simultaneous column and row generation for large scale linear programs with column dependent rows in this paper we develop a simultaneous column and row generation algorithm that could be applied to a general class of large scale linear programming problems these problems typically arise in the context of linear programming formulations with exponentially many variables the defining property for these formulations is a set of linking constraints which are either too many to be included in the formulation directly or the full set of linking constraints can only be identified if all variables are generated explicitly due to this dependence between columns and rows we refer to this class of linear programs as problems with column dependent rows to solve these problems we need to be able to generate both columns and rows on the fly within an efficient solution approach we emphasize that the generated rows are structural constraints and distinguish our work from the branch and cut and price framework we first characterize the underlying assumptions for the proposed column and row generation algorithm these assumptions are general enough and cover all problems with column dependent rows studied in the literature up until now to the best of our knowledge we then introduce in detail a set of pricing subproblems which are used within the proposed column and row generation algorithm this is followed by a formal discussion on the optimality of the algorithm to illustrate our approach the paper is concluded by applying the proposed framework to the multi stage cutting stock and the quadratic set covering problems

optimal multi way number partitioning the np hard number partitioning problem is to separate a multiset s of n positive integers into k subsets such that the largest sum of the integers assigned to any subset is minimized the classic application is scheduling a set of n jobs with different runtimes on k identical machines such that the makespan the elapsed time to complete the schedule is minimized the two way number partitioning decision problem is one of the original 21 problems that richard karp proved np complete it is also one of garey and johnson s six fundamental np complete problems and the only one based on numbers this article explores algorithms for solving multi way number partitioning problems optimally we explore previous algorithms as well as our own algorithms which fall into three categories sequential number partitioning snp a branch and bound algorithm binary search improved bin completion bsibc a bin packing algorithm and cached iterative weakening ciw an iterative weakening algorithm we show experimentally that for large random numbers snp and ciw are state of the art algorithms depending on the values of n and k both algorithms outperform the previous state of the art by up to seven orders of magnitude in terms of runtime

hermitian completions of band matrices and applications linear fractional descriptions are given for hermitian completions of partial banded matrices with prescribed cholesky signatures an inertia related notion the results are applied to obtain a linear fractional description for completions of partial triangular matrices with prescribed numbers of singular values 1

exact algorithm for minimising the number of setups in the one dimensional cutting stock problem the cutting stock problem is that of finding a cutting of stock material to meet demands for small pieces of prescribed dimensions while minimising the amount of waste because changing over from one cutting pattern to another involves significant setups an auxiliary problem is to minimise the number of different patterns that are used the pattern minimisation problem is significantly more complex but it is of great practical importance in this paper we propose an integer programming formulation for the problem that involves an exponential number of binary variables and associated columns each of which corresponds to selecting a fixed number of copies of a specific cutting pattern the integer program is solved using a column generation approach where the subproblem is a nonlinear integer program that can be decomposed into multiple bounded integer knapsack problems at each node of the branch and bound tree the linear programming relaxation of our formulation is made tighter by adding super additive inequalities branching rules are presented that yield a balanced tree incumbent solutions are obtained using a rounding heuristic the resulting branch and price and cut procedure is used to produce optimal or approximately optimal solutions for a set of real life problems

a generic exact solver for vehicle routing and related problems major advances were recently obtained in the exact solution of vehicle routing problems vrps sophisticated branch cut and price bcp algorithms for some of the most classical vrp variants now solve many instances with up to a few hundreds of customers however adapting and reimplementing those successful algorithms for other variants can be a very demanding task this work proposes a bcp solver for a generic model that encompasses a wide class of vrps it incorporates the key elements found in the best existing vrp algorithms ng path relaxation rank 1 cuts with limited memory path enumeration and rounded capacity cuts all generalized through the new concepts of packing set and elementarity set the concepts are also used to derive a branching rule based on accumulated resource consumption and to generalize the ryan and foster branching rule extensive experiments on several variants show that the generic solver has an excellent overall performance in many problems being better than the best specific algorithms even some non vrps like bin packing vector packing and generalized assignment can be modeled and effectively solved

finite dimensional wiener hopf equations and factorizations of matrices abstract let a be the algebra of all n n matrices over the real or complex numbers let a be the subalgebra of upper triangular matrices and a the subalgebra of strictly lower triangular matrices denote by p the projection of a onto a with kernel a in this paper we investigate the wiener hopf equation p ax y where y a is given and x a is a solution

bc opt a branch and cut code for mixed integer programs a branch and cut mixed integer programming system called bc opt is described incorporating most of the valid inequalities that have been used or suggested for such systems namely lifted 0 1 knapsack inequalities 0 1 gub knapsack and integer knapsack inequalities flowcover and continuous knapsack inequalities path inequalities for fixed charge network flow structure and gomory mixed integer cuts the principal development is a set of interface routines allowing these cut routines to generate cuts for new subsets or aggregations of constraints

using branch and price and cut to solve origin destination integer multicommodity flow problems we present a column generation model and branch and price and cut algorithm for origin destination integer multicommodity flow problems the origin destination integer multicommodity flow problem is a constrained version of the linear multicommodity flow problem in which flow of a commodity defined in this case by an origin destination pair may use only one path from origin to destination branch and price and cut is a variant of branch and bound with bounds provided by solving linear programs using column and cut generation at nodes of the branch and bound tree because our model contains one variable for each origin destination path for every commodity the linear programming relaxations at nodes of the branch and bound tree are solved using column generation i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality we devise a new branching rule that allows columns to be generated efficiently at each node of the branch and bound tree then we describe cuts cover inequalities that can be generated at each node of the branch and bound tree these cuts help to strengthen the linear programming relaxation and to mitigate the effects of problem symmetry we detail the implementation of our combined column and cut generation method and present computational results for a set of test problems arising from telecommunications applications we illustrate the value of our branching rule when used to find a heuristic solution and compare branch and price and branch and price and cut methods to find optimal solutions for highly capacitated problems

the modified integer round up property of the one dimensional cutting stock problem a linear integer minimization problem ip has the modified integer round up property mirup if the optimal value of any instance of ip is not greater than the optimal value of the corresponding lp relaxation problem rounded up plus one the aim of this paper is to investigate numerically whether the mirup holds for the one dimensional cutting stock problem the computational experiments carried out with a lot of randomly generated instances of the one dimensional cutting stock problem show that for all instances an integer solution fulfills the mirup moreover in most cases the optimal value equals the round up optimal value of the corresponding lp relaxation similarly the approach proposed to verify the mirup is usable as a new heuristic procedure for solving one dimensional cutting stock problems this heuristic always leads to very good solutions being optimal in the most cases

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

strip generation algorithms for constrained two dimensional two staged cutting problems abstract the constrained two dimensional cutting c tdc problem consists of determining a cutting pattern of a set of n small rectangular piece types on a rectangular stock plate of length l and width w as to maximize the sum of the profits of the pieces to be cut each piece type i i 1 n is characterized by a length l i a width w i a profit or weight c i and an upper demand value b i the upper demand value is the maximum number of pieces of type i which can be cut on rectangle l w in this paper we study the two staged fixed orientation c tdc noted fc 2tdc it is a classical variant of the c tdc where each piece is produced in the final cutting pattern by at most two guillotine cuts and each piece has a fixed orientation we solve the fc 2tdc problem using several approximate algorithms that are mainly based upon a strip generation procedure we evaluate the performance of these algorithms on instances extracted from the literature

shift invariant subspaces factorization and interpolation for matrices i the canonical case abstract this article begins a program of classifying subspaces of block matrices that are invariant under multiplication on the right by the algebra of block upper or lower triangular matrices this gives a unifying point of view to several general kinds of factorization problems including lu qr and cholesky factorization a second application is to some matrix completion problems such as that of filling in the upper triangular entries of a partially defined matrix to make the result of norm less than or equal to 1 included is a linear fractional parametrization of the set of all solutions of this problem

a two objective mathematical model without cutting patterns for one dimensional assortment problems this paper considers a one dimensional cutting stock and assortment problem one of the main difficulties in formulating and solving these kinds of problems is the use of the set of cutting patterns as a parameter set in the mathematical model since the total number of cutting patterns to be generated may be very huge both the generation and the use of such a set lead to computational difficulties in solution process the purpose of this paper is therefore to develop a mathematical model without the use of cutting patterns as model parameters we propose a new two objective linear integer programming model in the form of simultaneous minimization of two contradicting objectives related to the total trim loss amount and the total number of different lengths of stock rolls to be maintained as inventory in order to fulfill a given set of cutting orders the model does not require pre specification of cutting patterns we suggest a special heuristic algorithm for solving the presented model the superiority of both the mathematical model and the solution approach is demonstrated on test problems

integer linear programming models for 2 staged two dimensional knapsack problems we are given a unique rectangular piece of stock material s with height h and width w and a list of m rectangular shapes to be cut from s each shape s type i i 1 m is characterized by a height a width a profit and an upper bound ub i indicating the maximum number of items of type i which can be cut we refer to the two dimensional knapsack tdk as the problem of determining a cutting pattern of s maximizing the sum of the profits of the cut items in particular we consider the classical variant of tdk in which the maximum number of cuts allowed to obtain each item is fixed to 2 and we refer to this problem as 2 staged tdk 2tdk for the 2tdk problem we present two new integer linear programming models we discuss their properties and we compare them with other formulations in terms of the lp bound they provide finally both models are computationally tested within a standard branch and bound framework on a large set of instances from the literature by reinforcing them with the addition of linear inequalities to eliminate symmetries

positive semidefinite completions of partial hermitian matrices abstract we classify the ranks of positive semidefinite completions of hermitian band matrices and other partially specified hermitian matrices with chordal graphs and specified main diagonals completing a partially specified matrix means filling in the unspecified entries

a branch and cut and price algorithm for one dimensional stock cutting and two dimensional two stage cutting abstract the one dimensional cutting stock problem 1d csp and the two dimensional two stage guillotine constrained cutting problem 2d 2cp are considered in this paper the gilmore gomory models of these problems have very strong continuous relaxations providing a good bound in an lp based solution approach in recent years there have been several efforts to attack the one dimensional problem by lp based branch and bound with column generation called branch and price and by general purpose c hv a tal g omory cutting planes in this paper we investigate a combination of both approaches i e the lp relaxation at each branch and price node is strengthened by c hv a tal g omory and g omory mixed integer cuts the branching rule is that of branching on variables of the gilmore gomory formulation tests show that for 1d csp general purpose cuts are useful only in exceptional cases however for 2d 2cp their combination with branching is more effective than either approach alone and mostly better than other methods from the literature

integer linear programming models for the skiving stock problem abstract we consider the one dimensional skiving stock problem which is strongly related to the dual bin packing problem find the maximum number of items with minimum length l that can be constructed by connecting a given supply of m n smaller item lengths l 1 l m with availabilities b 1 b m for this optimization problem we present three new models the arcflow model the onestick model and a model of kantorovich type and investigate their relationships especially regarding their respective continuous relaxations to this end numerical computations are provided as a main result we prove the equivalence between the arcflow model the onestick approach and the existing pattern oriented standard model in particular this equivalence is shown to hold for the corresponding continuous relaxations too

spanning tree extensions of the hadamard fischer inequalities abstract all possible graph theoretic generalizations of a certain sort for the hadamard fischer determinantal inequalities are determined these involve ratios of products of principal minors which dominate the determinant furthermore the cases of equality in these inequalities are characterized and equality is possible for every set of values which can occur for the relevant minors this relates recent work of the authors on positive definite completions and determinantal identities when applied to the same collections of principal minors earlier generalizations give poorer more difficult to compute bounds than the present inequalities thus this work extends and in a certain sense completes a series of generalizations of hadamard fischer begun in the 1960s

a comparative study of the arcflow model and the one cut model for one dimensional cutting stock problems abstract we consider the one dimensional cutting stock problem which consists in determining the minimum number of given large stock rolls that has to be cut to satisfy the demands of certain smaller item lengths besides the standard pattern based approach of gilmore and gomory containing an exponential number of variables several pseudo polynomial formulations were proposed in the last decades much research has dealt with arcflow models their relationship to the standard model and possible reduction methods whereas the one cut approach has not attracted that much scientific interest yet in this paper we aim to compare both alternative formulations from a theoretical and numerical point of view as a theoretical main contribution we constructively prove the equivalence of the continuous relaxations of the one cut model the arcflow model and the pattern based model in particular the relationship between the one cut model and the pattern based model has remained an open question since the one cut approach was proposed moreover in order to make a computational comparison of the arcflow model and the one cut model we present how reduction methods partly originating from arcflow considerations can successfully be transferred to the one cut context furthermore we derive relations between the numbers of variables and constraints in both models and investigate their influences in numerical simulations

computational study of a column generation algorithm for bin packing and cutting stock problems this paper reports on our attempt to design an efficient exact algorithm based on column generation for the cutting stock problem the main focus of the research is to study the extend to which standard branch and bound enhancement features such as variable fixing the tightening of the formulation with cutting planes early branching and rounding heuristics can be usefully incorporated in a branch and price algorithm we review and compare lower bounds for the cutting stock problem we propose a pseudo polynomial heuristic we discuss the implementation of the important features of the integer programming column generation algorithm and in particular the implementation of the branching scheme our computational results demonstrate the efficiency of the resulting algorithm for various classes of bin packing and cutting stock problems

a schur analysis approach to minimum distance problems abstract some minimum distance problems in hilbert space are to be transformed into associated matrix optimization problems which is their turn are studied using schur parametrization of nonnegative hermitian block matrices this enables a clear geometrical insight into the structure of the feasible region and a complete description of the set of all solutions

finding rough and fuzzy rough set reducts with sat feature selection refers to the problem of selecting those input features that are most predictive of a given outcome a problem encountered in many areas such as machine learning pattern recognition and signal processing in particular solution to this has found successful application in tasks that involve datasets containing huge numbers of features in the order of tens of thousands which would otherwise be impossible to process further recent examples include text processing and web content classification rough set theory has been used as such a dataset pre processor with much success but current methods are inadequate at finding globally minimal reductions the smallest sets of features possible this paper proposes a technique that considers this problem from a propositional satisfiability perspective in this framework globally minimal subsets can be located and verified

tighter relaxations for the cutting stock problem in the cutting stock problem csp a given order for smaller pieces has to be cut from larger stock material in such a way that the number of stock material needed is minimal based on the classical integer linear programming model the common solution technique consists of solving the corresponding continuous relaxation problem followed by several heuristics which construct integer solutions in many cases an optimal solution can be obtained quickly in this way but for instances which do not possess the integer round up property the optimality of the solution obtained cannot be verified by means of the lp bound in order to overcome this non satisfactory situation two tighter relaxations of the csp are proposed and results of theoretical and numerical investigations are presented

model validation of multirate systems from time domain experimental data the model validation problem using time domain experimental data is studied for multirate linear fractional uncertain models in this note as a technical tool the caratheodory fejer cf interpolation problem with a nest operator constraint is first investigated this problem is itself of interest mathematically and has potential applications in addressing other problems in control signal processing and circuit theory a necessary and sufficient solvability condition for this interpolation problem is given the validation tests are then presented based on this condition and the lifting technique tractable convex optimization methods can be used to solve the validation problems

lp models for bin packing and cutting stock problems abstract we review several linear programming lp formulations for the one dimensional cutting stock and bin packing problems namely the models of kantorovich gilmore gomory onecut models as in the dyckhoff stadtler approach position indexed models and a model derived from the vehicle routing literature we analyse some relations between the corresponding lp relaxations and their relative strengths and refer how to derive branching schemes that can be used in the exact solution of these problems using branch and price

the circulant rational covariance extension problem the complete solution the rational covariance extension problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion problem to construct an infinite dimensional positive definite toeplitz matrix the northwest corner of which is given the circulant rational covariance extension problem considered in this paper is a modification of this problem to partial stochastic realization of periodic stationary process which are better represented on the discrete unit circle rather than on the discrete real line the corresponding matrix completion problem then amounts to completing a finite dimensional toeplitz matrix that is circulant another important motivation for this problem is that it provides a natural approximation involving only computations based on the fast fourier transform for the ordinary rational covariance extension problem potentially leading to an efficient numerical procedure for the latter the circulant rational covariance extension problem is an inverse problem with infinitely many solutions in general each corresponding to a bilateral arma representation of the underlying periodic process in this paper we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them a procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented

a branch and price algorithm for the two dimensional vector packing problem with piecewise linear cost function abstract the two dimensional vector packing problem with piecewise linear cost function 2dvpp plc models a practical packing problem faced by many companies that use courier services we propose a branch and price algorithm to solve the 2dvpp plc exactly the column generation procedure is a key component that affects the performance of a branch and price algorithm the pricing problem exhibits an interesting structure that allows us to decompose it into subproblems that form a lattice we explore dominance relations on the lattice and design an efficient algorithm for the pricing problem experimental results show that our branch and price algorithm is capable of solving 2dvpp plc test instances effectively

inheritance principles for chordal graphs abstract we prove two results concerning positive completions of partial positive matrices first we give a parametrization of all positive invertible completions of a partial positive matrix having as support a chordal graph along a fixed increasing chordal sequence then a formula for the determinant of each completion is obtained in terms of the associated parameters as an application we obtained a proof of a conjecture of w w barrett c r johnson and l rodman regarding an inheritance principle for chordal graphs

reduction operations in fuzzy or valued constraint satisfaction in the constraint satisfaction problem csp knowledge about a relation on n variables is given in the form of constraints on subsets of the variables a solution to the csp is simply an instantiation of the n variables which satisfies all the constraints the fuzzy constraint satisfaction problem fcsp is a generalisation of the csp to the case in which the constraints are not categorical but represent preferences in the form of fuzzy sets the fcsp is then an optimisation problem in the space of all possible instantiations of the n variables it has potentially many more applications in real world problems than the csp when the aim is to maximise the value of the most violated constraint then the fcsp can be solved by solving a logarithmic number of csps fuzzy k consistency can even be established by an algorithm whose worst case complexity is identical to that of an optimal k consistency algorithm for csps when the operator used to aggregate constraint values is strictly monotonic as is the case in max csp for example the classic operation of arc consistency has to be completely redefined we show that an ordered version of arc consistency exists for all fcsps with a strictly monotonic aggregation operator and that in the case of max csp it can be established by an algorithm whose worst case complexity is identical to that of the optimal arc consistency algorithm for csps neighbourhood substitution in csps can be generalised to fuzzy neighbourhood substitution in fcsps whether the aggregation operator is strictly monotonic or idempotent we give an algorithm for applying fuzzy neighbourhood substitutions until convergence based on the corresponding algorithm for csps

completion of partial matrices to contractions an n by m partially specified complex matrix is called a partial contraction if every rectangular submatrix consisting entirely of specified entries is itself a contraction necessary and sufficient condition are given for the pattern of specified entries such that any n by m partial contraction with this pattern may be completed to a full n by m contraction

average weight controlled bin oriented heuristics for the one dimensional bin packing problem abstract bin oriented heuristics for one dimensional bin packing problem construct solutions by packing one bin at a time several such heuristics consider two or more subsets for each bin and pack the one with the largest total weight these heuristics sometimes generate poor solutions due to a tendency to use many small items early in the process to address this problem we propose a method of controlling the average weight of items packed by bin oriented heuristics constructive heuristics and an improvement heuristic based on this approach are introduced additionally reduction methods for bin oriented heuristics are presented the results of an extensive computational study show that 1 controlling average weight significantly improves solutions and reduces computation time of bin oriented heuristics 2 reduction methods improve solutions and processing times of some bin oriented heuristics and 3 the new improvement heuristic outperforms all other known complex heuristics in terms of both average solution quality and computation time

an empirical study of phase transitions in binary constraint satisfaction problems abstract an empirical study of randomly generated binary constraint satisfaction problems reveals that for problems with a given number of variables domain size and connectivity there is a critical level of constraint tightness at which a phase transition occurs at the phase transition problems change from being soluble to insoluble and the difficulty of problems increases dramatically a theory developed by williams and hogg 44 and independently developed by smith 37 predicts where the hardest problems should occur it is shown that the theory is in close agreement with the empirical results except when constraint graphs are sparse

factorizations of operator matrices abstract this note gives explicit factorizations of a 2 2 operator matrix as a product of an upper triangular operator matrix and an involutory unitary or j unitary operator matrix a pattern is given for construction of factorizations of this kind

minimizing conflicts a heuristic repair method for constraint satisfaction and scheduling problems abstract the paper describes a simple heuristic approach to solving large scale constraint satisfaction and scheduling problems in this approach one starts with an inconsistent assignment for a set of variables and searches through the space of possible repairs the search can be guided by a value ordering heuristic the min conflicts heuristic that attempts to minimize the number of constraint violations after each step the heuristic can be used with a variety of different search strategies we demonstrate empirically that on the n queens problem a technique based on this approach performs orders of magnitude better than traditional backtracking techniques we also describe a scheduling application where the approach has been used successfully a theoretical analysis is presented both to explain why this method works well on certain types of problems and to predict when it is likely to be most effective

risk assessment of serious crime with fuzzy random theory this paper presents a novel approach for assessing the potential risk of serious crime events e g terrorist attack the modelling and assessment of such risk is carried out under uncertain circumstances because of both the randomness and fuzziness inherent in crime data the approach is based on fuzzy random theory that complements probability theory with an additional dimension of imprecision this allows for potential loss caused by a crime to be expressed as a fuzzy random variable crime risk is therefore estimated as the mean chance of a fuzzy random event where the resulting loss reaches a given confidence level the concept of the average loss per unit of time is also introduced in order to calculate the rate at which the loss may increase due to possible crime events the work is compared with typical existing approaches and supported with examples throughout that illustrate its utility

approximation of linear operators in the 2 norm abstract the problems of approximating linear operators in the 2 induced norm which are 1 finite dimensional unstructured and 2 infinite dimensional structured hankel have been solved the solutions of these two problems exhibit striking similarities these similarities suggest the search of a unifying framework for the approximation of linear operators in the 2 induced norm

mihaly matrix completions moments and sums of hermitian squares by mihaly bakonyi and hugo j woerdeman princeton series in applied mathematics princeton university press 2011 xii 518 pp hardback isbn 978 0 691 12889 4

a maximum entropy principle in the general framework of the band method abstract a maximum entropy principle is developed in the general framework of the band method known maximum entropy principles appear as corollaries and new ones are derived

dualistic differential geometry of positive definite matrices and its applications to related problems abstract analysis on the set of positive definite matrices is involved in many engineering problems we develop a differential geometric theory of the set of positive definite matrices by means of a specific class of connections introduced into it consequently various dualistic aspects of the set are elucidated next using the theory we derive new results and interesting geometrical interpretations for matrix approximation positive definite matrix completion and linear matrix inequality problems

a note on extensions of band matrices with maximal and submaximal invertible blocks abstract completions or extensions of band matrices with invertible maximal and submaximal blocks are considered linear fractional descriptions for invertible minimal rank and lu ul decomposable extensions are derived self adjoint and toeplitz cases are included

an integer programming model for two and three stage two dimensional cutting stock problems abstract in this paper an integer programming model for two dimensional cutting stock problems is proposed in the problems addressed it is intended to cut a set of small rectangular items of given sizes from a set of larger rectangular plates in such a way that the total number of used plates is minimized the two stage and three stage exact and non exact problems are considered other issues are also addressed as the rotation of items the length of the cuts and the value of the remaining plates the new integer programming model can be seen as an extension of the one cut model proposed by dyckhoff for the one dimensional cutting stock problem in the proposed model each decision variable is associated with cutting one item from a plate or from a part of a plate resulting from previous cuts residual plates comparative computational results of the proposed model and of models from the literature are presented and discussed

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

strictly contractive and positive completions for block matrices abstract this paper deals with the following two problems of completion of block matrices 1 let a ij i j q be given matrices find additional matrices a ij such that the block matrix a a ij n i 1 m j 1 has norm less than one 2 let b ij j i q be given matrices find additional b ij such that the completion b b ij n i j 1 is positive definite the analysis is based on a study of certain linear fractional maps this approach leads to an explicit description of all solutions by linear fractional maps of which the coefficients are given directly in terms of the given data the maximum entropy principle appears as a corollary of a general formula for the determinant of a completion special attention is paid to the toeplitz case

unitary dilation approach to contractive matrix completion in this paper a new method to construct all contractive matrix completion to a block triangular matrix is given this new method uses only elementary matrix operations and has the advantages of better numerical reliability and ease in implementation on computers

a new model for complete solutions to one dimensional cutting stock problems abstract this new model resolves the non linearity in the one dimensional cutting stock problem between pattern variables and pattern run lengths by a novel use of 0 1 variables the model does not require pre specification of cutting patterns it explicitly includes practical constraints such as pattern cardinality constraints minimum run length constraints and sequence related constraints the resultant mip is relatively compact and its application to four problem instances using a commercial mip package is presented

schur analysis of some completion problems abstract the complete schur analysis for contractive block matrices is given together with its connections with positiveness these are used for reobtaining some results on completion problems sometimes in a more general setting these completion problems include positive completions contractive and isometric completions and the realizability of upper triangular contractions

maximum entropy kernels for system identification bayesian nonparametric approaches have been recently introduced in system identification scenario where the impulse response is modeled as the realization of a zero mean gaussian process whose covariance kernel has to be estimated from data in this scheme quality of the estimates crucially depends on the parametrization of the covariance of the gaussian process a family of kernels that have been shown to be particularly effective in the system identification framework is the family of diagonal correlated dc kernels maximum entropy properties of a related family of kernels the tuned correlated tc kernels have been recently pointed out in the literature in this technical note we show that maximum entropy properties indeed extend to the whole family of dc kernels the maximum entropy interpretation can be exploited in conjunction with results on matrix completion problems in the graphical models literature to shed light on the structure of the dc kernel in particular we prove that the dc kernel admits a closed form factorization inverse and determinant these results can be exploited both to improve the numerical stability and to reduce the computational complexity associated with the computation of the dc estimator

dynamic programming and strong bounds for the 0 1 knapsack problem two new algorithms recently proved to outperform all previous methods for the exact solution of the 0 1 knapsack problem this paper presents a combination of such approaches where in additi on valid inequalities are generated and surrogate relaxed and a new initial core problem is adopted the algorithm is able to solve all classical test instances with up to 10 000 variables in less than 0 2 seconds on a hp9000 735 99 computer the c language implementation of the algorithm is available on the internet

friendly bin packing instances without integer round up property it is well known that the gap between the optimal values of bin packing and fractional bin packing if the latter is rounded up to the closest integer is almost always null known counterexamples to this for integer input values involve fairly large numbers specifically the first one was derived in 1986 and involved a bin capacity of the order of a billion later in 1998 a counterexample with a bin capacity of the order of a million was found in this paper we show a large number of counterexamples with bin capacity of the order of a hundred showing that the gap may be positive even for numbers which arise in customary applications the associated instances are constructed starting from the petersen graph and using the fact that it is fractionally but not integrally 3 edge colorable

a cutting plane algorithm for the one dimensional cutting stock problem with multiple stock lengths abstract a cutting plane approach combining chvatal gomory cutting planes with column generation is generalized for the case of multiple stock lengths in the one dimensional cutting stock problem appropriate modifications of the column generation procedure and the rounding heuristic are proposed a comparison with the branch and price method for the problem with one stock type and representative test results are reported

fuzzy rr dfcsp and planning constraint satisfaction is a fundamental artificial intelligence technique for knowledge representation and inference however the formulation of a static constraint satisfaction problem csp with hard imperative constraints is insufficient to model many real problems fuzzy constraint satisfaction provides a more graded viewpoint priorities and preferences are placed on individual constraints and aggregated via fuzzy conjunction to obtain a satisfaction degree for a solution to the problem this paper examines methods for solving an important instance of dynamic flexible constraint satisfaction dfcsp combining fuzzy csp and restriction relaxation based dynamic csp fuzzy rrdfcsp this allows the modelling of complex situations where both the set of constraints may change over time and there is flexibility inherent in the definition of the problem this paper also presents a means by which classical planning can be extended via fuzzy sets to enable flexible goals and preferences to be placed on the use of planning operators a range of plans can be produced trading compromises made versus the length of the plan the flexible planning operators are close in definition to fuzzy constraints hence through a hierarchical decomposition of the planning graph the work shows how flexible planning reduces to the solution of a set of fuzzy rrdfcsps

the stochastic trim loss problem the cutting stock problem csp is one of the most fascinating problems in operations research the problem aims at determining the optimal plan to cut a number of parts of various length from an inventory of standard size material so to satisfy the customers demands the deterministic csp ignores the uncertain nature of the demands thus typically providing recommendations that may result in overproduction or in profit loss this paper proposes a stochastic version of the csp which explicitly takes into account uncertainty using a scenario based approach we develop a two stage stochastic programming formulation the highly non convex nature of the model together with its huge size prevent the application of standard software we use a solution approach designed to exploit the specific problem structure encouraging preliminary computational results are provided

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar

the maximum distance problem and band sequences abstract we solve the following problem for 1 j k n and j k m let a jk be a given complex number with a kj jk we wish to find linearly independent vectors x 1 x n such that x k x j a jk for j k m and such that the distance from x k to the linear span of x 1 x k 1 is maximal for 2 k n we construct and characterize all such sequences of vectors our solution leads naturally to the class of m band sequences of vectors in an inner product space we study these sequences and characterize their equivalence classes under unitary transformations