traces of sobolev functions on fractal type sets and characterization of extension domains abstract we describe traces of sobolev functionsu w1 p r n 1 r nin terms of sobolev spaces on metric spaces 7 our results apply to smooth submanifolds fractal subsets as well as to open subsets of r n in particular if 0 r nis a john domain then we characterize thosew1 p functions which can be extended tow1 p r n if is uniform then this result implies jones extension theorem 14 in the case of traces on fractal subsets our results are related to those of jonsson and wallin 16

a combinatorial approach to level of repair analysis abstract this paper presents an approach to optimise level of repair decisions taking into account submodular properties of standard life cycle cost functions which include fixed and variable costs it proposes an integer programming formulation to solve level of repair problems for multi echelon multi indenture level systems the method converges quickly to the optimum solution relying on heuristics to obtain tight bounds for a subsequent branch and bound procedure a software package called level of repair optimisation model lorom was developed to implement the branch and bound method that does not rely on linear programming relaxations this approach is rather generic and can be applied to a wide class of problems with convex total cost functions such as plant location problems or transportation problems with fixed costs

a lagrangian heuristic for the capacitated plant location problem with single source constraints abstract in this paper we look at the lagrangian relaxations of the capacitated plant location problem with single source constraints in terms of the bounds and solution techniques we propose a heuristic based on a lagrangian relaxation of the problem we also show how this heuristic can be used to solve other combinatorial problems that are related to this problem such as the concentrator location problem and the bin packing problem

majorization relations for hadamard products abstract we settle affirmatively the conjecture of johnson and bapat on the hadamard product a b of positive definite matrices a b i k n i a b ab k 1 2 n

variable neighborhood search principles and applications abstract systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization called variable neighborhood search vns we present a basic scheme for this purpose which can easily be implemented using any local search algorithm as a subroutine its effectiveness is illustrated by solving several classical combinatorial or global optimization problems moreover several extensions are proposed for solving large problem instances using vns within the successive approximation method yields a two level vns called variable neighborhood decomposition search vnds modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed vns svns heuristic finally we show how to stabilize column generation algorithms with help of vns and discuss various ways to use vns in graph theory i e to suggest disprove or give hints on how to prove conjectures an area where metaheuristics do not appear to have been applied before

on the coordination of product and by product flows in two level distribution networks model formulations and solution procedures abstract we study the problem of coordinating product and by product flows in a two level distribution network in this problem a single product is produced at multiple plants in order to fulfill customer demands furthermore in the production process a by product is produced that must be transported to and processed at a post processing unit the problem objective is to determine suitable locations for plants and post processing units from a predetermined set of feasible locations such that total costs are minimal here total costs consist of fixed costs associated with opening plants and post processing units and variable costs associated with transport of product and by product we discuss alternative mathematical model formulations for this problem and derive lower and upper bounding procedures the results from computational experiments in which we analyse the numerical behaviour of these procedures are also reported

a heuristic lagrangean algorithm for the capacitated plant location problem abstract lagrangean techniques have been widely applied to the uncapacitated plant location problem and in some cases they have proven to be successfull even when capacitated problems with additional constraints are taken into account in our paper we study the application of these techniques to the capacitated plant location problem when the model considered is a pure integer one several lagrangean decompositions are considered and for some of them heuristic algorithms have been designed to solve the resulting lagrangean subproblems the heuristics consisting of a two phase procedure the first location phase defines a set of multipliers from the analysis of the dual lp relaxation and makes a choice of the plants considering the resulting subproblems as a particular case of the general assignment problems several heuristics have been studied for this second phase based either on a decomposition of knapsack type subproblems through a definition of a set of penalties or of looking into the duality gap and trying to reduce it computational experience is reported

combining very large scale and ilp based neighborhoods for a two level location problem abstract in this paper we tackle a generalization of the single source capacitated facility location problem in which two sets of facilities called intermediate level and upper level facilities have to be located the dimensioning of the intermediate set the assignment of clients to intermediate level facilities and of intermediate level facilities to upper level facilities must be optimized as well such problem arises for instance in telecommunication network design in fact in hierarchical networks the traffic arising at client nodes often have to be routed through different kinds of facility nodes which provide different services we propose a heuristic approach based on very large scale neighborhood search to tackle the problem in which both ad hoc algorithms and general purpose solvers are applied to explore the search space we report on experimental results using datasets from the capacitated location literature such results show that the approach is promising and that integer linear programming based neighborhoods are significantly effective

inequalities involving unitarily invariant norms and operator monotone functions let be a unitarily invariant norm on matrices for matrices a b x with a b positive semidefinite and x arbitrary we prove that the function t atxb1 t r a1 txbt r is convex on 0 1 for each r 0 this convexity result interpolates the matrix cauchy schwarz inequality a1 2xb1 2 r 2 ax r xb r due to r bhatia and c davis linear algebra appl 223 224 1995 119 and also it generalizes a w marshall and i olkin s pacific j math 15 1965 241 result that the condition number as a s is increasing in s 0 we prove that if f t is a nonnegative operator monotone function on 0 and is a normalized unitarily invariant norm then f x f x for every matrix x the special case when f t tr 0 r 1 is used to consider the monotonicity of p ap bp 1 p as well as p ap bp 1 p furthermore we obtain some norm inequalities of holder and minkowski types related to the expression a p b p 1 p for example comparisons are made between c a d b and a p b p 1 p c q d q 1 q where p 1 q 1 1

weak differentiability of bv functions on stratified groups we prove the almost everywhere approximate differentiability of functions with bounded variation in stratified groups and we study their approximate discontinuity set we introduce functions of bounded higher order variation and obtain a weak version of alexandroff differentiability theorem in this context we present a nontrivial class of functions with second order bounded variation arising from inf convolution formula of a suitable cost function

upper and lower bounds for the single source capacitated location problem abstract the single source capacitated location problem is considered given a set of potential locations and the plant capacities it must be decided where and how many plants must be open and which clients must be assigned to each open plant a lagrangean relaxation is used to obtain lower bounds for this problem upper bounds are given by lagrangean heuristics followed by search methods and by one tabu search metaheuristic computational experiments on different sets of problems are presented

exploiting unsupervised and supervised constraints for subspace clustering data in many image and video analysis tasks can be viewed as points drawn from multiple low dimensional subspaces with each subspace corresponding to one category or class one basic task for processing such kind of data is to separate the points according to the underlying subspace referred to as subspace clustering extensive studies have been made on this subject and nearly all of them use unconstrained subspace models meaning the points can be drawn from everywhere of a subspace to represent the data in this paper we attempt to do subspace clustering based on a constrained subspace assumption that the data is further restricted in the corresponding subspaces e g belonging to a submanifold or satisfying the spatial regularity constraint this assumption usually describes the real data better such as differently moving objects in a video scene and face images of different subjects under varying illumination a unified integer linear programming optimization framework is used to approach subspace clustering which can be efficiently solved by a branch and bound bb method we also show that various kinds of supervised information such as subspace number outlier ratio pairwise constraints size prior and etc can be conveniently incorporated into the proposed framework experiments on real data show that the proposed method outperforms the state of the art algorithms significantly in clustering accuracy the effectiveness of the proposed method in exploiting supervised information is also demonstrated

an exact method for the two echelon single source capacitated facility location problem facility location problems form an important class of integer programming problems with applications in the telecommunication distribution and transportation industries in this paper we are conc

hybrid scatter search and path relinking for the capacitated p median problem in this paper we propose heuristic algorithms for the capacitated p median problem in the first one solutions are obtained with scatter search whereas the second algorithm uses path relinking a third approach that combines path relinking with scatter search is also analyzed the grasp methodology is used to generate the initial reference set both for scatter search and path relinking computational experiments have been carried out with different data sets in order to evaluate the performance of the approaches in general scatter search outperforms path relinking but the use of path relinking prior to scatter search has shown to enhance the performance of scatter search specially in terms of the computation times required due to the improvement of the initial reference set the obtained results that have been compared with previous approaches show the efficiency of the methods both in terms of the quality of the solutions found and of the required computation times even for very large instances

lagrangean heuristics for location problems abstract in this paper we present a framework for developing lagrangean heuristics heuristics based upon lagrangean relaxation and subgradient optimisation with respect to location problems computational results are given for four different location problems p median uncapacitated warehouse location capacitated warehouse location and capacitated warehouse location with single source constraints these results indicate that the framework presented in this paper is robust i e it gives good quality solutions for each of these different location problems

a multi exchange heuristic for the single source capacitated facility location problem we present a very large scale neighborhood vlsn search algorithm for the capacitated facility location problem with single source constraints the neighborhood structures are induced by customer multi exchanges and by facility moves we consider both traditional single customer multi exchanges detected on a suitably definedcustomer improvement graph and more innovative multicustomer multi exchanges detected on afacility improvement graph dynamically built through the use of a greedy scheme computational results for some benchmark instances are reported that demonstrate the effectiveness of the approach for solving large scale problems a further test on real data involving an italian factory is also presented

norm inequalities related to operator monotone functions let a b be positive semidefinite matrices and cdot any unitarily invariant norm on the space of matrices we show f a f b geq f a b for any non negative operator monotone function f t on 0 infty and g a g b leq g a b for non negative increasing function g t on 0 infty with g 0 0 and g infty infty whose inverse function is operator monotone

a branch and bound algorithm for the multi level uncapacitated facility location problem abstract this paper discusses the problem of locating facilities over all levels in a multi level distribution system where commodities are shipped from origin level facilities to destination points via a number of intermediate level facilities the objective is to determine the optimal set of facilities to open for each distribution level which minimizes the total distribution costs including the fixed costs associated with opening them the problem is formulated as a mixed integer linear program with the so called tight or disaggregated constraints by which in our branch and bound solution method the dual based scheme of dual ascent procedure by erlenkotter and bilde and krarup is successfully applied in addition a heuristic primal descent procedure which improves integer solutions already obtained and some efficient node simplification procedure are introduced though our approach can be applied to any multi level case computational experiments are conducted mostly for two level problems such that our approach can be compared with the other approaches computational experience with some small scale three level problems is also reported on

the trace problem for vector fields satisfying hormander s condition trace theorems are proved for non isotropic sobolev and l p lipschitz spaces defined by vector fields satisfying hormander s bracket condition of order 2 it is shown that the loss of regularity by traces is the same as in the classical case

submultiplicativity vs subadditivity for unitarily invariant norms abstract let a b be nonzero positive semidefinite matrices we prove that ab a b a b a b a b a b a b a b for any unitarily invariant norm with diag 1 0 0 1 some related inequalities are derived

trace theorems for vector fields in the setting of carnot caratheodory spaces we prove some trace theorems for sobolev functions we consider the trace on a non characteristic surface for hormander vector fields of step r ge 1 and the trace on the boundary of a class of domains in the grushin plane

a repeated matching heuristic for the single source capacitated facility location problem facility location models form an important class of integer programming problems with application in many areas such as the distribution and transportation industries an important class of solution methods for these problems are so called lagrangean heuristics which have been shown to produce high quality solutions and which are at the same time robust the general facility location problem can be divided into a number of special problems depending on the properties assumed in the capacitated location problem each facility has a specific capacity on the service it provides we describe a new solution approach for the capacitated facility location problem when each customer is served by a single facility the approach is based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied the method generates feasible solutions in each iteration in contrast to lagrangean heuristics where problem dependent heuristics must be used to construct a feasible solution numerical results show that the approach produces solutions which are of similar and often better than those produced using the best lagrangean heuristics

uncapacitated facility location general solution procedure and computational experience abstract this paper presents a general model and dual based branch and bound solution procedure for finding optimal solutions to the single echelon two echelon and multi activity uncapacitated facility location problems the problem is formulated as an arborescent fixed charge network arbnet programming model which visually displays the problem s physical and mathematical structure and generalized modeling capabilities alternative strategies for implementing the dual based solution procedure are discussed and the results of computational experiments are given to document their relative efficiency the computational results indicate that the proposed dual based solution procedure effectively solves realistic sized single echelon two echelon and multi activity location problems this research takes another step toward the development of an efficient general purpose decision support system for distribution system planning

a multiperiod two echelon multicommodity capacitated plant location problem abstract in this paper we deal with a facility location problem where one desires to establish facilities at two different distribution levels by selecting the time periods our model intends to minimize the total cost for meeting demands for all the products specified over the planning horizon at various customer locations while satisfying the capacity requirements of the production plants and intermediate warehouses we address this problem by means of a formulation as a mixed integer programming problem a lagrangean relaxation is proposed to solve the problem together with a heuristic procedure that constructs feasible solutions of the original problem from the solutions at the lower bounds obtained by the relaxed problems computational tests are provided showing the good performance of this approach for a wide range of problems

a branch and bound algorithm for the two level uncapacitated facility location problem with some side constraints abstract this paper deals with the problem of locating at minimum total costs both plants and warehouses in a two level distribution system where commodities are delivered from plants to customers either directly or indirectly via warehouses some side constraints are imposed on the problem which represent the adjunct relationship of some warehouses to a certain plant proposed is an efficient branch and bound solution procedure employing a set of new devices for lower bounds and simplifications which are obtained by exploiting the submodularity of the objective function and the special structure of the side constraints computational experiments with fifteen test problems are provided

a heuristic for bilp problems the single source capacitated facility location problem abstract in the single source capacitated facility location problem sscflp each customer has to be assigned to one facility that supplies its whole demand the total demand of customers assigned to each facility cannot exceed its capacity an opening cost is associated with each facility and is paid if at least one customer is assigned to it the objective is to minimize the total cost of opening the facilities and supply all the customers in this paper we extend the kernel search heuristic framework to general binary integer linear programming bilp problems and apply it to the sscflp the heuristic is based on the solution to optimality of a sequence of subproblems where each subproblem is restricted to a subset of the decision variables the subsets of decision variables are constructed starting from the optimal values of the linear relaxation variants based on variable fixing are proposed to improve the efficiency of the kernel search framework the algorithms are tested on benchmark instances and new very large scale test problems computational results demonstrate the effectiveness of the approach the kernel search algorithm outperforms the best heuristics for the sscflp available in the literature it found the optimal solution for 165 out of the 170 instances with a proven optimum the error achieved in the remaining instances is negligible moreover it achieved on 100 new very large scale instances an average gap equal to 0 64 computed with respect to a lower bound or the optimum when available the variants based on variable fixing improved the efficiency of the algorithm with minor deteriorations of the solution quality

an exact algorithm for the capacitated facility location problems with single sourcing facility location problems are often encountered in many areas such as distribution transportation and telecommunication we describe a new solution approach for the capacitated facility location problem in which each customer is served by a single facility an important class of heuristic solution methods for these problems are lagrangian heuristics which have been shown to produce high quality solutions and at the same time be quite robust a primal heuristic based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied is incorporated into the lagrangian heuristic finally a branch and bound method based on the lagrangian heuristic is developed and compared computationally to the commercial code cplex the computational results indicate that the proposed method is very efficient

aircrew schedule generation using repeated matching we consider the aircrew scheduling problem of determining tours of duty tods for aircrews given a set of sectors or flights requiring regular crews a regular crew consists of two crew members but by including supplementary crew a third pilot on some sectors it is possible to extend duty periods to generate more cost efficient tods a related problem is thus to generate tods for these third pilots but the sectors requiring a third pilot are not known in advance to solve these two related problems simultaneously we apply a heuristic procedure that solves a sequence of matching problems i e a repeated matching algorithm numerical results based on the solution of a real problem show that this approach is a valid and efficient method for solving the aircrew scheduling problem especially when there is the option of using supplementary crew to extend duty periods

multi level facility location as the maximization of a submodular set function abstract in this paper we model the multi level uncapacitated facility location problem as two different combinatorial optimization problems the first model is the classical representation of the problem which uses a set of vertices as combinatorial objects to represent solutions whereas in the second model we propose the use of a set of paths an interesting observation is that the real valued set function associated with the first combinatorial problem does not satisfy the submodular property whereas the set function associated with the second problem does satisfy this property this illustrates the fact that submodularity is not a property intrinsic to an optimization problem but rather to its mathematical representation

algorithms for the vehicle routing and scheduling problems with time window constraints this paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints given the intrinsic difficulty of this problem class approximation methods seem to offer the most promise for practical size problems after describing a variety of heuristics we conduct an extensive computational study of their performance the problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems the percentage of customers with time windows their tightness and positioning and the scheduling horizon we found that several heuristics performed well in different problem environments in particular an insertion type heuristic consistently gave very good results

a hypergraph multi exchange heuristic for the single source capacitated facility location problem abstract in this paper we introduce a large scale neighborhood search procedure for solving the single source capacitated facility location problem sscflp the neighborhood structures are induced by innovative split multi customer multi exchanges where clusters of customers assigned to one facility can be moved simultaneously to multiple destination facilities and vice versa to represent these exchanges we use two types of improvement hypergraphs the improvement hypergraphs are built dynamically and the moving customers associated with each hyperedge are selected by solving heuristically a suitably defined mixed integer program we develop a hypergraph search framework including forward and backward procedures to identify improving solutions efficiently our proposed algorithm can obtain improving moves more quickly and even find better solutions than a traditional multi exchange heuristic ahuja orlin pallottino scuttella 2004 in addition when compared with the kernel search algorithm guastaroba speranza 2014 which at present is the most effective for solving sscflp our algorithm is not only competitive but can find better solutions or even the best known solution to some very large scale benchmark instances from the literature

efficient production distribution system design the production distribution system design is an integral part of the general supply chain design this paper proposes a novel solution methodology for this problem that is based on lagrangean relax

a lagrangean relax and cut approach for the two stage capacitated facility location problem abstract the two stage capacitated facility location problem is to find the optimal locations of depots to serve customers with a given demand the optimal assignment of customers to depots and the optimal product flow from plants to depots in order to solve this problem a lagrangean heuristic is proposed which is based on the relaxation of the capacity constraints the resulting lagrangean subproblem is an uncapacitated facility location problem with an aggregate capacity constraint and can be solved efficiently by branch and bound methods the best bound available from this relaxation is computed by means of a weighted dantzig wolfe decomposition approach feasible solutions are constructed from those of the lagrangean subproblems by applying simple reassignment procedures furthermore the relaxation is strengthened by adding and dualizing valid inequalities which are violated by the fractional primal solution of the dual master program computational results and comparisons with some other bounds and heuristics for this problem are presented

log majorization and complementary golden thompson type inequalities abstract we obtain a log majorization result for power means of positive semidefinite matrices this implies matrix norm inequalities for unitarily invariant norms which are considered as complementary to the golden thompson one other log majorization results are also obtained we give logarithmic trace inequalities and determinant inequalities as applications of our log majorizations

the multi level uncapacitated facility location problem is not submodular abstract unlike the uncapacitated facility location problem submodularity does not hold for the multi level uncapacitated facility location problem this point needs to be clarified as submodularity is assumed in some published papers and may lead to erroneous procedures to tackle the problem

optimizing over the split closure the polyhedron defined by all the split cuts obtainable directly i e without iterated cut generation from the lp relaxation p of a mixed integer program mip is termed the elementary or rank 1 split closure of p this paper deals with the problem of optimizing over the elementary split closure this is accomplished by repeatedly solving the following separation problem given a fractional point say x find a rank 1 split cut violated by x or show that none exists following caprara and letchford 17 we formulate this separation problem as a nonlinear mixed integer program that can be treated as a parametric mixed integer linear program pmilp with a single parameter in the objective function and the right hand side we develop an algorithmic framework to deal with the resulting pmilp by creating and maintaining a dynamically updated grid of parameter values and use the corresponding mixed integer programs to generate rank 1 split cuts our approach was implemented in the coin or framework using cplex 9 0 as a general purpose mip solver we report our computational results on well known benchmark instances from miplib 3 0 and several classes of structured integer and mixed integer problems our computational results show that rank 1 split cuts close more than 98 of the duality gap on 15 out of 41 mixed integer instances from miplib 3 0 more than 75 of the duality gap can be closed on an additional 10 instances the average gap closed over all 41 instances is 72 78 in the pure integer case rank 1 split cuts close more than 75 of the duality gap on 13 out of 24 instances from miplib 3 0 on average rank 1 split cuts close about 72 of the duality gap on these 24 instances we also report results on several classes of structured problems capacitated versions of warehouse location single source facility location p median fixed charge network flow multi commodity network design with splittable and unsplittable flows and lot sizing the fraction of the integrality gap closed varies for these problem classes between 100 and 67 we also gathered statistics on the average coefficient size absolute value of the disjunctions generated they turn out to be surprisingly small