capacity constraints across nests in assortment optimization under the nested logit model we consider assortment optimization problems when customers choose according to the nested logit model and there is a capacity constraint limiting the total capacity consumption of all products offered in all nests when each product consumes one unit of capacity our capacity constraint limits the cardinality of the offered assortment for the cardinality constrained case we develop an efficient algorithm to compute the optimal assortment when the capacity consumption of each product is arbitrary we give an algorithm to obtain a 4 approximate solution we show that we can compute an upper bound on the optimal expected revenue for an individual problem instance by solving a linear program in our numerical experiments we consider problem instances involving products with arbitrary capacity consumptions comparing the expected revenues from the assortments obtained by our 4 approximation algorithm with the upper bounds on the optimal expected revenues our numerical results indicate that the 4 approxim

further results on invariance of the eigenvalues of matrix products involving generalized inverses abstract it is shown that if m n n n and p m complex matrices a b and c do not satisfy at least one of the range inclusions r a c r b and r ca r b then for each complex number there is a choice of a generalized inverse b such that this number is an eigenvalue of the product ab c combined with criteria derived by baksalary and puntanen 1990 and by baksalary and pukkila 1992 this result implies that the invariance of the magnitude largest eigenvalue and or the magnitude smallest nonzero eigenvalue of ab c respect to the choice of b is equivalent to the invariance of the set of all eigenvalues

contractions of low dimensional lie algebras theoretical background of continuous contractions of finite dimensional lie algebras is rigorously formulated and developed in particular known necessary criteria of contractions are collected and new criteria are proposed a number of requisite invariant and semi invariant quantities are calculated for wide classes of lie algebras including all low dimensional lie algebras an algorithm that allows one to handle one parametric contractions is presented and applied to low dimensional lie algebras as a result all one parametric continuous contractions for both the complex and real lie algebras of dimensions not greater than 4 are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases levels and colevels of low dimensional lie algebras are discussed in detail properties of multiparametric and repeated contractions are also investigated

spectrum and trace invariance criterion and its statistical applications abstract necessary and sufficient conditions are established for the set of nonzero eigenvalues of the product b a to be invariant with respect to the choice of a generalized inverse b it turns out that the same conditions constitute a criterion for the invariance of trace b a the algebraic result derived is then applied to investigate some properties of canonical correlations in the context of the general gauss markov model m y x 2 v in particular the results of watson 1967 and puntanen 1987 are generalized and a new characterization of the best linear unbiased estimator of x under the model m is established

graded contractions of the gell mann graded sl 3 c the gell mann grading one of the four gradings of sl 3 c that cannot be further refined is considered as the initial grading for the graded contraction procedure using the symmetries of the gell mann grading the system of contraction equations is reduced and solved each non trivial solution of this system determines a lie algebra which is not isomorphic to the original algebra sl 3 c the resulting 53 contracted algebras are divided into two classes the first is represented by the algebras which are also continuous inonu wigner contractions the second is formed by the discrete graded contractions

the pauli matrices in n dimensions and finest gradings of simple lie algebras of type a n 1 properties of the lie algebra gl n c are described for a basis which is a generalization of the 2 2 pauli matrices the 3 3 case is described in detail the remarkable properties of that basis are the grading of the lie algebra it offers each grading subspace is one dimensional and the matrix group it generates it is a finite group with the center of sl n c as its commutator group

approximation algorithms for combinatorial fractional programming problems we are concerned with a combinatorial optimization problem which has the ratio of two linear functions as the objective function this type of problems can be solved by an algorithm that uses an auxiliary problem with a parametrized linear objective function because of its combinatorial nature however it is often difficult to solve the auxiliary problem exactly in this paper we propose an algorithm which assumes that the auxiliary problems are solved only approximately and prove that it gives an approximate solution to the original problem of which the accuracy is at least as good as that of approximate solutions to the auxiliary problems it is also shown that the time complexity is bounded by the square of the computation time of the approximate algorithm for the auxiliary problem as an example of the proposed algorithm we present a fully polynomial time approximation scheme for the fractional 0 1 knapsack problem

realizations of real low dimensional lie algebras using a new powerful technique based on the notion of megaideal we construct a complete set of inequivalent realizations of real lie algebras of dimension no greater than four in vector fields on a space of an arbitrary finite number of variables our classification amends and essentially generalizes earlier works on the subject

parametric approaches to fractional programs the fractional program p is defined by maxf x g x subject tox x a class of methods for solving p is based on the auxiliary problem q with a parameter maxf x g x subject tox x starting with two classical methods in this class the newton method and the binary search method a number of variations are introduced and compared among the proposed methods the modified binary search method is theoretically interesting because of its superlinear convergence and the capability to provide an explicit interval containing the optimum parameter value bar lambda computational behavior is tested by solving fractional knapsack problems and quadratic fractional programs the interpolated binary search method seems to be most efficient while other methods also behave surprisingly well

approximating a class of combinatorial problems with rational objective function in the late seventies megiddo proposed a way to use an algorithm for the problem of minimizing a linear function a 0 a 1 x 1 a n x n subject to certain constraints to solve the problem of minimizing a rational function of the form a 0 a 1 x 1 a n x n b 0 b 1 x 1 b n x n subject to the same set of constraints assuming that the denominator is always positive using a rather strong assumption hashizume et al extended megiddo s result to include approximation algorithms their assumption essentially asks for the existence of good approximation algorithms for optimization problems with possibly negative coefficients in the linear objective function which is rather unusual for most combinatorial problems in this paper we present an alternative extension of megiddo s result for approximations that avoids this issue and applies to a large class of optimization problems specifically we show that if there is an approximation for the problem of minimizing a nonnegative linear function subject to constraints satisfying a certain increasing property then there is an approximation 1 approximation for the problem of minimizing maximizing a nonnegative rational function subject to the same constraints our framework applies to covering problems and network design problems among others

twisted cocycles of lie algebras and corresponding invariant functions abstract we consider finite dimensional complex lie algebras using certain complex parameters we generalize the concept of cohomology cocycles of lie algebras a special case is generalization of 1 cocycles with respect to the adjoint representation so called derivations parametric sets of spaces of cocycles allow us to define complex functions which are invariant under lie isomorphisms such complex functions thus represent useful invariants we show how they classify three and four dimensional lie algebras as well as how they apply to some eight dimensional one parametric nilpotent continua of lie algebras these functions also provide necessary criteria for existence of 1 parametric continuous contraction

on the identification of a lie algebra given by its structure constants i direct decompositions levi decompositions and nilradicals abstract methods are given for identifying a lie algebra l given by its structure constants the identification involves a transformation to a canonical basis in which the structure becomes obvious thus decomposable lie algebras are already decomposed into direct sums of indecomposable ones an indecomposable lie algebra that is not simple or solvable has its radical exhibited a solvable algebra has its nilradical displayed the methods all lead to simple algorithms that have been implemented as computer programs involving some symbolic manipulations

a note on invariance of the eigenvalues singular values and norms of matrix products involving generalized inverses abstract it is shown that the set of singular values of the product ab c as well as any norm of this product is invariant with respect to the choice of a generalized inverse b if and only if ab c is invariant itself this is in contrast with the corresponding invariance property of the set of eigenvalues of ab c for which the invariance of ab c is not necessary

a general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into one in this paper we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function examples of such problems include those in which the objective function is a product or ratio of two linear functions parallel machine scheduling problems with the makespan objective robust versions of weighted multiobjective optimization problems and assortment optimization problems with logit choice models the main idea behind our approximation schemes is the construction of an approximate pareto optimal frontier of the functions that constitute the given objective using this idea we give the first fully polynomial time approximation schemes for the max min resource allocation problem with a fixed number of agents combinatorial optimization problems in which the objective function is the sum of a fixed number of ratios of linear functions or the product of a fixed number of linear functions and assor

fractional programming applications and algorithms abstract the paper deals with decision problems that give rise to the optimization of ratios subject to constraints these so called fractional programs have been treated in a considerable number of papers it is attempted to survey applications as well as solution methods in linear quadratic and concave convex fractional programming in the first part we give a detailed outline on the major areas of applications of fractional programming following a brief review on some basic theoretical results like duality relations in the second part we then discuss different algorithmic approaches a primal a dual and a parametric solution method is considered here we try to determine for each method the kind of fractional programs that are solved by it suitably

an algorithm to determine the isomorphism classes of 4 dimensional complex lie algebras abstract we give an algorithm to determine the isomorphism classes of 4 dimensional complex lie algebras from the representation theoretic viewpoint for this purpose we give the gl v irreducible decomposition of the polynomial ring of the space 2 v v v c 4 up to degree 3 and show that intrinsic concepts defined by the vanishing of these covariants are sufficient to distinguish the isomorphism classes as an application we describe the variety of 4 dimensional lie algebras and their degenerations in a comparatively simple form by introducing a new family of normal forms of 4 dimensional lie algebras that are just fitted for these purposes

the inverse parametric knapsack problem abstract this paper deals with parametric knapsack problems where the costs resp weights are replaced by linear functions depending on a parameter t the aim is to find the smallest parameter t such that the optimal solution value of the knapsack problem is equal to a prespecified solution value for this inverse parametric problem pseudopolynomial algorithms are developed and search methods making use of the special properties of the parametric value function are constructed using computational experiments the behaviour of these algorithms is investigated and the favourable practical performance of different search methods is exhibited

invariance properties of a triple matrix product involving generalized inverses abstract given complex valued matrices a b and c of appropriate dimensions this paper investigates certain invariance properties of the product axc with respect to the choice of x where x is a generalized inverse of b different types of generalized inverses are taken into account the purpose of the paper is three fold first to review known results scattered in the literature second to demonstrate the connection between invariance properties and the concept of extremal ranks of matrices and third to add new results related to the topic

ratio combinatorial programs abstract we consider here a combinatorial optimization problem where the objective function is a ratio of two linear functions one way of solving this problem is to solve repeatedly an auxiliary problem with a parameterized linear objective function in this paper we relate this method to newton s method for solving equations and present a modification of this algorithm which solves the ratio problem approximate ly by repeatedly solving the auxiliary linear problems with the same accuracy we demonstrate by reporting results of extensive computational experiments that the above modified algorithm for the approximate knapsack problem performs extremely well in practice

graded contractions of the pauli graded sl 3 c the lie algebra sl 3 c is considered in the basis of generalized pauli matrices corresponding grading is the pauli grading here it is one of the four gradings of the algebra which cannot be further refined the set s of 48 contraction equations for 24 contraction parameters is solved our main tools are the symmetry group of the pauli grading of sl 3 c which is essentially the finite group sl 2 z3 and the induced symmetry of the system s a list of all equivalence classes of solutions of the contraction equations is provided among the solutions 175 equivalence classes are non parametric and 13 solutions depend on one or two continuous parameters providing a continuum of equivalence classes and subsequently continuum of non isomorphic lie algebras solutions of the contraction equations of pauli graded sl 3 c are identified here as specific solvable lie algebras of dimensions up to 8 earlier algorithms for identification of lie algebras given by their structure constants had to be made more efficient in order to distinguish non isomorphic lie algebras encountered here resulting lie algebras are summarized in tabular form there are 88 indecomposable solvable lie algebras of dimension 8 77 of them being nilpotent there are 11 infinite sets of parametric lie algebra which still deserve further study

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

some remarks on partial orderings of nonnegative definite matrices abstract baksalary and pukelsheim 1990 investigated partial orderings of nonnegative definite matrices some additional remarks are given in this note

classification of orbit closures of 4 dimensional complex lie algebras abstract let l n c be the variety of complexn dimensional lie algebras the groupgln c acts on it via change of basis an orbito under this action consists of all structures isomorphic to the aim of this paper is to give a complete classification of orbit closures of 4 dimensional lie algebras i e determining all o where l 4 c starting with a classification of complex lie algebras of dimensionn 4 we study the behavior of several lie algebra invariants under degeneration i e under transition to the orbit closure as a corollary we will show that all degenerations in l 3 c can be realized via a one parameter subgroup but this is not the case in l 4 c

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown