geometry of free loci and factorization of noncommutative polynomials abstract the free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces z n f x m n k g det f x 0 the main theorem of this article shows that f is irreducible if and only if z n f is eventually irreducible a key step in the proof is an irreducibility result for linear pencils arising from this is a free singularity locus nullstellensatz for noncommutative polynomials apart from consequences to factorization in a free algebra the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry

two level linear programming decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which often exist in practice multilevel programming models partition control over decision variables among ordered levels within a hierarchical planning structure a planner at one level of the hierarchy may have his objective function and set of feasible decisions determined in part by other levels however his control instruments may allow him to influence the policies at other levels and thereby improve his own objective function this paper examines the special case of the two level linear programming problem geometric characterizations and algorithms are presented with some examples the goal is to demonstrate the tractability of such problems and motivate a wider interest in their study

thompson s group f is not almost convex abstract we show that thompson s group f does not satisfy cannon s almost convexity condition ac n for any positive integer n with respect to the standard generating set with two elements to accomplish this we construct a family of pairs of elements at distance n from the identity and distance 2 from each other which are not connected by a path lying inside the n ball of length less than k for increasingly large k our techniques rely upon fordham s method for calculating the length of a word in f and upon an analysis of the generators geometric actions on the tree pair diagrams representing elements of f

multiple bottleneck assignment problem we consider the multiple bottleneck assignment problem which subsumes the well known min sum and bottleneck assignment problems the problem arises in the context of flexible manufacturing systems where the objective is to maximize the throughput of a production system with several flow shops running in parallel to produce a product the problem is known to be strongly np hard we propose a new algorithm for obtaining sharp lower bounds to the optimal objective value computational experiments are conducted to show the improvement over existing methods

a solution method for the static constrained stackelberg problem via penalty method this note presents a new solution method for the static constrained stackelberg problem through our approach the stackelberg problem is completely transformed into a one level unconstrained problem such that the newly introduced overall augmented objective function is minimized with repect to the leader s and the follower s variables jointly it can be proved that a sequence of solutions to the transformed problems converges to the solution of the original problem when the penalty parameters are updated

integer programming models for the multidimensional assignment problem with star costs abstract we consider a variant of the multidimensional assignment problem map with decomposable costs in which the resulting optimal assignment is described as a set of disjoint stars this problem arises in the context of multi sensor multi target tracking problems where a set of measurements obtained from a collection of sensors must be associated to a set of different targets to solve this problem we study two different formulations first we introduce a continuous nonlinear program and its linearization along with additional valid inequalities that improve the lower bounds second we state the standard map formulation as a set partitioning problem and solve it via branch and price these approaches were put to test by solving instances ranging from tripartite to 20 partite graphs of 4 to 30 nodes per partition computational results show that our approaches are a viable option to solve this problem a comparative study is presented

a geometric characterisation of the quadratic min power centre abstract for a given set of nodes in the plane the min power centre is a point such that the cost of the star centred at this point and spanning all nodes is minimised the cost of the star is defined as the sum of the costs of its nodes where the cost of a node is an increasing function of the length of its longest incident edge the min power centre problem provides a model for optimally locating a cluster head amongst a set of radio transmitters however the problem can also be formulated within a bicriteria location model involving the 1 centre and a generalised fermat weber point making it suitable for a variety of facility location problems we use farthest point voronoi diagrams and delaunay triangulations to provide a complete geometric description of the min power centre of a finite set of nodes in the euclidean plane when cost is a quadratic function this leads to a new linear time algorithm for its construction when the convex hull of the nodes is given we also provide an upper bound for the performance of the centroid as an approximation to the quadratic min power centre finally we briefly describe the relationship between solutions under quadratic cost and solutions under more general cost functions

a conjectured property of hermitian pencils abstract if a b is any pair of hermitian matrices the power of dividing det i xa yb will be given by the number of basic singular summands in the pair contrary to conjecture this power can be greater than one even when the pair is unitarily irreducible

the mixed integer linear bilevel programming problem a two person noncooperative game in which the players move in sequence can be modeled as a bilevel optimization problem in this paper we examine the case where each player tries to maximize the individual objective function over a jointly constrained polyhedron the decision variables are variously partitioned into continuous and discrete sets the leader goes first and through his choice may influence but not control the responses available to the follower for two reasons the resultant problem is extremely difficult to solve even by complete enumeration first it is not possible to obtain tight upper bounds from the natural relaxation and second two of the three standard fathoming rules common to branch and bound cannot be applied fully in light of these limitations we develop a basic implicit enumeration scheme that finds good feasible solutions within relatively few iterations a series of heuristics are then proposed in an effort to strike a balance between accuracy and speed the computational results suggest that some compromise is needed when the problem contains more than a modest number of integer variables

solving airline crew scheduling problems by branch and cut the crew scheduling problem is one that has been studied almost continually for the past 40 years but all prior approaches have always approximated the problem of finding an optimal schedule for even the smallest of an airline s fleets the problem is especially important today since costs for flying personnel of major u s carriers have grown and now often exceed 1 3 billion a year and are the second largest item next to fuel cost of the total operating cost of major u s carriers thus even small percentage savings amount to substantial dollar amounts we present a branch and cut approach to solving to proven optimality large set partitioning problems arising within the airline industry we first provide some background related to this important application and then describe the approach for solving representative problems in this problem class the branch and cut solver generates cutting planes based on the underlying structure of the polytope defined by the convex hull of the feasible integer points and incorporates these cuts into a tree search algorithm that uses automatic reformulation procedures heuristics and linear programming technology to assist in the solution numerical experiments are reported for a sample of 68 large scale real world crew scheduling problems these problems include both pure set partitioning problems and set partitioning problems with side constraints these base constraints represent contractual labor requirements and have heretofore not been represented explicitly in the construction of crew schedules thus making it impossible to provide any measure of how far the obtained solution was from optimality an interesting result of obtaining less costly schedules is that the crews themselves are happier with the schedules because they spend more of their duty time flying than waiting on the ground

an efficient point algorithm for a linear two stage optimization problem this paper presents an algorithm using sensitivity analysis to solve a linear two stage optimization problem the underlying theory rests on a set of first order optimality conditions that parallel the kuhn tucker conditions associated with a one dimensional parametric linear program the solution to the original problem is uncovered by systematically varying the parameter over the unit interval and solving the corresponding linear program finite convergence is established under nondegenerate assumptions the paper also discusses other solution techniques including branch and bound and vertex enumeration and gives an example highlighting their computational and storage requirements by these measures the algorithm presented here has an overall advantage finally a comparison is drawn between bicriteria and bilevel programming and underscored by way of an example

assignment problems a golden anniversary survey abstract having reached the 50th golden anniversary of the publication of kuhn s seminal article on the solution of the classic assignment problem it seems useful to take a look at the variety of models to which it has given birth this paper is a limited survey of what appear to be the most useful of the variations of the assignment problem that have appeared in the literature over the past 50 years the intention here is not to identify every such paper of which there have been hundreds nor to identify the best solution procedure for each variation rather the intention is to identify what these variations are and what they are called so as to make it easier for a researcher trying to develop some variation of the assignment problem for a particular application to find the relevant literature

simultaneous similarity of matrices abstract in this paper we solve completely and explicitly the long standing problem of classifying pairs of nxn complex matrices a b under a simultaneous similarity roughly speaking the classification decomposes to a finite number of steps in each step we consider an open algebraic set then we construct a finite number of rational functions in the entries of a and b whose values are constant on all pairs similar to a b the values of the functions phi sub i a b i equals 1 s determine a finite number of similarity classes

cone types and geodesic languages for lamplighter groups and thompson s group f we study languages of geodesics in lamplighter groups and thompson s group f we show that the lamplighter groups ln have infinitely many cone types have no regular geodesic languages and have 1 counter context free and counter geodesic languages with respect to certain generating sets we show that the full language of geodesics with respect to one generating set for the lamplighter group is not counter but is context free while with respect to another generating set the full language of geodesics is counter and context free in thompson s group f with respect to the standard finite generating set we show there are infinitely many cone types and that there is no regular language of geodesics we show that the existence of families of seesaw elements with respect to a given generating set in a finitely generated infinite group precludes a regular language of geodesics and guarantees infinitely many cone types with respect to that generating set

computational difficulties of bilevel linear programming we show using small examples that two algorithms previously published for the bilevel linear programming problem blp may fail to find the optimal solution and thus must be considered to be heuristics a proof is given that solving blp problems is np hard which makes it unlikely that there is a good exact algorithm

set packing relaxations of some integer programs this paper is about set packing relaxations of combinatorial optimization problems associated with acyclic digraphs and linear orderings cuts and multicuts and set packings themselves families of inequalities that are valid for such a relaxation as well as the associated separation routines carry over to the problems under investigation

a counterexample to kippenhahn s conjecture on hermitian pencils abstract a counterexample is constructed to a conjecture of kippenhahn math nachr 6 193 288 1951 52 a pair of hermitian 8 8 matrices h k is found such that 1 h k generate m 8 c and 2 the minimal polynomial of the pencil xh yk has degree 4 recent work of h shapiro e g linear algebra appl 43 201 221 1982 has established the conjecture for n n matrices n 5

tame combing and almost convexity conditions we give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation but which are not minimally almost convex on a standard generating set namely we explicitly construct such combings for thompson s group f and the baumslag solitar groups bs 1 p with p 3 in order to make this construction for thompson s group f we significantly expand the understanding of the cayley complex of this group with respect to the standard finite presentation in particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2 cells adjacent to edges that do not lie on normal form paths

a note on an efficient point algorithm for a linear two stage optimization problem the efficient point algorithm proposed by j f bard for the computation of the solution of the linear two stage optimization problem does not always converge to the desired solution a counterexample is provided and the reasons for this lack of convergence are discussed

combinatorial properties of thompson s group f we study some combinatorial consequences of blake fordham s theorems on the word metric of thompson s group f in the standard two generator presentation we explore connections between the tree pair diagram representing an element w of f its normal form in the infinite presentation its word length and minimal length representatives of it we estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates in addition we explore some properties of the cayley graph of f with respect to the two generator finite presentation namely we exhibit the form of dead end elements in this cayley graph and show that it has no deep pockets finally we discuss a simple method for constructing minimal length representatives for strictly positive or negative words

a new computational method for stackelberg and min max problems by use of a penalty method this paper is concerned with the stackelberg problem and the min max problem in competitive systems the stackelberg approach is applied to the optimization of two level systems where the higher level determines the optimal value of its decision variables parameters for the lower level so as to minimize its objective while the lower level minimizes its own objective with respect to the lower level decision variables under the given parameters meanwhile the min max problem is to determine a min max solution such that a function maximized with respect to the maximizer s variables is minimized with respect to the minimizer s variables this problem is also characterized by a parametric approach in a two level scheme new computational methods are proposed here that is a series of nonlinear programming problems approximating the original two level problem by application of a penalty method to a constrained parametric problem in the lower level are solved iteratively it is proved that a sequence of approximated solutions converges to the correct stackelberg solution or the min max solution some numerical examples are presented to illustrate the algorithms

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

metrics and embeddings of generalizations of thompson s group f the distance from the origin in the word metric for generalizations f p of thompson s group f is quasi isometric to the number of carets in the reduced rooted tree diagrams representing the elements of f p this interpretation of the metric is used to prove that every f p admits a quasiisometric embedding into every f q and also to study the behavior of the shift maps under these embeddings the first example of a finitely presented infinite simple group was discovered by r j thompson in 1965 this group g arose in the study of logic and it has many interesting descriptions in 1974 higman 10 used a description as the group of automorphisms of the jonsson tarski algebra to generalize g to an infinite family of groups g has another description as a group of homeomorphisms of the cantor set it has subgroups which may be described as homeomorphism groups of different objects for instance the circle and the unit interval among the people studying these generalizations are brown 3 and bieri and strebel in a set of unpublished notes 1 in this paper we will concentrate on the groups f p which correspond to the groups fp in 3 these families of groups have also been extensively studied by brin and guzman 2 the group f 2 commonly known as thompson s group f can be regarded as the group of piecewise linear orientation preserving homeomorphisms of the interval 0 1 which have breakpoints only at dyadic points and whose slopes where defined are powers of two in 1984 brown and geoghegan 4 found it to be the first example of a finitely presented infinite dimensional torsion free fp group this fact has been extended to all f p in 3 and studied by stein 11 where generalizations to more general groups of homeomorphisms with general rational breakpoint sets are considered cleary 7 has studied these properties for groups of piecewise linear homeomorphisms with irrational breakpoints and slopes in 5 burillo established the relationship between the word metric of thompson s group f and an estimate of the distance derived from the unique normal form of the elements this algebraic estimate is quasi isometric to the word metric and was used to prove that some subgroups are nondistorted in f in this paper we find a geometric estimate of the word metric in terms of rooted tree diagrams see 6 show that this is quasi isometric to the word metric and use this interpretation to prove that some embeddings of f p into f q including some of those outlined in 2 are quasi isometric the interpretations of these embeddings in terms of received by the editors september 25 1998 and in revised form august 11 1999 2000 mathematics subject classification primary 20f65 secondary 20f05 20f38 20e99 05c25 c 2000 american mathematical society

the mixed vertex packing problem we study a generalization of the vertex packing problem having both binary and bounded continuous variables called the mixed vertex packing problem mvpp the well known vertex packing model arises as a subproblem or relaxation of many 0 1 integer problems whereas the mixed vertex packing model arises as a natural counterpart of vertex packing in the context of mixed 0 1 integer programming we describe strong valid inequalities for the convex hull of solutions to the mvpp and separation algorithms for these inequalities we give a summary of computational results with a branch and cut algorithm for solving the mvpp and using it to solve general mixed integer problems

wheel inequalities for stable set polytopes we introduce new classes of valid inequalities called wheel inequalities for the stable set polytopepg of a graphg each wheel configuration gives rise to two such inequalities the simplest wheel configuration is an odd subdivisionw of a wheel and for these we give necessary and sufficient conditions for the wheel inequality to be facet inducing forpw generalizations arise by allowing subdivision paths to intersect and by replacing the hub of the wheel by a clique the separation problem for these inequalities can be solved in polynomial time

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

on weighting two criteria with a parameter in combinatorial optimization problems abstract two criteria in a combinatorial problem are often combined in a weighted sum objective using a weighting parameter between 0 and 1 for special problem types e g when one of the criteria is a bottleneck value efficient algorithms are known that solve for a given value of the weighting parameter we transform the underlying enumeration method into a parametric algorithm solving simultaneously for all values of the weighting parameter efficient implementations are presented for combinatorial problems with criteria as balanced optimization min sum with min max and min sum with balanced optimization considering the spanning tree the linear assignment and the single machine scheduling problem further the new algorithmic scheme can easily incorporate a trade off of the criteria by means of penalty functions again without consequences for the algorithm and its complexity order

on two level optimization decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which have often existed in practice similarly tractable game theoretic approaches have not allowed for a continuous solution space in which one player s decision restricts the feasible choices of another this correspondence offers a formulation to help model this type of decision making hierarchy characterizations and algorithmic approaches for a class of two level problems are discussed

a variation of the assignment problem abstract in this paper a meaningful variant of the cost minimizing assignment problem with the objective of minimizing the total cost of assignment plus an additional supervisory cost which depends on the total time of completion of the project is formulated an efficient method of finding an optimal solution to such a problem is presented with a numerical example to illustrate the same

equality of higher numerical ranges of matrices and a conjecture of kippenhahn on hermitian pencils let mn be the algebra of all n n complex matrices for 1 k n the kth numerical range of a mn is defined by wk a 1 k jk 1xj axj x1 xk is an orthonormal set in n it is known that tr a n wn a wn 1 a w1 a we study the condition on a under which wm a wk a for some given 1 m k n it turns out that this study is closely related to a conjecture of kippenhahn on hermitian pencils a new class of counterexamples to the conjecture is constructed based on the theory of the numerical range

a modified simplex approach for solving bilevel linear programming problems abstract a method for solving bilevel linear programming problems is proposed the inner problem is represented by its kuhn tucker conditions and the complementary slackness equations are moved to the outer objective as a penalty function the global optimum of the resulting quadratic programme is reached by iterative application of a modified simplex algorithm

solving combinatorial problems with combined min max min sum objective and applications the purpose of this paper is to introduce and study a new class of combinatorial optimization problems in which the objective function is the algebraic sum of a bottleneck cost function min max and a linear cost function min sum general algorithms for solving such problems are described and general complexity results are derived a number of examples of application involving matchings paths and cutsets matroid bases and matroid intersection problems are examined and the general complexity results are specialized to each of them the interest of these various problems comes in particular from their strong relation to other important and difficult combinatorial problems such as weighted edge coloring of a graph optimum weighted covering with matroid bases optimum weighted partitioning with matroid intersections etc another important area of application of the algorithms given in the paper is bicriterion analysis involving a min max criterion and a min sum one

a bilevel bottleneck programming problem this paper studies a bilevel programming problem with linear constraints and in which the objective functions at both levels are concave bottleneck functions which are to be minimized the problem is a non convex programming problem it is shown that an optimal solution to the problem is attainable at an extreme point of the underlying region the outer level objective function values are ranked in increasing order until a value is reached one of the solutions corresponding to which is feasible for the problem this solution is then the required global optimal solution

the wigner distribution of n arbitrary observables we study a generalization of the wigner function to arbitrary tuples of hermitian operators we show that for any collection of hermitian operators a1 an and any quantum state there is a unique joint distribution on rn with the property that the marginals of all linear combinations of the ak coincide with their quantum counterparts in other words we consider the inverse radon transform of the exact quantum probability distributions of all linear combinations we call it the wigner distribution because for position and momentum this property defines the standard wigner function we discuss the application to finite dimensional systems establish many basic properties and illustrate these by examples the properties include the support the location of singularities positivity the behavior under symmetry groups and informational completeness

quasi isometrically embedded subgroups of thompson s groupf abstract the goal of this paper is to study subgroups of thompson s group f which are isomorphic to f z n and f f a result estimating the norm of an element of thompson s group is found and this estimate is used to prove that these particular subgroups are quasi isometrically embedded

a solution method for the linear static stackelberg problem using penalty functions a new solution technique is presented for the linear constrained static stackelberg problem for a given value of x the leader s decision vector the follower is at its rational reaction set when the duality gap of the second level problem becomes zero the outer problem is solved by appending to the leader s objective a function that minimizes the duality gap of the follower s problem this structure leads to the decomposition of the composite problem into a series of linear programs leading to an efficient algorithm it is proved that optimality is reached for an exact penalty function and the method is illustrated with some examples

a survey of canonical forms and invariants for unitary similarity abstract matrices a and b are said to be unitarily similar if u au b for some unitary matrix u this expository paper surveys results on canonical forms and invariants for unitary similarity the first half gives a detailed description of methods developed by several authors brenner littlewood mitchell mcrae radjavi serge chuk and benedetti and cragnolini using inductively defined reduction procedures to transform matrices to canonical form the matrix is partitioned and successive unitary similarities applied to reduce the submatrices to some nice form at each stage one refines the partition and restricts the set of permissible unitary similarities to those that preserve the already reduced blocks the process ends in a finite number of steps producing both the canonical form and the subgroup of the unitary group that preserves that form depending on the initial step various canonical forms may be defined the method can also be used to define canonical forms relative to certain subgroups of the unitary group and canonical forms for finite sets of matrices under simultaneous unitary similarity the remainder of the paper surveys results on unitary invariants and other topics related to unitary similarity such as the specht pearcy trace invariants the numerical range and unitary reducibility

conflict graphs in solving integer programming problems abstract we report on the use of conflict graphs in solving integer programs a conflict graph represents logical relations between binary variables we develop algorithms and data structures that allow the effective and efficient construction management and use of dynamically changing conflict graphs our computational experiments show that the techniques presented work very well

hermitian pencils with a cubic minimal polynomial let a be an n n complex matrix and write a h ik where h and k are hermitian matrices we show that if the minimal polynomial of the pencil xh yk has degree 3 then there is a unitary matrix u such that u 1au is block diagonal with blocks of size 3 3 or smaller this is a special case of a conjecture made by kippenhahn in 1951

on a conjecture of kippenhahn about the characteristic polynomial of a pencil generated by two hermitian matrices i abstract let a be an n n matrix write a h ik where i 2 1 and h and k are hermitian let f x y z det zi xh yk we first show that a pair of matrices over an algebraically closed field which satisfy quadratic polynomials can be put into block upper triangular form with diagonal blocks of size 1 1 or 2 2 via a simultaneous similarity this is used to prove that if f x y z g x y z n 2 where g has degree 2 then for some unitary matrix u the matrix u au is the direct sum of n 2 copies of a 2 2 matrix a 1 where a 1 is determined up to unitary similarity by the polynomial g x y z we use the connection between f x y z and the numerical range of a to investigate the case where f x y z has the form z ax y r g x y z s where g x y z is irreducible of degree 2