∞ ∞ ++ ++ immediate from (i). Then the following holds: (iv) The relation U(p¯) ∈ Θ(1) is immediate from (i) supp¯>0U(p¯) = limp¯→∞U(p¯) = 1/λ∞ and limp¯→∞U(p¯) = 1/λ∞, as shown in (i). To prove that supp¯>0E(p¯)=limp¯→0+E(p¯)=1/(cid:107)T(0)(cid:107)b. E(p¯) ∈ Θ(1/p¯), recall that, from the equivalence of (ii) (∀p¯∈R++) norms in finite dimensional spaces, there exists a scalar  β ∈ R such that (∀p¯> 0) (cid:107)P(p¯)(cid:107) ≤ β(cid:107)P(p¯)(cid:107) = βp¯, p¯/(cid:107)T(0)(cid:107) , if p¯≤(cid:107)T(0)(cid:107) /λ ++ ∞ a a a ∞ U(p¯)≤ which implies (∀p¯ > 0) P(p¯) ≤ βp¯1. Now we use the 1/λ , otherwise. ∞ bound in (iii), the monotonicity of the norm (cid:107) · (cid:107) , and a (iii) (∀p¯∈R++)E(p¯)≤min{1/(cid:107)T(0)(cid:107)b,α/(λ∞ p¯)}, where the monotonicity and scalability properties of standard α ∈ R++ is any scalar satisfying (∀x ∈ RN) (cid:107)x(cid:107)a ≤ interference functions to verify that (∀p¯ > 1) α/(λ∞p¯) ≥ α(cid:107)x(cid:107)b (suchascalaralwaysexistsbecauseoftheequiv- E(p¯) = 1/(cid:107)T(P(p¯))(cid:107)b ≥ 1/(cid:107)T(βp¯1)(cid:107)b ≥ 1/(p¯(cid:107)T(β1)(cid:107)b), alence of norms in finite dimensional spaces). which implies E(p¯)∈Θ(1/p¯) as p¯→∞. (cid:3) (iv) U(p¯)∈Θ(1) and E(p¯)∈Θ(1/p¯) as p¯→∞, where big thetaΘisdefinedasinthestandardfamilyofBachmann- Proposition 3(ii) motivates the definition of the following Landau notations. operating point, which is an improvement over that in [4] because the utility bounds in Proposition 3 are tight, and we Proof: (i) The function U is monotonically increasing by consider a larger class of utility maximization problems: Fact 2(iii) and lim U(p¯) = 1/λ by Proposition 2(iii), p¯→∞ ∞ so sup U(p¯)=lim U(p¯)=1/λ . p¯>0 p¯→∞ ∞ Definition 5 If the assumptions of Proposition 3 are valid, Now, let (p¯n)n∈N ⊂ R++ be an arbitrary sequence such we say that the network operates in the low power regime if that lim p¯ = 0. To prove that lim E(p¯) = n→∞ n p¯→0+ p¯ ≤ p¯ or in the high power regime if p¯ > p¯ , where the T T lim 1/(cid:107)T(P(p¯)(cid:107) = 1/(cid:107)T(0)(cid:107) , we only have to p¯→0+ b b power budget p¯ :=(cid:107)T(0)(cid:107) /λ is the transition point. T a ∞ show that lim 1/(cid:107)T(P(p¯ ))(cid:107) = 1/(cid:107)T(0)(cid:107) . By n→∞ n b b Fact 2(ii), we have limn→∞(cid:107)P(p¯n)(cid:107)a = limn→∞p¯n = Inpractice,thetransitionpointisthepowerbudgetinwhich 0, and thus limn→∞P(p¯n) = 0. Therefore, by conti- networks are typically transitioning from a regime where nuity and positivity of T, we obtain limn→∞E(p¯n) = the performance is limited by noise to a regime where the limn→∞1/(cid:107)T(P(p¯n))(cid:107)b = 1/(cid:107)T(0)(cid:107)b < ∞. The equality performance is limited by interference. sup E(p¯) = lim E(p¯) = 1/(cid:107)T(0)(cid:107) is now immedi- p¯>0 p¯→0+ b IV. NUMERICALEXAMPLEANDFINALREMARKS ate from Lemma 2, and the proof of (i) is complete. (ii) By Fact 2(ii), positivity of U and T, and monotonicity We apply the above results to the utility maximization of T, (cid:107)·(cid:107) , and P, we have problemdescribedin[2].Briefly,theobjectiveistomaximize a the minimum downlink rate in an OFDMA-based network, (∀p¯>0) 0<U(p¯)(cid:107)T(0)(cid:107) ≤U(p¯)(cid:107)T(P(p¯))(cid:107) =p¯, a a and the optimization variables are the rates, the transmit and thus (∀p¯ > 0) U(p¯) ≤ p¯/(cid:107)T(0)(cid:107) . Furthermore, by power, and the load (i.e., the fraction of resource blocks used a (i), we also have (∀p¯ > 0) U(p¯) ≤ 1/λ . Combining for transmission) at the base stations. As shown in [2], this ∞ 106 1010 s) n bits/ bits/J] 109 e i105 y [ 108 at nc Utility (max-min r104 Transition pointUB(op¹u)nd l-energy efficie1111000567 EBo(p¹u)nd 10130-6 10-5 10-4 10-3 10-2 10-1 100 101 10140-6 10-5 10-4 10-3 10-2 10-1 100 101 Power budget p¹ [Watts] Power budget p¹ [Watts] Fig.1. Networkutilityasafunctionofthepowerbudgetp¯fortheproblem Fig.2. Energyefficiencyasafunctionofthepowerbudgetp¯fortheproblem describedin[4,Sect.V-B]. describedin[4,Sect.V-B]. problem can be written in the canonical form in Problem 2. [2] R. L. G. Cavalcante, M. Kasparick, and S. Stanćzak, “Max-min utility optimizationinloadcoupledinterferencenetworks,”IEEETrans.Wireless In the simulation, we use the same dense network used to Commun.,acceptedforpublication. produce [2, Fig. 2], with the only difference that here the [3] R.SunandZ.-Q.Luo,“Globallyoptimaljointuplinkbasestationasso- noise power spectral density is fixed to -154 dBm/Hz, and ciation and power control for max-min fairness,” in IEEE International ConferenceonAcoustics,SpeechandSignalProcessing(ICASSP),2014, we vary the power budget. For brevity, we refer the readers pp.454–458. to [2, Sect. V.B] for further details. Fig. 1 shows the utility [4] R. L. G. Cavalcante and S. Stanćzak, “Fundamental properties of obtained in the simulations together with the utility bound solutions to utility maximization problems,” submitted (available at https://arxiv.org/abs/1610.01988). in Proposition 3(ii) (for completeness, we show the energy [5] U.Krause,“ConcavePerron–Frobeniustheoryandapplications,”Nonlin- efficiency in Fig. 2). All conditional eigenvalue problems earAnalysis:Theory,Methods&Applications,vol.47,no.3,pp.1457– have been solved with the fixed point iteration in (2). In 1466,2001. [6] R. D. Yates, “A framework for uplink power control in cellular radio Fig. 1, we clearly observe that, for a power budget above the systems,”IEEEJ.Select.AreasCommun.,vol.13,no.7,pp.pp.1341– transition point, increasing the budget by orders of magnitude 1348,Sept.1995. brings only marginal gains in utility, an indication that the [7] A. Auslender and M. Teboulle, Asymptotic Cones and Functions in OptimizationandVariationalInequalities. NewYork:Springer,2003. performanceislimitedbyinterference.Incontrast,forapower [8] R. L. G. Cavalcante, Y. Shen, and S. Stanćzak, “Elementary properties budgetbelowthetransitionpoint,thegaininutilityiscloseto ofpositiveconcavemappingswithapplicationstonetworkplanningand linear,whichisanindicationthattheperformanceislimitedby optimization,”IEEETrans.SignalProcessing,vol.64,no.7,pp.1774– 1873,April2016. noise.WealsonotethatthesimpleboundinProposition3(ii), whichrequiresthesolutionofonlyoneconditionaleigenvalue problem, provides us with a good estimate of the actual network performance for any value of p¯. This observation is not necessarily surprising because the bound for U is asymptoticallysharpasp¯→0+andasp¯→∞.Similarresults have been obtained for different network utility maximization problems (e.g., [3] [4, Sect. IV]), and they will be reported elsewhere. REFERENCES [1] C. J. 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