an algorithm for global minimization of linearly constrained concave quadratic functions we present an algorithm for the global minimization of a quadratic function x y xtqx htx dty over a polytope x y rn k ax by b x 0 y 0 where x h rn y d rk b rm and where a and b are m n and m k matrices respectively and q is an n n symmetric positive definite matrix we first consider the case where k 0 and construct a tight parallelpiped r containing by using an arbitrary set of q conjugate directions and by solving 2n linear programs we show that the convex envelope of with respect to r is linear and obtain an explicit formula for it we then describe a branch and bound algorithm in which the linearity of the convex envelope allows efficient lower bounding for the subproblems for each subproblem we obtain the linear convex envelope of over a smaller parallelepiped r with facets parallel to those of r then we obtain a lower bound by minimizing this convex envelope over r if is the optimal objective value for each e 0 the algorithm generates a sequence of feasible points zj with nonincreasing function values j and a nondecreasing sequence j of lower bounds to moreover if e 0 there exist j0 such that j j e for all j j0 and if e 0 j j converges to 0 the results are then generalized to the case where k is nonzero and possibly much larger than n preliminary computational results are also presented

statistical mechanics of neocortical interactions canonical momenta indicatorsof electroencephalography a series of papers has developed a statistical mechanics of neocortical interactions smni deriving aggregate behavior of experimentally observed columns of neurons from statistical electrical chemical properties of synaptic interactions while not useful to yield insights at the single neuron level smni has demonstrated its capability in describing large scale properties of short term memory and electroencephalographic eeg systematics the necessity of including nonlinear and stochastic structures in this development has been stressed sets of eeg and evoked potential data were fit collected to investigate genetic predispositions to alcoholism and to extract brain signatures of short term memory adaptive simulated annealing asa a global optimization algorithm was used to perform maximum likelihood fits of lagrangians defined by path integrals of multivariate conditional probabilities canonical momenta indicators cmi are thereby derived for an individual s eeg data the cmi give better signal recognition than the raw data and can be used to advantage as correlates of behavioral states these results give strong quantitative support for an accurate intuitive picture portraying neocortical interactions as having common algebraic or physics mechanisms that scale across quite disparate spatial scales and functional or behavioral phenomena i e describing interactions among neurons columns of neurons and regional masses of neurons

mixed integer bilinear programming problems this paper addresses a class of problems called mixed integer bilinear programming problems these problems are identical to the well known bilinear programming problems with the exception that one set of variables is restricted to be binary valued and they arise in various production location allocation and distribution application contexts we first identify some special cases of this problem which are relatively more readily solvable even though their continuous relaxations are still nonconvex for the more general case we employ a linearization technique and design a composite lagrangian relaxation implicit enumeration cutting plane algorithm extensive computational experience is provided to test the efficacy of various algorithmic strategies and the effects of problem data on the computational effort of the proposed algorithm

a piecewise linearization framework for retail shelf space management models abstract managing shelf space is critical for retailers to attract customers and optimize profits this article develops a shelf space allocation optimization model that explicitly incorporates essential in store costs and considers space and cross elasticities a piecewise linearization technique is used to approximate the complicated nonlinear space allocation model the approximation reformulates the non convex optimization problem into a linear mixed integer programming mip problem the mip solution not only generates near optimal solutions for large scale optimization problems but also provides an error bound to evaluate the solution quality consequently the proposed approach can solve single category shelf space management problems with as many products as are typically encountered in practice and with more complicated cost and profit structures than currently possible by existing methods numerical experiments show the competitive accuracy of the proposed method compared with the mixed integer nonlinear programming shelf space model several extensions of the main model are discussed to illustrate the flexibility of the proposed methodology

statistical mechanics of neocortical interactions path integral evolution of short term memory previous papers in this series of statistical mechanics of neocortical interactions smni have detailed a development from the relatively microscopic scales of neurons up to the macroscopic scales as recorded by electroencephalography eeg requiring an intermediate mesocolumnar scale to be developed at the scale of minicolumns approx 10 sup 2 neurons and macrocolumns approx 10 sup 5 neurons opportunity was taken to view smni as sets of statistical constraints not necessarily describing specific synaptic or neuronal mechanisms on neuronal interactions on some aspects of short term memory stm e g its capacity stability and duration a recently developed c language code pathint provides a non monte carlo technique for calculating the dynamic evolution of arbitrary dimension subject to computer resources nonlinear lagrangians such as derived for the two variable smni problem here pathint is used to explicitly detail the evolution of the smni constraints on stm

semidefinite relaxations for quadratically constrained quadratic programming a review and comparisons at the intersection of nonlinear and combinatorial optimization quadratic programming has attracted significant interest over the past several decades a variety of relaxations for quadratically constrained quadratic programming qcqp can be formulated as semidefinite programs sdps the primary purpose of this paper is to present a systematic comparison of sdp relaxations for qcqp using theoretical analysis it is shown that the recently developed doubly nonnegative relaxation is equivalent to the shor relaxation when the latter is enhanced with a partial first order relaxation linearization technique these two relaxations are shown to theoretically dominate six other sdp relaxations a computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations while providing relaxation gaps averaging 3 as opposed to gaps of up to 19 for weaker relaxations on 700 randomly generated problems with up to 60 variables an sdp relaxation derived from lagrangian relaxation after the addition of redundant nonlinear constraints to the primal achieves gaps averaging 13 in a few cpu seconds

linearization strategies for a class of zero one mixed integer programming problems this paper is concerned with a new linearization strategy for a class of zero one mixed integer programming problems that contains quadratic cross product terms between continuous and binary variables and between the binary variables themselves this linearization scheme provides an equivalent mixed integer linear programming problem which yields a tighter continuous relaxation than that obtainable via the alternative linearization techniques available in the literature moreover the proposed technique provides a unifying framework in the sense that all the alternate methods lead to formulations that are accessible through appropriate surrogates of the constraints of the new linearized formulation extensions to various other types of mixed integer nonlinear programming problems are also discussed

feature selection in simple neurons how coding depends on spiking dynamics the relationship between a neuron s complex inputs and its spiking output defines the neuron s coding strategy this is frequently and effectively modeled phenomenologically by one or more linear filters that extract the components of the stimulus that are relevant for triggering spikes and a nonlinear function that relates stimulus to firing probability in many sensory systems these two components of the coding strategy are found to adapt to changes in the statistics of the inputs in such a way as to improve information transmission here we show for two simple neuron models how feature selectivity as captured by the spike triggered average depends on both the parameters of the model and the statistical characteristics of the input

a branch and cut algorithm for nonconvex quadratically constrained quadratic programming we present a branch and cut algorithm that yields in finite time a globally e optimal solution with respect to feasibility and optimality of the nonconvex quadratically constrained quadratic programming problem the idea is to estimate all quadratic terms by successive linearizations within a branching tree using reformulation linearization techniques rlt to do so four classes of linearizations cuts depending on one to three parameters are detailed for each class we show how to select the best member with respect to a precise criterion the cuts introduced at any node of the tree are valid in the whole tree and not only within the subtree rooted at that node in order to enhance the computational speed the structure created at any node of the tree is flexible enough to be used at other nodes computational results are reported that include standard test problems taken from the literature some of these problems are solved for the first time with a proof of global optimality

comprehension categories and the semantics of type dependency abstract a comprehension category is defined as a functor p e b satisfying a a cod p is a fibration and b f is cartesian in e implies that p f is a pullback in b this notion captures many structures which are used to describe type dependency like display map categories taylor 1986 hyland and pitts 1989 and lamarche 1988 categories with attributes cartmell 1978 and moggi 1991 d categories ehrhard 1988 and comprehensive fibrations pavlovic 1990 it also captures comprehension as occurring in topos theory and as described by lawvere s 1970 hyperdoctrines this paper is meant as an introduction to these comprehension categories a comprehension category will be called closed if it has appropriate dependent products and sums a few examples of closed comprehension categories will be described here more of them may be found in jacobs 1991 applications occur in jacobs 1991 and jacobs et al 1991

long range memory and non markov statistical effects in human sensorimotor coordination in this paper the non markov statistical processes and long range memory e ects in human sensorimotor coordination are investigated the theoretical basis of this study is the statistical theory of non stationary discrete non markov processes in complex systems phys rev e 62 6178 2000 the human sensorimotor coordination was experimentally studied by means of standard dynamical tappingtest on the g roup of 32 youngpeoples with tap numbers up to 400 this test was carried out separately for the right and the left hand according to the degree of domination of each brain hemisphere the numerical analysis of the experimental results was made with the help of power spectra of the initial time correlation function the memory functions of low orders and the 7rst three points of the statistical spectrum of non markovity parameter our observations demonstrate that with the regard to results of the standard dynamic tapping test it is possible to divide all examinees into 7ve di erent dynamic types we have introduced the con ict coe cient to estimate quantitatively the order disorder e ects underlying life systems the last one re ects the existence of disbalance between the nervous and the motor human coordination the suggested classi7cation of the neurophysiological activity represents the dynamic generalization of the well known neuropsychological types and provides the new approach in a modern neuropsychology c

scheduling under linear constraints abstract we introduce a parallel machine scheduling problem in which the processing times of jobs are not given in advance but are determined by a system of linear constraints the objective is to minimize the makespan i e the maximum job completion time among all feasible choices this novel problem is motivated by various real world application scenarios we discuss the computational complexity and algorithms for various settings of this problem in particular we show that if there is only one machine with an arbitrary number of linear constraints or there is an arbitrary number of machines with no more than two linear constraints or both the number of machines and the number of linear constraints are fixed constants then the problem is polynomial time solvable via solving a series of linear programming problems if both the number of machines and the number of constraints are inputs of the problem instance then the problem is np hard we further propose several approximation algorithms for the latter case

a symmetrical linear maxmin approach to disjoint bilinear programming the disjoint bilinear programming problem can be reformulated using two distinct linear maxmin programming problems there is a simple bijection between the optimal solutions of the reformulations and the bilinear problem moreover the number of local optima of the reformulations is less or equal to that of the bilinear problem in that sense the reformulations do not introduce additional complexity necessary optimality conditions complementarity and monotonicity of both reformulations are used to obtain a finitely convergent and exact branch and bound algorithm for the bilinear problem determining if the optimal value of a bilinear programming problem is bounded is shown to be strongly np complete the proposed algorithm may be used to answer this question

bidimensional packing by bilinear programming we consider geometric problems in which rectangles have to be packed into identical squares these problems turn out to be very hard in practice and ilp formulations in which variables specify the coordinates in the packing perform very poorly while most methods developed so far are based on simple geometric considerations a recent landmark result of fekete and schepers suggests to put these geometric aspects aside and use the most advanced tools for the one dimensional case in this paper we make additional progress in this direction especially on the basic question does a given set of rectangles fit into a square which turns out to be the bottleneck of all the approaches known given a set of rectangles and the associated convex hull of rectangle subsets that fit into a square we derive a wide class of valid inequalities for this convex hull from a complete description of the two knapsack polytopes associated with the widths and the heights of the rectangles respectively in addition we illustrate how to solve the associated separation problem as a bilinear program for which we develop a solution method that turns out to be fast in practice and show that the integer solutions that satisfy all these constraints generally correspond to vertices of the original convex hull for the benchmark instances in the literature the same tools are used to derive lower bounds for the two dimensional bin packing problem corresponding to the determination of an optimal pair of so called dual feasible functions these lower bounds in many cases equal those obtained by the customary set covering formulation for which column generation is very hard but are computable within a time that is smaller by some orders of magnitude this allows us to close a few of the benchmark instances in the literature

a decomposition algorithm for a discrete location allocation problem this paper considers a discrete location allocation problem for simultaneously determining the location of a given number of capacitated facilities on certain predesignated sites and the allocation of their products among a fixed number of customers the objective is to determine the location and subsequent allocation that minimizes the total cost of construction or location production and transportation the formulation of this model is a specially structured generally nonconvex problem however we specify certain sufficient conditions for the solution to two network structured linear programs to provide an optimal solution to this problem for other situations we present an implicit enumeration algorithm within the framework of benders decomposition method which fully exploits the structure of the problem this method is an improvement over traditional lagrangian approaches computational experience is provided to demonstrate the practicality of this algorithm for reasonable size problems as well as its superiority over alternative implicit enumeration schemes this paper also sheds light on the general implementation of benders decomposition technique

generalized bilinear programming part i models applications and linear programming relaxation abstract the evolution of the bilinear programming problem is reviewed and a new more general model is discussed the model involves two decision vectors and reduces to a linear program when one of the decision vectors is fixed the class of problems under consideration contains traditional bilinear programs general quadratic programs and bilinearly constrained and quadratically constrained extensions of these problems we describe how several important applications from the literature including the multiple modular design problem can be modeled as generalized bilinear programs finally we derive a linear programming relaxation that can be used as a subproblem in algorithmic solution schemes based on outer approximation and branch and bound

revisiting the categorical interpretation of dependent type theory we show that hofmann s and curien s interpretations of martin lof s type theory which were both designed to cure a mismatch between syntax and semantics in seely s original interpretation in locally cartesian closed categories are related via a natural isomorphism as an outcome we obtain a new proof of the coherence theorem needed to show the soundness after all of seely s interpretation

socp relaxation bounds for the optimal subset selection problem applied to robust linear regression abstract this paper deals with the problem of finding the globally optimal subset of h elements from a larger set of n elements in d space dimensions so as to minimize a quadratic criterion with an special emphasis on applications to computing the least trimmed squares estimator ltse for robust regression the computation of the ltse is a challenging subset selection problem involving a nonlinear program with continuous and binary variables linked in a highly nonlinear fashion the selection of a globally optimal subset using the branch and bound bb algorithm is limited to problems in very low dimension typically d 5 as the complexity of the problem increases exponentially with d we introduce a bold pruning strategy in the bb algorithm that results in a significant reduction in computing time at the price of a negligeable accuracy lost the novelty of our algorithm is that the bounds at nodes of the bb tree come from pseudo convexifications derived using a linearization technique with approximate bounds for the nonlinear terms the approximate bounds are computed solving an auxiliary semidefinite optimization problem we show through a computational study that our algorithm performs well in a wide set of the most difficult instances of the ltse problem

a simplicial branch and bound algorithm for solving quadratically constrained quadratic programs we propose a branch and bound algorithm for solving nonconvex quadratically constrained quadratic programs the algorithm is novel in that branching is done by partitioning the feasible region into the cartesian product of two dimensional triangles and rectangles explicit formulae for the convex and concave envelopes of bilinear functions over triangles and rectangles are derived and shown to be second order cone representable the usefulness of these new relaxations is demonstrated both theoretically and computationally

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

concavity cuts for disjoint bilinear programming we pursue the study of concavity cuts for the disjoint bilinear programming problem this optimization problem has two equivalent symmetric linear maxmin reformulations leading to two sets of concavity cuts we first examine the depth of these cuts by considering the assumptions on the boundedness of the feasible regions of both maxmin and bilinear formulations we next propose a branch and bound algorithm which make use of concavity cuts we also present a procedure that eliminates degenerate solutions extensive computational experiences are reported sparse problems with up to 500 variables in each disjoint sets and 100 constraints and dense problems with up to 60 variables again in each sets and 60 constraints are solved in reasonable computing times

a new technique for generating quadratic programming test problems this paper describes a new technique for generating convex strictly concave and indefinite bilinear or not quadratic programming problems these problems have a number of properties that make them useful for test purposes for example strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an exponential number of local minima with distinct function values and indefinite and jointly constrained bilinear problems with nonextreme global minima can be generated unlike most existing methods our construction technique does not require the solution of any subproblems or systems of equations in addition the authors know of no other technique for generating jointly constrained bilinear programming problems

a probabilistic model for minmax regret in combinatorial optimization in this paper we propose a new probabilistic model for minimizing the anticipated regret in combinatorial optimization problems with distributional uncertainty in the objective coefficients the interval uncertainty representation of data is supplemented with information on the marginal distributions as a decision criterion we minimize the worst case conditional value at risk of regret the proposed model includes the interval data minmax regret model as a special case for the class of combinatorial optimization problems with a compact convex hull representation a polynomial sized mixed integer linear program is formulated when a the range and mean are known and b the range mean and mean absolute deviation are known and a mixed integer second order cone program is formulated when c the range mean and standard deviation are known for the subset selection problem of choosing k elements of maximum total weight out of a set of n elements the probabilistic regret model is shown to be solvable in polynomial time in the instances a and b above this extends the current known polynomial complexity result for minmax regret subset selection with range information only

white noise susceptibility and critical slowing in neurons near spiking threshold we present mathematical and simulation analyses of the below threshold noisy response of two biophysically motivated models for excitable membrane due to h r wilson a squid axon resonator and a human cortical neuron integrator when stimulated with a low intensity white noise superimposed on a dc control current both membrane types generate voltage fluctuations that exhibit critical slowing down that is the voltage responsiveness to noisy input currents grows in amplitude while slowing in frequency as the membrane approaches spiking threshold from below we define threshold unambiguously as that dc current that renders a zero real eigenvalue for the jacobian matrix for the integrator neuron and for the resonator neuron as the dc current that gives a complex eigenvalue pair whose real part is zero using a linear ornstein uhlenbeck analysis we give exact small noise expressions for the variance power spectrum and correlation function of the voltage fluctuations and we derive the scaling laws for the divergence of susceptibility and correlation times for approach to threshold we compare these predictions with numerical simulations of the nonlinear stochastic equations and demonstrate that provided the white noise perturbations are kept sufficiently small the linearized theory works well these predictions should be testable in the laboratory using a current clamped cell configuration if confirmed then the proximity of a neuron to its spike transition point can be judged by measuring its subthreshold susceptibility to white noise stimulation we postulate that such temporally correlated fluctuations could provide a means of subthreshold signaling via gap junction connections with neighboring neurons

low nonconvexity rank bilinear matrix inequalities algorithms and applications in robust controller and structure designs a branch and bound bb algorithm for solving a general class of bilinear matrix inequality bmi problems is proposed first linear matrix inequality lmi constraints are incorporated into bmi constraints in a special way to take advantage of useful information on nonconvex terms then the nonconvexity of the bmi is centralized in coupling constraints so that when the latter are omitted we get a relaxed lmi problem for computing lower bounds as in our previous developments the branching is performed in a reduced dimensional space of complicating variables this makes the approach practical even with a large dimension of overall variables applications of the algorithm to several test problems of robust control are discussed

subthreshold dynamics of a single neuron from a hamiltonian perspective we use hamilton s equations of classical mechanics to investigate the behavior of a cortical neuron on the approach to an action potential we use a two component dynamic model of a single neuron due to wilson with added noise inputs we derive a lagrangian for the system from which we construct hamilton s equations the conjugate momenta are found to be linear combinations of the noise input to the system we use this approach to consider theoretically and computationally the most likely manner in which such a modeled neuron approaches a firing event we find that the firing of a neuron is a result of a drop in inhibition due to a temporary increase in negative bias of the mean noise input to the inhibitory control equation moreover we demonstrate through theory and simulation that on average the bias in the noise increases in an exponential manner on the approach to an action potential in the hamiltonian description an action potential can therefore be considered a result of the exponential growth of the conjugate momenta variables pulling the system away from its equilibrium state into a nonlinear regime

the identity type weak factorisation system we show that the classifying category c t of a dependent type theory t with axioms for identity types admits a non trivial weak factorisation system we provide an explicit characterisation of the elements of both the left class and the right class of the weak factorisation system this characterisation is applied to relate identity types and the homotopy theory of groupoids

jointly constrained bilinear programs and related problems an overview abstract linear complementarity quadratic programming and separably constrained bilinear programming problems are all special cases of jointly constrained bilinear programs in this paper we review current theory and procedures for bilinear programs and related problems the key notion of convex envelopes that drives the general procedure is explored and some minor extensions to known results are presented extensions and directions for future research are also discussed

design of planar articulated mechanisms using branch and bound this paper considers an optimization model and a solution method for the design of two dimensional mechanical mechanisms the mechanism design problem is modeled as a nonconvex mixed integer program which allows the optimal topology and geometry of the mechanism to be determined simultaneously the underlying mechanical analysis model is based on a truss representation allowing for large displacements for mechanisms undergoing large displacements elastic stability is of major concern we derive conditions modeled by nonlinear matrix inequalities which guarantee that a stable equilibrium is found and that buckling is prevented the feasible set of the design problem is described by nonlinear differentiable and non differentiable constraints as well as nonlinear matrix inequalities to solve the mechanism design problem a branch and bound method based on convex relaxations is developed to guarantee convergence of the method two different types of convex relaxations are derived the relaxations are strengthened by adding valid inequalities to the feasible set and by solving bound contraction sub problems encouraging computational results indicate that the branch and bound method can reliably solve mechanism design problems of realistic size to global optimality

global optimization of water distribution networks through a reduced space branch and bound search a branch and bound approach to the problem of optimal design of water distribution networks is presented global optimum is reached through the generation of convergent sequences of upper and lower bounds the relaxations responsible for the lower bounds correspond to linear programming problems formulated through the enlargement of the original feasible region by outer approximations of the constraints although formulated within an arc based framework the proposed scheme does not apply the branching process to all flow variables by utilizing the mass conservation principle only a reduced set of variables is assigned for branching the algorithm was applied to three variants of a classical problem from the literature comparisons with previous results indicates a faster convergence to the optimum fewer linear programming problems solved in several situations

jointly constrained biconvex programming this paper presents a branch and bound algorithm for minimizing the sum of a convex function in x a convex function in y and a bilinear term in x and y over a closed set such an objective function is called biconvex with biconcave functions similarly defined the feasible region of this model permits joint constraints in x and y to be expressed the bilinear programming problem becomes a special case of the problem addressed in this paper we prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point the algorithm is proven to converge to a global solution of the nonconvex program we discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs for which the algorithm has been implemented

the inverse optimal value problem this paper considers the following inverse optimization problem given a linear program a desired optimal objective value and a set of feasible cost vectors determine a cost vector such that the corresponding optimal objective value of the linear program is closest to the desired value the above problem referred here as the inverse optimal value problem is significantly different from standard inverse optimization problems that involve determining a cost vector for a linear program such that a pre specified solution vector is optimal in this paper we show that the inverse optimal value problem is np hard in general we identify conditions under which the problem reduces to a concave maximization or a concave minimization problem we provide sufficient conditions under which the associated concave minimization problem and correspondingly the inverse optimal value problem is polynomially solvable for the case when the set of feasible cost vectors is polyhedral we describe an algorithm for the inverse optimal value problem based on solving linear and bilinear programming problems some preliminary computational experience is reported

univariate parameterization for global optimization of mixed integer polynomial problems abstract this paper presents a new relaxation technique to globally optimize mixed integer polynomial programming problems that arise in many engineering and management contexts using a bilinear term as the basic building block the underlying idea involves the discretization of one of the variables up to a chosen accuracy level teles j p castro p m matos h a 2013 multiparametric disaggregation technique for global optimization of polynomial programming problems j glob optim 55 227 251 by means of a radix based numeric representation system coupled with a residual variable to effectively make its domain continuous binary variables are added to the formulation to choose the appropriate digit for each position together with new sets of continuous variables and constraints leading to the transformation of the original mixed integer non linear problem into a larger one of the mixed integer linear programming type the new underestimation approach can be made as tight as desired and is shown capable of providing considerably better lower bounds than a widely used global optimization solver for a specific class of design problems involving bilinear terms

functional graphical models graphical models have attracted increasing attention in recent years especially in settings involving high dimensional data in particular gaussian graphical models are used to model the conditio

linear quadratic and bilinear programming approaches to the linear complementarity problem abstract traditional pivoting procedures for solving the linear complementarity problem can only guarantee convergence for problems having well defined structures recently optimization procedures based on linear quadratic and bilinear programming have been developed to extend the class of problems that can be solved efficiently these latter approaches are the focus of this paper the strengths and weaknesses of each of the approaches are discussed the linear programming approach advanced by mangasarian is the most efficient once an appropriate objective function is found this requires the solution of a system of linear and bilinear equations that is easily solvable only in some cases extensions to this approach due to shiau show some promise but are still limited to special cases of the general problem the quadratic programming approaches discussed here are restricted to specialized procedures for the complementarity problem one proposed by cheng is based on the levitin poljak gradient projection method and the other due to cirina is based on karush kuhn tucker theory both are only successful on some problems the two bilinear programming algorithms discussed are the most general for any problem they are guaranteed to find at least one solution or conclude that none exist one is a specialization of the recent biconvex programming algorithm of al khayyal and falk and the other is an entirely new implicit enumeration procedure

life space foam a medium for motivational and cognitive dynamics abstract general stochastic dynamics developed in a framework of feynman path integrals have been applied to lewinian field theoretic psychodynamics k lewin field theory in social science university of chicago press chicago 1951 k lewin resolving social conflicts and field theory in social science american psychological association washington 1997 m gold a kurt lewin reader the complete social scientist american psychological association washington 1999 resulting in the development of a new concept of life space foam lsf as a natural medium for motivational and cognitive psychodynamics according to lsf formalisms the classic lewinian life space can be macroscopically represented as a smooth manifold with steady force fields and behavioral paths while at the microscopic level it is more realistically represented as a collection of wildly fluctuating force fields loco motion paths and local geometries and topologies with holes a set of least action principles is used to model the smoothness of global macro level lsf paths fields and geometry to model the corresponding local micro level lsf structures an adaptive path integral is used defining a multi phase and multi path multi field and multi geometry transition process from intention to goal driven action application examples of this new approach include but are not limited to information processing motivational fatigue learning memory and decision making

solving mixed integer bilinear problems using milp formulations in this paper we examine a mixed integer linear programming reformulation for mixed integer bilinear problems where each bilinearterm involves the product of a nonnegative integer variable and a nonnegative continuous variable this reformulation is obtained by first replacing a general integer variable with its binary expansion and then using mccormick envelopes to linearize the resulting product of continuous and binary variables we present the convex hull of the underlying mixed integer linear set the effectiveness of this reformulation and associated facet defining inequalities are computationally evaluated on five classes of instances

note on solving linear complementarity problems as jointly constrained bilinear programs abstract we investigate refinements of an existing nonconvex programming algorithm that exploits special structure of linear complementarity problems we construct a linear program having an n n working basis which is equivalent to the subproblems generated by the algorithm that generally require 3n 3n basis matrices

two stage quadratic integer programs with stochastic right hand sides we consider two stage quadratic integer programs with stochastic right hand sides and present an equivalent reformulation using value functions we propose a two phase solution approach the first phase constructs value functions of quadratic integer programs in both stages the second phase solves the reformulation using a global branch and bound algorithm or a level set approach we derive some basic properties of value functions of quadratic integer programs and utilize them in our algorithms we show that our approach can solve instances whose extensive forms are hundreds of orders of magnitude larger than the largest quadratic integer programming instances solved in the literature

robust nonparametric tests of general linear model coefficients a comparison of permutation methods and test statistics statistical inference in neuroimaging research often involves testing the significance of regression coefficients in a general linear model in many applications the researcher assumes a model of the form y x z e where y is the observed brain signal and x and z contain explanatory variables that are thought to be related to the brain signal the goal is to test the null hypothesis h0 0 with the nuisance parameters included in the model several nonparametric permutation methods have been proposed for this problem and each method uses some variant of the f ratio as the test statistic however recent research suggests that the f ratio can produce invalid permutation tests of h0 0 when the e terms are heteroscedastic i e have non constant variance which can occur for a variety of reasons this study compares the classic f test statistic to the robust w wald test statistic using eight different permutation methods the results reveal that permutation tests using the f ratio can produce accurate results when the errors are homoscedastic but high false positive rates when the errors are heteroscedastic in contrast permutation tests using the w test statistic produced valid results when the errors were homoscedastic and asymptotically valid results when the errors were heteroscedastic in the situation with homoscedastic errors permutation tests using the w statistic showed slightly reduced power compared to the f statistic but the difference disappeared as the sample size n increased consequently the w test statistic is recommended for robust nonparametric hypothesis tests of regression coefficients in neuroimaging research

on the choice of step size in subgradient optimization abstract this paper recommends some procedures for the selection of step sizes in the context of subgradient optimization the first of these procedures is developed in detail in this study and is a theoretically convergent scheme this method has two phases the first phase is designed to accelerate the solution procedure towards an optimal solution while the second phase helps to close in on an optimal solution a second technique recommended is a simple minded scheme which although not theoretically convergent seems to be computationally very efficient these two methods are shown to compare favorably with held wolfe and crowder s scheme for prescribing step sizes we also suggest some modifications of the latter scheme to make it computationally more efficient

global minimization of large scale constrained concave quadratic problems by separable programming the global minimization of a large scale linearly constrained concave quadratic problem is considered the concave quadratic part of the objective function is given in terms of the nonlinear variablesx r n while the linear part is in terms ofy r k for large scale problems we may havek much larger thann the original problem is reduced to an equivalent separable problem by solving a multiple cost row linear program with 2n cost rows the solution of one additional linear program gives an incumbent vertex which is a candidate for the global minimum and also gives a bound on the relative error in the function value of this incumbent ana priori bound on this relative error is obtained which is shown to be 0 25 in important cases if the incumbent is not a satisfactory approximation to the global minimum a guaranteede approximate solution is obtained by solving a single liner zero one mixed integer programming problem this integer problem is formulated by a simple piecewise linear underestimation of the separable problem

simultaneous convexification of bilinear functions over polytopes with application to network interdiction we study the simultaneous convexification of graphs of bilinear functions g k x y y intercal a k x over x in ensuremath xi x in 0 1 n e x geq f and y in delta m y in r m 1 intercal y leq 1 we propose a constructive procedure to obtain a linear description of the convex hull of the resulting set this procedure can be applied to derive convex and concave envelopes of certain bilinear functions to study unary expansions of integer variables in mixed integer bilinear sets and to obtain convex hulls of sets with complementarity constraints exploiting the structure of xi the procedure naturally yields stronger linearizations for bilinear terms in a variety of practical settings in particular we demonstrate the effectiveness of the approach by strengthening the traditional dual formulation of network interdiction problems and report encouraging preliminary numerical results

generalized bilinear programming an application in farm management abstract in this paper we analyze a model which arises from considering manure and fertilizer aspects in the farm management modelling for cattle husbandry modelling these aspects may lead to a multi extremal problem with bilinear terms solution techniques for this problem are discussed and a specific branch and bound procedure is outlined an example illustrates the multi extremal structure of the problem and the solution methods mentioned

mixed integer quadratic programming this paper considers mixed integer quadratic programs in which the objective function is quadratic in the integer and in the continuous variables and the constraints are linear in the variables of both types the generalized benders decomposition is a suitable approach for solving such programs however the program does not become more tractable if this method is used since benders cuts are quadratic in the integer variables a new equivalent formulation that renders the program tractable is developed under which the dual objective function is linear in the integer variables and the dual constraint set is independent of these variables benders cuts that are derived from the new formulation are linear in the integer variables and the original problem is decomposed into a series of integer linear master problems and standard quadratic subproblems the new formulation does not introduce new primary variables or new constraints into the computational steps of the decomposition algorithm

exploiting special structures in constructing a hierarchy of relaxations for 0 1 mixed integer problems a new hierarchy of relaxations is presented that provides a unifying framework for constructing a spectrum of continuous relaxations spanning from the linear programming relaxation to the convex hull representation for linear mixed integer 0 1 problems this hierarchy is an extension of the reformulation linearization technique rlt of sherali and adams 1990 1994a and is particularly designed to exploit special structures specifically inherent special structures are exploited by identifying particular classes of multiplicative factors that are applied to the original problem to reformulate it as an equivalent polynomial programming problem and subsequently this resulting problem is linearized to produce a tighter relaxation in a higher dimensional space this general framework permits us to generate an explicit hierarchical sequence of tighter relaxations leading up to the convex hull representation a similar hierarchy can be constructed for polynomial mixed integer 0 1 problems additional ideas for further strengthening rlt based constraints by using conditional logical implications as well as relationships with sequential lifting are also explored several examples are presented to demonstrate how underlying special structures including generalized and variable upper bounding covering partitioning and packing constraints as well as sparsity can be exploited within this framework for some types of structures low level relaxations are exhibited to recover the convex hull of integer feasible solutions

a note on the solution of bilinear programming problems by reduction to concave minimization we present a new algorithm for solving the bilinear programming problem by reduction to concave minimization this algorithm is finite does not assume the boundedness of the constraint set and uses an efficient procedure for checking whether a concave function is bounded below on a given halfline some preliminary computational experience with a computer code for implementing the algorithm on a microcomputer is also reported

accelerating benders decomposition algorithmic enhancement and model selection criteria this paper proposes methodology for improving the performance of benders decomposition when applied to mixed integer programs it introduces a new technique for accelerating the convergence of the algorithm and theory for distinguishing good model formulations of a problem that has distinct but equivalent mixed integer programming representations the acceleration technique is based upon selecting judiciously from the alternate optima of the benders subproblem to generate strong or pareto optimal cuts this methodology also applies to a much broader class of optimization algorithms that includes dantzig wolfe decomposition for linear and nonlinear programs and related cutting plane type algorithms that arise in resource directive and price decomposition when specialized to network location problems this cut generation technique leads to very efficient algorithms that exploit the underlying structure of these models in discussing the proper formulation of mixed integer programs we suggest criteri