ON THE ARITHMETICGEOMETRIC MEAN INEQUALITY AND ITS RELATIONSHIP TO LINEAR PROGRAMMING, BAHMAN KALANTARI


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1 ON THE ARITHMETICGEOMETRIC MEAN INEQUALITY AND ITS RELATIONSHIP TO LINEAR PROGRAMMING, MATRIX SCALING, AND GORDAN'S THEOREM BAHMAN KALANTARI Abstract. It is a classical inequality that the minimum of the ratio of the (weighted) arithmetic mean to the geometric mean of a set of positive variables is equal to one, and is attained at the center of the positivity cone. While there are numerous proofs of this fundamental homogeneous inequality, in the presence of an arbitrary subspace, and/or the replacement of the arithmetic mean with an arbitrary linear form, the new minimization is a nontrivial problem. We prove a generalization of this inequality, also relating it to linear programming, to the diagonal matrix scaling problem, as well as to Gordan's theorem. Linear programming is equivalent to the search for a nontrivial zero of a linear or positive semidenite quadratic form over the nonnegative points of a given subspace. The goal of this paper is to present these intricate, surprising, and signicant relationships, called scaling dualities, and via an elementary proof. Also, to introduce two conceptually simple polynomialtime algorithms that are based on the scaling dualities, signicant bounds, as well as Nesterov and Nemirovskii's machinery of selfconcordance. The algorithms are simultaneously applicable to linear programming, to the computation of a separating hyperplane, to the diagonal matrix scaling problem, and to the minimization of the arithmeticgeometric mean ratio over the positive points of an arbitrary subspace. The scaling dualities, the bounds, and the algorithms are special cases of a much more general theory on convex programming, developed by the author. For instance, via the scaling dualities semidenite programming is a problem dual to a generalization of the classical tracedeterminant ratio minimization over the positive denite points of a given subspace of the Hilbert space of symmetric matrices. Key words. Convexity. Arithmeticgeometric mean inequality, Linear programming, Gordan's theorem, Diagonal matrix scaling, Department of Computer Science, Rutgers University, New Brunswick, NJ
2 2 BAHMAN KALANTARI. Introduction. The classical (weighted) arithmeticgeometric mean inequality can be viewed as a statement on the minimization of the ratio of the two (weighted) means over the positive orthant: the minimum is attained at the center of this cone, any positive scalar multiple of the vector of ones. While there are numerous proofs of this fundamental homogeneous inequality, see e.g. Hardy, Littlewood, and Polya [5], Bullen, Mitrinovic, and Vasic [3], Mitrinovic, Pecaric, and Fink [8], and Alzer []; in the presence of a proper subspace not containing the center, the new minimization is a nontrivial problem. A more general version of the above problem is the minimization of the ratio of an arbitrary linear function to the geometric mean, over the positive points of an arbitrary subspace. Still an even more general version is when the ratio is replaced with that of an arbitrary positive semidenite quadratic form, to the square of the geometric mean. On the other hand linear programming can be formulated as the problem of computing a nontrivial zero of a linear function over the intersection of the nonnegative orthant and a given subspace of the Euclidean space, or proving that such zero does not exist. This problem is in fact Karmarkar's canonical linear programming [5]. A more general, more interesting, and more important problem than Karmarkar's canonical LP problem is when the linear form is replaced with a positive semidenite quadratic form, considered in Kalantari [7]. Indeed, as proved in [7], by removing the positive semideniteness assumption, the zeronding problem becomes NPcomplete. Linear programming is also equivalent to the problem of testing if the convexhull of a given set of points in the Euclidean space contains the origin. A more extended version of this problem is the following: determine if the convexhull of a given set of points contains the origin; otherwise, compute a separating hyperplane. In fact the latter problem calls for an algorithmic proof of Gordan's theorem. Another problem of interest is the diagonal matrix scaling problem. Given a positive semidenite symmetric matrix, this is the problem of computing a positive denite diagonal matrix such that pre and post multiplication of the given matrix by the computed diagonal matrix results in a matrix whose row sums (hence column sums) equal prescribed positive numbers; or proving that such diagonal matrix does not exist. The case of diagonal scaling of matrices with nonnegative entries, not necessarily positive semidenite, has been a problem of interest for a very long time. The matrix scaling problem can also be stated in the presence of a subspace. In this more general case of the problem, the diagonal matrix is replaced with the product of a diagonally scaled orthogonal projection matrix, and a positive denite diagonal matrix induced by a positive point of the underlying subspace. A diagonal scaling problem can also be dened for the case of linear forms over a given subspace. This turns out to be a problem dual to Karmarkar's canonical LP. The goal of this paper is to present the intricate, surprising, and signicant relationships between the above stated problems, in the form dualities, called scaling dualities, and via an elementary proof. Also, to introduce two conceptually simple polynomialtime algorithms that are based on the scaling dualities, several signicant bounds, as well as Nesterov and Nemirovskii's machinery of selfconcordance. The algorithms are simultaneously applicable to linear programming, to the computation of a separating hyperplane, to the diagonal matrix scaling problem, and to the minimization of the arithmeticgeometric mean ratio over the positive points of a given subspace. The scaling dualities, the bounds, and the algorithms are special cases of a much more general theory on convex programming, developed by the author (see Kalantari [3]). For instance, via the scaling dualities semidenite programming is a problem dual to a generalization of the classical tracedeterminant ratio minimization over the positive denite points of a given subspace of the Hilbert space of symmetric matrices. Indeed, all the theorems to be stated in this paper, as well as the two algorithms can be shown to hold for analogous problems dened with respect to the cone of positive semidenite symmetric matrices. Although the proof of general scaling dualities is nontrivial, the proof of those presented in this paper are established via elementary means. The simplicity of the proofs is due
3 AGMean Inequality, LP Matrix Scaling, Gordan's Theorem 3 to the very special nature of the underlying problems, i.e., linearity or quadraticity of the homogeneous objective function, as well as symmetric properties of the underlying cone, the nonnegative orthant. In x 2, we preset the main theorems. In x 3 and x 4, we present their proofs. In x 5, we describe the two algorithms, state their complexities, the main ingredients that result in the algorithms, and their application in the derivation of the claimed complexity results. One of the major ingredient is the scaling dualities. Other ingredients are signicant bounding theorems that will be stated without proof. The proof of these bounds is given in [3] and in much more generality. Another ingredient is a theorem that combines two fundamental but most basic properties from Nesterov and Nemirovskii's theory on selfconcordance, [9]. In our concluding remarks, x 6, we briey describe generalization of the theorems and the algorithms, in particular with respect to corresponding problems dened over the cone of positive semidenite matrices. 2. The arithmeticgeometric mean inequality, linear programming, Gordan's separation theorem, and matrix scaling. Let K = fx 2 < n : x 0g, the nonnegative orthant, and P K = fx 2 < n : x > 0g, the positive orthant. Let 2 K n be an arbitrary vector of weights, and = i= i. The classical (weighted) arithmeticgeometric mean inequality is the following (2.) T x Qn i= x i i = ; 8x 2 K : Moreover, the minimum is attained at any positive scalar multiple of e = (; : : :; ) T. Let W be a subspace of < n. If W is a proper subspace, we will assume that W = fx 2 < n : Ax = 0g, where A is a given m n matrix of rank m. Let S = fx 2 < n : kxk = g. We will assume that d 0 2 W \ K \ S is available. Let (x) = 2 xt Qx, where Q is an n n symmetric matrix, also assumed to be positive semidenite, i.e., x T Qx 0, for all x 2 < n. Let (2.2) F (x) =? nx i= i ln x i : Linear programming is equivalent to the problem of testing if has a nontrivial zero over W \ K, see Kalantari [7]. Indeed, the latter problem is more general and more important than Karmarkar's canonical linear programming problem [5], which is the following problem: given c 2 < n, determine if c T x = 0, for some x 2 W \ K, x 6= 0. It is easy to see that has a desired zero if and only if = 0, where (2.3) = minf(x) : x 2 W \ K \ Sg: We shall refer to the problem of testing if = 0 as homogeneous programming (HP). Let the logarithmic potential, and the homogeneous potential be dened as (2.4) respectively. Let (2.5) (x) = (x) + F (x); X (x) = (x) exp(? 2 F (x)) = (x) n i= x2 i= i = inff (x) : x 2 W \ K g; X = inffx (x) : x 2 W \ K g: ; The scaling problem (SP) is to determine if =?, and if is nite, to compute its value, together with a corresponding minimizer d. The Homogeneous scaling problem (HSP) is to determine if X =?, and if X is nite, to compute its value, together with a corresponding minimizer d. Since (x) is convex and F (x) is strictly convex, (x) is strictly convex.
4 4 BAHMAN KALANTARI For a given d 2 K, let (2.6) D = r 2 F (d)?=2 = diag( d d p ; ; p n ); n (2.7) e = D? d = (p ; ; p n ) T ; (2.8) ;d (x) = (D x) = 2 xt D QD x; (2.9) ;d(x) = (D x) = ;d (x) + F (x) + F (d): Let P ;d be the orthogonal projection operator onto the subspace W ;d = D? W = fx 2 <n : AD x = 0g. Thus, if W = < n, then P ;d = I, the identity matrix. Otherwise, (2.0) P ;d = I? D A T (AD 2 AT )? AD : The algebraic scaling problem (ASP) is to test the solvability of the scaling equation (SE) : (2.) P ;d r ;d (e ) = P ;d D QD e = e ; d 2 W \ K : Theorem 2.. The following statements are equivalent: () : > 0. (2) : 9 d 2 W \ K such that (d) > 0, and X (x) X (d), 8 x 2 W \ K. (3) : 9 d 2 W \ K such that (x) (d), 8 x 2 W \ K. (4) : 9 d 2 W \ K such that P e;d D e QD e e = P e;d. (5) : 9 d 2 W \ K such that P e;d D e QD e e > 0. (6) : 9 d 2 W \ K such that P ;d D QD e = e. (7) : 9 d 2 W \ K such that kp ;d D QD e? e k < min = minf ; : : :; n g. (8) : 9 d 2 W \ K such that P ;d D QD e > 0. (9) : 9 d 2 W \ K such that P 2 ;dd 2QD 2 =, where 2 = ( 2 ; ; 2 n) T. Moreover, (3), (4), and (6) have a unique and common solution d, and d also satises (2). The following is an obvious but important corollary of Theorem 2.. Corollary 2.2. (Scaling dualities for positive semidenite quadratic forms) Either = 0, or condition (i) is true, i = 2; : : :; 9; but not both. The equivalence of the statements of Theorem 2. justies the naming of the corresponding dualities of Corollary 2.2 as scaling dualities. From the algorithmic point of view it turns out that the dualities that concern diagonal scaling are the more important dualities. Indeed, Theorem 2. can be stated in more generality, where (x) can be an arbitrary convex homogeneous function. The corresponding proof however becomes more involved than the simple proofs presented in this paper. For the proof of some of the corresponding parts of Theorem 2., as derived for general
5 AGMean Inequality, LP Matrix Scaling, Gordan's Theorem 5 convex homogeneous functions, see Kalantari []. We mention that linearly constrained convex quadratic programming can be formulated as the problem of computing a nontrivial root of the homogeneous function 2 xt Qx=e T x + c T x over W \ K, for some positive semidenite Q, and c 2 < n, see Kalantari [3]. In fact, most of the scaling dualities implied by Theorem 2. can be stated in much more generality, where the underlying cone is an arbitrary closed convex pointed cone, and an arbitrary convex form dened over this cone (see x 5). Our study of what we now call scaling dualities, and their algorithmic signicance began in Kalantari [6], and was continued in [7][]. We emphasize that the goal of the present paper is to present these signicant dualities for the very special problems considered, and via an elementary proof. Also, to introduce two conceptually simple, but very capable polynomialtime algorithms that are motivated by these dualities. Indeed, the linear programming/matrix scaling algorithm of Khachiyan and Kalantari [6] is also based on a scaling duality, implied by Gordan's theorem (see [3]). Remark. When W = < n, Corollary 2.2 implies that either there exists x 0; x 6= 0, such that Qx = 0, or there exists d > 0 such that Qd > 0; but not both. It is easy to show that this duality is equivalent to Gordan's theorem, see Kalantari []. Gordan's theorem (see Dantzig [2], Schrijver [20]) is the following: given a real m n matrix B, either there exists x 0; x 6= 0, such that Bx = 0, or there exists y such that B T y > 0; but not both. If the latter condition occurs, the vector y induces a hyperplane separating the column vectors of B from the origin. If we let Q = B T B, and Qd > 0, then Bd can be taken to be y. We will discuss the complexity of computing a desired zero, or a separating hyperplane in x 5. Remark 2. When W = < n, Corollary 2.2 implies that either there exists x 0; x 6= 0, such that Qx = 0, or given any > 0, there exists d > 0 such that D e QD e e = ; but not both. When = e, this is the quasi doublystochastic scaling problem, a problem which has been of considerable interest for matrices of nonnegative entries, see e.g. Marshal and Olkin [7], Kalantari []. Another statement implied by this corollary is that > 0 if and only if given any > 0, there exists d > 0, such that D QD e = e. In other words, there exists a diagonal scaling of Q such that e is a xedpoint. Moreover, the latter result is true if and only if there exits a diagonal scaling of Q such that D 2QD 2 =, i.e., itself is a xed point. The case of D e QD e e = e scaling was shown to be polynomially solvable in [6] via a simple analysis, see also [3]. But in fact all other cases considered in this paper, including matrix scaling over a given subspace, are polynomially solvable. However, the polynomialtime solvability of the subspace constrained case is considerably demanding, requiring several important and nontrivial auxiliary results (see x 5). The following theorem will be shown to be a consequence of Theorem 2.. It in particular implies a generalization of the familiar (weighted) arithmeticgeometric mean inequality. Let c 2 < n be arbitrary, and dene (2.2) f (x) = c T x Q n i= xi= i ; g (x) = c T x? nx i= i ln x i Theorem 2.3. The following statements are equivalent: ( 0 ) : c T x > 0, 8 x 2 W \ K, x 6= 0. (2 0 ) : 9 d 2 W \ K such that c T d > 0, and f (x) f (d), 8x 2 W \ K. (3 0 ) : 9 d 2 W \ K such that g (x) g (d), 8x 2 W \ K. (4 0 ) : 9 d 2 W \ K such that P e;d D e c = P e;d. (5 0 ) : 9 d 2 W \ K such that P e;d D e c > 0. (6 0 ) : 9 d 2 W \ K such that P ;d D c = e.
6 6 BAHMAN KALANTARI (7 0 ) : 9 d 2 W \ K such that kp ;d D c? e k < min = minf ; : : :; n g. (8 0 ) : 9 d 2 W \ K such that P ;d D c > 0. (9 0 ) : 9 d 2 W \ K such that P 2 ;dd 2c =, where 2 = ( 2 ; ; 2 n) T. Moreover, (3 0 ), (4 0 ), and (6 0 ) have a unique and common solution d, and d also satises (2 0 ). The following is an obvious but important corollary of Theorem 2.3. Corollary 2.4. (Scaling dualities for linear forms) Either c T x = 0, for some x 2 W \ K, x 6= 0, or condition (i) is true, i = 2 0 ; : : :; 9 0 ; but not both. Remark 3. For i = 2 0, Corollary 2.4 implies that either Karmarkar's canonical LP is solvable, or the potential function f (x) is unbounded from below; but not both. In fact Karmarkar's algorithm, [5], can be viewed as an algorithmic proof of this fact, see also Kalantari [6], [9]. Another duality implied by this corollary is that either Karmarkar's canonical LP is solvable, or there there exists d 2 W \ K such that P e;d D e c > 0, but not both. This duality which implies Gordan's theorem gives rise to a simple variation of Karmarkar's algorithm (see Kalantari [0]), a polynomialtime algorithm that actually establishes this duality. and 3. Proof of Theorem 2.. Proof. We will prove the theorem by proving the following circle of implications () =) (2) =) (3) =) (4) =) (6) =) (7) =) (8) =) (); () () (5); (6) () (9): () =) (2) : Suppose > 0. Consider the set W \K \S, S = fx 2 E : kxk = g. Since X (x) approaches innity as x approaches a boundary point of W \ K \ S, its inmum is attained at some point, say, x. We claim that x must be the minimum of X (x) over W \ K. Otherwise, there exists x 2 W \ K such that X (x) < X (x ). But as X is homogeneous of degree zero, we get X (x=kxk) = X (x) < X (x ), a contradiction. (2) =) (3) : Suppose there exists d 2 W \ K such that (d) > 0, and X (x) X (d), for all x 2 W \ K. From the rst order optimality condition, d must be a stationary point of X, i.e., (3.) P rx (d) = 0; where P = P e;e = I? A T (AA T )? A. Dierentiating X (x), we get (3.2) rx (x) = r(x) + 2 (x)rf (x) exp( 2 F (x)): For any positive real, we have (3.3) r(x) = r(x); rf (x) = rf (x): From (3.), (3.2), and (3.3) we conclude that for any positive real, we have (3.4) P r(d) + 2 (2 (d))( rf (d)) = 0:
7 AGMean Inequality, LP Matrix Scaling, Gordan's Theorem 7 In particular, since (d) > 0, we can let = p =2(d). Then, setting d = d, we have (3.5) P r (d ) = 0: Equivalently, this implies (3.6) r (d ) = A T v; for some vector v of Lagrange multipliers. Since is convex, for all x 2 W \ K we have (3.7) (x) (d ) + r (d ) T (x? d ) = (d ) + v T A(x? d ) = (d ); hence (3) is satised. (3) =) (4) : Suppose d satises (3). Then P r (d) = 0. Equivalently, (3.8) r (d) = QD e e? D? e = A T v; for some v. Multiplying the latter equation by D e, we get (3.9) D e QD e e? = D e A T v: Multiply the above by AD e, we can solve for v to get (3.0) v = (A T D 2 e A)? AD e (D e QD e? ): Substituting the above in the previous equation, we get (3.) P e;d (D e QD e? ) = 0: Thus, (4) is valid. (4) =) (6) : If d satises (4), then it follows that r (d) = A T v, for some v. Equivalently, (3.2) r (d) = QD e? D? e = A T v: Multiplying the latter equation by D, we get (3.3) D QD e? e = D A T v: Multiply the above by AD, we can solve for v to get (3.4) v = (A T D 2 A)? AD (D QD e? e ): Substituting the above in the previous equation, we get (3.5) P ;d (D QD e? e ) = 0: Since AD e = Ad = 0, P ;d e = e. But this implies (6) is valid. (6) =) (7) : This is immediate. (7) =) (8) : Suppose that d satises (7). Assume that a given y 2 < n satises the inequality ky? k < min. Then, jy i? i j < min, for all i = ; : : :; n. Thus, it follows that y > 0. Hence (8) must hold.
8 8 BAHMAN KALANTARI (8) =) () : Suppose that d satises (8), and = 0. Since Q is positive semidenite, there exist x 2 W \ K, x 6= 0, such that (3.6) Qx = 0; Ax = 0: Let w = D? x. Note that w 0, w 6= 0. Since P ;dw = w, and D e = d, we have (3.7) w T P ;d D QD e = w T D QD e = x T Qd: On the one hand, since Qx = 0, we must have x T Qd = 0. On the other hand, since P ;d D QD e > 0, and w 0, w 6= 0, we get x T Qd > 0, a contradiction. Thus, if (8) holds, then so does (). () =) (5) : Since () implies (4), given any 2 K, we can select = e. Then, there exists d 2 W \K such that P e;d D e QD e e = P e;d e = e. Thus, (5) holds. (5) =) () : Suppose that d satises (5). Then, d satises (8) for = e. Hence, > 0. (6) =) (9) : This implication and and its converse are trivial. To prove the remaining claims of the theorem, we need to observe that since is strictly convex, the minimizer d of, if it exists, must be unique. We have already seen that if d satises (4) or (6), then it must satisfy P r (d) = 0. Thus, d is the minimizer of. Hence it satises (3) and is unique. Next we show that d also satises (2). Suppose that d 0 satises (2). Then, P rx (d 0 ) = 0. Since (d 0 ) > 0, some positive scalar multiple of d 0, say d 0 satises P r (d 0 ) = 0. But then we must have d 0 = d. Since X (x) = X(x), for all x 2 W \ K, d also satises (2). 4. Proof of Theorem 2.3. Proof. Assume that ( 0 ) holds. Let Q = cc T. Then, the corresponding is positive. From Theorem 2. we conclude that for each i = 2; : : :; 9, condition (i) is satised. (2) =) (2 0 ) : Since X (x) = f 2 (x), and ct x > 0, for x 2 W \ K, x 6= 0, (2 0 ) follows. (3) =) (3 0 ) : Suppose that d satises (3). Then cc T d? D e? = AT v, for some vector of lagrange multipliers, v. Since = c T d > 0, from the above equation we get c? D? e = A T v, where d = d=, v = v=. But this implies that d is the minimizer of g (x) = c T x + F (x). (4) =) (4 0 ) : Suppose d satises (4), i.e., P e;d D e cc T D e e = P e;d. Since = c T d > 0, ^d = d lies in W \ K. Since P e;d = P e; ^d, we get P e; ^d ^D e c = P e; ^d. Analogously to the above, it follows that for i = 5? 8, if d satises (i), then ^d = (c T d)d satises (i 0 ). (8 0 ) =) ( 0 ) : Follows from similar argument given for proof of the corresponding part of Theorem 2.. (6 0 ) =) (9 0 ) : This and its converse are trivial. The proof of the remaining parts follow in a similar fashion as in Theorem Two polynomialtime algorithms for HP, SP, HSP, and ASP. Here we will describe two polynomialtime algorithms for solving approximate version of these problems. These algorithms are conceptually easy. However, their analysis requires the use of scaling dualities, signicant bounds, as well as some results from Nesterov and Nemirovskii's machinery of selfconcordance. First, we dene  approximate version of the four problems. Next, we describe the two algorithms. We will then state without proof the main ingredients, other than the scaling dualities already proved. Finally, we will justify how the main ingredients result in the stated complexity results. For the proof of the stated theorems and in much more generality, see Kalantari [3].
9 AGMean Inequality, LP Matrix Scaling, Gordan's Theorem 9 Given 2 (0; ], HP, SP, HSP, and ASP are dened as follows. HP is to compute d 2 W \K \S such that (d), or proving that such point does not exist. Given 2 K, ASP is to test the solvability of kp ;d D QD e? e k <. SP is to compute, if it exists, a point d 2 W \ K such that (d)?. HSP is to compute, if it exists, a point d 2 W \ K such that X (d)=x exp(). In particular, if = min, then from Theorem 2. the solution of ASP will result in a point d 2 W \ K such that P ;d D QD e > 0. For the convexhull problem (see Remarks and 3) such a point gives rise to a separating hyperplane. Let u =?P r (d 0 ), where d 0 is a given point in W \ K \ S. For each t 2 (0; ), dene Given d 2 W \ K, let f (t) ;d f (t) (x) = t(x) + tut x + F (x): (t) (x) = f (D x) = t ;d (x) + tu T Dx + F (x) + F (d): Also, given d 2 W \ K, let y be the Newton direction with respect to, i.e., the solution to the equation P r 2 (d)y =?P r (d), where P is the orthogonal projection operator onto W. The Newton decrement is dened as (d) = [y T r 2 (d)y] =2. Let 2 [ ; ), where = 2? p 3. The Newton iterate is d 0 = NEW ( ; d) = d + (d)y; (d) = Analogously, corresponding to f (t) 8 >< >: ; if (d) > +(d) ;?(d) (d)(3?(d)) if (d) 2 [ ; ]; ; if (d) <., one denes y t, t (d), and d 0 t = NEW (f (t) ; d). Consider the following PotentialReduction algorithm: PotentialReduction: Initialization. Let d = d 0. Iterative Step. Replace d with d 0 = NEW ( ; d) and repeat. Theorem 5.. (PotentialReduction Complexity Theorem, Kalantari [3]) Assume that e. Let be in (0; ). Consider the PotentialReduction algorithm, and let = exp[ 2 (d 0 )? + ln ], and 2 R = supfexp(? 2 F (x)) : x 2 W \ K ; kxk = g. If = 0, the number of iterations to solve HP is O( ln R ). If > 0, the number of iterations to solve SP, HSP, or ASP is O( ln R kqk + ln ln ). Consider the following PathFollowing algorithm, where t 2 (0; ) is an appropriately selected number: PathFollowing: Initialization. Let t =, d = d 0. Phase I. While t > t, replace (d; t) with (d 0 t ; t0 ), where d 0 t absolute positive constant (can be taken equal to =9). Phase II. Replace d with d 0 = NEW (f (t) ; d) and repeat. (t) = NEW (f ; d), t0 = t exp(?c = p ), c an Theorem 5.2. (PathFollowing Complexity Theorem, Kalantari [3]) Assume e. Let be in (0; ). Consider the PathFollowing algorithm.
10 0 BAHMAN KALANTARI If = 0, the number of iterations to solve HP is O( p ln kuk iterations to solve ASP is O( p ln kuk is O( p ln kuk + ln ln kqk ). + ln ln kqk). If > 0, the number of + ln ln kqk), and the number of iterations to solve SP or HSP Remark 4. The above algorithms also apply to the case of an arbitrary > 0, not necessarily satisfying e. Essentially all is needed is to replace with 0 = = min. These algorithm can be used to solve linear programs, to compute separating hyperplanes, to diagonally scale positive semidenite symmetric matrices over an arbitrary subspace, or to compute the minimum of the ratio of an arbitrary linear (positive semidenite form) to the geometric mean (square of the geometric mean) over the positive points of an arbitrary subspace. The implementation of the above two algorithms for SP, HSP, and ASP requires the computation of a lower bounds on, assuming that it is positive. For the computation of such a lower bound for the case where Q has rational or algebraic entries, see [2]. These algorithms can also be applied to solve Karmarkar's canonical LP. All is needed is to replace the linear form c T x with (c T x) 2. In particular, taking = e, the PathFollowing algorithm gives an O( p n ln nkuk ) iteration complexity for solving approximate version of Karmarkar's problem. As usual, over the rational inputs can be taken to be O(2?L ), where L is the size of LP (see [2] for dierent notions of size, over the rationals or the algebraic numbers). The PathFollowing algorithm not only results in the bestknown iteration complexity for solving LP (see e.g. [4]), while being one of the simplest of such algorithms, but also is an algorithm that is capable of solving more general diagonal matrix scaling problems than that considered in [6], as well as having the capability of solving SP and HSP. We now explain the main ingredients, other than the scaling dualities already proved, that result in the above two complexity theorems. The following theorem characterizes basic properties of Newton's method (see [9], Theorem 2.2.2), and main properties of the parametric family f (t) (see [9], Theorem 3..): Theorem 5.3. (Nesterov and Nemirovskii [9]) Given d 2 W \ K, d 0 = NEW ( ; d) 2 W \ K, and 8 >< >: (d 0 )? (d)?((d)? ln( + (d))); if (d) > ; (d 0 ) (6(d)? 4 2 (d)? ); if (d) 2 [ ; ]; (d 0 ) 2 (d) ; if (d) < (?(d)) 2. If (d) < 3, then = inff (x) : x 2 W \ K g >? and (d)?!2 ((d))( +!((d)) ;!((d)) =? (? 3(d)) =3 : 2(?!((d)) Given t 2 (0; ), all the above apply to f (t). Moreover, suppose that t(d) =4, then t 0(d) =4, where t 0 = t exp(?c p ). Now we state some other signicant results. Theorem 5.4. (Kalantari [3], x 8) Assume that e. Suppose that d 2 W \K satises (d) < =2. Then, kqk (d) kp ;d r ;d (e )k = O (d) : Suppose that > 0. Given any t 2 (0; ], suppose that d 2 W \ K satises t (d) <. Then, kqk kuk kp ;d rf (t) ;d (e )k = O t (d) :
11 AGMean Inequality, LP Matrix Scaling, Gordan's Theorem In either case ( > 0, or 0), given any t 2 (0; ], suppose that d 2 W \ K is a point obtained via Phase I of the PathFollowing algorithm, and t (d) < =2. Then, kp ;d rf (t) ;d (e )k = O kqk t (d)( t p ) c : Theorem 5.5. (Kalantari [3], x 9) Let the kth iterate of the PotentialReduction algorithm be denoted by d k. If (d k ), then d k kd k R exp? 2k ; =? ln( + ): k Theorem 5.6. (Kalantari [3], x 9) Suppose that > 0. Then, for all x 2 W \ K we have X (x) ln X 2 (x)? : The following theorem is the nal ingredient, a signicant consequence of the scaling dualities: Theorem 5.7. (Kalantari [3], x 0) Let be a number in (0; ]. Given t 2 (0; ], suppose that d 2 W \ K satises kp ;d rf (t) ;d (e )k. Let ^d = p td. If kp 2 ; ^d r ; ^d (e )k, then ( kdk d ) C()t, where C() = [2 + (p + )]kuk 2 : 2 Now we describe how Theorem 2., and Theorems imply the desired complexity theorems, Theorem 5., and Theorem 5.2. We will rst describe this for Theorem 5.. From Theorem 2., HP is solvable (i.e. = 0) if and only if the other three problems are not. If = 0, Theorem 5.5 gives the desired complexity bound to solve HP. If > 0, since (d k =kd k k), Theorem 5.5 implies that in O( ln R ) iterations we obtain a point x 2 W \ K such that (x) <. Let x k be the subsequent iterates of the PotentialReduction algorithm, where x 0 = x. From Theorem 5.3 it follows that (5.) (x k ) r 2k ; r = (? ) 2 : The above together with the rst bound in Theorem 5.4 implies that the number of iterations, k, to get a point x k satisfying ASP, is O(ln ln kqk ). Hence the claimed complexity for solving ASP. From Theorem 5.3 it also follows that (see [3]) once we have obtained a point x k such that (x k ) < minf ; (? 3!( ))g, we have (5.2) (x k )? : This together with the bound in Theorem 5.6 gives the desired complexity for solving SP, or HSP. We will next derive the complexity result of Theorem 5.2. From Theorem 5.3 it follows that given any t 2 (0; ], since (d 0 ) = 0, the number of iterations, k t, to obtain d 2 W \K such that t (d) <, satises (5.3) O p ln ( t ) :
12 2 BAHMAN KALANTARI Suppose that = 0, then from Theorem 2. we must have kp ; ^d r ; ^d (e )k min. Theorem 5.7 it suces to compute d 2 W \ K such that From this, and (5.4) kp ;d rf (t) d (e d )k 2 min; where t satises (5.5) = C( min) : t Thus, the number of iterations of Phase I will equal O( p ln kuk ), and at the termination of this phase we have a point x 2 W \ K such that t (x) <. Let x 0 = x; x ; : : :; x k be the sequence of iterates of Phase II. From Theorem 5.3, it follows that t (x k ) r 2k. From this and the third upper bound given in Theorem 5.4, it follows that the number of iterations of Phase II to solve HP is O(ln ln kqk?p =c ). Hence the claimed combined complexity of the two phases. Suppose that > 0. To solve ASP, from Theorem 5.7 it suces to compute d 2 W \ K such that (5.6) kp ;d rf (t) d (e d )k 2 ; where t satises (5.7) t = 2C() : Thus, the number of iterations of Phase I is O( p ln kuk ). At the termination of this phase we have a point x 2 W \ K such that t (x) <. Let x 0 = x; x ; : : :; x k be the sequence of iterates of Phase II. From the second bound of Theorem 5.4, and since t (x k ) r 2k, the number of iterations of Phase II is O(ln ln kqk kuk ). Hence the claimed combined complexity. To solve SP, or HSP, we rst solve ASP, i.e., compute a point d > 0 such that kp ;d D QD e? e k <, in O( p ln kuk ) iterations. We then replace d with NEW ( ; d), and repeat this step. The number of latter iterations can be estimated as it was done for the PotentialReduction algorithm. 6. Concluding remarks. In this paper we have proved several dualities relating a generalization of the classical arithmetic geometric mean inequality to linear programming, to Gordan's theorem, and to several diagonal matrix scaling problems. These dualities, called scaling dualities, although were derived via elementary means are remarkable and signicant from the theoretical and algorithmic point of view. Many of the scaling dualities can be stated with respect to much more general convex programming problems, relating their equivalent HP formulation to problems that can be viewed as analogues of SP, HSP, and ASP, considered in this paper. The latter three problems are genuinely dual to the HP formulation of convex programming problems, i.e., HP is solvable if and only if the other three are not. Indeed, as shown in [3], given an arbitrary closed convex pointed cone K in a nite dimensional space, a logarithmically homogeneous barrier F (x) for the interior of K, a subspace W, and a homogeneous function with homogeneous degree p > 0, the scaling equation can be dened as P d r d (e d ) = e d, or P d r 2 d (e d ) = (p? )? e d, if p 6= ; where d 2 W \ K, D = r 2 F (d)?=2, d (x) = (Dx), e d = D? d, and P d the orthogonal projection onto W d = D? W. For instance, semidenite programming can be formulated as the problem of testing the existence of a nontrivial zero of tr(cx), over an arbitrary subspace of the space of symmetric matrices, and its intersection with the cone of positive semidenite symmetric matrices, K = S + n, where c is a given symmetric matrix, and tr() is the trace function. Now suppose that given 2 K, we take F (x) =? ln det(x ), where x is dened as follows. Firstly, given a matrix x 2 K, the matrix
13 AGMean Inequality, LP Matrix Scaling, Gordan's Theorem 3 exp(x) = P j= xj =j! is welldened and lies in K. If x is a diagonal positive denite matrix, dene ln x as the matrix diag(ln x ; : : :; lnx n ). If x is an arbitrary positive denite matrix, dene ln x = U x ln x U T x, where x = U x x U T x, with U x unitary, and x the diagonal matrix of eigenvalues of x. Dene x as the matrix exp( =2 ln x =2 ). Now if for a given d 2 K we dene D = r 2 F (d)?=2, and e = I the identity matrix, and D e the operator that maps x 2 S + n to d =2 xd =2, then all the results proved or presented in this paper apply, verbatim. These include all the complexity theorems, and the auxiliary theorems that imply them. The quantity min will stand for the minimum eigenvalue of. These results are the subject of a forthcoming paper, [4]. In particular, in [4] we prove the following inequality which is a generalization of the arithmeticgeometric mean inequality as well as the wellknown tracedeterminant inequality: let be an arbitrary matrix in K, the cone of positive denite symmetric matrices, and = tr(). Then, tr(x) (det(x )) = ; 8x 2 K : REFERENCES [] H. Alzer, A proof of arithmeticgeometric mean inequality, Amer. Math. Monthly, 03 (996) 585. [2] G.B. Dantzig, Linear Programming and Extensions (Princeton University Press, Princeton, New Jersey, 963). [3] P.S. Bullen, D.S. Mitrinovic, and P.M. Vasic, Means and Their Inequalities (Reidel, Dordrecht, 988). [4] C.C. Gonzaga, Pathfollowing methods for linear programming, SIAM Review, 34 (992) [5] G.H. Hardy, J.E. Littlewood, and G. Polya, Inequalities (Cambridge University Press, Cambridge, 952). [6] B. Kalantari, Karmarkar's algorithm with improved steps, Math. Programming, 46 (990) [7] B. Kalantari, Canonical problems for quadratic programming and projective methods for their solution, Contemporary Mathematics, 4 (990) [8] B. Kalantari, Derivation of a generalized and strengthened Gordan theorem from generalized Karmarkar potential and logarithmic barrier functions, Technical Report LCSRTR2, Department of Computer Science, Rutgers University, New Brunswick, NJ, 989. [9] B. Kalantari, Generalization of Karmarkar's algorithm to convex homogeneous functions, Oper. Res. Lett., (992) [0] B. Kalantari, A simple polynomial time algorithm for a convex hull problem equivalent to linear programming, Combinatorics Advances, Kluwer Academic Publishers (995) [] B. Kalantari, A Theorem of the alternative for multihomogeneous functions and its relationship to diagonal scaling of matrices, Linear Algebra and its Applications, 236 (996) 24. [2] B. Kalantari and M.R. EmamyK, On linear programming and matrix scaling over the algebraic numbers, Linear Algebra and its Applications, 262 (997) [3] B. Kalantari, Scaling dualities and selfconcordant homogeneous programming in nite dimensional spaces, Technical Report DCSTR 359, Department of Computer Science, Rutgers University, New Brunswick, NJ, 998. [4] B. Kalantari, On the tracedeterminant inequality, semidenite programming, and matrix scaling over the cone matrices, forthcoming. [5] N. Karmarkar, A new polynomial time algorithm for linear programming, Combinatorica, 4 (984) [6] L. Khachiyan and B. Kalantari, Diagonal matrix scaling and linear programming, SIAM J. Optim., 4 (992) [7] A.W. Marshall and I. Olkin, Scaling of matrices to achieve specied row and column sums, Numer. Math., 2 (968) [8] D.S. Mitrinovic, J.E. Pecaric, and A.M. Fink, Classical and New Inequalities in Analysis (Kluwer Academic Publishers, Dordrecht, 993). [9] Y. Nesterov and A.S. Nemirovskii, InteriorPoint Polynomial Algorithms in Convex Programming (SIAM, Philadelphia, PA, 994). [20] A. Schrijver, Theory of Linear and Integer Programming (John Wiley and Sons, New York, 986).
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