the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

reachable set of operators on controlled open quantum systems in a markovian environment abstract in this article we propose a framework studying all possible operations one can generate at any given time on quantum systems in a markovian environment a complete characterization of the operations one can simulate at any given time on a qubit in a unital markovian environment is given and possible extensions on general cases are discussed

a filled function method for finding a global minimizer on global integer optimization the paper gives a definition of the filled function for nonlinear integer programming this definition is modified from that of the global convexized filled function for continuous global optimization a filled function with only one parameter which satisfies this definition is presented we also discuss the properties of the proposed function and give a filled function method to solve the nonlinear integer programming problem the implementation of the algorithm on several test problems is reported with satisfactory numerical results

the collapse transition of linear polymers on fractal lattices the authors show that for linear polymers a collapse transition exists on several fractal lattices and obtain exact results for the critical exponents at this transition a rod like phase is found in some cases at intermediate temperatures between the swollen phase and the collapsed phase they introduce infinitesimal recursion relations with correlation function rescaling as the formal limit of a class of fractals which give a better approximation to euclidean 2d lattices the gyration radius exponent at the transition temperature lies in the range nu 1 0 546 or 0 010 in good agreement with a recent transfer matrix calculation the possible relevance of anisotropy at the collapse transition is discussed

free energy of magnetic model systems at the fractal to euclidian crossover abstract we prove that the free energy per spin of the nearest neighbour classical n vector spin model situated on the sierpinski gasket sg family of fractals tends to the free energy per spin of the same model situated on the standard triangular lattice when the linear size b of the sg fractal generator tends to infinity this finding can be regarded as the first step in the analytical investigation of the spin system thermodynamics at the fractal to euclidean crossover

controllability properties of affine systems this paper deals with transitivity controllability of affine families of vector fields on a finite dimensional vector space v in particular we focus on affine families whose corresponding families of linear fields are transitive on v 0 and which in addition have no fixed points in v we show that such families are necessarily transitive on v and we also show that they remain transitive under small perturbations in general however affine families need not remain transitive under small perturbations for example small affine perturbations of transitive linear systems are not necessarily transitive since any affine system mathcal f naturally defines a system mathcal f r of right invariant vector fields on the semi direct product of v with gl v we also investigate transitivity properties mathcal f r our result is that if mathcal f is an affine family satisfying the preceding conditions then mathcal f r generates the full lie algebra on the semi direct product

time minimal control of dissipative two level quantum systems the generic case the objective of this article is to complete preliminary results from the work of bonnard and sugny 2009 and sugny et al 2007 concerning the time minimal control of dissipative two level quantum systems whose dynamics is governed by the lindblad equation the extremal system is described by a 3d hamiltonian depending upon three parameters we combine geometric techniques with numerical simulations to deduce the optimal solutions

controllability properties for finite dimensional quantum markovian master equations various notions from geometric control theory are used to characterize the behavior of the markovian master equation for n level quantum mechanical systems driven by unitary control and to describe the structure of the sets of reachable states it is shown that the system can be accessible but neither small time controllable nor controllable in finite time in particular if the generators of quantum dynamical semigroups are unital then the reachable sets admit easy characterizations as they monotonically grow in time the two level case is treated in detail

characterization of majorization monotone quantum dynamics in this technical note the author studies the dynamics of open quantum system in markovian environment the author gives necessary and sufficient conditions for such dynamics to be majorization monotone which are those dynamics always mixing the states

critical exponents of the self avoiding walks on a family of finitely ramified fractals the authors have studied the self avoiding walks saw on a family of finitely ramified fractals the first member b 2 of the family is the two dimensional sierpinski gasket while the last member b infinity appears to be a wedge of the homogeneous triangular lattice by means of the exact renormalisation group transformations they have calculated the critical exponents alpha nu and gamma and the connectivity constant mu of saw on each member of a sequence 2 or b or 8 of the studied fractal family the obtained exact results are compared with the recent phenomenological proposals and with the results believed to be exact in the case of a homogeneous two dimensional lattice

a filled function method for finding a global minimizer of a function of several variables the concept of a filled function is introduced we construct a particular filled function and analyze its properties an algorithm for global minimization is generated based on the concept and properties of the filled function some typical examples with 1 to 10 variables are tested and computational results show that in most cases this algorithm works better than the tunneling algorithm the advantages and disadvantages are analyzed and further research directions are discussed

phase transitions on fractals i quasi linear lattices magnetic spin models and resistor networks are studied on certain self similar fractal lattices which are described as quasi linear because they share a significant property of the line finite portions can be isolated from the rest by removal of two points sites in all cases there is no long range order at finite temperature the transition at zero temperature has a discontinuity in the magnetisation and the associated magnetic exponent is equal to the fractal dimensionality d when the lattice reduces to a non branching curve the thermal exponent v 1 y is equal to d when the lattice is a branching curve y is related respectively to the dimensionality of the single channel segments of the curve for the ising model or to the exponent describing the resistivity for models with continuous spin symmetry

a class of generalized filled functions with improved computability the filled function method ffm is an approach to find the global minimum of multimodal functions the computability of conventional filled functions is limited as they are defined on either exponential or logarithmic terms this paper proposes a new filled function that does not have such disadvantages numerical experiments on typical testing functions showed that the new function is superior to the conventional one

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

modeling and control of quantum systems an introduction the scope of this work is to provide a self contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems as well as a brief survey on the main approaches to control synthesis while part of the existing theory especially in the open loop setting stems directly from classical control theory most notably geometric control and optimal control a number of tools specifically tailored for quantum systems have been developed since the 1980s in order to take into account their distinctive features the probabilistic nature of atomic scale physical systems the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback design problems the relevant dynamical models for both closed and open quantum systems are presented along with the main results on their controllability and stability a brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies

transfer matrix studies of branching fractals abstract we introduce the method of transfer matrices to study physical models defined on branching fractal lattices in this context we derive matricial recurrence relations connecting the matrices associated with successive fractal generations these relations depend only on the geometry of the fractal and may be used for any model explicit results are obtained for ferromagnetic and antiferromagnetic ising models on branching koch curves and sierpinski gasket

transfer matrix solution of the ising model on the koch curve a transfer matrix approach is used to calculate the partition function of the ising model on the koch curve in zero magnetic field it is possible to obtain an exact analytic result in the presence of a field the problem becomes more difficult a procedure based on an expansion about h 0 is developed which shows that the spontaneous magnetisation vanishes identically and which makes it possible to give the zero field susceptibility as a power series in a small temperature dependent parameter

is measurement based feedback still better for quantum control systems in this paper we put forward a fundamental question concerning feedback control of quantum systems is measurement based feedback control still better than open loop control in contrast to the classical control theory the answer is far from obvious this is because measurement based feedback needs measurement to reduce the system uncertainty whereas the measurement on a quantum system will inevitably increase the system uncertainty in turn in fact there is a complicated tradeoff between the uncertainty introduced and the information gained by the measurement on a quantum system to investigate this fundamental problem we will only focus on a typical model of coherent control mode with and without the decoherence term in the paper by establishing some fundamental limits on the performances of both the open loop and measurement based feedback controls we will demonstrate via simulation that the measurement based feedback control of quantum systems is still superior to the open loop control in some sense for the typical model under consideration

reachable set of bilinear control systems with time varying drift in this article we give a complete characterization of the reachable set at all times for a class of bilinear control systems with time varying drift and unbounded control amplitude these results are of fundamental interest in geometric control theory and have important applications to control of coupled spins in solid state nmr spectroscopy

an approximate algorithm for nonlinear integer programming abstract this paper suggests an approximate algorithm designed to solve nonlinear integer problems this algorithm belongs to the class of component algorithms of feasible integer directions the search for a feasible integer direction is done on the basis of a linear approximation of the objective function and the constraints at the integer points under consideration theoretical analysis is presented as well as experimental investigation using the algorithm for test examples taken from the literature

impact of positivity and complete positivity on accessibility of markovian dynamics we consider a two dimensional quantum control system evolving under an entropy increasing irreversible dynamics in the semigroup form considering a phenomenological approach to the dynamics we show that the accessibility property of the system depends on whether its evolution is assumed to be positive or completely positive in particular we characterize the family of maps having different accessibilities and show the impact of that property on observable quantities by means of a simple physical model

exact renormalization group treatment of the piecewise directed random walks on fractals abstract an exact evaluation of the critical exponent v associated with the mean squared end to end distance of the piecewise directed random walks is presented in the case of the complete family of the sierpinski gasket type of fractals

finding discrete global minima with a filled function for integer programming the filled function method is an approach to find the global minimum of multidimensional functions this paper proposes a new definition of the filled function for integer programming problem a filled function which satisfies this definition is presented furthermore we discuss the properties of the filled function and design a new filled function algorithm numerical experiments on several test problems with up to 50 integer variables have demonstrated the applicability and efficiency of the proposed method

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

a critical review of discrete filled function methods in solving nonlinear discrete optimization problems many real life problems can be modeled as nonlinear discrete optimization problems such problems often have multiple local minima and thus require global optimization methods due to high complexity of these problems heuristic based global optimization techniques are usually required when solving large scale discrete optimization or mixed discrete optimization problems one of the more recent global optimization tools is known as the discrete filled function method nine variations of the discrete filled function method in literature are identified and a review on theoretical properties of each method is given some of the most promising filled functions are tested on various benchmark problems numerical results are given for comparison

the globally convexized filled functions for global optimization a new filled function the globally convexized filled function is proposed the value of this function tends to positive infinity as the norm of the variable vector tends to infinity and it has no stationary point in the region in which the objective function value is greater than or equal to the least value found so far but it does have a minimizer not only along the radius direction from the minimizer with the least objective function value found so far like filled function found earlier by ge in the region in which the objective function value is less than the least value found so far but with a prefixed minimizer how to adjust parameters in the filled function is considered such a filled function can be appropriately combined with stochastic stopping rules proposed by boender rinnooy kan and timmer the computational results show that this algorithm is quite efficient and reliable

a new discrete filled function method for finding global minimizer of the integer programming in this paper a new discrete filled function method is proposed for finding a global minimizer of integer programming problems only one parameter is included in the proposed filled function and it does not need to be adjusted further when it is taken as large as possible moreover the current local minimizer obtained by minimizing the proposed filled function will be one of the local minimizers of the original problem and it is better than the minimizers found previously thus it is not necessary to use a local search method to the original function as a result the computation cost of the proposed discrete filled function method is relatively low numerical results demonstrate the effectiveness of the proposed method

frustration and correlations in fractals abstract a technique for obtaining exact thermodynamic and correlation functions of regular fractals is illustrated by a development for the ising system on the triangular sierpinsky gasket a generalised hyperscaling statement is obtained the technique is then applied to the fully frustrated antiferromagnetic case where the exact ground state entropy per spin is shown to be s 0 0 4930 this result shows that the introduction of regularly disposed holes into the triangular lattice does not relieve frustration as is commonly supposed relations completely determining the correlation functions are obtained that for the zero field pair correlation function of vertex spins at separation r is g 1 2 r g 2 r g 1 together with a relation which sets the correlation length scale exact asymptotic expressions are derived for ferromagnetic correlations in both zero and non zero field the zero field zero temperature correlation length of the fully frustrated system is shown to be 0 79 representing extremely rapid decay of correlations

a continuous approach to nonlinear integer programming this paper is concerned with nonlinear integer programming problems unconstrained and constrained as well as mixed the problems are transformed into nonlinear global optimization problems and then solved by the filled function transformation method this approach should be efficient since it has been shown that the filled function transformation method is efficient in solving global optimization problems with large numbers up to 30 2 5 in the present examples of local minimizers however the sense of efficient as having polynomial complexity in the worst or average case is not suitable for nonlinear integer programming problems since the complexity of a nonlinear programming algorithm is in general nonpolynomial

criteria for reachability of quantum states we address the question of which quantum states can be inter converted under the action of a time dependent hamiltonian in particular we consider the problem as applied to mixed states and investigate the difference between pure and mixed state controllabilities introduced in previous work we provide a complete characterization of the eigenvalue spectrum for which the state is controllable under the action of the symplectic group we also address the problem of which states can be prepared if the dynamical lie group is not sufficiently large to allow the system to be controllable

time minimal control of dissipative two level quantum systems the integrable case the objective of this article is to apply recent developments in geometric optimal control to analyze the time minimum control problem of dissipative two level quantum systems whose dynamics is governed by the lindblad equation we focus our analysis on the case where the extremal hamiltonian is integrable

controllability of quantum mechanical systems by root space decomposition of su n the controllability property of the unitary propagator of an n level quantum mechanical system subject to a single control field is described using the structure theory of semisimple lie algebras sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases

several filled functions with mitigators several filled functions were reported to seek the global minimum of multimodal functions of multiple variables this paper proposes an alternative formulation that may reduce the negative definite effect of the hessian of a filled function proposed before furthermore a class of mitigators is defined and applied to improve the computational characteristics of filled functions results of numerical experiments on typical testing functions are also reported