bounding the gap between the mccormick relaxation and the convex hull for bilinear functions we investigate how well the graph of a bilinear function b 0 1 n rightarrow mathbb r b 0 1 n r can be approximated by its mccormick relaxation in particular we are interested in the smallest number c such that the difference between the concave upper bounding and convex lower bounding functions obtained from the mccormick relaxation approach is at most c times the difference between the concave and convex envelopes answering a question of luedtke namazifar and linderoth we show that this factor c cannot be bounded by a constant independent of n more precisely we show that for a random bilinear function b we have asymptotically almost surely c geqslant sqrt n 4 c n 4 on the other hand we prove that c leqslant 600 sqrt n c 600n which improves the linear upper bound proved by luedtke namazifar and linderoth in addition we present an alternative proof for a result of misener smadbeck and floudas characterizing functions b for which the mccormick relaxation is equal to the convex hull

convex extensions and envelopes of lower semi continuous functions we define a convex extension of a lower semi continuous function to be a convex function that is identical to the given function over a pre specified subset of its domain convex extensions are not necessarily constructible or unique we identify conditions under which a convex extension can be constructed when multiple convex extensions exist we characterize the tightest convex extension in a well defined sense using the notion of a generating set we establish conditions under which the tightest convex extension is the convex envelope then we employ convex extensions to develop a constructive technique for deriving convex envelopes of nonlinear functions finally using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of x y over a rectangle and prove that the extensions theory provides convex relaxations that are much tighter than the relaxation provided by the classical outer linearization of bilinear terms

explicit hybrid model predictive control this article presents an algorithm for the exact solution of explicit hybrid model predictive control problems of time invariant discrete time mixed logical dynamical systems using multiparametric programming this control problem is formulated as a multiparametric mixed integer quadratic programming problem where the initial states of the system are treated as parameters in conjunction with decomposition type or branch and bound type approaches the proposed solution strategy first creates affine relaxations of nonconvex critical regions that result from the comparison of two quadratic objective functions these relaxations are then used to generate an affine outer approximation for the critical regions upon termination the bounded space of the initial states is partitioned into possibly nonconvex critical regions and corresponding optimal control laws

identification of transport coefficient models in convection diffusion equations a rigorous method is presented for the systematic identification of the structure and the parameters of transport coefficient models in three dimensional transient convection diffusion systems using high resolution measurement data the transport is represented by a convection term with known convective velocity and a diffusion term with an unknown generally state dependent transport coefficient the identification of a transport coefficient model constitutes an ill posed highly nonlinear inverse problem in our previous work karalashvili et al siam j sci comput 30 2008 pp 3249 3269 we presented a novel incremental identification method which decomposes this inverse problem into easier to handle inverse subproblems this way the incremental identification method not only allows for the identification of the structure and the parameters of the model but also supports the rigorous decision making on the best suited transport model structure due to the decomposition approach the identified transport model structure and parameters are subject to errors to cope with the error propagation inherent in the incremental method the present work suggests a model correction procedure as a supplement to the incremental identification method of our previous work which results in a transport model of higher precision the correction refers to both the model structure and parameters no a priori knowledge on the unknown transport model structure is necessary the identification approach is numerically illustrated for a three dimensional transient convection diffusion equation which has its origin in the modeling and simulation of energy transport in a laminar wavy film flow

convex envelopes generated from finitely many compact convex sets we consider the problem of constructing the convex envelope of a lower semi continuous function defined over a compact convex set we formulate the envelope representation problem as a convex optimization problem for functions whose generating sets consist of finitely many compact convex sets in particular we consider nonnegative functions that are products of convex and component wise concave functions and derive closed form expressions for the convex envelopes of a wide class of such functions several examples demonstrate that these envelopes reduce significantly the relaxation gaps of widely used factorable relaxation techniques

convex envelopes for edge concave functions deterministic global optimization algorithms frequently rely on the convex underestimation of nonconvex functions in this paper we describe the structure of the polyhedral convex envelopes of edge concave functions over polyhedral domains using geometric arguments an algorithm for computing the facets of the convex envelope over hyperrectangles in 3 is described sufficient conditions are described under which the convex envelope of a sum of edge concave functions may be shown to be equivalent to the sum of the convex envelopes of these functions

concave extensions for nonlinear 0 1 maximization problems a well known linearization technique for nonlinear 0 1 maximization problems can be viewed as extending any polynomial in 0 1 variables to a concave function defined on 0 1 n some properties of this standard concave extension are investigated polynomials for which the standard extension coincides with the concave envelope are characterized in terms of integrality of a certain polyhedron or balancedness of a certain matrix the standard extension is proved to be identical to another type of concave extension defined as the lower envelope of a class of affine functions majorizing the given polynomial

selecting optimal alternatives and risk reduction strategies in decision trees in this paper we conduct a quantitative analysis for a strategic risk management problem that involves allocating certain available failure mitigating and consequence alleviating resources to reduce the failure probabilities of system safety components and subsequent losses respectively together with selecting optimal strategic decision alternatives to minimize the risk or expected loss in the event of a hazardous occurrence using a novel decision tree optimization approach to represent the cascading sequences of probabilistic events as controlled by key decisions and investment alternatives the problem is modeled as a nonconvex mixed integer 0 1 factorable program we develop a specialized branch and bound algorithm in which lower bounds are computed via tight linear relaxations of the original problem that are constructed by utilizing a polyhedral outer approximation mechanism in concert with two alternative linearization schemes having different levels of tightness and complexity we also suggest three alternative branching schemes each of which is proven to guarantee convergence to a global optimum for the underlying problem extensive computational results and sensitivity analyses are presented to provide insights and to demonstrate the efficacy of the proposed algorithm

a trust region algorithm with a worst case iteration complexity of mathcal o epsilon 3 2 o 3 2 for nonconvex optimization we propose a trust region algorithm for solving nonconvex smooth optimization problems for any overline epsilon in 0 infty 0 the algorithm requires at most mathcal o epsilon 3 2 o 3 2 iterations function evaluations and derivative evaluations to drive the norm of the gradient of the objective function below any epsilon in 0 overline epsilon 0 this improves upon the mathcal o epsilon 2 o 2 bound known to hold for some other trust region algorithms and matches the mathcal o epsilon 3 2 o 3 2 bound for the recently proposed adaptive regularisation framework using cubics also known as the arc algorithm our algorithm entitled trace follows a trust region framework but employs modified step acceptance criteria and a novel trust region update mechanism that allow the algorithm to achieve such a worst case global complexity bound importantly we prove that our algorithm also attains global and fast local convergence guarantees under similar assumptions as for other trust region algorithms we also prove a worst case upper bound on the number of iterations function evaluations and derivative evaluations that the algorithm requires to obtain an approximate second order stationary point

relaxations of factorable functions with convex transformable intermediates we propose to strengthen standard factorable relaxations of global optimization problems through the use of functional transformations of intermedi ate expressions in particular we exploit convex transformability of the component functions of factorable programs as a tool in the generation of bounds we define suit able forms of transforming functions and assess theoretically and computationally the sharpness of the resulting relaxations in comparison to existing schemes

large deviation approach to the randomly forced navier stokes equation the random forced navier stokes equation can be obtained as a variational problem of a proper action by virtue of incompressibility the integration over transverse components of the fields allows to cast the action in the form of a large deviation functional since the hydrodynamic operator is nonlinear the functional integral yielding the statistics of fluctuations can be practically computed by linearizing around a physical solution of the hydrodynamic equation we show that this procedure yields the dimensional scaling predicted by k41 theory at the lowest perturbative order where the perturbation parameter is the inverse reynolds number moreover an explicit expression of the prefactor of the scaling law is obtained

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

deterministic and stochastic global optimization techniques for planar covering with ellipses problems abstract problems of planar covering with ellipses are tackled in this work ellipses can have a fixed angle or each of them can be freely rotated deterministic global optimization methods are developed for both cases while a stochastic version of the method is also proposed for large instances of the latter case numerical results show the effectiveness and efficiency of the proposed methods

renormalization group operator product expansion and anomalous scaling in models of turbulent advection recent progress on the anomalous scaling in models of turbulent heat and mass transport is reviewed with the emphasis on the approach based on the field theoretic renormalization group rg and operator product expansion ope in that approach the anomalous scaling is established as a consequence of the existence in the corresponding field theoretic models of an infinite number of dangerous composite fields operators with negative critical dimensions which are identified with the anomalous exponents this allows one to calculate the exponents in a systematic perturbation expansion similar to the e expansion in the theory of critical phenomena the rg and ope approach is presented in a self contained way for the example of a passive scalar field temperature concentration of an impurity etc advected by a self similar gaussian velocity ensemble with vanishing correlation time the so called kraichnan s rapid change model where the anomalous exponents are known up to order o e3 effects of anisotropy compressibility and the correlation time of the velocity field are discussed passive advection by non gaussian velocity field governed by the stochastic navier stokes equation and passively advected vector e g magnetic fields are considered

a decomposition method for minlps with lipschitz continuous nonlinearities many mixed integer optimization problems are constrained by nonlinear functions that do not possess desirable analytical properties like convexity or factorability or cannot even be evaluated exactly this is e g the case for many problems constrained by differential equations or for models that rely on black box simulation runs for these problem classes we present analyze and test algorithms that solve mixed integer problems with lipschitz continuous nonlinearities our theoretical results depend on the assumptions made on the in exactness of function evaluations and on the knowledge of lipschitz constants if lipschitz constants are known we prove finite termination at approximate globally optimal points both for the case of exact and inexact function evaluations if only approximate lipschitz constants are known we prove finite termination and derive additional conditions under which infeasibility can be detected a computational study for gas transport problems and an academic case study show the applicability of our algorithms to real world problems and how different assumptions on the constraint functions up or downgrade the practical performance of the methods

glauber dynamics of the sk model theory and simulations in the high temperature phase the low temperature glauber dynamics of the sherrington kirkpatrick spin glass model is studied using a combination of theory and computer simulations the theoretical approach follows the spirit of mori s continuous fraction expansion the predicted amplitude of the long time decay of the time dependent equilibrium spin spin correlation function agrees well with the glauber dynamics simulation results the theory predicts a universal dynamic exponent of in the low temperature phase the simulations cannot distinguish between a universal exponent and a temperature dependent one

an outer approximation algorithm for a class of mixed integer nonlinear programs an outer approximation algorithm is presented for solving mixed integer nonlinear programming problems of a particular class linearity of the integer or discrete variables and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure based on principles of decomposition outer approximation and relaxation the proposed algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed integer linear master program convergence and optimality properties of the algorithm are presented as well as a general discussion on its implementation numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper finally a theoretical comparison with generalized benders decomposition is presented on the lower bounds predicted by the relaxed master programs

cutting planes for optimization of convex functions over nonconvex sets we derive linear inequality characterizations for sets of the form mathop rm conv x q in mathbb r d times mathbb r q ge q x x in mathbb r d mathop rm int p where q is convex and differentiable and p subset mathbb r d we show that in several cases our characterization leads to polynomial time separation algorithms that operate in the original space of variables in particular when q is a positive definite quadratic and p is a polyhedron or an ellipsoid

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

non gaussian invariant measures for the majda model of decaying turbulent transport the problem of turbulent transport of a scalar field by a random velocity field is considered the scalar field amplitude exhibits rare but very large fluctuations whose typical signature is fatter than gaussian tails for the probability distribution of the scalar the existence of such large fluctuations is related to clustering phenomena of the lagrangian paths within the flow this suggests an approach to turn the large deviation problem for the scalar field into a small deviation or small ball problem for some appropriately defined process measuring the spreading with time of the lagrangian paths here such a methodology is applied to a model proposed by majda consisting of a white in time linear shear flow and some generalizations of it where the velocity field has finite or even infinite correlation time the non gaussian invariant measure for the reduced scalar field is derived and in particular it is shown that the one point distribution of the scalar has stretched exponential tails with a stretching exponent depending on the parameters in the model different universality classes for the scalar behavior are identified which all other parameters being kept fixed display a one to one correspondence with an exponent measuring time persistence effects in the velocity field c 2001 john wiley sons inc

instanton filtering for the stochastic burgers equation we address the question of whether one can identify instantons in direct numerical simulations of the stochastically driven burgers equation for this purpose we first solve the instanton equations using the chernykh stepanov method 2001 phys rev e 64 026306 these results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations using this approach we can extract the entire time history of the instanton evolution which allows us to identify the different phases predicted by the direct method of chernykh and stepanov with remarkable agreement

dual adaptive model predictive control abstract we present an adaptive dual model predictive controller dmpc that uses current and future parameter estimation errors to minimize expected output error by optimally combining probing for uncertainty reduction with control of the nominal model our novel approach relies on orthonormal basis function models to derive expressions for the predicted distributions for the output and unknown parameters conditional on the future input sequence propagating the exact future statistics enables reformulating the original stochastic problem into a deterministic equivalent that illustrates the dual nature of the optimal control but is nonlinear and nonconvex we further reformulate the nonlinear deterministic problem to pose an equivalent quadratically constrained quadratic programming qcqp problem that state of the art algorithms can solve efficiently providing the exact solution to the probabilistically constrained finite horizon dual control problem the adaptive dmpc solves this qcqp at each sampling time on a receding horizon the adaptation is a result of updating the parameter estimates used by the dmpc to decide the control input the paper demonstrates the application of dmpc to a single input single output siso system with unknown parameters in the simulation example the parameter estimates converge quickly and the probing vanishes with increasing accuracy and precision of the estimates improving the future control performance

an efficient global algorithm for single group multicast beamforming consider the single group multicast beamforming problem where multiple users receive the same data stream simultaneously from a single transmitter the problem is np hard and all existing algorithms for the problem either find suboptimal approximate or local stationary solutions in this paper we propose an efficient branch and bound algorithm for the problem that is guaranteed to find its global solution to the best of our knowledge our proposed algorithm is the first tailored global algorithm for the single group multicast beamforming problem simulation results show that our proposed algorithm is computationally efficient albeit its theoretical worst case iteration complexity is exponential with respect to the number of receivers and it significantly outperforms a state of the art general purpose global optimization solver called baron our proposed algorithm provides an important benchmark for performance evaluation of existing algorithms for the same problem by using it as the benchmark we show that two state of the art algorithms semidefinite relaxation algorithm and successive linear approximation algorithm work well when the problem dimension i e the number of antennas at the transmitter and the number of receivers is small but their performance deteriorates quickly as the problem dimension increases

a global minlp approach to symbolic regression symbolic regression methods generate expression trees that simultaneously define the functional form of a regression model and the regression parameter values as a result the regression problem can search many nonlinear functional forms using only the specification of simple mathematical operators such as addition subtraction multiplication and division among others currently state of the art symbolic regression methods leverage genetic algorithms and adaptive programming techniques genetic algorithms lack optimality certifications and are typically stochastic in nature in contrast we propose an optimization formulation for the rigorous deterministic optimization of the symbolic regression problem we present a mixed integer nonlinear programming minlp formulation to solve the symbolic regression problem as well as several alternative models to eliminate redundancies and symmetries we demonstrate this symbolic regression technique using an array of experiments based upon literature instances we then use a set of 24 minlps from symbolic regression to compare the performance of five local and five global minlp solvers finally we use larger instances to demonstrate that a portfolio of models provides an effective solution mechanism for problems of the size typically addressed in the symbolic regression literature

error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube we consider the problem of minimizing a polynomial on the hypercube 0 1 n and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the positivstellensatz of schmudgen math ann 289 1991 pp 203 206 the main tool we employ is bernstein approximations of polynomials which also gives constructive proofs and degree bounds for positivity certificates on the hypercube

explicit convex and concave envelopes through polyhedral subdivisions in this paper we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper rectangle these envelopes are obtained by identifying polyhedral subdivisions of the hyper rectangle over which the envelopes can be constructed easily in particular we use these techniques to derive in closed form the concave envelopes of concave extendable supermodular functions and the convex envelopes of disjunctive convex functions

optimization as an analysis tool for human complex problem solving we present a problem class of mixed integer nonlinear programs minlps with nonconvex continuous relaxations which stem from economic test scenarios that are used in the analysis of human complex problem solving in a round based scenario participants hold an executive function a posteriori a performance indicator is calculated and correlated to personal measures such as intelligence working memory or emotion regulation altogether we investigate 2088 optimization problems that differ in size and initial conditions based on real world experimental data from 12 rounds of 174 participants the goals are twofold first from the optimal solutions we gain additional insight into a complex system which facilitates the analysis of a participant s performance in the test second we propose a methodology to automatize this process by providing a new criterion based on the solution of a series of optimization problems by providing a mathematical optimization model and this methodology we disprove the ass

analysis of milp techniques for the pooling problem the pq relaxation for the pooling problem can be constructed by applying mccormick envelopes for each of the bilinear terms appearing in the so called pq formulation of the pooling problem this relaxation can be strengthened by using piecewise linear functions that over and under estimate each bilinear term although there is a significant amount of empirical evidence to show that such piecewise linear relaxations which can be written as mixed integer linear programs milps yield good bounds for the pooling problem to the best of our knowledge no formal result regarding the quality of these relaxations is known in this paper we prove that the ratio of the upper bound obtained by solving piecewise linear relaxations objective function is maximization to the optimal objective function value of the pooling problem is at most n where n is the number of output nodes furthermore for any 0 and for any piecewise linear relaxation there exists an instance where the ratio of the relaxation value t

on convex envelopes for bivariate functions over polytopes in this paper we discuss convex envelopes for bivariate functions satisfying suitable assumptions over polytopes we first propose a technique to compute the value and a supporting hyperplane of the convex envelope over a general two dimensional polytope through the solution of a three dimensional convex subproblem with continuously differentiable constraint functions then for quadratic functions as well as for some polynomial and rational ones again satisfying suitable assumptions we show how the same computations can be carried out through the solution of a single semidefinite 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

on minimal valid inequalities for mixed integer conic programs we study disjunctive conic sets involving a general regular closed convex full dimensional and pointed cone x1d4a6 such as the nonnegative orthant the lorentz cone or the positive semidefinite cone in a unified framework we introduce x1d4a6 minimal inequalities and show that under mild assumptions these inequalities together with the trivial cone implied inequalities are sufficient to describe the convex hull we focus on the properties of x1d4a6 minimal inequalities by establishing algebraic necessary conditions for an inequality to be x1d4a6 minimal this characterization leads to a broader algebraically defined class of x1d4a6 sublinear inequalities we demonstrate a close connection between x1d4a6 sublinear inequalities and the support functions of convex sets with a particular structure this connection results in practical ways of verifying x1d4a6 sublinearity and or x1d4a6 minimality of inequalities our study generalizes some of the results from the mixed integer linear case it is well known that the minimal inequalities for

exact quadratic convex reformulations of mixed integer quadratically constrained problems we propose a solution approach for the general problem qp of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints the resolution is based on the reformulation of the original problem qp into an equivalent quadratic problem whose continuous relaxation is convex so that it can be effectively solved by a branch and bound algorithm based on quadratic convex relaxation we concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible furthermore we extend our method to the case of mixed integer quadratic problems with the following restriction all quadratic sub functions of purely continuous variables are already convex finally we illustrate the different results of the article by small examples and we present some computational experiments on pure integer and mixed integer instances of qp most of the considered instances with up to 53 variables can be solved by our approach combined with the use of cplex

strong socp relaxations for the optimal power flow problem this paper proposes three strong second order cone programming socp relaxations for the ac optimal power flow opf problem these three relaxations are incomparable to each other and two of them are incomparable to the standard sdp relaxation of opf extensive computational experiments show that these relaxations have numerous advantages over existing convex relaxations in the literature i their solution quality is extremely close to that of the standard sdp relaxation the best one is within 99 96 of the sdp relaxation on average for all the ieee test cases and consistently outperforms previously proposed convex quadratic relaxations of the opf problem ii the solutions from the strong socp relaxations can be directly used as a warm start in a local solver such as ipopt to obtain a high quality feasible opf solution and iii in terms of computation times the strong socp relaxations can be solved an order of magnitude faster than the standard sdp relaxation for example one of the proposed s

mccormick based relaxations of algorithms theory and implementation for the global optimization of a wide class of algorithms is presented via convex affine relaxations the basis for the proposed relaxations is the systematic construction of subgradients for the convex relaxations of factorable functions by mccormick math prog 10 1976 pp 147 175 similar to the convex relaxation the subgradient propagation relies on the recursive application of a few rules namely the calculation of subgradients for addition multiplication and composition operations subgradients at interior points can be calculated for any factorable function for which a mccormick relaxation exists provided that subgradients are known for the relaxations of the univariate intrinsic functions for boundary points additional assumptions are necessary an automated implementation based on operator overloading is presented and the calculation of bounds based on affine relaxation is demonstrated for illustrative examples two numerical examples for the global optimization of algorithms are presented in both examples a parameter estimation problem with embedded differential equations is considered the solution of the differential equations is approximated by algorithms with a fixed number of iterations

error bounds for monomial convexification in polynomial optimization convex hulls of monomials have been widely studied in the literature and monomial convexifications are implemented in global optimization software for relaxing polynomials however there has been no study of the error in the global optimum from such approaches we give bounds on the worst case error for convexifying a monomial over subsets of this implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately our main error bounds depend primarily on the degree of the monomial making them easy to compute since monomial convexification studies depend on the bounds on the associated variables in the second part we conduct an error analysis for a multilinear monomial over two different types of box constraints as part of this analysis we also derive the convex hull of a multilinear monomial over

quantifying double mccormick when using the standard mccormick inequalities twice to convexify trilinear monomials as is often the practice in modeling and software there is a choice of which variables to group first for the important case in which the domain is a nonnegative box we calculate the volume of the resulting relaxation as a function of the bounds defining the box in this manner we precisely quantify the strength of the different possible relaxations defined by all three groupings in addition to the trilinear hull itself as a by product we characterize the best double mccormick relaxation we wish to emphasize that in the context of spatial branch and bound for factorable formulations our results do not only apply to variables in the input formulation our results apply to monomials that involve auxiliary variables as well so our results apply to the product of any three possibly complicated expressions in a formulation

statistics of a passive scalar advected by a large scale two dimensional velocity field analytic solution steady statistics of a passive scalar advected by a random two dimensional flow of an incompressible fluid is described in the range of scales between the correlation length of the flow and the diffusion scale this corresponds to the so called batchelor regime where the velocity is replaced by its large scale gradient the probability distribution of the scalar in the locally comoving reference frame is expressed via the probability distribution of the line stretching rate the description of line stretching can be reduced to a classical problem of the product of many random matrices with a unit determinant we have found the change of variables that allows one to map the matrix problem onto a scalar one and to thereby prove the central limit theorem for the stretching rate statistics the proof is valid for any finite correlation time of the velocity field whatever the statistics of the velocity field the statistics of the passive scalar averaged over time locally in space is shown to approach gaussian statistics with increase in the p eclet number pe the pumping to diffusion scale ratio the first nsimultaneous correlation functions are expressed via the flux of the square of the scalar and only one factor depending on the velocity field the mean stretching rate which can be calculated analytically in limiting cases non gaussian tails of the probability distributions at finite pe are found to be exponential

mathematical programming techniques in water network optimization abstract in this article we survey mathematical programming approaches to problems in the field of drinking water distribution network optimization among the predominant topics treated in the literature we focus on two different but related problem classes one can be described by the notion of network design while the other is more aptly termed by network operation the basic underlying model in both cases is a nonlinear network flow model and we give an overview on the more specific modeling aspects in each case the overall mathematical model is a mixed integer nonlinear program having a common structure with respect to how water dynamics in pipes are described finally we survey the algorithmic approaches to solve the proposed problems and we discuss computation on various types of water networks

strong valid inequalities for orthogonal disjunctions and bilinear covering sets in this paper we derive a closed form characterization of the convex hull of a generic nonlinear set when this convex hull is completely determined by orthogonal restrictions of the original set although the tools used in this construction include disjunctive programming and convex extensions our characterization does not introduce additional variables we develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed we demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed integer polyhedral sets and finding the convex hull of certain mixed pure integer bilinear sets we then extend the utility of the convexification tool to relaxing nonconvex inequalities which are not naturally disjunctive by providing sufficient conditions for establishing the convex extension property over the non negative orthant we illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non negative orthant although we illustrate our results primarily on bilinear covering sets they also apply to more general polynomial covering sets for which they yield new tight relaxations

instanton for the kraichnan passive scalar problem we consider high order correlation functions of the passive scalar in the kraichnan model using the instanton formalism we find the scaling exponents ensuremath zeta n of the structure functions s n for n ensuremath gg 1 under the additional condition d ensuremath zeta 2 ensuremath gg 1 where d is the dimensionality of space at nl n c where n c d ensuremath zeta 2 2 2 ensuremath ensuremath zeta 2 the exponents are ensuremath zeta n ensuremath zeta 2 4 2n ensuremath n 2 n c while at ng n c they are n independent ensuremath zeta n ensuremath zeta 2 n c 4 we also estimate n dependent factors in s n particularly their behavior at n close to n c

a polyhedral branch and cut approach to global optimization a variety of nonlinear including semidefinite relaxations have been developed in recent years for nonconvex optimization problems their potential can be realized only if they can be solved with sufficient speed and reliability unfortunately state of the art nonlinear programming codes are significantly slower and numerically unstable compared to linear programming software in this paper we facilitate the reliable use of nonlinear convex relaxations in global optimization via a polyhedral branch and cut approach our algorithm exploits convexity either identified automatically or supplied through a suitable modeling language construct in order to generate polyhedral cutting planes and relaxations for multivariate nonconvex problems we prove that if the convexity of a univariate or multivariate function is apparent by decomposing it into convex subexpressions our relaxation constructor automatically exploits this convexity in a manner that is much superior to developing polyhedral outer approximators for the original function the convexity of functional expressions that are composed to form nonconvex expressions is also automatically exploited root node relaxations are computed for 87 problems from globallib and minlplib and detailed computational results are presented for globally solving 26 of these problems with baron 7 2 which implements the proposed techniques the use of cutting planes for these problems reduces root node relaxation gaps by up to 100 and expedites the solution process often by several orders of magnitude

integer programming subject to monomial constraints we investigate integer programs containing constraints of the type prod i in i x i alpha i b due to the number theoretic nature of these constraints standard methods based on linear algebra cannot be applied directly instead we present a reformulation resulting in integer programs with linear constraints and polynomial objective functions using prime decompositions of the right hand sides b moreover we show that minimizing a linear objective function with nonnegative coefficients over bivariate constraints is possible in polynomial time

convergent upper bounds in global minimization with nonlinear equality constraints we address the problem of determining convergent upper bounds in continuous non convex global minimization of box constrained problems with equality constraints these upper bounds are important for the termination of spatial branch and bound algorithms our method is based on the theorem of miranda which helps to ensure the existence of feasible points in certain boxes then the computation of upper bounds at the objective function over those boxes yields an upper bound for the globally minimal value a proof of convergence is given under mild assumptions an extension of our approach to problems including inequality constraints is possible

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

a polyhedral study of binary polynomial programs we study the polyhedral convex hull of a mixed integer set x1d4ae defined by a collection of multilinear equations over the unit hypercube such sets appear frequently in the factorable reformulation of mixed integer nonlinear optimization problems in particular the set x1d4ae represents the feasible region of a linearized unconstrained binary polynomial optimization problem we define an equivalent hypergraph representation of the mixed integer set x1d4ae which enables us to derive several families of facet defining inequalities structural properties and lifting operations for its convex hull in the space of the original variables our theoretical developments extend several well known results from the boolean quadric polytope and the cut polytope literature paving a way for devising novel optimization algorithms for nonconvex problems containing multilinear sub expressions

some results on the strength of relaxations of multilinear functions we study approaches for obtaining convex relaxations of global optimization problems containing multilinear functions specifically we compare the concave and convex envelopes of these functions with the relaxations that are obtained with a standard relaxation approach due to mccormick the standard approach reformulates the problem to contain only bilinear terms and then relaxes each term independently we show that for a multilinear function having a single product term this approach yields the convex and concave envelopes if the bounds on all variables are symmetric around zero we then review and extend some results on conditions when the concave envelope of a multilinear function can be written as a sum of concave envelopes of its individual terms finally for bilinear functions we prove that the difference between the concave upper bounding and convex lower bounding functions obtained from the mccormick relaxation approach is always within a constant of the difference between the concave and convex envelopes these results along with numerical examples we provide give insight into how to construct strong relaxations of multilinear functions

kinetic theory approach to the sk spin glass model with glauber dynamics a new method to analyse glauber dynamics of the sherrington kirkpatrick sk spin glass model is presented the method is based on ideas used in the classical kinetic theory of fluids it is applied to study spin correlations in the high temperature phase of the sk model at zero external field the zeroth order theory is equivalent to a disorder dependent local equilibrium approximation its predictions agree well with computer simulation results the first order theory involves coupled evolution equations for the spin correlations and the dynamic excess parts of the local field distributions it qualitatively accounts for the error made in the zeroth approximation

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

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

evolution of a scalar gradient s probability density function in a random flow the evolution of the probability density function pdf of the passive scalar gradient is studied in the limit of large peclet and prandtl numbers in d dimensions without diffusion the closed fokker planck equation can be derived and solved analytically leading to a number of conclusions in particular it allows the description of the restoration of the rotational symmetry and enables one to distinguish different regimes of evolution and different intervals of pdf behavior

a milp formulation for generalized geometric programming using piecewise linear approximations abstract generalized geometric programming ggp problems are converted to mixed integer linear programming milp problems using piecewise linear approximations our approach is to approximate a multiple term log sum function of the form log x 1 x 2 x n in terms of a set of linear equalities or inequalities of log x 1 log x 2 and log x n where x 1 x n are strictly positive the advantage of this approach is its simplicity and readiness to implement and solve using commercial milp solvers while milp problems in general are no easier than ggp problems this approach is justified by the phenomenal progress of computing power of both personal computers and commercial milp solvers the limitation of this approach is discussed along with numerical tests

polyhedral approximation in mixed integer convex optimization generalizing both mixed integer linear optimization and convex optimization mixed integer convex optimization possesses broad modeling power but has seen relatively few advances in general purpose solvers in recent years in this paper we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class our approach is based on constructing polyhedral outer approximations of the convex constraints resulting in a global solution by solving a finite number of mixed integer linear and continuous convex subproblems the key advance we present is to strengthen the polyhedral approximations by constructing them in a higher dimensional space in order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming dcp and for generality and ease of implementation we use conic representations of the convex constraints although our framework requires a manual translation of existing models into dcp form after performing this transformation on the minlplib2 benchmark library we were able to solve a number of unsolved instances and on many other instances achieve superior performance compared with state of the art solvers like bonmin scip and artelys knitro

an error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution we study the minimization of fixed degree polynomials over the simplex this problem is well known to be np hard as it contains the maximum stable set problem in graph theory as a special case in this paper we consider a rational approximation by taking the minimum over the regular grid which consists of rational points with denominator r for given r we show that the associated convergence rate is o 1 r 2 for quadratic polynomials for general polynomials if there exists a rational global minimizer over the simplex we show that the convergence rate is also of the order o 1 r 2 our results answer a question posed by de klerk laurent and sun math program 151 2015 pp 433 457 and improves on previously known o 1 r bounds in the quadratic case

global optimization in hilbert space we propose a complete search algorithm for solving a class of non convex possibly infinite dimensional optimization problems to global optimality we assume that the optimization variables are in a bounded subset of a hilbert space and we determine worst case run time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set because these run time bounds are independent of the number of optimization variables and in particular are valid for optimization problems with infinitely many optimization variables we prove that the algorithm converges to an varepsilon suboptimal global solution within finite run time for any given termination tolerance varepsilon 0 finally we illustrate these results for a problem of calculus of variations

robust goal programming using different robustness echelons via norm based and ellipsoidal uncertainty sets abstract the notion of robust goal programming rgp using cardinality constrained robustness via interval based uncertainty was first examined over a decade ago since then the rgp methodology has not been widely researched specifically when considering different uncertainty sets to implement within this context this paper compares interval based and norm based uncertainty sets using cardinality constrained robustness strict robustness using ellipsoidal uncertainty sets is also examined in the rgp realm the aforementioned methods are demonstrated for a simple instance from the literature and the results are summarized conclusions are made regarding the proposed rgp models when likened to a similar rgp model seen in the literature further the suitability of each rgp model is offered when a decision maker s risk preference or computing availability are taken into consideration inferences are made regarding the effectiveness of each uncertainty set in the context of solutions that are relatively unaffected by data uncertainty that is robust solutions

global optimization of mixed integer nonlinear programs a theoretical and computational study this work addresses the development of an efficient solution strategy for obtaining global optima of continuous integer and mixed integer nonlinear programs towards this end we develop novel relaxation schemes range reduction tests and branching strategies which we incorporate into the prototypical branch and bound algorithm in the theoretical algorithmic part of the paper we begin by developing novel strategies for constructing linear relaxations of mixed integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties we then use lagrangian linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming this theory leads to new range reduction schemes including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch and bound tree finally we incorporate these relaxation and reduction strategies in a branch and bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems in the computational part of the paper we describe our implementation discussing wherever appropriate the use of suitable data structures and associated algorithms we present computational experience with benchmark separable concave quadratic programs fractional 0 1 programs and mixed integer nonlinear programs from applications in synthesis of chemical processes engineering design just in time manufacturing and molecular design

rigorous estimates of the tails of the probability distribution function for the random linear shear model in previous work majda and mclaughlin and majda computed explicit expressions for the 2nth moments of a passive scalar advected by a linear shear flow in the form of an integral over r n in this paper we first compute the asymptotics of these moments for large moment number we are able to use this information about the large n behavior of the moments along with some basic facts about entire functions of finite order to compute the asymptotics of the tails of the probability distribution function we find that the probability distribution has gaussian tails when the energy is concentrated in the largest scales as the initial energy is moved to smaller and smaller scales we find that the tails of the distribution grow longer and the distribution moves smoothly from gaussian through exponential and stretched exponential we also show that the derivatives of the scalar are increasingly intermittent in agreement with experimental observations and relate the exponents of the scalar derivative to the exponents of the scalar

outer product free sets for polynomial optimization and oracle based cuts this paper introduces cutting planes that involve minimal structural assumptions enabling the generation of strong polyhedral relaxations for a broad class of problems we consider valid inequalities for the set s cap p where s is a closed set and p is a polyhedron given an oracle that provides the distance from a point to s we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron these cuts are generated from convex forbidden zones or s free sets derived from the oracle we also consider the special case of polynomial optimization accordingly we develop a theory of outer product free sets where s is the set of real symmetric matrices of the form xx t all maximal outer product free sets of full dimension are shown to be convex cones and we identify several families of such sets these families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently computational experiments demonstrate the promise of our approach

two complementary descriptions of intermittency we describe two complementary formalisms designed for the description of the probability density function pdf of the gradients of turbulent fields the first approach we call it adiabatic describes the pdf at the values much less than dispersion the second instanton approach gives the tails of the pdf at the values of the gradient much larger than dispersion together both approaches give a satisfactory description of gradient pdfs as illustrated here by an example of a passive scalar advected by a one dimensional compressible random flow

dynamic linear response of the sk spin glass coupled microscopically to a bath the dynamic linear response theory of a general ising model weakly coupled to a heat bath is derived by employing the quantum statistical theory of mori treating the hamiltonian of the spin bath coupling as a perturbation and applying the markovian approximation both the dynamic susceptibility and the relaxation function are expressed in terms of the static susceptibility and the static internal field distribution function for the special case of the sk spin glass this internal field distribution can be related to the solutions of the tap equations in the entire temperature region application of this new relation and the use of numerical solutions of the modified tap equations lead for finite but large systems to explicit results for the distribution function and for dynamic linear response functions a detailed discussion is presented which includes finite size effects due to the derived temperature dependence of the onsager casimir coefficients a frequency dependent shift of the cusp temperature of the real part of the dynamic susceptibility is found

a new framework to relax composite functions in nonlinear programs in this paper we devise new relaxations for composite functions which improve the prevalent factorable relaxations without introducing additional variables by exploiting the inner function structure we outer approximate inner functions using arbitrary under and over estimators and then convexify the outer function over a polytope p which models the ordering relationships between the inner functions and their estimators and utilizes bound information on the inner functions as well as on the estimators we show that there is a subset q of p with significantly simpler combinatorial structure such that the separation problem of the graph of the outer function over p is polynomially equivalent via a fast combinatorial algorithm to that of its graph over q we specialize our study to consider the product of two inner functions with one non trivial underestimator for each inner function for the corresponding polytope p we show that there are eight valid inequalities besides the four mccormick inequalities which improve the factorable relaxation finally we show that our results generalize to simultaneous convexification of a vector of outer functions

convergence analysis for lasserre s measure based hierarchy of upper bounds for polynomial optimization we consider the problem of minimizing a continuous function f over a compact set mathbf k k we analyze a hierarchy of upper bounds proposed by lasserre siam j optim 21 3 864 885 2011 obtained by searching for an optimal probability density function h on mathbf k k which is a sum of squares of polynomials so that the expectation int mathbf k f x h x dx kf x h x dx is minimized we show that the rate of convergence is no worse than o 1 sqrt r o 1 r where 2r is the degree bound on the density function this analysis applies to the case when f is lipschitz continuous and mathbf k k is a full dimensional compact set satisfying some boundary condition which is satisfied e g for convex bodies the rth upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d and if all moments of order up to 2r d 2r d of the lebesgue measure on mathbf k k are known which holds for example if mathbf k k is a simplex hypercube or a euclidean ball

universal long time properties of lagrangian statistics in the batchelor regime and their application to the passive scalar problem we consider the transport of dynamically passive quantities in the batchelor regime of a smooth in space velocity field for the case of arbitrary temporal correlations of the velocity we formulate the statistics of relevant characteristics of lagrangian motion this allows us to generalize many results obtained previously for strain ensuremath delta correlated in time thus answering a question about the universality of these results

irreducible memory function for dissipative stochastic systems with detailed balance abstract we show quite generally that the memory function for purely dissipative stochastic systems with detailed balance can be reduced further and can be expressed in terms of the irreducible memory function this generalizes the similar finding for the smoluchowski equation the irreducible memory function is intimately connected with the lifetime renormalization and is shown to provide a useful starting point for introducing approximations such as the mode coupling approximation

conic mixed integer rounding cuts a conic integer program is an integer programming problem with conic constraints many problems in finance engineering statistical learning and probabilistic optimization are modeled using conic constraints here we study mixed integer sets defined by second order conic constraints we introduce general purpose cuts for conic mixed integer programming based on polyhedral conic substructures of second order conic sets these cuts can be readily incorporated in branch and bound algorithms that solve either second order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch and bound tree central to our approach is a reformulation of the second order conic constraints with polyhedral second order conic constraints in a higher dimensional space in this representation the cuts we develop are linear even though they are nonlinear in the original space of variables this feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed integer programming the reformulation also allows the use of polyhedral methods for conic integer programming we report computational results on solving unstructured second order conic mixed integer problems as well as mean variance capital budgeting problems and least squares estimation problems with binary inputs our computational experiments show that conic mixed integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed integer programs and hence improving their solvability