Solving Fractional Polynomial Problems by Polynomial Optimization Theory

1 Solving Fractional Polynomial Problems by Polynomial Optimizati on Th eory Andrea Pizzo, Student Member , I EEE , Alessio Zappon e , Senior Member , IEEE , Luca Sanguinetti, S enior Member , IEEE Abstract —This work aims to introduce the framew ork of polynomial optimization theory to solv e fractional polynomial problems (FPPs). Unlike other widely used optimization frame- works, t h e proposed one applies to a larg er class of FPP s, n ot necessarily defined by concav e and con vex functions. An iterativ e algorithm th at is prov ably conv erge nt and enjoys asymptotic optimality properties is proposed. Numerical results are used to validate its accuracy in the non-asymptotic regime when applied to the energy efficiency maximization in multiuser multiple-inp ut multiple-outp u t communication systems. I . I N T R O D U C T I O N C ONSIDER the fraction a l p olynom ial pro blem (FPP) r ⋆ = max x ∈X f ( x ) g ( x ) (1) with x = [ x 1 , . . . , x n ] T and X = { x ∈ R n | h i ( x ) ≥ 0 , i = 1 , . . . , m } (2) where f ( x ) , g ( x ) , h i ( x ) : R n → R are m ultiv ariate polyno - mial fun c tions and X ⊆ R n is a co mpact semialg ebraic set, not n e cessarily conv ex. Problem s of the for m in (1 ) arise in different areas o f signal processing, e.g., energy efficiency maximization [1], filter design [2], [3], remote sensing [4], and control the o ry [5]. More g enerally , [6] sh owed th at any n o n- linear func tion can be approxima ted using ration al f unctions, achieving better accur acy than with a trun cated T aylor series. The standard a pproach to tackle fractional p roblems is fractional prog ramming theory [ 7]. However , this theory pro - vides algor ithms with limited complexity only if f ( x ) and g ( x ) are co ncave and co nvex, respecti vely , an d the co nstraint function s { h i ( x ) } are conv ex. I f any of these assumption s is not fulfilled, subop timal method s are need ed. On e o f them is the alternating optimizatio n method [8], which decomposes the origina l prob lem in subpro blems who se solutio ns can be computed with affordable complexity . Howe ver, this is not always th e case f or (1), since f ( x ) , g ( x ) and { h i ( x ) } may not be conv ex or concave fun c tio ns even with respect to th e individual variables { x i } . Anoth e r po ssible ap proach is g i ven by sem idefinite relax ation [9] . This method, howe ver , applies only to the q u adratic case, and is op timal only when at m ost two constraints are enfor c ed. Finally , another approach to tackle non-conc ave fraction al prob lems is the framew ork of sequential fractio nal progr amming [10], [11 ] , e ven though it is in gen eral suboptimal and its application to th e case of general multiv a r iate polyno mial fu nctions is no t straightf o rward. A. Pizzo and L. Sanguinetti are with the Univ ersity of Pisa, Dipar- timento di Ingegneria dell’Informazione , Ita ly (a ndrea.piz zo@ing.uni pi.it, luca.sangu inetti @unipi.it). A. Zappone is with the Large Systems and Net works Group, Centrale Sup ´ elec, France (alessio.zappone@l 2s.centra lesupelec.fr) Motiv ated by this backgro u nd, this work aims at in troducin g to the field of sign a l pro cessing an altern ati ve approach based on polynomial optimization th eory [13], and more specifically on the so-c a lled sum-of-sq uares’ (SOSs) ref o rmulation [ 12]. Combined with th e classical fractiona l programm in g theor y [7], we show how the po lynomial optimiza tion theory can b e used to globally solve (1) as the o rder of the SOS refo r mula- tion gr ows to in finity . This is achieved without req uiring any a priori assumptio n on the con vexity or concavity of the inv olved function s. A few preliminaries on multivariate poly nomial theory in connection with the SOS method is provid e d in the Appen dix. Due to space lim itations, we limit ourselves to an introdu ctory discussion only . For a more compreh ensiv e overview on polynomial pro grammin g by the SOS method the reader is r eferred to [12 ], [13]. The de veloped fr amew ork is then ap plied to the m axi- mization of energy efficiency (EE), defined as the b enefit- cost ratio in terms of amount o f informatio n that can be reliably transferr ed per unit o f time, in multiuser m ultiple- input multiple- output (MU-M I MO) commun ication systems, over the total co nsumed power . Particularly , the optimization is carried out with respect to the numb er o f users and an- tennas [ 14], [1 5]. Nume rical results are used to sh ow tha t the de veloped framework closely approx imates the o ptimal configur ation ( obtained b y exhau stive search ) with affordab le complexity . I I . P RO P O S E D F R A M E W O R K The state-of-th e-art approach to solve (1) is the Din kelbach’ s algorithm [ 7], which oper ates as follows. 1 Algorithm 1 Dinkelbach ’ s a lgorithm Set k = 0 ; λ k = 0 ; λ k − 1 = − 1 ; 0 < ε < 1 ; while | λ k − λ k − 1 | ≥ ε d o x k = arg max x ∈X { p k ( x ) = f ( x ) − λ k g ( x ) } ; (3) F ( λ k ) = f ( x k ) − λ k g ( x k ) , λ k +1 = f ( x k ) g ( x k ) ; k = k + 1 ; end while Algorithm 1 conver ges to the glob al op tim um x ⋆ of (1) with sup er-linear conv ergence rate, but each iteratio n k re- quires to solve the no n -fraction al auxiliary problem in (3). Unfortu n ately , (3 ) is in general non-c o n vex in the settin g o f (1), wh ich makes the direc t implem entation of Algorithm 1 computatio nally u nfeasible. The aim of this work is to show 1 W ithout loss of generali ty , we assume that g ( x ) > 0 . If g ( x ) < 0 , one can alway s replace f ( x ) g ( x ) by f ( x ) g ( x ) g 2 ( x ) , w hich has a positi ve denominator . 2 how (3 ) can be glo bally solved wh en p k ( x ) is a generic (non - conv ex) p olynom ia l function. W e begin by reformula tin g (3) into its e pigraph for m: 2 r ⋆ λ = max x ∈ R n ,t ∈ R t sub ject to h i ( x ) ≥ 0 i = 1 , . . . , m h 0 ( x , t ) = p ( x ) − t ≥ 0 . (4) In order to solve (4), we resort to the SOS reform ulation [13], [2 7], who se basic id ea is to app roximate non- negati ve polyno mials as a sum of squar e s. Specifically , fo llowing [13] , [16], the first step of the method is to emb ed all constraint function s in ( 4) into the single con stra in t σ ( x ) + σ 0 ( x ) h 0 ( x , t ) + m X i =1 σ i ( x ) h i ( x ) ≥ 0 (5) wherein, σ ( x ) ∈ SOS ℓ , σ 0 ( x ) ∈ SOS ℓ − v , and σ i ( x ) ∈ SOS ℓ − deg ( h i ) , with SOS q denoting the set of all p olynomia ls of degree q that c an be written as a sum of squares, namely: SOS q = { p ( x ) , deg ( p ) ≤ q : p ( x ) = J X j =1 θ 2 j ( x ) , deg ( θ j ) ≤ ⌈ q / 2 ⌉} , whereas v = max { deg( f ) , deg( g ) , max i deg( h i ) } , and ℓ > v is the orde r of the SOS refor mulation. I t is interesting to observe that the representation in (5) can be viewed as a generalized Lagran gian function [13 ], [17] associated with the constrain ed o ptimization prob lem in (4), with the SOS polyno mials σ ( · ) and { σ i ( · ) } m i =0 playing the role of non- negativ e Lag r ange multip liers, as in tradition al duality theory [18, Ch. 5 ] . Next, based on (5), the f ollowing SOS r eformula tio n o f (4 ) is ob tained: r ⋆ sos ,ℓ = max x ∈ R n ,t ∈ R t sub ject to σ ( x ) + σ 0 ( x ) h 0 ( x , t ) + m X i =1 σ i ( x ) h i ( x ) ≥ 0 σ i ( x ) ∈ SOS ℓ − deg( h i ) i = 1 , . . . , m σ ( x ) ∈ SOS ℓ , σ 0 ( x ) ∈ SOS ℓ − v . (6) For gener al poly nomials, Pro b lem (6 ) is still difficult to solve since its constrain t function s might no t be concave. Nev ertheless, it can b e sh own th at Problem (6) and its dual have z e ro duality gap [13 ] and that the dual of Problem (6 ) can be cast as a semi-defin ite prog ram (SDP), which therefo re can be solved in polyn omial time by using standar d semi- definite program ming tools [18]. M oreover , for a sufficiently large ℓ the solu tion of Problem (6) is also the g lobal solution of Problem (4) (and he n ce o f Pro blem ( 3), too). Formally , denote by M d and { p α } the moment matrix 3 and th e vector of coord inates in th e mon omial ba se of polyn omial p ( · ) , by M d − deg( h i ) / 2 and { h i, α } the momen t matrix an d the vector of coo rdinates in the m onomial base o f polyn omial h i , for i = 1 , . . . , M . Th en, the following theo r em holds. 2 The subscript k is omitted hereafter for notational simpli city . 3 The definitio n of moment matrix of a polynomial and of polynomial expa nsion ov er the monomial base are formally introduced in the Appendix . Theorem 1. [13] The du al of Pr oblem ( 6 ) is the SDP r ⋆ mom ,d = min y ∈ R s n,d X α ∈ N n d p α y α sub ject to M d ( y )  0 M d −⌈ deg( h i ) / 2 ⌉ ( h i, α y )  0 i = 0 , . . . , m , y (0 ,..., 0) = 1 , (7) and for any SOS r eformulation o rder ℓ , str ong duality holds, i.e. r ⋆ sos ,ℓ = r ⋆ mom ,d [13, Theor em 4.2]. Mor e over , r ⋆ sos ,ℓ → r ⋆ λ when ℓ → ∞ . The final step of the procedur e is to recover th e optimal x ⋆ ∈ R n from th e glo b al solution o f (7 ), say y ⋆ ∈ R s n,d . Follo wing [13], [19] , if (7) is feasible, then th e momen t matrix is gu aranteed to have rank one and th erefore th ere exists one vector v such that M d ( y ⋆ ) = vv T . Finally , the optimal solution of ( 3) is fou nd to be equal to x ⋆ = z ⋆ , with z ⋆ = v (2 : n + 1) . Theorem 1 e nsures that we can appro ach th e global solution of (3) within any desired tolera nce, if the SOS refor mulation order ℓ is cho sen large en ough 4 . As a con sequence, the propo sed implementation of Algor ithm 1 in which (3) is solved in each iteration by solving (7), con verges to the globa l solution of Pr oblem (1). The comp utational comp lexity of Algor ithm 1 depen d s on the nu m ber of iterations re q uired to con verge and the computatio nal complexity to solve (3). The latter is formulated as an SDP in (7), which accounts for a total number of m + 1 LMIs. Eac h LMI includ e a system of s n,d single LMIs eac h of dimension s n,d × s n,d . T h us, the comp utational com p lexity of solvin g (7) throu g h, e.g ., interior poin t me th od, is in th e order of O  n 2 ms 3 n,d + nms 4 n,d  arithmetic oper a tions [20, Ch. 11]. Also, sinc e s n,d ≈ n d , the overall co mplexity g rows polyno mially 5 with bo th nu mber of primal variables n an d d (which depends on the order ℓ of the SOS reformulatio n), and linearly with the n umber of po lynomial constrain ts m . I I I . A P P L I C AT I O N : E N E R G Y E FFI C I E N C Y M A X I M I Z A T I O N The framework developed a b ove is applied next to solve an EE max imization pr oblem in cellular networks. A. Pr oblem statem e nt Inspired by [1 5], we look for th e optima l deployme nt of a cellular n etwork for maximal EE w h ile im posing the average signal-to-in terference- plus-noise ratio ( SINR) be larger th an a given con straint γ . T he optimization v ariables are the pilot reuse factor β , the number K of users per cell and the n umber M of antenna s at each base station. Th e op tim al β is proved to be such that the SINR constraint is satisfied w ith equality . 4 In practice, we do not need to solve (7) for increasin g value s of ℓ until con verge nce, but it is enough to solve it just once , for a large enough ℓ . The numerica l analysis in Section III shows that ℓ = 12 (i.e., d = 6 ) leads to global optimality for the considered proble m. 5 Although solving (7) s eems unpractica l for problem of large size, practical problems exhibit an affordabl e computat ional complexit y for modest value s of n and d . In addit ion, most polynomials have only a few nonzero monomial coef ficients and thus sparsi ty can be le veraged; see [21] and [22]. 3 EE ( K, M ) =  1 − K τ B 1 ( K,M ) γ M − B 2 ( K ) γ  B lo g 2 (1 + γ ) K C 0 +  C 1 + U τ  K + D 0 M + D 1 K M +  1 − K τ B 1 ( K,M ) γ M − B 2 ( K ) γ  K  U + A B lo g 2 (1 + γ )  (8) EE ( x 1 , x 2 ) = f (1 , 0) x 1 + f (2 , 0) x 2 1 + f (1 , 1) x 1 x 2 + f (2 , 1) x 2 1 x 2 + f (3 , 0) x 3 1 g (0 , 0) + g (1 , 0) x 1 + g (0 , 1) x 2 + g (2 , 0) x 2 1 + g (1 , 1) x 1 x 2 + g (0 , 2) x 2 2 + g (2 , 1) x 2 1 x 2 + g (1 , 2) x 1 x 2 2 + g (3 , 0) x 3 1 (9) T ABLE I: Polynomial coef ficients associated with the constraints in (11). Parame ter V alue Parame ter V alue h 1 (0 , 0) − γ τ  1 + 2 α − 2  h 1 (2 , 0) − γ  4 ( α − 2) 2 + 1 α − 1 + 2 α − 2  h 1 (1 , 0) − γ SNR 2 α − 2 − γ τ  1 + 2 α − 2   1 + 1 SNR  h 2 (0 , 0) γ SNR  2 α − 2 + 1 + 1 SNR  h 1 (0 , 1) τ h 2 (1 , 0) γ  4 ( α − 2) 2 + 1 α − 1 + 2 α − 2  + γ  1 + 2 α − 2   1 + 1 SNR  h 1 (1 , 1) − γ α − 1 h 2 (0 , 1) γ  1 α − 1 − 1  This y ields β ⋆ = B 1 ( K,M ) γ M − B 2 ( K ) γ where B 1 ( K, M ) and B 2 ( K ) are defined in [15 , Eqs. (19)- (20)]. 6 The optimization over ( K, M ) relies on th e f ollowing pro b lem [15, Eq. (22) ] max ( K,M ) ∈ R 2 EE( K, M ) sub ject to 1 ≤ B 1 ( K, M ) γ M − B 2 ( K ) γ ≤ τ K (10) where the objectiv e f unction E E( K , M ) is given in (8) with known param eters 7 U , A , B , {C i } , {D i } . In [15] the problem is tackled by an alternating max imization approac h, which is in general suboptimal. Here, Prob le m (10 ) is tackled by the considered polynom ial fram ew o rk. T o elaborate, we define x = [ K, M ] T ∈ R 2 and rewrite the optimization prob lem a s in (1), wh ic h yield s max x EE( x ) = f ( x ) g ( x ) sub ject to h i ( x ) ≥ 0 , i = 1 , 2 (11) where EE( x ) is in (9) an d h 1 ( x ) = h 1 (0 , 0) + h 1 (1 , 0) x 1 + h 1 (0 , 1) x 2 + h 1 (1 , 1) x 1 x 2 + h 1 (2 , 0) x 2 1 and h 2 ( x ) = h 2 (0 , 0) + h 2 (1 , 0) x 1 + h 2 (0 , 1) x 2 ; the coor d inates { h 1 α } , { h 2 α } f or α ∈ N 2 2 are shown in T able I and can be easily der i ved as done in the Example 1 giv en in Appendix . Next, we solve (11) by using the framework discussed in Section II. B. Numerica l va lidation Numerical resu lts are now u sed to validate the a c curacy o f Algorithm 1 when applied with a finite o rder ℓ o f the SOS reform u lation. The har dware coefficients are taken f rom [15], while we fix the SINR co nstraint to γ = 3 , the base station density to λ = 5 BS / km 2 and the average signal-to-no ise ratio (SNR) to 0 dB. At e a ch iteration of Algorithm 1, ( 3 ) is solved by using Y ALMIP [23 ] with the solver SDPT3 [24 ]. The code is available on line at https://github.com/lucasangu inetti/ 6 W e neglect the hardwa re impairments for the sak e of simplicity . 7 Those depend on a varie ty of fixed hardwa re coef fi cient s, whose typical v alues strongly depend on the actual hardware equipment and the state-of- the-art in circuit implementation ; details can be found in [15]. 0 0 200 2 400 600 0 4 10 20 800 30 40 6 K M EE [M bit / Joule] x ⋆ = (8 , 136) x ⋆ mom = (8 , 133) k = 0 k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 Fig. 1: EE (in Mbit/Joule) as a function of M and K . The optimal is computed either by mean s of the iterativ e Algorithm 1 with ( d = 6) ℓ = 12 (red square) or exhausti ve search (black t r i angle). 1 2 3 4 5 6 7 8 9 10 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Number of itera tions k Relat i ve error ǫ d = 2 d = 4 d = 6 d = 8 Fig. 2: Relative error ǫ of Algorithm 1 as a function of the number of iterations k and for dif ferent ℓ -order S OS reformulation in (6)–(7). EE- Polynomial- Theor y f or testing d ifferent network config- urations. Fig. 1 shows the E E (measu red in Mbit/Joule) as a function of M and K , obtain ed with a exhaustive search. Algo rithm 1 conv erges in less th an ten iteratio ns to the po int x ⋆ mom = ( ⌊ K ⋆ mom ⌉ , ⌊ M ⋆ mom ⌉ ) = (8 , 1 3 3) , which yields an e rror of ǫ = 1 − r ⋆ mom /r ⋆ λ ≈ 10 − 4 with respect to the glob al o ptimal x ⋆ = (8 , 13 5) . This is ach iev ed by using 4 d = 6( ℓ = 12) . Since n = 2 , we ha ve s =  6 4  = 15 . Th us, the overall com plexity in so lving (11) is roughly O  10 5  arithmetic operation s, which with a pro cessing unit op erating at 1 0 Gflops/s [14] takes 1 0 µs per iteratio n. Fig. 2 p lots th e relativ e er r or a s a f unction of the nu mber o f iteratio ns fo r d ∈ { 2 , 4 , 6 , 8 } , which means ℓ ∈ { 4 , 8 , 12 , 1 6 } . As it can b e seen, Algorithm 1 is provably conv ergent even f or small d a nd within f ew iterations. I V . C O N C L U S I O N S W e proposed a framework for fractional poly nomial op - timization b y employing SOS reformulation metho ds with in Dinkelbach’ s iterativ e algorithm . The p roposed appr o ach ap - plies to a wid e r set of prob lems th an com peting alternatives and enjoys optimality proper ties as the o rder ℓ of the SOS reform u lation g rows to in finity . The fr amew ork was applied to the EE ma x imization o f cellular network s, and with d = 6 , ( ℓ = 12 ) was shown to conv erge in fiv e iterations and exhibits near-optimal p erform ance when compared to an exhaustive search algorithm. I t sho u ld also be observed that the consid- ered fram ework could be app lied to power contro l pr oblems for EE maximization , upon expanding all non -polyno mial function s b y T aylor series or le veragin g th e appro ximation method f rom [ 6]. A P P E N D I X Consider n, v ∈ N , the po lynomial p ( x ) : R n → R , and define th e sets: N n v = { α ∈ N n : n X i =1 α i ≤ v } , | N n v | =  n + v v  = s n,v . (12) Every p ( x ) with degree v , i.e., deg ( p ) = v , may be u niquely written as a finite linear com bination of mo nomials with maximum degree less than o r equal to v [ 25] p ( x ) = X α ∈ N n v p α x α with x α = x α 1 1 x α 2 2 . . . x α n n ∈ R (13) with co e fficients p α ∈ R for α = [ α 1 , . . . , α n ] T ∈ N n v . Th e expression in (13) represents th e expansion o f the poly nomial p ( x ) over th e can onical monomial base, and the vector p α collects the c oordina te s of the expansion. While the represen - tation in (13 ) ap plies to any mu lti-variate polyn omial, if p ( x ) can be written as an SOS, then it also admits the repre sentation p ( x ) = m d ( x ) T Wm d ( x ) , W  0 (14) where m d ( x ) = [1 , x 1 , . . . , x n , x 2 1 , x 1 x 2 , . . . , x d n ] T ∈ R s n,d is the f ull mo nomial ba sis inclu ding all the mono mials up to degree d = ⌈ v / 2 ⌉ , and W is a p ositi ve semidefinite matrix. By rea rranging ( 14) as tr( m d ( x ) m d ( x ) T W ) and u sing the fact that W  0 we infer that it must hold m d ( x ) m d ( x ) T =      1 x 1 . . . x n x 1 x 2 1 . . . x 1 x n . . . . . . . . . . . . x n x 1 x n . . . x d n       0 (15) in order f or p ( x ) to be po siti ve. Elaboratin g fur th er on (15) fo llowing [13], [17], we introdu c e the so-c alled moment matrix rep r esentation of (15). Specifically , by a lin e a rization appro ach, each entry of the matrix in (15 ), i.e., x α x β ∈ R , is replaced by y α + β ∈ R , for α , β ∈ N n d . Denoting by y = [ y 00 ... 0 , y 10 ... 0 , . . . , y 00 ...d , . . . , y 00 ... 2 d ] T ∈ R s n,d the collection of the linearized variables, (15) is reform ulated as M d ( y ) =      y 00 ... 0 y 10 ... 0 . . . y 0 ...d y 10 ... 0 y 20 ... 0 . . . y 10 ...d . . . . . . . . . . . . y 00 ...d y 10 ...d . . . y 00 ... 2 d       0 , ( 16) which is by definitio n th e moment matrix of the po lynomial p ( x ) . SOS ref o rmulation tu rns out very u seful when it is ne e ded to check the non- negati vity of a multivariate functio n w ( x ) [12], which is in gen e r al an NP-hard proble m [26]. In th is context, it is normally easier to check whether w ( x ) can be reformulated as an SOS polyno mial, which clearly implies non-negativity . In th e spe c ial case of gen eric v - degree polyn omial func tions, for which w ( x ) = p ( x ) as in (13), the SOS set is co n vex an d the feasibility test reduces to solv ing a semidefinite program (SDP) [27, Lemma 3.1]. Clearly , although being an SOS implies non-n egati vity , the co ntrary does not usu a lly hold 8 and thus, in gener al, solving a pr oblem inv o king SOS r ather th a n non-n egati vity lea ds to a subop timal solution. Howev er , by increasing the degree of the SOS po lynomials that are used to represent p ( x ) , it is possible to appr o ach the optimal solution within any pred e fin ed to lerance ( see Theo rem 1). Example 1 . In ord er to g rasp th e potential of th e above framework, we provide h ere an easy (u nconstrain e d ) problem as an example. T o this end, conside r the following: min x ∈ R 2 p ( x ) = ( x 2 − 2) 2 + 2 x 2 1 + x 1 x 2 + 5 . (17) The solutio n to (17) is clearly x ⋆ = ( x 1 , x 2 ) = ( − 1 , 2) . T o use the fr a mew ork developed above, we first write down the set N 1 2 = [(0 , 0) , (1 , 0 ) , (0 , 1 ) , (2 , 0 ) , (1 , 1 ) , (0 , 2 )] T , (18) and then retrieve { p α } = [9 , 0 , − 4 , 2 , 1 , 2] T from p ( x ) in (17). Next, the moment m atrix o f p ( x ) is obtain ed as: M 1 ( y ) =   y 0 , 0 y 1 , 0 y 0 , 1 y 1 , 0 y 2 , 0 y 1 , 1 y 0 , 1 y 1 , 1 y 0 , 2   . (19) Then, exploitin g the result in Theorem 1, (17) can be ref or- mulated as th e dua l prob lem min y 9 y 0 , 0 − 4 y 0 , 1 + 2 y 2 , 0 + y 1 , 1 + 2 y 0 , 2 sub ject to M 1 ( y )  0 , (20) whose so lu tion is fou n d to be y ⋆ = (1 , − 1 , 2 , ∗ , ∗ , ∗ ) from which we obtain x ⋆ = ( − 1 , 2) . Thus, zero duality gap is shown. 8 If n = 1 , d = 2 or ( n, v ) = (2 , 4) , then the two definition coincide s [28]. 5 R E F E R E N C E S [1] A. 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