A Semidefinite Programming approach for minimizing ordered weighted averages of rational functions

A SEMIDEFINITE PR OGRAMMING APPR O A CH F OR MINIMIZING ORDERED WEIGHTED A VERA GES OF RA TIONAL FUNCTIONS V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO Abstract. This paper considers the problem of minimizing the ordered w eighted a verage (or ordered median) function of finitely man y rational functions o ver compact semi-algebraic sets. Ordered weigh ted av erages of rational functions are not, in general, neither rational functions nor the suprem um of rational functions so that current results av ailable for the minimization of rational functions cannot b e applied to handle these problems. W e prov e that the problem can be transformed into a new problem em b edded in a higher dimension space where it admits a conv enien t representation. This reformulation admits a hierarch y of SDP relaxations that approximates, up to an y degree of accuracy , the optimal v alue of those problems. W e apply this general framework to a broad family of contin uous location problems showing that some difficult problems (conv ex and non-conv ex) that up to date could only be solved on the plane and with Euclidean distance, can be reasonably solved with differen t ` p -norms and in an y finite dimension space. W e illustrate this methodology with some extensive computational results on location problems in the plane and the 3-dimension space. 1. Introduction W eighted Averaging (OW A) or Ordered Median F unction (OMF) op erators provide a parameterized class of mean type aggregation op erators (see [25, 44] and the references therein for further details). Man y notable mean operators such as the max, arithmetic a v erage, median, k-centrum, range and min, are mem- b ers of this class. They ha v e b een widely used in location theory and computational intelligence b ecause of their abilit y to represent flexible mo dels of mo dern logistics and linguistically expressed aggregation in- structions in artificial in telligence ([25] and [39, 40, 41, 42, 43, 44]). W eighted a v erages (or ordered median) of rational functions are not, in general, neither rational functions nor the supremum of rational functions so that current results av ailable for the minimization of rational functions are not applicable. In spite of its in trinsic interest, as far as we kno w, a common approac h for solving this family of problems is not a v ailable. Nevertheless, one can find in the literature different metho ds for solving particular instances of problems within this family , see e.g. [5, 6, 14, 25, 26, 27, 28, 29, 30, 31, 32, 34]. The first goal of this pap er is to dev elop a unified to ol for solving this class of optimization problems. In this line, we prov e that the general problem can b e transformed into a new problem embedded in a higher dimension space where it admits a conv enien t representation that allows to arbitrarily approximate or to solve it as a minimization problem ov er an adequate closed semi-algebraic set. Hence, our approac h go es b eyond a trivial adaptation of current theory . Regarding the applications, it is commonly agreed that ordered median lo cation problems are among the most imp ortant applications of OW A op erators. Con tin uous lo cation has ac hieved an imp ortant degree of maturit y . Witnesses of it are the large num ber of pap ers and researc h b o oks published within this field. In addition, this developmen t has b een also recognized by the mathematical communit y since the AMS co de 90B85 is reserved for this area of research. Contin uous lo cation problems app ear v ery often in economic mo dels of distribution or logistics, in statistics when one tries to find an estimator from a data set or in pure optimization problems where one lo oks for the optimizer of a certain function. F or a comprehensive o verview the reader is referred to [4] or [25]. Despite the fact that many contin uous location problems rely hea vily on a common framew ork, sp ecific solution approaches hav e been developed for each of the t ypical ob jective functions in lo cation theory (see for instance [4]). T o ov ercome this inflexibilit y and to w ork tow ards a unified approach to lo cation theory the so called Ordered Median Problem (OMP) w as dev elop ed (see [25] and references therein). Ordered median problems represent as sp ecial cases nearly 2010 Mathematics Subject Classific ation. 90B85 ; 90C22 ; 65K05 ; 12Y05 ; 46N10. Key wor ds and phr ases. Con tinuous lo cation ; Ordered median problems ; Semidefinite programming ; Moment problem. 1 2 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO all classical ob jectiv e functions in lo cation theory , including the Median, Cen tDian, cen ter and k-centra. More precisely , the 1-facilit y ordered median problem in the plane can b e formulated as follows: A v ector of w eigh ts ( λ 1 , . . . , λ n ) is giv en. The problem is to find a location for a facility that minimizes the w eigh ted sum of distances where the distance to the closest p oin t to the facility is m ultiplied by the weigh t λ n , the distance to the second closest, b y λ n − 1 , and so on. The distance to the farthest p oint is multiplied by λ 1 . Man y lo cation problems can b e formulated as the ordered 1-median problem by selecting appropriate w eights. F or example, the v ector for which all λ i = 1 is the unw eighted 1-median problem, the problem where λ n = 1 and all others ar e equal to zero is the 1-cen ter problem, the problem where λ 1 = . . . = λ k = 1 and all others are equal to zero is the k -centrum. Minimizing the range of distances is achiev ed by λ 1 = 1, λ n = − 1 and all others are zero. Despite its full generality , the main drawbac k of this framework is the difficult y of solving the problems with a unified to ol. There hav e b een some successful approac hes that are no w a v ailable whenever the framework space is either discrete (see [2, 22, 30]) or a netw ork (see [11], [12] or [24]). Nevertheless, the contin uous case has b een, so far, only partially cov ered. There hav e b een some attempts to o vercome this drawbac k and there are now ada ys some av ailable metho dologies to tackle these problems, at least in the plane and with Euclidean norm. In Drezner [3] and Drezner and Nick el [5, 6] the authors present t wo differen t approac hes. The first one uses a contin uous branc h and b ound metho d based on triangulations (BTST) and the second one on a D-C decomp osition for the ob jective function that allo w solving the problems on the plane. More recen tly , Ro driguez-Chia et al. [34] also address the particular case of the k -centrum problem and using geometric arguments develop a b etter algorithm applicable only for that problem on the plane and Euclidean distances. Quoting the conclusions of the authors of [5]: “A l l our exp eriments wer e c onducte d for Euclide an distanc es. As futur e r ese ar ch we suggest to test these algorithms on pr oblems (even the same pr oblems) b ase d on other distanc e me asur es. (...) Solving k-dimensional pr oblems by a similar appr o ach r e quir es the c onstruction of k-dimensional V or onoi diagr ams which is extr emely c omplic ate d.” Therefore, the c hallenge is to design a common approac h also to solve the ab ov e men tioned family of lo cation problems, for different distances and in an y finite dimension. This is essentially the second goal of this pap er. In our wa y , we hav e addressed the more general problem that consists of the minimization of the OW A op erator of a finite num ber of rational functions o v er closed semialgebraic sets that is the first goal of this paper. Th us, our second goal is to solve a general class of con tin uous lo cation problems using the general approac h mentioned abov e for the minimization of O W A rational functions and to sho w the p o w erfulness of this metho dology . Of course, we know that the problem in its full generality is N P − har d since it includes general instances of conv ex minimization. Therefore, we cannot exp ect to obtain p olynomial algorithms for this class of problems. Rather, we will apply a new metho dology first prop osed by Lasserre [16], that provides a hierarc hy of s emidefinite problems that conv erge to the optimal solution of the original problem, with the prop ert y that eac h auxiliary problem in the pro cess can b e solv ed in p olynomial time. The pap er is organized in 5 sections. The first one is our in tro duction. In the second section and for the sake of completeness, we recall some general results on the Theory of Moments and Semidefinite Programming (SDP) that will b e useful in the rest of the pap er. Section 3 considers what we call the MOMRF problem whic h consists of minimizing the or der e d me dian function of finitely many rational functions ov er a compact basic semi-algebraic set. In the spirit of the momen t approach developed in Lasserre [16, 18] for p olynomial optimization and later adapted by Jib etean and De Klerk [10], w e define a hierarch y of semidefinite relaxations (in short SDP relaxations). Each SDP relaxation is a semidefinite program which, up to arbitrary (but fixed) precision, can b e solv ed in p olynomial time and the monotone sequence of optimal v alues asso ciated with the hierarch y conv erges to the optimal v alue of MOMRF . Sometimes the con v ergence is finite and a sufficien t condition p ermits to detect whether a certain relaxation in the hierarc h y is exact (i.e. provides the optimal v alue), and to extract optimal solutions (theoretical b ounds on the relaxation order for the exact results can b e found in [35, 36]). Section 4 considers a general family of lo cation problems that is built from the problem MOMRF but whic h do es not actually fits under the same form ulation b ecause the ob jective functions are not quotients of polynomials. Nevertheless, w e pro ve that under a certain reformulation one can define another hierarch y of SDP that fulfils conv ergence prop erties ‘` a la Lasserre’. This approach is applicable to lo cation problems with any ` p -norm ( p ∈ Q ) and in an y finite dimension space. W e exploit the sp ecial structure of these problems to find a blo ck diagonal reform ulation that reduces the sizes of the SDP relaxations and allows to solv e larger instances. MINIMIZING ORDERED WEIGHTED A VERAGING OF RA TIONAL FUNCTIONS 3 Our computational tests are presen ted in Section 5. W e analyze five families of problems, namely , W eb er, cen ter, k -centrum, trimmed-mean and range. There we show that conv ergence is rather fast and very high accuracy is ac hiev ed in all cases, even with the first feasible relaxation. (W e observ e that for lo cation problems with Euclidean distances that relaxation order is r = 2.) The pap er ends with some conclusions and an outlo ok for further researc h. 2. Preliminaries In this section we recall the main definitions and results on the moment problem and semidefinite programming that will b e useful for the developmen t through this pap er. W e use standard notation in the field (see e.g. [20]). W e denote b y R [ x ] the ring of real polynomials in the v ariables x = ( x 1 , . . . , x n ), and b y R [ x ] d ⊂ R [ x ] the space of p olynomials of degree at most d ∈ N (here N denotes the set of nonnegativ e integers). W e also denote by B = { x α : α ∈ N n } a canonical basis of monomials for R [ x ], where x α = x α 1 1 · · · x α n n , for an y α ∈ N n . F or any sequence indexed in the canonical monomial basis B , y = ( y α ) α ∈ N n ⊂ R , let L y : R [ x ] → R b e the linear functional defined, for an y f = P α ∈ N n f α x α ∈ R [ x ], as L y ( f ) := P α ∈ N n f α y α . The moment matrix M d ( y ) of order d asso ciated with y , has its rows and columns indexed by ( x α ) and M d ( y )( α, β ) := L y ( x α + β ) = y α + β , for | α | , | β | ≤ d. Note that the momen t matrix is  n + d n  ×  n + d n  and that there are  n +2 d n  y α v ariables. F or g ∈ R [ x ] (= P γ g γ ∈ N n x γ ), the lo c alizing matrix M d ( g , y ) of order d asso ciated with y and g , has its ro ws and columns indexed by ( x α ) and M d ( g , y )( α , β ) := L y ( x α + β g ( x )) = P γ g γ y γ + α + β , for | α | , | β | ≤ d . Definition 1. L et y = ( y α ) ⊂ R b e a se quenc e indexe d in the c anonic al monomial b asis B . We say that y has a r epr esenting me asur e supp orte d on a set K ⊆ R n if ther e is some finite Bor el me asur e µ on K such that y α = Z K x α dµ ( x ) , for al l α ∈ R n . The main assumption that is needed to impose when one wan ts to assure the con v ergence of the SDP relaxations for solving p olynomial optimization problems (see for instance [19, 20]) was introduced by Putinar [33] and it is stated as follows. Putinar’s Prop erty . L et { g 1 , . . . , g l } ⊂ R [ x ] and K := { x ∈ R d : g j ( x ) ≥ 0 , : j = 1 , . . . , ` } a b asic close d semialgebr aic set. Then, K satisfies Putinar’s pr op erty if ther e exists u ∈ R [ x ] such that: (1) { x : u ( x ) ≥ 0 } ⊂ R n is c omp act, and (2) u = σ 0 + P ` j =1 σ j g j , for some σ 1 , . . . , σ l ∈ Σ[ x ] . (This expr ession is usual ly c al le d a Putinar’s r epr esentation of u over K ). Being Σ[ x ] ⊂ R [ x ] the subset of p olynomials that ar e sums of squar es. Note that Putinar’s prop ert y is equiv alent to impose that the quadratic polynomial M − P n i =1 x 2 i has a Putinar’s representation ov e r K . W e observe that Putinar’s prop erty implies compactness of K . It is easy to see that Putinar’s prop erty holds if either { x : g j ( x ) ≥ 0 } is compact for some j , or all g j are affine and K is compact. F urthermore, Putinar’s prop ert y is not restrictiv e at all, since any semialgebraic set K for whic h is kno wn that P n i =1 x 2 i ≤ M holds for some M > 0 and for all x ∈ K , K = K ∪ { g l +1 ( x ) := M − P n i =1 x 2 i ≥ 0 } v erifies Putinar’s prop ert y . The imp ortance of Putinar’s prop ert y stems from the following result: Theorem 2 (Putinar [33]) . L et { g 1 , . . . , g l } ⊂ R [ x ] and K := { x ∈ R d : g j ( x ) ≥ 0 , : j = 1 , . . . , ` } satisfying Putinar’s pr op erty. Then: (1) Any f ∈ R [ x ] which is strictly p ositive on K has a Putinar’s r epr esentation over K . (2) y = ( y α ) has a r epr esenting me asur e on K if and only if M d ( y )  0 , and M d ( g j , y )  0 , for al l j = 1 , . . . , l and d ∈ N . (Her e, the symb ol  0 stands for semidefinite p ositive matrix.) 4 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO The following result that app ears in [10] and [15] will b e also imp ortan t for the dev elopment in the next sections. Lemma 3. L et K ⊂ R d b e c omp act and let p, q b e c ontinuous with q > 0 on K . L et M ( K ) b e the set of finite Bor el me asur es on K and let P ( K ) ⊂ M ( K ) b e its subset of pr ob ability me asur es on K . Then min µ ∈P ( K ) R K p dµ R K q dµ = min ϕ ∈M ( K ) { Z K p dϕ : Z K q dϕ = 1 } = min µ ∈P ( K ) Z K p q dµ = min x ∈ K p ( x ) q ( x ) . 3. Minimizing the ordered weighted a verage of finitel y many ra tional functions Let K ⊂ R d b e a basic semi-algebraic set defined as K := { x ∈ R d : g j ( x ) ≥ 0 , j = 1 , . . . , ` } for g 1 , . . . , g ` ∈ R [ x ]. Let us in tro duce the function OM( x ) = P m k =1 λ k ( x ) f ( k ) ( x ), for some rational functions ( f j ) ⊂ R [ x ], b eing f k = p k /q k rational functions with p k , q k ∈ R [ x ], λ k ( x ) ∈ R [ x ], and f ( k ) ( x ) ∈ { f 1 ( x ) , . . . , f m ( x ) } suc h that f (1) ( x ) ≥ f (2) ( x ) ≥ · · · ≥ f ( m ) ( x ) for x ∈ R n . W e assume that K satisfies Putinar’s prop erty and that q k > 0 on K , for ev ery k = 1 , . . . , m . Consider the following problem: (OMRP 0 λ ) ρ λ := min x { OM( x ) : x ∈ K } , Asso ciated with the ab ov e problem we in tro duce an auxiliary problem. F or eac h i = 1 , . . . , m , j = 1 , . . . , m consider the decision v ariables w ij that mo del for each x ∈ K w ij =  1 if f i ( x ) = f ( j ) ( x ), 0 otherwise. . No w, w e consider the problem: ρ λ = min x,w m X j =1 λ j ( x ) m X i =1 f i ( x ) w ij (OMRP λ ) s.t. m X j =1 w ij = 1 , for i = 1 , . . . , m, (1) m X i =1 w ij = 1 , for j = 1 , . . . , m, w 2 ij − w ij = 0 , for i, j = 1 , . . . , m, m X i =1 w ij f i ( x ) ≥ m X i =1 w ij +1 f i ( x ) , j = 1 , . . . , m, (2) m X i =1 m X j =1 w 2 ij ≤ m, (3) w ij ∈ R , for i, j = 1 , . . . , m, x ∈ K . (4) The first set of constraints ensures that for each x , f i ( x ) is sorted in a unique position. The second set ensures that the j th p osition is only assigned to one rational function. The next constrain ts are added to assure that w ij ∈ { 0 , 1 } . The fourth one states that f (1) ( x ) ≥ · · · ≥ f ( m ) ( x ). The last set of constrain ts ensures the satisfaction of Putinar’s prop erty of the new feasible region. (Note that this last set of constrain ts are redundan t but it is conv enien t to add them for a b etter description of the feasible set.) These tw o problems, (OMRP 0 λ ) and (OMRP λ ) satisfy the following relationship. Theorem 4. L et x b e a fe asible solution of (OMRP 0 λ ) then ther e exists a solution ( x, w ) for (OMRP λ ) such that their obje ctive values ar e e qual. Conversely, if ( x, w ) is a fe asible solution for (OMRP λ ) then ther e exists a solution x for (OMRP 0 λ ) having the same obje ctive value. In p articular % λ = ˆ % λ . MINIMIZING ORDERED WEIGHTED A VERAGING OF RA TIONAL FUNCTIONS 5 Pr o of. Let ¯ x b e a feasible solution of (OMRP 0 λ ). Then, it clearly satisfies that ¯ x ∈ K . In addition, let σ b e the p ermutation of (1 , . . . , m ) suc h that f σ (1) ( ¯ x ) ≥ f σ (2) ( ¯ x ) ≥ . . . ≥ f σ ( m ) ( ¯ x ). T ak e, w ij =  1 if i = σ ( j ) , 0 otherwise . Clearly , ( ¯ x, w ) satisfy the constraints in (1-4). Indeed, for an y i P m j =1 w ij = w iσ − 1 ( i ) = 1. Analogously , for an y j , P m i =1 w ij = w σ ( j ) ,j = 1. By its o wn definition, w only tak es 0 , 1 v alues and thus, w 2 ij − w ij = 0 for all i, j and P m i,j w 2 ij ≤ m . Finally , to prov e that ( x, w ) satisfies (2), w e observe, w.l.o.g., that for any j there exist i ∗ and ˆ i such that σ ( j ) = i ∗ and σ ( j + 1) = ˆ i . Hence,: m X i =1 w ij f i ( ¯ x ) = w i ∗ j f σ ( j ) ( ¯ x ) ≥ w ˆ ij +1 f σ ( j +1) ( ¯ x ) = m X i =1 w ij +1 f i ( ¯ x ) . Moreo ver, O M λ ( ¯ x ) = m X j =1 λ j ( x ) m X i =1 f i ( ¯ x ) w ij . Con versely , if ( ¯ x, w ) is a feasible solution of (OMRP λ ) then, clearly ¯ x is feasible of (OMRP 0 λ ) and by the ab ov e, O M λ ( ¯ x ) = P m j =1 λ j ( x ) P m i =1 f i ( ¯ x ) w ij .  Then, w e observe that f i = p i /q i for each i = 1 , . . . , m . Therefore, the constrain t P m i =1 w ij f i ( x ) ≥ P m j =1 w ij +1 f i ( x ) can b e written as a p olynomial constraint as m X i =1 w ij p i ( x ) m Y k 6 = i q k ( x ) ≤ m X i =1 w ij +1 p i ( x ) m Y k 6 = i q k ( x ) j = 1 , . . . , m. Let us denote by K the basic closed semi-algebraic set that defines the feasible region of (OMRP λ ). Lemma 5. If K ⊂ R m satisfies Putinar’s pr op erty then K ⊂ R n + m 2 satisfies Putinar’s pr op erty. Pr o of. Since K satisfies Putinar’s prop erty , the quadratic p olynomial x 7→ u ( x ) := M − k x k 2 2 can b e written as u ( x ) = σ 0 ( x ) + P p j =1 σ j ( x ) g j ( x ) for some s.o.s. polynomials ( σ j ) ⊂ Σ[ x ]. Next, consider the p olynomial ( x, w ) 7→ r ( x, w ) = M + m − k x k 2 2 − m X i =1 m X j =1 w 2 ij . Ob viously , its level set { ( x, w ) ∈ R n × m 2 : r ( x, z ) ≥ 0 } ⊂ R n + m 2 is compact and moreov er, r can b e written in the form r ( x, w ) = σ 0 ( x ) + p X j =1 σ j ( x ) g j ( x ) + 1 × g ( x,w ) def ining K z }| { ( m − m X i =1 m X j =1 w 2 ij ) , for appropriate s.o.s. p olynomials ( σ 0 j ) ⊂ Σ[ x, w ]. Therefore K satisfies Putinar’s property , the desired result.  No w, we observe that the ob jectiv e function of (OMRP λ ) can b e written as a quotien t of p olynomials in R [ x, w ]. Indeed, take (5) p λ ( x, w ) = m X j =1 λ j ( x ) m X i =1 w ij p i ( x ) m Y k 6 = i q k ( x ) and q λ ( x, w ) = m Y k =1 q k ( x ) . Then, (6) m X j =1 λ j ( x ) m X i =1 f i ( x ) w ij = p λ ( x, w ) q λ ( x, w ) . 6 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO Then, we can transform Problem (OMRP λ ) in an infinite dimension linear program on the space of Borel measures defined on K . Prop osition 6. L et K ⊂ R n + m 2 b e the close d b asic semi-algebr aic set define d by the c onstr aints (1-4). Consider the infinite-dimensional optimization pr oblem P λ : b ρ λ = min x,w  Z K p λ dµ : Z K q λ dµ = 1 , µ ∈ M ( K )  , b eing p λ , q λ ∈ R [ x, w ] as define d ab ove. Then ρ λ = b ρ λ . Pr o of. It follows by applying Lemma 3 to the reformulation of (OMRP λ ) with the ob jectiv e function written using p λ and q λ in (5).  The reader may note the great generality of this class of problems. Dep ending on the choice of the p olynomial weigh ts λ we get different classes of problems. Among then, we emphasize the imp ortant instances given by: (1) λ = (1 , 0 , . . . , 0 , 0) which corresp onds to minimize the maximum of a finite num ber of rational functions, (2) λ = (1 , ( k ) . . ., 1 , 0 , . . . , 0) which corresp onds to minimize the sum of the k -largest rational functions ( k -centrum) (3) λ = (0 , ( k 1 ) . . . , 0 , 1 , . . . , 1 , 0 , ( k 2 ) . . . , 0) which mo dels the minimization of the ( k 1 , k 2 )-trimmed mean of m rational functions,... (4) λ = (1 , α , . . . , α ) which corresp onds to the α -centdian, i.e. minimizing the c on v ex combination of the sum and the maximum of the set of rational functions. (5) λ = (1 , . . . , − 1) which corresp onds to minimize the range of a set of rational functions. Remark 7. Pr oblem OMRP 0 λ c an b e e asily extende d to de al with the minimization of the or der e d me dian function of a finite numb er of other or der e d me dian of r ational functions. The r e ader may observe that this c an b e done by p erforming a similar tr ansformation to the one in (OMRP λ ) and thus lifting the original pr oblem into a higher dimension sp ac e. 3.1. Some remark able sp ecial cases. The ab ov e general analysis extends the general theory of Lasserre to the case of ordered weigh ted a verages of rational functions. Notice that this approac h go es b eyond a trivial adaptation of that theory since ordered w eigh ted av erages of rational functions are not, in general, neither rational functions nor the supremum of rational functions so that current results cannot be applied to handle these problems. Ho wev er, one can transform the problem into a new problem embedded in a higher dimension space where it admits a representation that can b e cast in the minimization of another rational function in a con venien t closed semi-algebraic set. Needless to say that the n um b er of indeterminates increases with resp ect to the original one. This may b ecome a problem in particular implemen tations due to the current state of semidefinite solvers. In some important particular cases that ha ve been extensiv ely been considered in the field of Op erations Researc h the ab ov e approach can b e further simplified as w e will show in the following. One of this cases, the minimization of the maximum of finitely man y rational functions, has b een already analyzed by Laraki and Lasserre [15]. W e will show that the approach in [15] is also a particular case of the analysis that w e presen t in the following. F or the rest of this subsection we will restrict ourselv es, for the sak e of readability , to the case of scalar (real) lambda weigh ts. W e will b egin with the case of λ = (1 , ( k ) . . ., 1 , 0 . . . , 0), for 1 ≤ k ≤ m . Note that for the case k = 1 we will recov er the case analyzed in [15], the case k = m is trivial since it reduces to minimize the ov erall sum and the remaining cases are not y et kno wn. W e are in terested in finding the minimum of the sum of the k -largest v alues { f 1 ( x ) , . . . , f m ( x ) } for all x ∈ K , b eing a closed basic semi-algebraic set. In other w ords, for an y k , k = 1 , . . . , m − 1, w e wish to solv e the problem: % := min x ∈ K S k ( x ) := k X j =1 f ( j ) ( x ) . MINIMIZING ORDERED WEIGHTED A VERAGING OF RA TIONAL FUNCTIONS 7 W e observe that for a given x , w e ha v e: S k ( x ) = k X j =1 f ( j ) ( x ) = max { m X j =1 v j f j ( x ) : m X j =1 v j = k , 0 ≤ v j ≤ 1 , ∀ j } . Therefore, by duality in linear programming: S k ( x ) = min { k t + m X j =1 r j : t + r j ≥ f j ( x ) , r j ≥ 0 , ∀ j } . Finally , we consider the problem: ˆ % := min k t + m X j =1 r j s.t. t + r j ≥ f j ( x ) , j = 1 , . . . , m (kC) r j ≥ 0 , j = 1 , . . . , m, x ∈ K . Let us denote by K the basic closed semi-algebraic set that defines the feasible region of (kC). Lemma 8. If K ⊂ R n satisfies Putinar’s pr op erty then K ⊂ R n + m +1 satisfies Putinar’s pr op erty. Mor e- over % = ˆ % . Pr o of. Since we hav e assumed K to b e compact, for any j = 1 , . . . , m , there exist LB j , U B j suc h that for an y x ∈ K , LB j ≤ f j ( x ) ≤ U B j . Let us denote LB = min j =1 ..m LB j and U B = max j =1 ..m U B j . Consider an arbitrary k , 1 ≤ k ≤ m − 1 and an arbitrary (but fixed) ¯ x ∈ K . Without loss of generality , assume that f m ( ¯ x ) ≥ . . . ≥ f 1 ( ¯ x ). W e define the function g ( t ) := min { k t + m X j =1 r j : t + r j ≥ f j ( ¯ x ) , r j ≥ 0 , ∀ j = 1 , .., m } . Clearly , g is piecewise linear and con v ex; and it attains its minimum on any p oin t of the in terv al I k = ( f k +1 ( ¯ x ) , f k ( ¯ x )]. Indeed, observe that for any t ∈ I k , the slop e of g (i.e. its deriv ativ e with resp ect to t ) is n ull since: g ( t ) = k t + k X j =1 ( f j ( ¯ x ) − t ) = k X j =1 f j ( ¯ x ) = S k ( ¯ x ) . ¿F rom the ab o v e, w e observ e that % = min x ∈ K S k ( x ) = min x ∈ K min { g ( t ) : k t + m X j =1 r j : t + r j ≥ f j ( x ) , r j ≥ 0 , ∀ j = 1 , .., m } = ˆ %. It remains to prov e that K , the feasible region of problem (kC), satisfies Putinar’s condition. First, w e observ e from the argumen t ab o ve that in order to obtain the minim um v alue of the function g , for an y k = 1 , .., m − 1 and any x ∈ K , w e only need to consider the range t ∈ ( f ( m ) ( x ) , f (1) ( x )]. Hence, the o verall range for t can b e restricted to LB ≤ t ≤ U B . On the other hand, for any x ∈ K , the constraints 0 ≤ r j ≤ f j ( x ) − t set the range of the v ariable r j . Hence 0 ≤ r j ≤ U B j − LB , ∀ j = 1 , . . . , m. Including the constrain ts, LB ≤ t ≤ U B , 0 ≤ r j ≤ U B j − LB , ∀ j = 1 , . . . , m , in the definition of K do es not change the v alue of ˆ % and makes the feasible set compact. Thus, satisfying Putinar’s condition.  8 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO This approac h extends also to the more general case of non-increasing monotone lambda-weigh ts, i.e. λ 1 ≥ λ 2 ≥ ... ≥ λ m ≥ λ m +1 := 0 (Note that we define an artificial λ m +1 to be equal to 0). In this case the problem to b e solv ed is: % λ := min x ∈ K M O M λ ( x ) := m X j =1 λ j f ( j ) ( x ) . W e observe that for a fixed x ∈ K , we can write the ob jective function as: M O M λ ( x ) = m X j =1 ( λ j − λ j +1 ) S j ( x ) . Then, we introduce the problem ˆ % λ := min m X k =1 ( λ k − λ k +1 ) S k ( x ) (7) t k + r kj ≥ f j ( x ) , j, k = 1 , . . . , m, r kj ≥ 0 , j, k = 1 , . . . , m, x ∈ K . Let us denote by K the basic closed semi-algebraic set that defines the feasible region of the Problem (7). No w, based in the previous lemma, it is straightforw ard to chec k the following result. Lemma 9. If K ⊂ R n satisfies Putinar’s pr op erty then K ⊂ R n + m 2 + m satisfies Putinar’s pr op erty. Mor e over % λ = ˆ % λ . Another class of problems that can also b e analyzed giving rise to a more compact formulation that the one in the general approach (OMRP λ ) is the trimmed mean problem. A trimmed mean ob jectiv e app ears for λ = ( k 1 z }| { 0 , . . . , 0 , 1 , . . . , 1 , k 2 z }| { 0 , . . . , 0). This family of problems has attracted a lot of atten tion in last times in the field of location analysis b ecause of its connections to robust solution concepts. Its rationale rests on the trimmed mean concepts in statistics where the extreme observ ations ( outliers ) are remo v ed to compute the central estimates ( me an ) of a sample. Thus, we are lo oking for a p oint x ∗ that minimizes the sum of the central functions, once w e ha ve excluded the k 2 smallest and the k 1 largest. F ormally , the problem is: % = min x ∈ R n n − k 2 X i = k 1 +1 f ( i ) ( x ) . No w, we observe that P n − k 2 i = k 1 +1 f ( i ) ( x ) = S n − k 2 ( x ) − S k 1 ( x ). Therefore, using the ab ov e transformation w e ha v e: S k 1 ( x ) = = max { m X j =1 v j f j ( x ) : m X j =1 v j = k 1 , 0 ≤ v j ≤ 1 , ∀ j } , S n − k 2 ( x ) = min { ( n − k 2 ) t + m X j =1 r j : t + r j ≥ f j ( x ) , r j ≥ 0 , ∀ j } . MINIMIZING ORDERED WEIGHTED A VERAGING OF RA TIONAL FUNCTIONS 9 Th us, using b oth reformulations the trim-mean problem results in: ˆ % := min ( n − k 2 ) t + m X j =1 r j − m X j =1 v j f j ( x ) s.t. m X j =1 v j = k 1 , t + r j ≥ f j ( x ) , j = 1 , . . . , m, (kT r) r j ≥ 0 , j = 1 , . . . , m, v j ( v j − 1) = 0 , j = 1 , . . . , m, x ∈ K . Let us denote by K the basic closed semi-algebraic set that defines the feasible region of (kT r). Lemma 10. If K ⊂ R n satisfies Putinar’s pr op erty then K ⊂ R n + m +1 satisfies Putinar’s pr op erty. Mor e over % = ˆ % . Remark 11. We observe that the sp e cial formulations for k-c entrum (kC) and trim-me an (kT r) ar e sp e cial ly suitable for hand ling these two classes of pr oblems. First of al l, we note that if k 1 = 0 the pr oblem r e duc es to a k 2 -c entrum, variables v j ar e not ne e de d and formulation (kT r) simplifies exactly to (kT r) . Se c ond, we p oint out that b oth formulations take advantage of the sp e cial structur e of the c onsider e d pr oblems and thus they ar e simpler than the gener al formulation (OMRP λ ) applie d to these pr oblems. A ctual ly, the numb er of variables in (kC) , for solving the k-c entrum pr oblem (r esp. (kT r) for solving the trim-me an pr oblem), is m + n + 1 (r esp. 2 m + d + 1 ) while the numb er of variables for the same pr oblem using (OMRP λ ) is m 2 + n . This r e duction is r emarkable due to the curr ent status of SDP solvers which ar e not at a pr ofessional level. In spite of that, those pr oblems, wher e no sp e cial structur e is known or it c annot b e exploite d, c an also b e tackle d using the gener al formulation (OMRP λ ) at the pric e of using lar ger numb er of variables. 3.2. A con v ergence result of semidefinite relaxations ‘ ` a la Lasserre’. W e are now in p osition to define the hierarch y of semidefinite relaxations for solving the MOMRF problem. Let y = ( y α ) b e a real sequence indexed in the monomial basis ( x β w γ ) of R [ x, w ] (with α = ( β , γ ) ∈ N n × N m 2 ). Let p λ ( x, w ) and q λ ( x, w ) b e defined as in (5). Let h 0 ( x, w ) := p λ ( x, w ), and denote ξ j := d (deg g j ) / 2 e , ν j := d (deg h j ) / 2 e and ν 0 j := d (deg h 0 j ) / 2 e where { g 1 , . . . , g ` } are the polynomial constraints that define K and { h 1 , . . . , h m } and { h 0 1 , . . . , h 0 m } are, resp ectiv ely , the p olynomial constraints (2) and (3) in K \ K , resp ectively . Let us denote by I (0) = { 1 , . . . , n } and I ( j ) = { ( j, k ) } k =1 ,...,m , for all j = 1 , . . . , m . With x ( I (0)), w ( I ( j )) we refer, resp ectiv ely , to the monomials x , w indexed only by subsets of elements in the sets I (0) and I ( j ), resp ectively . Then, for g k , with k = 1 , . . . , ` , let M r ( y , I (0)) (respectively M r ( g k y , I (0))) b e the moment (resp. localizing) submatrix obtained from M r ( y ) (resp. M r ( g k y )) retaining only those ro ws and columns indexed in the canonical basis of R [ x ( I (0))] (resp. R [ x ( I (0))]). Analogously , for h j and h 0 j , j = 1 , . . . , m , as defined in (2) and (3), resp ectiv ely , let M r ( y , I (0) ∪ I ( j ) ∪ I ( j + 1)) (resp ectively M r ( h j y , I (0) ∪ I ( j ) ∪ I ( j + 1)), M r ( h 0 j y , I (0) ∪ I ( j ) ∪ I ( j + 1)) ) b e the moment (resp. lo calizing) submatrix obtained from M r ( y ) (resp. M r ( h j y ), M r ( h 0 j y )) retaining only those rows and columns indexed in the canonical basis of R [ x ( I (0)) ∪ w ( I ( j )) ∪ w ( I ( j + 1))] (resp. R [ x ( I (0)) ∪ w ( I ( j )) ∪ w ( I ( j + 1))]). 10 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO F or r ≥ max { r 0 , ν 0 } where r 0 := max k =1 ,...,` ξ k , ν 0 := max { max j =1 ,...,m ν j , max j =1 ,...,m ν 0 j } , we introduce the follow- ing hierarch y of semidefinite programs: ( Q r ) min y L y ( p λ ) s . t . M r ( y , I (0))  0 , M r − ξ k ( g k y , I (0))  0 , k = 1 , . . . , `, M r ( y , I (0) ∪ I ( j ) ∪ I ( j + 1))  0 , j = 1 , . . . , m, M r − ν j ( h j y , I (0) ∪ I ( j ) ∪ I ( j + 1))  0 , j = 1 , . . . , m, M r − ν 0 j ( h 0 j y , I (0) ∪ I ( j ) ∪ I ( j + 1))  0 , j = 1 , . . . , m, L y ( P m i =1 w ij − 1) = 0 , j = 1 , . . . , m, L y ( P m j =1 w ij − 1) = 0 , i = 1 , . . . , m, L y ( w 2 ij − w ij ) = 0 , i, j = 1 , . . . , m, L y ( q λ ) = 1 , with optimal v alue denoted inf Q r (and min Q r if the infimum is attained). Theorem 12. L et K ⊂ R n + m 2 (c omp act) b e the fe asible domain of (OMRP λ ) . Then, with the notation ab ove: (a) inf Q r ↑ ρ λ as r → ∞ . (b) L et y r , b e an optimal solution of the SDP r elaxation ( Q r ). If rank M r ( y r , I (0)) = rank M r − r 0 ( y r , I (0)) rank M r ( y r , I (0) ∪ I ( j ) ∪ I ( j + 1)) = rank M r − ν 0 ( y r , I (0) ∪ I ( j ) ∪ I ( j + 1)) j = 1 , . . . , m (8) and if rank(M r ( y ∗ , I (0) ∪ ( I ( k ) ∪ I ( k + 1)) ∩ ( I ( j ) ∪ I ( j + 1)))) = 1 for al l j 6 = k then min Q r = ρ λ . Mor e over, let ∆ j := { ( x ∗ ( j ) , w ∗ ( j )) } b e the set of solutions obtaine d by the c ondition (8). Then, every ( x ∗ , w ∗ ) such that ( x ∗ i , w ∗ i ) i ∈ I ( j ) = ( x ∗ ( j ) , w ∗ ( j )) for some ∆ j is an optimal solution of Pr oblem MOMRF . Pr o of. The conv ergence of the semidefinite relaxation ( Q r ) was prov ed by Jib etean and De Klerk [10] for a general rational function ov er a closed semialgebraic set. Here, w e use that result applied to the rational function in (6). Moreov er, the index set of the indeterminates in the feasible set giv en by constraints (1)-(4) admits the decomp osition I ( k ), k = 0 . . . , m that satisfies the running intersection property (see [17, (1.3)]) and therefore, the result follows by combining Theorem 3.2 in [17] and the results in [10].  The abov e theorem allo ws us to appro ximate and solv e the original problem MOMRF up to an y degree of accuracy by solving blo c k diagonal (sparse) SDP programs whic h are conv ex programs for each fixed relaxation order r and that can b e solv ed with a v ailable op en source solv ers as SeDuMi, SDP A, SDPT3 [13], etc. 4. Generalized Loca tion Problems with ra tional objective functions This sections considers a w ide family of con tin uous lo cation problems that has attracted a lot of attention in the recen t literature of lo cation analysis but for which there are not common solution approaches. The c hallenge is to design a common resolution approach to solve them for different distances and in an y finite dimension. W e are given a set A = { a 1 , . . . , a n } ⊂ R d endo wed with an ` τ -norm (here ` τ stands for the norm k x k τ =  P d i =1 | x i | τ  1 τ , for all x ∈ R d ); and a feasible domain K ⊂ R d , closed and semi-algebraic. The goal is to find a p oint x ∗ ∈ K ⊂ R d minimizing some globalizing function of the distances to the set A . Here, we consider that the globalizing function is rather general and that it is given as a rational function. Some well-kno wn examples are listed b elo w (see e.g. [1], [3], [9], [21] or [25]) : • f ( u 1 , . . . , u n ) = n X i 0 for all j . W e shall define the dep endence of f j to the decision v ariable x ∈ R d via u = ( u 1 , . . . , u n ), where u i : R d 7→ R , u i ( x ) := k x − a i k τ , i = 1 , . . . , n . Therefore, the j -th comp onent of the ordered median ob jective function of our problems reads as: ˜ f j ( x ) : R d 7→ R x 7→ ˜ f j ( x ) := f j ( k x − a 1 k τ , . . . , k x − a n k τ ) . Consider the following problem: ( LOCOMRF ) ρ λ := min x { m X j =1 λ j ( x ) ˜ f ( j ) ( x ) : x ∈ K } , where: • K ⊆ R n satisfies Putinar’s prop erty , • τ := r s , r, s ∈ N , r ≥ s and g cd ( r , s ) = 1. This problem do es not reduce to the family MOMRF considered ab ov e since the dependence on the decision v ariable x is not given in the form of p olynomials. Note that ` τ -norms are not, in general, p olynomials. T o av oid this inconv enience, we introduce the following auxiliary problem. ρ λ = min x,w,u,v m X j =1 λ j ( x ) m X i =1 f i ( u ) w ij (9) s.t. m X j =1 w ij = 1 , for i = 1 , . . . , m, m X i =1 w ij = 1 , for j = 1 , . . . , m, m X i =1 w ij f i ( u ) ≥ m X i =1 w ij +1 f i ( u ) , j = 1 , . . . , m, w 2 ij − w ij = 0 , for i, j = 1 , . . . , m, v s kl ≥ ( x k − a kl ) r , k = 1 , . . . , n, l = 1 , . . . , d, (10) v s kl ≥ ( a kl − x l ) r , k = 1 , . . . , n, l = 1 , . . . , d, (11) u r k = ( d X l =1 v kl ) s , k = 1 , . . . , n, m X i,j =1 w 2 ij ≤ m, w ij ∈ R , ∀ i, j = 1 , . . . , m, v kl ∈ R , u k ∈ R , k = 1 , . . . , n, l = 1 , . . . , d, x ∈ K . 12 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO W e note in passing that the ab ov e problem simplifies for those cases where r is even. In these cases, we can replace the tw o sets of constrain ts, namely (10) and (11) b y the simplest constraint v s kl = ( x k − a kl ) r , ∀ k , l. This reformulation reduces by ( n × d ) the num b er of constraints defining the feasible set. Moreov er, these constrain ts do not induce semidefinite constrain ts in the mom en t approach but linear matrix inequalities whic h are easier to handle. F ollowing the same scheme of the pro of in Theorem 4 we get the following result, whose pro of is left to the reader. Theorem 13. L et x b e a fe asible solution of ( LOCOMRF ) then ther e exists a solution ( x, u, v , w ) for (9) such that their obje ctive values ar e e qual. Conversely, if ( x, u, v , w ) is a fe asible solution for (9) then ther e exists a solution ( x ) for ( LOCOMRF ) having the same obje ctive value. In p articular % λ = % λ . Mor e over, if K ⊂ R d satisfies Putinar’s pr op erty then K ⊂ R d + m 2 + n ( d +2) also satisfies Putinar’s pr op erty. No w, we can prov e a conv ergence result that allows us to solve, up to an y degree of accuracy , the ab o v e class of problems. Let y = ( y α ) b e a real sequence indexed in the monomial basis ( x β u γ v δ w ζ ) of R [ x, u, v , w ] (with α = ( β , γ , δ, ζ ) ∈ N d × N n × N nd × N m 2 ). Let h 0 ( x, u, v , w ) := p λ ( x, u, v , w ), and denote ξ j := d (deg g j ) / 2 e and ν j := d (deg h j ) / 2 e , where { g 1 , . . . , g ` } , and { h 1 , . . . , h 3 m + m 2 +2 n ( d +1)+1 } are, resp ectiv ely , the p olynomial constraints that define K and K \ K in (9). F or r ≥ r 0 := max { max k =1 ,...,` ξ k , max j =0 ,...,l +3 m + m 2 +1 ν j } , introduce the hierarch y of semidefinite programs: ( Q r ) min y L y ( p λ ) s . t . M r ( y )  0 , M r − ξ k ( g k , y )  0 , k = 1 , . . . , `, M r − ν j ( h j , y )  0 , j = 1 , . . . , 3 m + m 2 + 1 , L y ( q λ ) = 1 , with optimal v alue denoted inf Q r (and min Q r if the infimum is attained). Theorem 14. L et K ⊂ R d + m 2 + n ( d +2) (c omp act) b e the fe asible domain of Pr oblem (9). L et Q r b e the semidefinite pr o gr am ( Q r ). Then, with the notation ab ove: (a) inf Q r ↑ ρ λ as r → ∞ . (b) L et y r b e an optimal solution of the SDP r elaxation Q r in ( Q r ). If rank M r ( y r ) = rank M r − r 0 ( y r ) = t then min Q r = ρ λ and one may extr act t p oints ( x ∗ ( k ) , u ∗ ( k ) , v ∗ ( k ) , w ∗ ( k )) t k =1 ⊂ K , al l glob al minimizers of the MOMRF pr oblem. Pr o of. The conv ergence of the semidefinite relaxation Q r w as prov ed by Jib etean and De Klerk [10] for a general rational function o ver a closed semialgebraic set. Here, we apply this result applied to the rational function in (6) and therefore, the result follows.  Here, w e also observe that one can exploit the blo ck diagonal structure of the problem since there is a sparsit y pattern in the v ariables of form ulation (9). The reader ma y note that the only monomials that app ear in that formulation are of the form x α u β i Q m j =1 v γ j ij for all i = 1 , . . . , m . Hence, a result similar to Theorem 14 also holds for the hierarch y ( Q r ) of SDP applied to the lo cation problem. Nevertheless, al- though w e ha v e used it in our computational test, w e do not giv e specific details for the sake of presentation and b ecause of the similarit y with Theorem 14. Example 15. We il lustr ate the ab ove r esults with an instanc e of the wel l-known Web er pr oblem with ` 3 -norm and for 20 r andom demand p oints in R 3 . L et A = { (0 . 0758 , 0 . 0540 , 0 . 5308) , (0 . 7792 , 0 . 9340 , 0 . 1299) , (0 . 5688 , 0 . 4694 , 0 . 0119) , (0 . 3371 , 0 . 1622 , 0 . 7943) , (0 . 3112 , 0 . 5285 , 0 . 1656) , (0 . 6020 , 0 . 2630 , 0 . 6541) , (0 . 6892 , 0 . 7482 , 0 . 4505) , (0 . 0838 , 0 . 2290 , 0 . 9133) , (0 . 1524 , 0 . 8259 , 0 . 5383) , (0 . 9961 , 0 . 0782 , 0 . 4427) , (0 . 1066 , 0 . 9619 , 0 . 0046) , (0 . 7749 , 0 . 8173 , 0 . 8687) , (0 . 0844 , 0 . 3998 , 0 . 2599) , (0 . 8000 , 0 . 4314 , 0 . 9106) , (0 . 1818 , 0 . 2638 , 0 . 1455) , (0 . 1361 , 0 . 8693 , 0 . 5797) , (0 . 5499 , 0 . 1450 , 0 . 8530) , (0 . 5499 , 0 . 1450 , 0 . 8530) , (0 . 4018 , 0 . 0760 , 0 . 2399) , (0 . 1233 , 0 . 1839 , 0 . 2400) } . MINIMIZING ORDERED WEIGHTED A VERAGING OF RA TIONAL FUNCTIONS 13 Then, the pr oblem c onsists of min X a ∈ A k x − a k 3 s.t. x ∈ R 3 . The fe asible r e gion of the first SDP r elaxation of this pr oblem, which in this c ase is r = 2 , c ontains 20 moment matric es of size 36 × 36 , 160 lo c alizing matric es of size 8 × 8 and 36 e quality c onstr aints. The exact optimal solution is given by x = (0 . 426397 , 0 . 438730 , 0 . 455857) with optimal value f = 8 . 729976 . We get with our appr o ach, using SDPT3 [13] , an optimal solution x ∗ = (0 . 426397 , 0 . 438730 , 0 . 455857) , for the first r elaxation of the pr oblem with optimal value f ∗ = 8 . 729976 . Thus, the r elative err or is  = | f ∗ − f | f = 2 . 199595 × 10 − 13 . F or the same set of p oints, we c onsider a mo dific ation of the ab ove pr oblem by adding an extr a nonc onvex c onstr aint: min X a ∈ A k x − a k 3 s.t. x 2 1 − 2 x 2 2 − 2 x 2 3 ≥ 0 , x ∈ R 3 . The exact optimal solution of this pr oblem is e x = (0 . 562304 , 0 . 266296 , 0 . 295262) with optimal value e f = 10 . 109333 . The r e ader may note that the original solution ¯ x is not fe asible for the new pr oblem. Using our appr o ach, again for the first r elaxation or der, we get x ∗∗ = (0 . 562304 , 0 . 266296 , 0 . 295262) with optimal value f ∗∗ = 10 . 109333 . Henc e, the r elative err or in this c ase is e  = | f ∗∗ − e f | e f = 5 . 801151 × 10 − 9 . We show in Figur e 1 the fe asible r e gion of our pr oblem as wel l as the demand p oints and the optimal solutions (the exact and the ones obtaine d with our r elaxe d formulations) of the pr oblems. The demand p oints in A ar e r epr esente d by ’ ∗ ’, the optimal solution, x ∗ , of the SDP r elaxation without the nonc onvex c onstr aint by ’  ’ and the optimal solution, x ∗∗ , of the SDP r elaxation with the nonc onvex c onstr aint is depicte d by ’ • ’. Figure 1. F easible region, demand p oints and optimal solutions of Example 15. 14 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO In the following, we will apply this general metho dology to get the reform ulation of the most standard problems in Lo cation Analysis (see Nick el and Puerto [25]) that will b e later the basis of our computa- tional exp eriments: minisum (W eb er) and minimax (ce n ter), k -centrum, ( k 1 , k 2 )-trimmed mean and range problems. 4.1. W eb er or median problem. In the standard v ersion of the W eb er problem, we are given a set of demand p oints { a 1 , . . . , a n } in R d and a set of non-negativ e weigh ts ω 1 , . . . , ω n and one lo oks for a p oint x ∗ minimizing the weigh ted Euclidean distance from the demand p oint. In other w ords, the problem is: min x ∈ R d n X i =1 ω i k x − a i k 2 . This problem has b een largely studied in the literature of Lo cation Analysis and p erhaps its most well- kno wn algorithm is the so called W eiszfeld algorithm (see [38]). This problem is a con v ex one and W eiszfeld algorithm is a gradien t type iterative algorithmic scheme for which several conv ergence results are kno wn. Here, w e observe that this problem corresp onds to a very particular choice of the elements in ( LOCOMRF ): λ = (1 , . . . , 1), f i ( u ) = ω i u and r = 2, s = 1. F urthermore, the general formulation ( LOCOMRF ) simpli- fies since there is no actual sorting. Therefore, we can a void many of our instrumental v ariables, namely , the problem can b e cast in to the form: min n X i =1 ω i z i s.t. z 2 i = d X j =1 ( x j − a ij ) 2 , i = 1 , . . . , n, d X j =1 x 2 j + z 2 i ≤ M , i = 1 , . . . , n, ( WP ) z i ≥ 0 , i = 1 , . . . , n, x ∈ R d . 4.2. The minimax or cen ter problem. The minimax location problem looks for the location of a serv er x ∈ R d that minimizes the maximum weigh ted distance to a given set of demands points { a 1 , . . . , a n } in R d . F ormally , the problem can b e stated as: min x ∈ R d max i =1 ,...,n ω i k x − a i k 2 , for some weigh ts ω 1 , . . . , ω n ≥ 0. Once more, this problem has b een extensiv ely analyzed in the literature of Lo cation Analysis and the most well-kno wn algorithms to solve it are those b y Elzinga-Hearn (only v alid in R 2 with Euclidean distance) and Dy er [8, 7] and Megiddo [23] whic h are polynomial in fixed dimension. Again, w e observ e that this problem corresp onds to a very particular choice of the elemen ts in ( LOCOMRF ): λ = (1 , 0 , . . . , 0), f i ( u ) = ω i u and r = 2, s = 1. In this case, the general form ulation ( LOCOMRF ) simplifies and therefore, w e can av oid many of our instrumen tal v ariables, namely , the problem can b e formulated as: MINIMIZING ORDERED WEIGHTED A VERAGING OF RA TIONAL FUNCTIONS 15 min t s.t. z 2 i = d X j =1 ( x j − a ij ) 2 , i = 1 , . . . , n, ω i z i ≤ t, i = 1 , . . . , n, d X j =1 x 2 j + z 2 i + t 2 ≤ M , i = 1 , . . . , n, ( CP ) t, z i , ≥ 0 , i = 1 , . . . , n, x ∈ K . 4.3. The k-cen trum problem. The k -cen trum location problem consists of finding the p oint x ∗ that minimizes the sum of the k largest distances with resp ect to a given set of demands p oints { a 1 , . . . , a n } in R d . F ormally , the problem can b e stated as: min x ∈ R d max i =1 ,...,k d ( i ) ( x ) , where d ( i ) ( x ) = k x − a σ ( i ) k for a permutation σ suc h that d σ (1) ( x ) ≥ . . . ≥ d σ ( n ) ( x ). This problem has be en considered in sev eral pap ers and textb o oks (see [25], [4]). Curren tly , there exist few approaches to solve it in the plane (i.e. d = 2) and with the Euclidean norm that do not extend further to higher dimension nor other norms (see [5, 6, 34]). The ob jective function of this problem is described b y a v ector of λ -parameters λ = ( k z }| { 1 , . . . , 1 , 0 , . . . , 0), f i ( u ) = u , r = 2, s = 1. Using the result in the reformulation (kC) the problem can b e restated as: ˆ % := min k t + n X i =1 r i s.t. z 2 i = d X j =1 ( x j − a ij ) 2 , i = 1 , . . . , n, t + r i ≥ z i , i = 1 , . . . , n, ( kCP ) d X j =1 x 2 j + z 2 i + r 2 i ≤ M , i = 1 , . . . , n, t, r i , z i ≥ 0 , i = 1 , . . . , n, x ∈ R d . 4.4. The ( k 1 , k 2 ) -trimmed-mean problem. The ( k 1 , k 2 )-trimmed-mean lo cation problem lo oks for a p oin t x ∗ that minimizes the sum of the central distances, once we hav e excluded the k 2 closest and the k 1 furthest. F ormally , the problem is: min x ∈ R d n − k 2 X i = k 1 +1 d ( i ) ( x ) , where d ( i ) ( x ) = k x − a σ ( i ) k 2 for a p ermutation σ such that d σ (1) ( x ) ≥ . . . ≥ d σ ( n ) ( x ). This problem has b een considered in several pap ers and textb o oks (see [25], [4]). Currently , there exists tw o approaches to solve it in the plane (i.e. d = 2) and with the Euclidean norm that do not extend further to higher dimension nor other norms (see [5, 6]). The ob jectiv e function of this problem, in terms of the elements in ( LOCOMRF ), is describ ed by a vector of λ -parameters λ = ( k 1 z }| { 0 , . . . , 0 , 1 , . . . , 1 , k 2 z }| { 0 , . . . , 0), f i ( u ) = u , r = 2, s = 1. Here, we could apply the general form ulation derived from ( LOCOMRF ). Nev ertheless, that approac h needs many decision v ariables which affects the sizes of the problems to b e handled. Rather than 16 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO the general formulation, we presen t here an alternative problem, based on (kT r), which takes adv antage of the particular structure of this problem and reduces the n umber of v ariables needed for its representation. W e consider the problem: min ( n − k 2 ) t + n X i =1 r i − n X i =1 u i z i s.t. z 2 i = d X j =1 ( x j − a ij ) 2 , i = 1 , . . . , n, n X i =1 u i = k 1 , u i ( u i − 1) = 0 , i = 1 , . . . , n, t + r i ≥ z i , i = 1 , . . . , n, ( TMP ) d X j =1 x 2 j + z 2 i + t 2 + u 2 i + r 2 i ≤ M , i = 1 , . . . , n, z i , r i , u i , t ≥ 0 , i = 1 , . . . , n, x ∈ R d . 4.5. The range problem. The last problem that we address in our computational exp eriments is the range lo cation problem. This problem consists of minimizing the difference (range) b etw een the maximum and minimum distances with respect to a given set of demands p oints { a 1 , . . . , a n } in R d (see [5, 6, 25]). F ormally , the problem can be stated as: min x ∈ R d  max i =1 ,...,n k x − a i k 2 − min i =1 ,...,n k x − a i k 2  . This problem corresponds to the following choice of the elements in ( LOCOMRF ): λ = (1 , 0 , . . . , 0 , − 1) , f i ( u ) = u and r = 2, s = 1. A simplified reformulation of the problem reduces to: min z − t s.t. z 2 i = d X j =1 ( x j − a ij ) 2 , i = 1 , . . . , n, t ≤ z i ≤ z , i = 1 , . . . , n, ( RP ) d X j =1 x 2 j + z 2 i + t 2 + z 2 ≤ M , i = 1 , . . . , n, t, z , z i ≥ 0 , i = 1 , . . . , n, x ∈ R d . 5. Comput a tional Experiments A series of computational exp eriments hav e b een p erformed in order to ev aluate the b ehavior of the prop osed metho dology . Programs hav e b een co ded in MA TLAB R2010b and executed in a PC with an In tel Core i7 pro cessor at 2x 2.93 GHz and 8 GB of RAM. The semidefinite programs hav e b een solved b y calling SDPT3 4.0[13]. W e run the algorithm for sev eral well-kno wn contin uous lo cation problems: W eb er problem, center problem, k-center problem, trimmed-mean problem and range problem. F or eac h of them, w e obtain the CPU times for computing solutions as well as the gap with resp ect to the optimal solution obtained with the battery of functions in optimset of MA TLAB or the implementation by [5, 6]. MINIMIZING ORDERED WEIGHTED A VERAGING OF RA TIONAL FUNCTIONS 17 With regard to computing the accuracy of an obtained solution, w e use the follo wing measure for the error (see [37]): (12)  ob j = | the optimal v alue of the SDP − fopt | max { 1 , fopt } , where fopt is the optimal ob jectiv e v alue for the problem obtained with the functions in optimset or the implemen tation b y [5, 6]. W e hav e organized our computational exp erimen ts in fiv e differen t problems types that coincide with those describ ed previously in sections 4.1-4.5. Our test problems are generated to b e comparable with previous results of some algorithms in the plane but, in addition, we also consider problems in R 3 . Thus, w e rep ort on randomly generated p oints on the unit square and in the unit cub e. Dep ending on the problem, we hav e b een able to solv e different problem sizes. In all problems, we could solve instances with at least 500 p oints for planar and 3-dimensional problems and with an av erage accuracy higher than 10 − 5 . (W e remark that for instance we could solv e instances of more than 1000 p oin ts for W eb er and center problems with high precisions.) Our goal is to present the results organized per problem type, framew ork space ( R 2 or R 3 ) and relaxation order. W e rep ort for W eb er problem on the first tw o relaxations to show that raising relaxation order one gains some extra precision (as exp ected) at the price of higher CPU times. In spite of that, the considered problems seems to b e v ery well-appro ximated even with the first relaxation (as shown by our results). F or this reason, we only rep ort results for relaxation order r = 2 for the remaining problem types, namely cen ter, k -cen trum, range and trim-mean. The results in our tables, for eac h size and problem type, are the av erage of ten runs. In all cases our tables are organized in the same w ay . Rows giv e the results for the different num ber of demand p oin ts considered in the problems. Column n stands for the num b er of points considered in the problem, CPU time is the av erage running time needed to solve each of the instances,  obj giv es the error measure (see 12). The final blo ck of 3 columns informs on the sizes of the SDP problems to b e solved: #Cols , #Rows and %NonZero represent, respectively , the n um b er of columns, ro ws and the p ercentage of nonzero entries of the constraint matrices of the problems to b e considered. W e tested problems with up to 500 demands points (except for W eb er problem where w e considered 1000 demands points) randomly generated in the unit square and the unit cube. W e mo ve n b etw een 10 and 500 (or 1000 for W eb er problem) and ten instances w ere generated for each v alue of n . The first relaxation of the problems was solved in all cases. F or the k-cen trum problem type w e considered three different k v alues to test the difficulty of problems with resp ect to that parameter, k = d 0 . 1 n e , d 0 . 5 n e , d 0 . 9 n e (tables 4 and 5). T ables 1-7 show the av erages CPU times and gaps obtained. T able 1 summarizes the results of the W eb er problems. W e remark that problems with up to 1000 demand points on the plane are solv ed with the first relaxation in few seconds and with accuracy higher than 10 − 4 . Raising the relaxation order, we impro ve accuracy till 10 − 6 at the cost of m ultiplying CPU time b y a factor of 8. T able 2 refers to W eb er problem in the 3 d space. Results are similar although precision is higher when considering the second relaxation order. T able 3 reports the results for the center problem on the plane and the 3 d -space. CPU times are slightly larger than for the W eb er problem but accuracy are also b etter sp ecially for sizes up to 100 demand points. T ables 4 and 5 are dev oted to sho w the behavior of our approac h for three different k v alues of the k -centrum problem (T able 4 in R 2 and T able 5 in R 3 ). W e observ e that for small v alues of k , i.e. k = d 0 . 1 n e or d 0 . 5 n e the k -centrum is sligh tly harder than for v alues closer to n . The remaining factors behav e similarly to those in W eb er or cen ter problems. T able 6 rep orts the results for the range problem. The b ehavior of these problems is similar to that of the k centrum problems b oth in CPU time and accuracy . Finally , T able 7 summarizes the results for the trimmed-mean problems. These are the harder problems among the five considered problem t yp es. W e are able to solve similar sizes with similar accuracies using the first order relaxation. Ho w ever, CPU times are significantly higher than for the other problem types. These results sho w that this methodology can b e efficiently applied to solve medium to large sized lo cation problems. ¿F rom our tables we conclude that W eb er problem is the simplest one whereas the trimmed-mean problem is the hardest one, as exp ected. W e remark that CPU times increase linearly with the n umber of p oints in all problem types. A linear regression betw een these times and the n umber of p oints gives 18 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO a regression co efficient R -squared (co efficient of determination of the regression) greater than 0 . 98 for all the problems. Therefore, this sho ws a linear dep endence, up to the tested sizes, betw een problem sizes and CPU times for solving the corresp onding relaxations. Observ e that the sizes of the matrices in the SDP relaxations increase exp onen tially with the n umber of p oin ts. Nevertheless, the p ercentage of nonzero elemen ts in the constrain t matrices decreases very slo wly (h yp erb olically) when increasing the size (n um ber of p oints) of the problems. 6. Conclusions W e dev elop a unified to ol for minimizing w eighted ordered a v eraging of rational functions. This approac h go es beyond a trivial adaptation of the general theory of moments-sos since ordered weigh ted av erages of rational functions are not, in general, neither rational functions nor the supremum of rational functions so that current results cannot directly b e applied to handle these problems. As an imp ortan t application we cast a general class of contin uous lo cation problems within the minimization of O W A rational functions. W e rep ort computational results that show the p ow erfulness of this metho dology to solve medium to large con tinuous lo cation problems. This new approac h solves a broad class of con vex and non con v ex con tin uous lo cation problems that, up to date, w ere only partially solv ed in the specialized literature. W e ha v e tested this methodology with some medium to large size standard ordered median lo cation problems in the plane and in the 3-dimensional space. Our goal was not to comp ete with previous algorithms since most of them are either problem sp ecific or only applicable for planar problems. How ev er, in all cases we obtained reasonable CPU times and high accuracy results ev en with first relaxation order. 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The Or der e d Weighte d Aver aging Op er ators: The ory and Applic ations , Kluw er: Norwell, MA. 20 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO First Relaxation ( r = 2) Second Relaxation ( r = 3) n CPU time  ob j #Cols #Rows %NonZero CPU time  ob j #Cols #Rows %NonZero 10 0.63 0.00191774 1420 214 0.780% 2.45 0.00008689 6200 587 0.279% 20 1.03 0.00079178 2840 414 0.403% 5.67 0.00002648 12400 1147 0.143% 30 1.03 0.00062061 4260 614 0.272% 8.94 0.00002065 18600 1707 0.096% 40 1.57 0.00082654 5680 814 0.205% 11.43 0.00000992 24800 2267 0.072% 50 2.12 0.00015842 7100 1014 0.165% 13.29 0.00000269 31000 2827 0.058% 60 2.31 0.00027699 8520 1214 0.137% 16.95 0.00000213 37200 3387 0.048% 70 2.72 0.00044228 9940 1414 0.118% 20.54 0.00000434 43400 3947 0.042% 80 3.03 0.00044249 11360 1614 0.103% 26.98 0.00000243 49600 4507 0.036% 90 3.38 0.00031839 12780 1814 0.092% 29.20 0.00000194 55800 5067 0.032% 100 3.92 0.00027367 14200 2014 0.083% 31.57 0.00000174 62000 5627 0.029% 150 6.12 0.00027644 21300 3014 0.055% 46.31 0.00000555 93000 8427 0.019% 200 8.36 0.00021865 28400 4014 0.042% 65.75 0.00000190 124000 11227 0.015% 250 10.42 0.00028088 35500 5014 0.033% 87.13 0.00000656 155000 14027 0.012% 300 12.19 0.00019673 42600 6014 0.028% 102.95 0.00001241 186000 16827 0.010% 350 14.63 0.00018747 49700 7014 0.024% 124.36 0.00000850 217000 19627 0.008% 400 17.25 0.00021381 56800 8014 0.021% 145.62 0.00000333 248000 22427 0.007% 450 20.37 0.00007970 63900 9014 0.019% 167.02 0.00000476 279000 25227 0.007% 500 22.03 0.00011803 71000 10014 0.017% 187.02 0.00000754 310000 28027 0.006% 600 28.11 0.00012725 85200 12014 0.014% 232.19 0.00000287 372000 33627 0.005% 700 33.47 0.00015215 99400 14014 0.012% 274.88 0.00000332 434000 39227 0.004% 800 39.50 0.00009879 113600 16014 0.010% 334.10 0.00000420 496000 44827 0.004% 900 45.31 0.00011740 127800 18014 0.009% 389.00 0.00000350 558000 50427 0.003% 1000 55.68 0.00012513 142000 20014 0.008% 443.13 0.00000351 620000 56027 0.003% T able 1. Computational results for planar W eb er problem and first and second relaxation. MINIMIZING ORDERED WEIGHTED A VERAGING OF RA TIONAL FUNCTIONS 21 First Relaxation ( r = 2) Second Relaxation ( r = 3) n CPU time  ob j #Cols #Rows %NonZero CPU time  ob j #Cols #Rows %NonZero 10 1.19 0.00112213 2900 384 0.442% 9.13 0.00000379 17100 1343 0.124% 20 1.84 0.00036619 5800 734 0.231% 23.89 0.00000000 34200 2603 0.064% 30 2.56 0.00019790 8700 1084 0.157% 28.97 0.00000000 51300 3863 0.043% 40 3.54 0.00011433 11600 1434 0.118% 45.19 0.00000000 68400 5123 0.033% 50 4.27 0.00008446 14500 1784 0.095% 58.34 0.00000001 85500 6383 0.026% 60 5.04 0.00019406 17400 2134 0.080% 66.09 0.00000000 102600 7643 0.022% 70 6.23 0.00009027 20300 2484 0.068% 77.67 0.00000000 119700 8903 0.019% 80 7.09 0.00018689 23200 2834 0.060% 90.86 0.00000000 136800 10163 0.016% 90 8.01 0.00010943 26100 3184 0.053% 124.89 0.00000000 153900 11423 0.015% 100 9.87 0.00005552 29000 3534 0.048% 164.37 0.00000008 171000 12683 0.013% 150 14.16 0.00004856 43500 5284 0.032% 211.02 0.00000000 256500 18983 0.009% 200 20.33 0.00003049 58000 7034 0.024% 275.02 0.00000000 342000 25283 0.007% 250 25.97 0.00005964 72500 8784 0.019% 429.67 0.00000014 427500 31583 0.005% 300 34.00 0.00004677 87000 10534 0.016% 501.09 0.00000006 513000 37883 0.004% 350 39.82 0.00004154 101500 12284 0.014% 588.29 0.00000007 598500 44183 0.004% 400 47.27 0.00005233 116000 14034 0.012% 746.70 0.00000011 684000 50483 0.003% 450 57.08 0.00003325 130500 15784 0.011% 762.54 0.00000000 769500 56783 0.003% 500 65.93 0.00002952 145000 17534 0.010% 1063.50 0.00000000 855000 63083 0.003% T able 2. Computational results for W eb er problem in R 3 and first and second relaxation. 22 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO R 2 R 3 n CPU time  ob j #Cols #Rows %NonZero CPU time  ob j #Cols #Rows %NonZero 10 0.95 0.00000002 3150 384 0.423% 1.90 0.00000001 5700 629 0.259% 20 1.78 0.00000001 6300 734 0.221% 4.05 0.00000000 11400 1189 0.137% 30 2.68 0.00000001 9450 1084 0.150% 6.24 0.00000008 17100 1749 0.093% 40 3.78 0.00000001 12600 1434 0.113% 8.96 0.00000000 22800 2309 0.071% 50 4.68 0.00000000 15750 1784 0.091% 12.05 0.00000000 28500 2869 0.057% 60 6.05 0.00000000 18900 2134 0.076% 16.63 0.00000000 34200 3429 0.048% 70 8.48 0.00000000 22050 2484 0.065% 18.84 0.00000002 39900 3989 0.041% 80 10.28 0.00000002 25200 2834 0.057% 28.08 0.00000000 45600 4549 0.036% 90 13.60 0.00000005 28350 3184 0.051% 32.16 0.00000000 51300 5109 0.032% 100 18.86 0.00000005 31500 3534 0.046% 38.78 0.00000291 57000 5669 0.029% 150 31.12 0.00002157 47250 5284 0.031% 59.19 0.00006902 85500 8469 0.019% 200 38.76 0.00013507 63000 7034 0.023% 82.01 0.00011298 114000 11269 0.014% 250 44.34 0.00027776 78750 8784 0.019% 111.64 0.00013810 142500 14069 0.012% 300 58.10 0.00033715 94500 10534 0.015% 124.47 0.00030316 171000 16869 0.010% 350 81.59 0.00047225 110250 12284 0.013% 170.43 0.00043926 199500 19669 0.008% 400 90.22 0.00048347 126000 14034 0.012% 172.05 0.00052552 228000 22469 0.007% 450 93.50 0.00047479 141750 15784 0.010% 242.66 0.00057288 256500 25269 0.006% 500 151.64 0.00066416 157500 17534 0.009% 226.73 0.00059268 285000 28069 0.006% T able 3. Computational results for center problem in R 2 and R 3 and first relaxations. MINIMIZING ORDERED WEIGHTED A VERAGING OF RA TIONAL FUNCTIONS 23 k = d 0 . 1 n e k = d 0 . 5 n e k = d 0 . 9 n e Sizes n CPU time  ob j CPU time  ob j CPU time  ob j #Cols #Rows #NonZero 10 2.64 0.00000630 2.76 0.00000081 2.59 0.00017665 6570 944 0.175% 20 6.43 0.00001375 6.15 0.00000298 5.30 0.00000545 13140 1854 0.089% 30 10.88 0.00000379 9.89 0.00000410 9.16 0.00000102 19710 2764 0.060% 40 15.89 0.00000717 16.33 0.00000090 12.22 0.00000122 26280 3674 0.045% 50 21.24 0.00000282 18.51 0.00000083 16.77 0.00000105 32850 4584 0.036% 60 25.77 0.00000077 25.41 0.00000283 20.21 0.00000806 39420 5494 0.030% 70 28.01 0.00000204 31.02 0.00000234 25.07 0.00000192 45990 6404 0.026% 80 37.25 0.00000085 31.48 0.00000044 30.66 0.00000220 52560 7314 0.023% 90 47.16 0.00000062 41.07 0.00000765 33.92 0.00000086 59130 8224 0.020% 100 53.68 0.00000084 41.42 0.00000065 39.49 0.00000188 65700 9134 0.018% 150 86.48 0.00000089 68.48 0.00000056 65.95 0.00000059 98550 13684 0.012% 200 123.02 0.00000056 96.40 0.00000075 88.10 0.00000275 131400 18234 0.009% 250 149.26 0.00003681 135.67 0.00000071 113.68 0.00000161 164250 22784 0.007% 300 180.38 0.00000408 161.84 0.00000081 146.22 0.00000349 197100 27334 0.006% 350 223.27 0.00003013 193.31 0.00003623 176.46 0.00000151 229950 31884 0.005% 400 260.27 0.00000079 225.07 0.00003689 201.01 0.00000376 262800 36434 0.005% 450 290.23 0.00004512 272.55 0.00000097 237.23 0.00000168 295650 40984 0.004% 500 345.93 0.00000224 310.19 0.00000119 269.99 0.00000200 328500 45534 0.004% T able 4. Computational results for planar k -centrum problems and first relaxation ( r = 2). 24 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO k = d 0 . 1 n e k = d 0 . 5 n e k = d 0 . 9 n e Sizes n CPU time  ob j CPU time  ob j CPU time  ob j #Cols #Rows %NonZero 10 7.06 0.00041340 5.85 0.00000039 6.05 0.00000168 10780 1469 0.114% 20 16.40 0.00000950 15.42 0.00000095 16.30 0.00000019 21560 2869 0.059% 30 27.63 0.00001682 23.72 0.00000028 27.12 0.00000132 32340 4269 0.039% 40 45.25 0.00000075 42.31 0.00000086 37.38 0.00000077 43120 5669 0.030% 50 54.39 0.00000282 53.66 0.00000026 51.94 0.00000087 53900 7069 0.024% 60 63.16 0.00000259 59.34 0.00000091 63.91 0.00000065 64680 8469 0.020% 70 85.17 0.00000144 81.32 0.00000258 74.24 0.00000079 75460 9869 0.017% 80 106.65 0.00000326 83.96 0.00000044 88.76 0.00000158 86240 11269 0.015% 90 114.38 0.00000209 93.85 0.00000100 103.56 0.00000092 97020 12669 0.013% 100 122.01 0.00000088 109.17 0.00000224 118.03 0.00000067 107800 14069 0.012% 150 235.10 0.00000073 211.54 0.00000890 187.51 0.00000135 161700 21069 0.008% 200 305.51 0.00002407 255.54 0.00007106 284.80 0.00000157 215600 28069 0.006% 250 403.89 0.00000519 348.32 0.00004300 357.79 0.00000143 269500 35069 0.005% 300 492.04 0.00046130 433.69 0.00007630 471.78 0.00000174 323400 42069 0.004% 350 529.61 0.00041229 484.87 0.00000058 448.60 0.00001791 377300 49069 0.003% 400 619.97 0.00000091 585.93 0.00000055 523.81 0.00000829 431200 56069 0.003% 450 705.99 0.00048727 693.77 0.00000037 580.06 0.00004327 485100 63069 0.003% 500 817.75 0.00012138 789.77 0.00000087 664.94 0.00000318 539000 70069 0.002% T able 5. Computational results for k -centrum problems in R 3 and first relaxation ( r = 2). MINIMIZING ORDERED WEIGHTED A VERAGING OF RA TIONAL FUNCTIONS 25 R 2 R 3 n CPU time  ob j #Cols #Rows %NonZero CPU time  ob j #Cols #Rows %NonZero 10 2.96 0.00007519 6060 629 0.252% 5.68 0.00001997 10080 965 0.164% 20 7.04 0.00001750 12120 1189 0.133% 18.45 0.00015758 20160 1805 0.088% 30 13.94 0.00098322 18180 1749 0.091% 35.37 0.00028187 30240 2645 0.060% 40 14.53 0.00002124 24240 2309 0.069% 35.77 0.00032049 40320 3485 0.045% 50 24.49 0.00004314 30300 2869 0.055% 65.80 0.00051293 50400 4325 0.037% 60 23.49 0.00047832 36360 3429 0.046% 59.19 0.00005082 60480 5165 0.031% 70 34.87 0.00003903 42420 3989 0.040% 68.46 0.00006841 70560 6005 0.026% 80 38.69 0.00026693 48480 4549 0.035% 79.54 0.00003016 80640 6845 0.023% 90 42.34 0.00042121 54540 5109 0.031% 90.76 0.00017468 90720 7685 0.021% 100 58.36 0.00052427 60600 5669 0.028% 97.26 0.00015535 100800 8525 0.019% 150 65.04 0.00021457 90900 8469 0.019% 159.41 0.00094711 151200 12725 0.012% 200 98.23 0.00041499 121200 11269 0.014% 197.66 0.00040517 201600 16925 0.009% 250 131.42 0.00033959 151500 14069 0.011% 274.14 0.00057559 252000 21125 0.007% 300 159.87 0.00014556 181800 16869 0.009% 322.21 0.00036845 302400 25325 0.006% 350 169.29 0.00003661 212100 19669 0.008% 393.80 0.00096204 352800 29525 0.005% 400 167.74 0.00123896 242400 22469 0.007% 361.12 0.00022448 403200 33725 0.005% 450 218.70 0.00207328 272700 25269 0.006% 513.55 0.00044016 453600 37925 0.004% 500 228.68 0.00438388 303000 28069 0.006% 554.94 0.00028013 504000 42125 0.004% T able 6. Computational results for range problem in R 2 and R 3 and first relaxation. 26 V ´ ICTOR BLANCO, SAF AE EL-HAJ-BEN-ALI, AND JUSTO PUER TO R 2 R 3 n CPU time  ob j #Cols #Rows %NonZero CPU time  ob j #Cols #Rows %NonZero 10 5.31 0.00017041 11760 1784 0.087% 14.09 0.00000197 18080 2669 0.059% 20 12.39 0.00000619 23520 3534 0.044% 33.85 0.00047792 36160 5269 0.030% 30 18.11 0.00020027 35280 5284 0.029% 49.16 0.00000670 54240 7869 0.020% 40 30.39 0.00035248 47040 7034 0.022% 73.13 0.00001450 72320 10469 0.015% 50 36.04 0.00181487 58800 8784 0.018% 98.17 0.00001624 90400 13069 0.012% 60 49.16 0.00085810 70560 10534 0.015% 131.38 0.00003143 108480 15669 0.010% 70 60.57 0.00012995 82320 12284 0.013% 161.25 0.00004420 126560 18269 0.009% 80 73.54 0.00092073 94080 14034 0.011% 188.51 0.00012265 144640 20869 0.008% 90 76.12 0.00040564 105840 15784 0.010% 203.06 0.00011847 162720 23469 0.007% 100 91.26 0.00218668 117600 17534 0.009% 220.68 0.00011032 180800 26069 0.006% 150 153.31 0.00814047 176400 26284 0.006% 400.37 0.00026203 271200 39069 0.004% 200 257.23 0.00032380 235200 35034 0.004% 552.19 0.00056138 361600 52069 0.003% 250 339.72 0.00051519 294000 43784 0.004% 659.01 0.00046219 452000 65069 0.002% 300 326.52 0.00225994 352800 52534 0.003% 884.40 0.00038481 542400 78069 0.002% 350 410.32 0.00047898 411600 61284 0.003% 955.53 0.00061467 632800 91069 0.002% 400 582.36 0.00047130 470400 70034 0.002% 1165.79 0.00058261 723200 104069 0.002% 450 631.58 0.00060180 529200 78784 0.002% 1931.76 0.00081711 813600 117069 0.001% 500 685.79 0.00079679 588000 87534 0.002% 9151.90 0.00063861 904000 130069 0.001% T able 7. Computational results for trimmed mean problem with k 1 = k 2 = d 0 . 20 n e in R 2 and R 3 and first relaxation. MINIMIZING ORDERED WEIGHTED A VERAGING OF RA TIONAL FUNCTIONS 27 Dep ar t amento de ´ Algebra, Universidad de Granada E-mail address : vblanco@ugr.es Dep ar t amento de Est ad ´ ıstica e Investigaci ´ on Opera tiv a, Universidad de Sevilla E-mail address : anasafae@gmail.com; puerto@us.es