Studia Scientiarum Mathematicarum Hungarica 42 (2), (2005) Communicated by D. Miklós

Transcription

1 Studia Scientiarum Mathematicarum Hungarica 4 (), 7 6 (5) A METHOD TO FIND THE BEST BOUNDS IN A MULTIVARIATE DISCRETE MOMENT PROBLEM IF THE BASIS STRUCTURE IS GIVEN G MÁDI-NAGY Communicated by D Miklós Abstract The multivariate discrete moment problem (MDMP) is to find the minimum and/or maximum of the expected value of a function of a random vector which has a discrete finite support The probability distribution is unknown, but some of the moments are given The MDMP has been initiated by Prékopa who developed a linear programming methodology to solve it The central results in this respect concern the structure of the dual feasible bases These bases provide us with bounds without any numerical difficulty (which is arising in the usual solution methods) In this paper we shortly summarize the properties of the above mentioned basis structures, and then we show a new method which allows us to get the basis corresponding to the best bound out of the known structures by optimizing independently on each variable We illustrate the efficiency of this method by numerical examples Introduction The multivariate discrete moment problem (MDMP) has been introduced and discussed in papers by Prékopa [, 3, 4] and by Mádi-Nagy and Prékopa [] We present the main results of that field The problem is formulated in connection with a random vector (X,, X s ) in the following way We assume that the support of X j is a known finite set Z j = {z j,, z jnj }, consists of distinct elements, and we define p i i s = P (X = z i,, X s = z sis ), i j n j, j =,, s, µ α α s = n i = n s i s= z α i z αs si s p i i s, where α,, α s are nonnegative integers The number µ α α s will be called the (α,, α s )-order moment of the random vector (X,, X s ), and the sum α + + α s will be called the total order of the moment Mathematics Subject Classification Primary 9C5 Key words and phrases Stochastic programming, linear programming, applied probability Partially supported by OTKA grants F-4639 and T-4734 in Hungary 8 696/$ c 5 Akadémiai Kiadó, Budapest

2 8 G MÁDI-NAGY Let Z = Z Z s and f(z), z Z be a function for which we will require some assumptions Let f i i s = f(z i,, z sis ) One way to formulate the multivariate discrete moment problem is the following min(max) n i = n s i s= f i i s p i i s subject to () n i = n s i s= z α i z αs si s p i i s = µ α α s for α j, j =,, s; α + α s m and for α j =, j =,, k, k +,, s, m α k m k, k =,, s; p i i s, all i,, i s In problem () the unknown variables are the p i i s, all other quantities are known This means that in addition to all moments of total order at most m, the at most m th k order moments (m k m) of the k th univariate marginal distribution are also known, k =,, s The above problem serve for bounding () E [ f(x,, X s ) ] under the given moment information We will use the notations of the compact matrix form of problem () (compatible with the notation of Mádi-Nagy and Prékopa []): min(max) f T p subject to (3) Âp = b p Let V min (V max ) designate the minimum (maximum) value in problem () Let further B ( B ) designate a dual feasible basis (ie, a basis for which

3 BEST BOUNDS IN A MULTIVARIATE DISCRETE MOMENT PROBLEM 9 the optimality condition is satisfied) for the minimization (maximization) problem Then, by linear programming theory, we know that (4) f Ṱ B p B V min E [ f(x,, X s ) ] V max f Ṱ B p B If B ( B ) is an optimal basis in the minimization (maximization) problem, then the first (last) inequality holds with equality We say that V min and V max are the sharp lower and upper bounds, respectively, for the expectation of f(x,, X s ) Consider the set of subscripts ( s (5) I = I j= I j ), where (6) I = { (i,, i s ) i j m, integers, j =,, s, i + + i s m } and (7) I j = { (i,, i s ) i j K j, i l = l j } K j = {k () j,, k ( K j ) j } {m, m +,, n j }, j =,, s Let us consider four different structures for K j : (8) K j even K j odd min u (j), u (j) +,, v (j), v (j) + m, u (j), u (j) +,, v (j), v (j) + max m, u (j), u (j) +,, v (j), v (j) +, n j u (j), u (j) +,, v (j), v (j) +, n j In order to make the paper self-contained we recall two theorems and two algorithms from earlier papers In what follows we will use the following Assumption The function f(z), z Z has nonnegative divided differences of total order m +, and in addition, in each variable z j it has nonnegative divided differences of order m j + := m + K j Theorem (Mádi-Nagy and Prékopa []) Let z j < z j < < z jnj, j =,, s Suppose that function f fulfils Assumption, where the set K j has one of the min structures in (8) Under these conditions the set of columns B of  in problem (3), with the subscript set I, is a dual feasible basis in the minimization problem (3), and (9) E [ f(x,, X s ) ] f Ṱ B p B

4 G MÁDI-NAGY If B is also a primal feasible basis in problem (3), then the inequality (9) is sharp In the next theorem we give both lower and upper bounds for the function f(z,, z s ), (z,, z s ) Z and the expectation E [ f(x,, X s ) ] Theorem (Mádi-Nagy and Prékopa []) Let z j > z j > > z jnj, j =,, s Suppose that function f fulfils Assumption, where K j has one of the structures in (8) that we specify below (a) If m + is even, K j is even and K j has the max structure in (8) or m + is even, K j is odd and K j has the min structure in (8), then the set of columns B in Â, corresponding to the subscripts I, is a dual feasible basis in the minimization problem (3) We also have the inequality () E [ f(x,, X s ) ] f Ṱ B p B (b) If m + is odd, K j is even and K j has the max structure in (8) or m + is odd, K j is odd and K j has the min structure in (8), then the basis B is dual feasible in the maximization problem (3) We also have the inequality () E [ f(x,, X s ) ] f Ṱ B p B The above two theorems yield dual feasible basis structures with the aid of the subscript set I defined in (5), ordering the elements of Z j s increasingly or decreasingly In the bivariate case (s = ), (still at Assumption ) we can give much more dual feasible bases corresponding to I, by suitable (not necessary increasing or decreasing) ordering of the variables In the following, we sketch these methods Detailed discussion with illustrative figures and examples can be found in Mádi-Nagy and Prékopa [] Consider first the case, where we want to construct lower bound We present an algorithm to find these sequences We may assume, without loss of generality, that the sets Z and Z are the following: Z = {,,, n }, Z = {,,, n } Min Algorithm (Mádi-Nagy and Prékopa []) Algorithm to find z,, z (m ) ; z,, z (m ) Step Initialize t =, q m, L = (,,, q ), U = ( n, n,, n (m q ) ) Let (z,, z (m ) ) = (arbitrary merger of the sets L, U) If U is even, then z =, l =, u = n, and if U is odd, then z = n, l =, u = n If t = m, then go to Step Otherwise go to Step

5 BEST BOUNDS IN A MULTIVARIATE DISCRETE MOMENT PROBLEM Step If z (m t) L, then let z (t+) = l t, l t+ = l t +, u t+ = u t, and if z (m t) U, then let z (t+) = u t, u t+ = u t, l t+ = l t Set t t + If t = m, then go to Step Otherwise repeat Step Step Stop, z,, z (m ) ; z,, z (m ) have been created Let,,, q, n,, n (m q ) be the numbers used to construct z, z,, z (m ) Then let (z jm, z j(m+),, z jnj ) = ( q j +, q j +,, n j (m q j ) ), j =, If m q j even, then K j should follow a minimum structure in (8), and if m q j odd, then K j should follow a maximum structure, j =, We have completed the construction of the dual feasible basis related to the subscript set I If we want to construct an upper bound, then only slight modification is needed in the above algorithm to find z,, z (m ) ; z,, z (m ) We only have to rewrite Step and keep the other steps unchanged, and then give the appropriate K j structures Max Algorithm (Mádi-Nagy and Prékopa []) Algorithm to find z,, z (m ) ; z,, z (m ) Step Initialize t =, q m, L = (,,, q ), U = ( n, n,, n (m q ) ) Let (z,, z (m ) ) = (arbitrary merger of the sets L, U) If U is odd, then z =, l =, u = n, and if U is even, then z = n, l =, u = n If t = m, then go to Step Otherwise go to Step, etc In case of the upper bound we have to choose K j the other way around as in case of the Min Algorithm If m q j even, then K j should follow a maximum structure, otherwise a minimum structure We have completed the construction of the dual feasible basis related to the subscript set I The new method In the previous section we showed dual feasible bases related to the index set I, where I was fixed, and we could find several I j, j =,, s, depending on which K j we have chosen among the related structures of (8) Considering the basic columns (which are assigned directly to the elements of Z, and just indirectly to the index set I), at Theorem and the fix I determines the elements of Z I, and the elements of Z Ij just depend on K j, j =,, s,

6 G MÁDI-NAGY in the bivariate Min and Max Algorithms the fixed I also determines the elements of Z I for a given z,, z (m ), and then the elements of Z Ij just depend on K j, j =, It means that in the above cases we can identify a basis by the K j sets j =,, s, unambiguously Below we present a method, by which we can find the K j sets independently, which correspond to the basis giving the best bound among the structures of Z I This method on one hand, clarifies the structure of the Z I - type bases, and on the other hand, gives an effective way to find the best bounds (and the related bases) yielded by the Min and Max Algorithms This second assertion is illustrated by numerical examples in the last section At first, let us define the investigated problem precisely: our aim is to find the set system K j, for which the inequality (9) (()) gives the best bound, ie, for which the objective function value is the highest in case of the minimum problem (lowest in case of the maximum problem): max(min) f T p subject to () Âp = b p the set of the Z I -type basic solutions Till now, in former works this problem was handled in the simplest way: the objective function value was calculated for all possible Z I -type bases and then the best bounds (and related bases) could be picked out This method seemed to be useful to give (not sharp, but usually close) bounds much faster than the dual method, without numerical difficulties, see Mádi-Nagy and Prékopa [] In some examples of this paper we could give suitable bounds in this manner even for cases, where the dual method did not yield useful bounds within acceptable time This phenomenon occured mostly because of the size of the problem In the following, we show how the closest Z I - type bounds can be found in much shorter way, which gives the possibility for bounding still greater sized problems At first, we convert the problem into a form, which fulfils some conditions We will see that the converted problem is equivalent with the original problem The conditions mentioned above are the following: (3) z j =, j =,, s and (4) f(z,, z s ) =

7 BEST BOUNDS IN A MULTIVARIATE DISCRETE MOMENT PROBLEM 3 If we consider an arbitrary problem(), which fulfils Assumption, then the function (5) f converted (z) = f (z + z,, z s + z s ) f (z,, z s ) on the domain (6) Z converted = Z (z,, z s ), (a) satisfies conditions (3) and condition (4), (b) has the same convexity as the original f (ie fulfils Assumption ) on the domain Z converted = Z (z,, z s ) Consider the coefficient matrix Âconverted fitting Z converted This matrix can be obtained from the original  by a nonsingular transformation; ie, there exists a nonsingular matrix D such that  converted = D (and  = D  converted ) Let bconverted = D b Then the converted problem is min(max) f converted T p subject to (7)  converted p = p b converted The above means that (7) has the same feasible set and the same dual feasible set (because of assertion (b)) Moreover, taking into account that e T p = for each solution, the related objective function value of the converted problem is always lower by the constant f (z,, z s ) This indeed means the equivalence of the original and the converted problems Hence if we solve () for the converted problem, then we get the optimal solution of () for the original problem, as well In the following we present our new method for problems which satisfy conditions (3), (4) This does not mean any restriction because a general problem () can be converted into an equivalent one which fulfils the conditions as we have shown in the previous paragraph

8 4 G MÁDI-NAGY For rewriting the problem to an appropriate form, let us consider the following index sets: (8) I int = { (i,, i s ) i j m, integer, j =,, s, i + + i s m }, I axes j = { (i,, i s ) i j n j, integer; i l = for l j } Fig Illustration of the index sets (8), where n = n = 9, m = 5 The elements of I int are designated by, the elements of I axes and I axes are designated by

9 BEST BOUNDS IN A MULTIVARIATE DISCRETE MOMENT PROBLEM 5 If we reorder the columns and rows of the constraint matrix of problem () according to the index sets above, we get a more perspicuous structure: s Z I int other columns b f T : f T I axes f T I axes s f T I int (9)  : µ z z n µ  Iaxes I axes I int z m zm n µ m z s z sns µ s  Iaxes s z ms s Is axes zms sn s I int µ ms  Iint I int p T : p p T I axes p Ṱ B : (p) B p T I axes s p T I int ( ) ( ) ( ) p T I axes B p T I axes s B p T I int B At problem (9) we have introduced some new notations, which help us in the following arguments The subscripts denote the columns of the matrix, while the superscripts refer to the rows Moreover (see later) the negative sign denotes the complement, ie, the columns and rows which are dropped out of the matrix p denotes an appropriate basic solution, while the p B vector consists of the components of the basis variables From the structure of the Z I bases follows that there is no basic variables in the last block of p T, hence these components have the value zero Between the rows p Ṱ and pt we referred by equality signs to the fact, B that all variables of p and p I int are also basic variables for each Z I -type basic solutions Considering that the last block of p has only zero values, we can drop out the related columns from (9):

10 6 G MÁDI-NAGY s Z I int b f T I axes f T I axes f T I axes s f T I int () µ = z z n µ  Iaxes µ I axes I axes I int z m zm n µ m z z n µ ÂIaxes I  Iaxes I int z m zm n µ m z s z sns µ ÂIaxes s  Iaxes s z ms s Is axes zms sn s I int µ I axes µ I axes s µ ms  Iint I int µ I int p p T I axes p T I axes p T I axes s p T I int Since the minor ÂIint is a square matrix, and at a basic solution we generate µ I int by the linear combination only of its columns, the vector p I I int are given by the formula below independently from the choices of sets K j : () p I int = (ÂIint I ) µ I int Considering that the value of p I int () in the following way: is constant, we can rewrite problem constant ( ) max(min) f T (,I axes,,i axes s ) p (,I axes,,i axes s ) + f T I intp I int subject to ()  (,I axes,,i axes s )p (,I axes,,i axes s ) = b A I intp I int = b

11 BEST BOUNDS IN A MULTIVARIATE DISCRETE MOMENT PROBLEM 7 p (,I axes,,i s axes ) = the corresponding part of a Z I -type basic solution In matrix form: s b f T I axes f T I axes f T I axes s (3) µ T p I int z z n I µ I axes p I int I int z m zm n z z n ÂIaxes I z m zm n µ I axes z s z sns s z ms s Is axes zms sn s µ I axes s ÂIaxes p I int I int ÂIaxes s p I int I int µ I int ÂIint I p I int = p p T I axes p T I axes p T I axes s In (3) there are only zeros in the rows corresponding to I int, thus we can drop out them Then we get the forthcoming equivalent problem: max(min) f T (,I axes,,i axes s ) p (,I axes,,i axes s ) subject to (4)  ( Iint ) (,I axes,,i s axes ) p (,I axes,,i s axes ) = b ( Iint ) p (,I axes,,i s axes ) = the corresponding part of a Z I -type basic solution Since f(z,, z s ) =, considering assumption (4), furthermore only the first component of the first column of  is different from zero, it is more

12 8 G MÁDI-NAGY convenient sorting the constraints in the following way: s b( I int ) f T I axes f T I axes f T I axes s b (5) z z n ÂIaxes I I axes b z m zm n z z n ÂIaxes I I axes b z m zm n z s z sns s Is Is axes b z ms s zms sn s p p T I axes p T I axes p T I axes s On one hand, the value of the objective function is independent from the value p, on the other hand, p has nonzero coefficient only in the first constraint in (5) Considering these, we can optimize the problem by taking into account only the variables corresponding to I,, I s and then calculating the value p by solving the equation of the first constraint, ie, subject to max(min) f T (I axes,,i s axes ) p (I axes,,i s axes ) (6) Â ( Iint ) (I axes,,i s axes ) p (I axes,,i s axes ) = b ( Iint ) p (I axes,,i s axes ) = the corresponding part of a Z I -type basic solution, (7) p = ( b ) (,, ) T p (I axes,,i s axes )

13 BEST BOUNDS IN A MULTIVARIATE DISCRETE MOMENT PROBLEM 9 Rewrite the objective function and the constraints of () in matrix form, skipping out the minors containing only zeros: s b( I int ) f T I axes f T I axes f T I axes s (8) z z n ÂIaxes I I axes b z m zm n z z n ÂIaxes I I axes b z m zm n z s z sns ÂIaxes s Is Is axes b z ms s zms sn s p T I axes p T I axes p T I axes s It can be seen that problem () splits up the following type of smaller subproblems: max(min) f T I j axesp Ij axes subject to (9) Â I j axes I j axesp Ij axes = b Iaxes j p Ij axes = the corresponding part of a Z I -type basic solution, j =,, s The last constraint above means that the subscript set of the basic variables of p Ij axes is the union of the part of I which contains the related axis and the set I j what is characterized by K j So far, we have explored the structure of problem () Taking the advantage of the fact that we can find the best basic solution component independently on each axes, we can write up the related new method The steps of the method are the following: (i) Convert the problem into the equivalent form, which satisfies (3) and (4) Then we start to solve this equivalent problem in the following way

14 G MÁDI-NAGY (ii) From equation () we know that p I int = (ÂIint I ) µ I int (iii) Using the definition in (): b = b ÂI intp I int Then we can write up problems (9) (iv) We solve problems (9) for j =,, s Let the optimal solutions be signed by p opt I j axes, j =,, s (v) From problem (): p opt = ( b ) (,, ) T p opt (I axes,,i s axes ) (vi) The other variables are zero, thus we have obtained the optimal solution of (): ( p opt ) T = ( ) p, (p opt (I axes,,i axes s )) T, (vii) This solution is also optimal solution of the original problem, just the related objective function value will be greater by f (z,, z s ) 3 Numerical Examples In this section we present the efficiency of the above method For the sake of simplicity we restrict ourselves to the bivariate case In the first part some examples of the paper of Mádi-Nagy and Prékopa [] are solved in several ways We give the best lower and upper bounds of the Min and Max Algorithms, and the related CPU times (assigned by CP U f in the table), when we simply calculate all of objective function values corresponding to the bases of the structures and then choose the best one among them (how we did it in former works) We also give the CPU s of the new method (CP U n ), applying it to the structures of the Min and Max Algorithms Comparing the running times we will see how faster and more effective the method of our paper is We also present the results of a numerically stable dual method to show that the (not sharp) bounds of the Min and Max Algorithms give a quite close results to the sharp bounds, within much shorter running time In the second part we show some examples for greater sized problems, where the dual method or the former method for the Min and Max Algorithm can not give results within a reasonable time Fortunately, the new method turns out to be fast enough to yield useable bounds We use programs written in Wolfram s Mathematica In connection with the Min and Max Algorithms on one hand we test all possible bases that we can obtain this way and choose the best ones (former method) and on the other hand we also find the best bounds by the new method In addition, we execute the dual algorithm starting from the bases ob-

15 BEST BOUNDS IN A MULTIVARIATE DISCRETE MOMENT PROBLEM tained by the use of Theorem (Theorem ), where K j = {m,, m j } (K j = {n j m j,, n j m}), j =, We use Bland s rule to avoid cycling Example The problem is taken from Prékopa, Vizvári and Regős [5] We have 4 events subdivided into two -element groups; X j equals the number of events that occur in the jth group, j =,, Z Z = {,, } {,, } We want to find bounds for the probability that at least one out of the 4 events occurs, ie P (X + X ) = E [ f(x, X ) ], where, if (z, z ) = (, ) (3) f(z, z ) =, otherwise Prékopa [3] has shown, that if m + is even (odd), then all divided differences (3) of total order m + are nonpositive (nonnegative) Consider the cross binomial moments of order (α,, α s ) (α,, α s are nonnegative integers): (3) S α α s = E [( X α ) ( Xs α s )] We can formulate the related binomial moment problem as an LP problem very similarly, as in case of the power moment problem It can be proved that, for a given objective function and support set, the power moment problem and the binomial moment problem can be transformed into each other by simple nonsingular transformations This means if we consider the matrices of the equality constraints there exists a nonsingular square matrix by which and by its inverse the two coefficient matrices can be transformed into the each other This implies that our new method also can be applied to binomial moment problems, cf Mádi-Nagy and Prékopa []

16 G MÁDI-NAGY Suppose that we know the following cross binomial moments: st nd group group That means that m = 4, m = m = 6 Below we present the results In the Lower (Upper) Bounds columns there are the best bounds of the Min (Max) Algorithm, while in the Dual Min/Max columns there are the sharp bounds obtained by the dual algorithm CPU times have been appended everywhere (in case of the Min and Max Algorithm for the former and for the new method, respectively) Lower CP U f CP U n Upper CP U f CP U n Dual Min CPU Dual Max CPU From the table above we can see the main tendencies: the dual method has yielded the sharp results but the bounds of the Min and Max Algorithms also have given a good approximation and we could find them within a much shorter time, especially by the use of our new method Example The following problem is taken from Mádi-Nagy and Prékopa [] Consider the function f(z, z ) = e z /+z /+z z /, the support set Z Z = {,, } {,, } and the same binomial moments as in Example We have obtained the following results In case of m = m = 6, m = 3: Lower CP U f CP U n Upper CP U f CP U n

17 BEST BOUNDS IN A MULTIVARIATE DISCRETE MOMENT PROBLEM 3 In case of m = m = 6, m = 4: Dual Min CPU Dual Max CPU Lower CP U f CP U n Upper CP U f CP U n Dual Min CPU Dual Max CPU The initial basis corresponding to the numerical value supplemented by the sign was obtained by the Max Algorithm, where (z,, z ) = (, 9, 8), K = {4, 5, 6, 7}, K = {, 6, 7, 8} In case of the next two examples we were enable to carry out the dual method by the use of CPLEX It always reported infeasibility of the primal problem, even though the moments that have used in the problem allow for feasibility, by construction Generally, our experiences show that the most popular LP packages quite often fail to solve discrete moment problems because of numerical instability This stresses the importance of using numerically very stable algorithms Example 3 This example is also taken from Mádi-Nagy and Prékopa [] Let Z = Z = {,, 4}, m = 6, m = m = 4, and generate the moments by the unique discrete distribution on Z We obtain: µ ij / / / / /3 4/3 49/ / /3 First, consider the bivariate utility function (3) f(z, z ) = log [ ( e αz +a )( e βz +b ) ], defined for where α, β are positive constants e αz +a >, e βz +b >,

18 4 G MÁDI-NAGY It is easy to see that f z j >, f z j <, 3 f z 3 j >, 4 f z 4 j <, 5 f z 5 j >,, j =,, f z z <, 3 f z z 3 f >, z z >, etc All even (odd) total order derivatives of the function are negative (positive) If we restrict the definition of the function to Z = Z Z, then it satisfies the conditions of Theorem and Assume that α = β = a = b = We get the following results: Lower CP U f CP U n Upper CP U f CP U n Dual Min CPU Dual Max CPU Secondly, consider the function (3) Then the bounds are the following Lower CP U f CP U n Upper CP U f CP U n Dual Min CPU Dual Max CPU The sign means that the dual algorithm does not solve the problem in acceptable time Finally, consider the function We have obtained the results: f(z, z ) = e z /5+z z /4+z /5 Lower CP U f CP U n Upper CP U f CP U n Dual Min CPU Dual Max CPU

19 BEST BOUNDS IN A MULTIVARIATE DISCRETE MOMENT PROBLEM 5 Example 4 The last problem we recalculate from Mádi-Nagy and Prékopa [] is the following Let Z = Z = {,, 9}, m = 6, m = m = 4 Consider the X, Y, Y random variables having Poisson distributions with λ parameters 3, 4, 5, respectively We generate the moments of the random vector ( min (X + Y, 9), min (X + Y, 9) ) and obtain µ ij µ 5 = , µ 6 = 66746, µ 5 = , µ 6 = Considering the bivariate utility function (3) with α = 75, β = 5, a =, b = 3, we get the following results: Lower CP U f CP U n Upper CP U f CP U n Dual Min CPU Dual Max CPU Considering the running times above, we can conclude, that for greater sized problems the numerically very stable dual method can not give result within a reasonable time However we can give approximations based on the Min and Max Algorithms by the application of our new method The following example will be the numerical representation of that, ie we will bound greater sized problems by the aid of the method of this paper Example 5 Let Z = Z = {,, }, m = 6, m = m = 4 First, generate the moments by the unique discrete distribution on Z We also consider the moments of the random vector ( min (X + Y, ), min (X + Y, ) ), where X, Y, Y are random variables having Poisson distributions with λ parameters 3, 4, 5, respectively The results of the new method, corresponding to the above moments at case of some functions, are shown below

20 6 G MÁDI-NAGY: A METHOD TO FIND THE BEST BOUNDS log [ ( e 75z + )( e 5z +3 ) ]: Moments Lower CP U n Upper CP U n Unique Poisson e z /+z z /5+z /4 : Moments Lower CP U n Upper CP U n Unique Poisson REFERENCES [] Mádi-Nagy, G and Prékopa, A, On Multivariate Discrete Moment Problems and Their Applications to Bounding Expectations and Probabilities, Mathematics of Operations Research, Vol 9, No (4), 9 58 MR 5e:9 [] Prékopa, A, Inequalities on Expectations Based on the Knowledge of Multivariate Moments, in: Shaked, M, Tong, Y L (Eds), Stochastic Inequalities, Institute of Mathematical Statistics, Lecture Notes Monograph Series, Vol (99), MR 94h:64 [3] Prékopa, A, Bounds on Probabilities and Expectations Using Multivariate Moments of Discrete Distributions, Studia Scientiarum Mathematicarum Hungarica 34 (998), MR k:68 [4] Prékopa, A, On Multivariate Discrete Higher Order Convex Functions and their Applications, RUTCOR Research Report (), 39 [5] Prékopa, A, Vizvári, B and Regős, G, A Method of Disaggregation for Bounding Probabilities of Boolean Functions of Events, RUTCOR Research Report (997), 97 (Received October 9, 3) DEPARTMENT OF DIFFERENTIAL EQUATIONS BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS MŰEGYETEM RAKPART 3 BUDAPEST H- HUNGARY gnagy@mathbmehu