isometric dilations for infinite sequences of noncommuting operators this paper develops a dilation theory for t n l an infinite sequence of noncommuting operators on a hilbert space when the matrix t1 t2 is a contraction a wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for tn i are obtained some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given this results are used to extend the sz nagy foia lifting theorem to this noncommutative setting this paper is a continuation of 5 and develops a dilation theory for an infinite sequence tn n of noncommuting operators on a hilbert space when en 1 tn tn i i is the identity on s many of the results and techniques in dilation theory for one operator 8 and also for two operators 3 4 are extended to this setting first we extend wold decomposition 8 4 to the case of an infinite sequence vn i of isometries with orthogonal final spaces in 2 we obtain a minimal isometric dilation for tn 1 by extending the schaffer construction in 6 4 using these results we give some theorems on the geometric structure of the space of the minimal isometric dilation finally we give some sufficient conditions on a sequence t7 n to be simultaneously quasi similar to a sequence rn n i of isometries on a hilbert space x with en rnrn n ix in 3 we use the above mentioned theorems to obtain the sz nagy foia5 lifting theorem 7 8 1 4 in our setting in a subsequent paper we will use the results of this paper for studying the characteristic function associated to a sequence tn lo 1 with ez 1 t t i throughout this paper a stands for the set 1 2 k k e n or the set n 1 2 received by the editors september 1 1987 1980 mathematics subject classification 1985 revision primary 47a20 secondary 47a45

technical note a langrangian algorithm for the multiple choice integer program in the multiple choice integer program the binary variables are partitioned into generalized upper bounding sets or special ordered sets we solve this problem using a branch and bound scheme designed to take advantage of this structure the basic procedure is enhanced by solving a lagrangean problem as an approximation further we use a variable reduction technique to eliminate many of the variables prior to problem solution computational experience is promising for randomly generated problems scheduling problems and budgeting problems tested

tensor algebras overc correspondences representations dilations andc envelopes tensor algebras overc correspondences are noncommutative generalizations of the disk algebra they contain as special cases analytic crossed products semi crossed products and popescu s noncommutative disk algebras in this paper a dilation theorem and a commutant lifting theorem for representations of tensor algebras is proved these are used to show that for certainc correspondences thec envelopes in the sense of arveson of the tensor algebras are the cuntz pimsner algebras of the correspondences

adaptive blind deconvolution of linear channels using renyi s entropy with parzen window estimation blind deconvolution of linear channels is a fundamental signal processing problem that has immediate extensions to multiple channel applications in this paper we investigate the suitability of a class of parzen window based entropy estimates namely renyi s entropy as a criterion for blind deconvolution of linear channels comparisons between maximum and minimum entropy approaches as well as the effect of entropy order equalizer length sample size and measurement noise on performance will be investigated through monte carlo simulations the results indicate that this nonparametric entropy estimation approach outperforms the standard bell sejnowski and normalized kurtosis algorithms in blind deconvolution in addition the solutions using shannon s entropy were not optimal either for super or sub gaussian source densities

nonsmooth ica contrast minimization using a riemannian nelder mead method this brief concerns the design and application of a riemannian nelder mead algorithm to minimize a hartley entropy based contrast function to reliably estimate the sources from their mixtures despite its nondifferentiability the contrast function is endowed with attractive properties such as discriminacy and hence warrants an effort to be effectively handled by a derivative free optimizer aside from tailoring the nelder mead technique to the constraint set namely oblique manifold the source separation results attained in an empirical study with quasi correlated synthetic signals and digital images are presented which favor the proposed method on a comparative basis

crame spl acute r rao and moment entropy inequalities for renyi entropy and generalized fisher information the moment entropy inequality shows that a continuous random variable with given second moment and maximal shannon entropy must be gaussian stam s inequality shows that a continuous random variable with given fisher information and minimal shannon entropy must also be gaussian the crame spl acute r rao inequality is a direct consequence of these two inequalities in this paper the inequalities above are extended to renyi entropy p sup th moment and generalized fisher information generalized gaussian random densities are introduced and shown to be the extremal densities for the new inequalities an extension of the crame spl acute r rao inequality is derived as a consequence of these moment and fisher information inequalities

computational comparison on the partitioning strategies in multiple choice integer programming abstract in this paper we review the theoretical background of two partitioning strategies the weighted mean method wmm and the reformulation and transformation technique rtt incorporated in the special ordered set branch and bound procedures leading to the global optimum in multiple choice integer programming mcip procedure flow is developed with a numerical example presented to illustrate the effect of these two partitioning strategies in the algorithm this procedure flow is then coded using ibm apl2 a total of 24 test problems are solved by this apl2 code for mcip to analyze the performance of wmm and rtt in mcip the preliminary computational results indicate that on the average rtt produces smaller branching trees than does wmm its performance tends to be better for those mcip problems having more multiple choice sets and a fewer average number of variables in each set however the average cpu time consumed by using rtt does not differ much from that consumed by using wmm

a derivative free riemannian powell s method minimizing hartley entropy based ica contrast even though the hartley entropy based contrast function guarantees an unmixing local minimum the reported nonsmooth optimization techniques that minimize this nondifferentiable function encounter computational bottlenecks toward this powell s derivative free optimization method has been extended to a riemannian manifold namely oblique manifold for the recovery of quasi correlated sources by minimizing this contrast function the proposed scheme has been demonstrated to converge faster than the related algorithms in the literature besides the impressive source separation results in simulations involving synthetic sources having finite support distributions and correlated images

wandering subspaces and inner operators abstract a noncommutative domain b h b h n generated by the certain class of n tuples 1 n of formal power series in noncommutative indeterminates z 1 z n admits a universal model m z m z 1 m z n acting on a model space h 2 in this paper we provide an explicit description of the wandering subspace w q q i 1 n m z i i g q where q is an arbitrary joint invariant subspace under the operators m z 1 i g m z n i g moreover we also provide a transfer function type characterization of the m z inner operators this extends to the noncommutative setting the corresponding result obtained by olofsson 7 8 when n 1 and by eschmeier 3 in the multivariable commutative case

a multiplier adjustment approach for the set partitioning problem we introduce an effective branch and bound algorithm for solving the set partitioning problem the new algorithm employs a new multiplier adjustment based bounding procedure and a complementary branching strategy which results in relatively small search trees computational results based on 20 moderately sized crew scheduling problems indicate that our new algorithm is on average 16 6 times faster than the popular code setpar the improvements are mainly due to the bounding procedure which is fast easy to use and provides tight lower bounds on average the bounds are 97 6 of the optimal objective value of the linear programming relaxation after only five iterations and 98 5 after ten iterations moreover the lower bounds are observed to be monotonically nondecreasing we also apply the technique of variable elimination which is very effective in reducing the size of the problems on average 89 of the variables are eliminated from the problem at the root node

note efficient heuristic algorithms for positive 0 1 polynomial programming problems we develop in this paper two types of heuristic methods for solving the positive 0 1 polynomial programming pp problem of finding a 0 1 vector x that maximizes ctx subject to fx b where c b 0 and f is an m vector of polynomials with non negative coefficients the various heuristics were first tested on randomly generated sparse problems of up to 50 variables and 50 constraints and their performance in terms of computational time and effectiveness was investigated the results were very encouraging optimal solutions were consistently obtained by some of the heuristic methods in over 50 of the problems solved the effectiveness was on the average better than 99 and no less than 96 5 the computational time using these heuristics is on the average 5 of the time required to solve the pp problems to optimality some results for very sparse problems with up to 1 000 variables and 200 constraints are also reported

the solution of large 0 1 integer programming problems encountered in automated cartography the placement of text on a map to maximize map legibility while avoiding graphic overplotting is a problem in combinatorial optimization this particular optimization problem can be formulated as a multiple choice integer programming problem and the integer programming problem has a structure that can be exploited using a lagrangian heuristic to obtain a cost effective solution even when tens of thousands of variables and thousands of constraints are involved this paper presents the mathematical formulation of the map label placement problem and describes a computer program implemented to solve it

free holomorphic functions on the unit ball of b h n abstract in this paper we continue the study of free holomorphic functions on the noncommutative ball b h n 1 x 1 x n b h n x 1 x 1 x n x n 1 2 1 where b h is the algebra of all bounded linear operators on a hilbert space h and n 1 2 or n several classical results from complex analysis have free analogues in our noncommutative setting we prove a maximum principle a naimark type representation theorem and a vitali convergence theorem for free holomorphic functions with operator valued coefficients we introduce the class of free holomorphic functions with the radial infimum property and study it in connection with factorizations and noncommutative generalizations of some classical inequalities obtained by schwarz and harnack the borel caratheodory theorem is extended to our noncommutative setting using a noncommutative generalization of schwarz s lemma and basic facts concerning the free holomorphic automorphisms of the noncommutative ball b h n 1 we obtain an analogue of julia s lemma for free holomorphic functions f b h n 1 b h m 1 we also obtain pick julia theorems for free holomorphic functions with operator valued coefficients we provide a noncommutative generalization of a classical inequality due to lindelof which turns out to be sharper then the noncommutative von neumann inequality finally we introduce a pseudohyperbolic metric on b h n 1 which is invariant under the action of the free holomorphic automorphism group of b h n 1 and turns out to be a noncommutative extension of the pseudohyperbolic distance on b n the open unit ball of c n in this setting we obtain a schwarz pick type lemma we also provide commutative versions of these results for operator valued multipliers of the drury arveson space

curvature invariant for hilbert modules over free semigroup algebras abstract we introduce a notion of relative curvature resp euler characteristic for finite rank contractive hilbert modules over c f n the complex free semigroup algebra generated by the free semigroup f n on n generators asymptotic formulas and basic properties for both the curvature and the euler characteristic are established in particular it is shown that the standard relative curvature invariant resp euler characteristic of a hilbert module h is a nonnegative number less than or equal to the rank of h and it depends only on the properties of the completely positive map t x ni 1 tixt i where t1 tn is the row contraction of not necessarily commuting operators uniquely determined by the c f n module structure of h moreover we prove that for every t 0 there is a hilbert module h such that curv h h t the module structure defined by the left creation operators on the full fock space f2 hn on n generators occupies the position of the rank one free module in the algebraic theory we obtain a complete description of the closed submodules resp quotients of the free hilbert module f2 hn and calculate their curvature invariant it is shown that the curvature is a complete invariant for the finite rank submodules of the free hilbert module f2 hn k where k is a finite dimensional hilbert space a noncommutative version of the gauss bonnet chern theorem from riemannian geometry is obtained for graded hilbert modules over c f n in particular it is proved that the curvature and euler characteristic coincide for certain classes of pure hilbert modules our investigation is based on noncommutative poisson transforms noncommutative dilation theory and harmonic analysis on fock spaces

blind separation of instantaneous mixture of sources based on order statistics in this paper we introduce a novel procedure for separating an instantaneous mixture of sources based on order statistics the method is derived in a general context of independence component analysis using a contrast function defined in term of the kullback leibler divergence or of the mutual information we introduce a discretized form of this contrast permitting its easy estimation through order statistics we show that the local contrast property is preserved and derive a global contrast exploiting only the information of the support of the distribution in case this support is finite some simulations are given illustrating the good performance of the method

range based ica using a nonsmooth quasi newton optimizer for electroencephalographic source localization in focal epilepsy independent component analysis ica aims at separating a multivariate signal into independent nongaussian signals by optimizing a contrast function with no knowledge on the mixing mechanism despite the availability of a constellation of contrast functions a hartley entropy based ica contrast endowed with the discriminacy property makes it an appealing choice as it guarantees the absence of mixing local optima fueled by an outstanding source separation performance of this contrast function in practical instances a succession of optimization techniques has recently been adopted to solve the ica problem nevertheless the nondifferentiability of the considered contrast restricts the choice of the optimizer to the class of derivative free methods on the contrary this letter concerns a riemannian quasi newton scheme involving an explicit expression for the gradient to optimize the contrast function that is differentiable almost everywhere furthermore the inexact line search insisting on the weak wolfe condition and a terminating criterion befitting the partly smooth function optimization have been generalized to manifold settings leaving the previous results intact the investigations with diversified images and the electroencephalographic eeg data acquired from 45 focal epileptic subjects demonstrate the efficacy of our approach in terms of computational savings and reliable eeg source localization respectively additional experimental results are available in the online supplement

on the risk of using renyi s entropy for blind source separation recently some researchers have suggested renyi s entropy in its general form as a blind source separation bss objective function this was motivated by two arguments 1 shannon s entropy which is known to be a suitable criterion for bss is a particular case of renyi s entropy and 2 some practical advantages can be obtained by choosing another specific value for the renyi exponent yielding to e g quadratic entropy unfortunately by doing so there is no longer guarantee that optimizing this generalized criterion would lead to recovering the original sources in this paper we show that renyi s entropy in its exact form i e out of any consideration about its practical estimation or computation might lead to not recovering the sources depending on the source densities and on renyi s exponent value this is illustrated on specific examples we also compare our conclusions with previous works involving renyi s entropies for blind deconvolution

school timetabling a case in large binary integer linear programming a timetabling problem is formulated as a large integer linear programming problem in 0 1 variables a solution method based on lagrangean relaxation coupled with subgradient optimization is presented the solution method also incorporates a branch and bound procedure which takes advantage of special ordered sets of variables we present computational results for a large timetabling problem involving 900 subjects in a one year graduate program

on the similarity of the entropy power inequality and the brunn minkowski inequality corresp the entropy power inequality states that the effective variance entropy power of the sum of two independent random variables is greater than the sum of their effective variances the brunn minkowski inequality states that the effective radius of the set sum of two sets is greater than the sum of their effective radii both these inequalities are recast in a form that enhances their similarity in spite of this similarity there is as yet no common proof of the inequalities nevertheless their intriguing similarity suggests that new results relating to entropies from known results in geometry and vice versa may be found two applications of this reasoning are presented first an isoperimetric inequality for entropy is proved that shows that the spherical normal distribution minimizes the trace of the fisher information matrix given an entropy constraint just as a sphere minimizes the surface area given a volume constraint second a theorem involving the effective radii of growing convex sets is proved

an accelerated covering relaxation algorithm for solving 0 1 positive polynomial programs the purpose of this note is to present an accelerated algorithm for solving 0 1 positive polynomial pp problems like our covering relaxation algorithm management science 1979 the accelerated algorithm is a cutting plane method which uses the linear set covering problem as a relaxation for pp however a unique and novel feature of the accelerated algorithm is that it attempts to generate cutting planes from heuristic solutions to the set covering problem whenever possible computational results reveal that this strategy of generating cutting planes has led to a significant reduction in the computational time required to solve a pp problem

free biholomorphic functions and operator model theory ii let f f1 fn be an n tuple of formal power series in noncommutative indeterminates z1 zn such that f 0 0 and the jacobian detjf 0 0 and let g g1 gn be its inverse with respect to composition we assume that f and g have nonzero radius of convergence and g is a bounded free holomorphic function on the open unit ball b h n 1 where b h is the algebra of bounded linear operators an a hilbert space h in this paper several results concerning the noncommutative multivariable operator theory on the unit ball b h n 1 are extended to the noncommutative domain bf h x b h n g f x x and f x 1 for an appropriate evaluation x f x we develop an operator model theory and dilation theory for bf h where the associated universal model is an n tuple mz1 mzn of left multiplication operators acting on a hilbert space h2 f of formal power series all the results of this paper have commutative versions

generalized assignment with nonlinear capacity interaction the utility of the generalized assignment problem gap is well known in regard to resource allocation problems arising in the areas of production planning scheduling and facility location this paper introduces an important generalization of the gap which is called the 0 1 generalized assignment problem with nonlinear capacity constraints nlgap and which allows for capacity interaction among tasks assigned to the same agent for example nlgap can be used to model the hierarchical production planning problem involving the assignment of product families to production facilities in the case where process changeover gives rise to nonlinear capacity interaction among product families assigned to be produced at the same facility other applications of nlgap occur readily throughout the areas of production planning and scheduling we define a branch and bound algorithm for the nlgap this algorithm draws upon the underlying structure of the problem by combining in a novel manner recent advances in both nonlinear 0 1 programming and bounding techniques for the gap a heuristic for obtaining approximate solutions to nlgap is also defined we discuss computational experience with both the exact algorithm and the heuristic the computational results demonstrate that the heuristic is extremely effective in obtaining near optimal solutions and that the branch and bound algorithm can solve to optimality nlgaps with 5 agents 20 tasks and over 1000 nonlinear terms per constraint

a genetic algorithm for the multiple choice integer program we present a genetic algorithm for the multiple choice integer program that finds an optimal solution with probability one though it is typically used as a heuristic general constraints are relaxed by a nonlinear penalty function for which the corresponding dual problem has weak and strong duality the relaxed problem is attacked by a genetic algorithm with solution representation special to the multiple choice structure nontraditional reproduction crossover and mutation operations are employed extensive computational tests for dual degenerate problem instances show that suboptimal solutions can be obtained with the genetic algorithm within running times that are shorter than those of the osl optimization routine

a minimum range approach to blind extraction of bounded sources in spite of the numerous approaches that have been derived for solving the independent component analysis ica problem it is still interesting to develop new methods when among other reasons specific a priori knowledge may help to further improve the separation performances in this paper the minimum range approach to blind extraction of bounded source is investigated the relationship with other existing well known criteria is established it is proved that the minimum range approach is a contrast and that the criterion is discriminant in the sense that it is free of spurious maxima the practical issues are also discussed and a range measure estimation is proposed based on the order statistics an algorithm for contrast maximization over the group of special orthogonal matrices is proposed simulation results illustrate the performances of the algorithm when using the proposed range estimation criterion

a characteristic operator function for the class of n hypercontractions we consider a class of bounded linear operators on hilbert space called n hypercontractions which relates naturally to adjoint shift operators on certain vector valued standard weighted bergman spaces on the unit disc in the context of n hypercontractions in the class c0 we introduce a counterpart to the so called characteristic operator function for a contraction operator this generalized characteristic operator function wn t is an operator valued analytic function in the unit disc whose values are operators between two hilbert spaces of defect type using an operator valued function of the form wn t we parametrize the wandering subspace for a general shift invariant subspace of the corresponding vector valued standard weighted bergman space the operator valued analytic function wn t is shown to act as a contractive multiplier from the hardy space into the associated standard weighted bergman space

mutual information approach to blind separation of stationary sources this paper presents a unified approach to the problem of blind separation of sources based on the concept of mutual information this concept is applied to the whole source sequences as stationary processes and thus provides a universal contrast applicable to both the instantaneous and convolutive mixture cases for practical implementation we introduce several degraded forms of this contrast computable from a finite dimensional distribution of the reconstructed source processes only from them we derive several sets of estimating equations generalizing those considered earlier

univariate parameterization for global optimization of mixed integer polynomial problems abstract this paper presents a new relaxation technique to globally optimize mixed integer polynomial programming problems that arise in many engineering and management contexts using a bilinear term as the basic building block the underlying idea involves the discretization of one of the variables up to a chosen accuracy level teles j p castro p m matos h a 2013 multiparametric disaggregation technique for global optimization of polynomial programming problems j glob optim 55 227 251 by means of a radix based numeric representation system coupled with a residual variable to effectively make its domain continuous binary variables are added to the formulation to choose the appropriate digit for each position together with new sets of continuous variables and constraints leading to the transformation of the original mixed integer non linear problem into a larger one of the mixed integer linear programming type the new underestimation approach can be made as tight as desired and is shown capable of providing considerably better lower bounds than a widely used global optimization solver for a specific class of design problems involving bilinear terms

bergman inner functions and m hypercontractions abstract we characterize operator valued bergman inner functions on the unit ball as functions admitting a suitable transfer function realization thus we extend corresponding one variable results of olofsson from the case of the unit disc to the unit ball at the same time we associate with each m hypercontraction t l h n a canonical bergman inner function w t and indicate a possible definition of a characteristic function t for t

parallel replacement under capital rationing constraints contrary to serial replacement parallel replacement problems require a decision maker to evaluate a portfolio of replacement decisions in each time period because of economic interdependencies among assets in this paper we describe a parallel replacement problem in which the economic interdependence among assets is caused by capital rationing the research was motivated by the experience gained from a vehicle fleet replacement study where solutions to serial replacement problems could not be implemented since they violated management s budget plan when firms use budgets to control their expenditures competition for the limited funds creates interdependent problems in this paper we formulate the problem as a zero one integer program and develop a branch and bound algorithm based on lagrangian relaxation methodology a multiplier adjustment method is developed to solve one lagrangian dual

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

complex valued filtering based on the minimization of complex error entropy in this paper we consider the training of complex valued filter based on the information theoretic method we first generalize the error entropy criterion to complex domain to present the complex error entropy criterion ceec due to the difficulty in estimating the entropy of complex valued error directly the entropy bound minimization ebm method is used to compute the upper bounds of the entropy of the complex valued error and the tightest bound selected by the ebm algorithm is used as the estimator of the complex error entropy then based on the minimization of complex error entropy mcee and the complex gradient descent approach complex valued learning algorithms for both the linear transverse filter and the nonlinear neural network are derived the algorithms are applied to complex valued linear filtering and complex valued nonlinear channel equalization to demonstrate their effectiveness and advantages

on the calculation of true and pseudo penalties in multiple choice integer programming abstract the general solution approach leading to the global optimum for multiple choice integer programming uses the branch and bound procedure designed for special ordered sets incorporated in such a solution procedure the partitioning strategy based on a weighted mean method is most often adopted to calculate the pseudo penalties on the partitioned subconstraints for branching and backtracking based on a reformulation and transformation technique we show that the calculation of true penalties of these partitioned subconstraints can be performed efficiently

a dynamic programming algorithm for multiple choice constraints abstract this report concerns a discrete mathematical programming problem in which the variables are binary or integer the objective function separable or factorable and the constraints are in either of two classes linear or multiple choice constraints the problem is solved using a dynamic programming approach with fathoming by bounds and by infeasibility

invariant subspaces and nevanlinna pick kernels abstract a theorem of beurling lax halmos represents a subspace m of h2 c d the hardy space of analytic functions with values in the hilbert space e and square summable power series invariant for multiplication by z as h2 f where f is a subspace of e and is an inner function with values in l f e when the hardy space is replaced by the hilbert space h k determined by a nevanlinna pick kernel k such as the dirichlet kernel or the row contraction kernel on the ball in c d the blh theorem survives with f an auxiliary hilbert space and a l f e valued function which is inner in the sense that the operator m of multiplication by is a partial isometry under mild additional hypotheses when e c mz the operator of multiplication by z is cellularly indecomposable and has the codimension one property however if m is invariant for mz m mz m need not be a cyclic subspace for mz restricted to m

branch and bound methods for multi item scheduling this paper presents two solution approaches for the multi item activity scheduling problem a problem of practical importance because it models many resource allocation and combinatorial problems both approaches are easily incorporated into commercially available linear programming lp based branch and bound codes such as ibm s mpsx370 mip many practitioners rely on such codes and will find these approaches useful the first approach is a specialization of a method for decomposing integer programs with block angular structure in the second approach the reduced costs in the solution to the usual lp relaxation are used to develop improved priorities for branching computational results on real applications are reported using mpsx 370 mip

an ideal column algorithm for integer programs with special ordered sets of variables one of the most frequently occurring integer programming structures is the one which has special ordered sets of variables included in multiple choice constraints for problems with this structure a set of ideal columns are defined from the linear programming relaxation of the integer program and a reduced integer program is formed by keeping only those columns within a specified distance from the ideal column conditions are established which guarantee when the optimal solution to the reduced problem is als optimal for the original problem when these conditions are not satisfied bounds on the optimal solution value are provided ideal columns are also used to establish weights for the special ordered set variables this procedure has been implemented through a control program written by the authors for mpsx 370 mip 370 computational results are given

an exact penalty function approach for nonlinear integer programming problems abstract nonlinear integer programming problems with bounded feasible sets are considered it is shown how the number of constraints in such problems can be reduced with the aid of an exact penalty function approach this approach can be used to construct an equivalent unconstrained problem or a problem with a constraint set which makes it easier to solve the application of this approach to various nonlinear integer programming problems is discussed

a branch and bound algorithm for solving the multiple choice knapsack problem abstract the multiple choice knapsack problem is a binary knapsack problem with the addition of disjoint multiple choice constraints we describe a branch and bound algorithm based on embedding glover and klingman s method for the associated linear program within a depth first search procedure a heuristic is used to find a starting dual feasible solution to the associated linear program and a pegging test is employed to reduce the size of the problem for the enumeration phase computational experience and comparisons with the code of nauss and an algorithm of armstrong et al for the same problem are reported