dc programming and dca thirty years of developments the year 2015 marks the 30th birthday of dc difference of convex functions programming and dca dc algorithms which constitute the backbone of nonconvex programming and global optimization in this article we offer a short survey on thirty years of developments of these theoretical and algorithmic tools the survey is comprised of three parts in the first part we present a brief history of the field while in the second we summarize the state of the art results and recent advances we focus on main theoretical results and dca solvers for important classes of difficult nonconvex optimization problems and then give an overview of real world applications whose solution methods are based on dca the third part is devoted to new trends and important open issues as well as suggestions for future developments

frozen percolation in two dimensions aldous math proc camb philos soc 128 465 477 2000 introduced a modification of the bond percolation process on the binary tree where clusters stop growing freeze as soon as they become infinite we investigate the site version of this process on the triangular lattice where clusters freeze as soon as they reach l infty diameter at least n for some parameter n we show informally speaking that in the limit n rightarrow infty the clusters only freeze in the critical window of site percolation on the triangular lattice hence the fraction of vertices that eventually i e at time 1 are in a frozen cluster tends to 0 as n goes to infinity we also show that the diameter of the open cluster at time 1 of a given vertex is with high probability smaller than n but of order n this shows that the process on the triangular lattice has a behaviour quite different from aldous process we also indicate which modifications have to be made to adapt the proofs to the case of the n parameter frozen bond percolation process on the square lattice this extends our results to the square lattice and answers the questions posed by van den berg et al random struct algorithms 40 220 226 2012

generalizations of the fkkm theorem and the ky fan minimax inequality with applications to maximal elements price equilibrium and complementarity abstract this paper generalizes the fan knaster kuratowski mazurkiewicz fkkm theorem of ky fan game theory and related topics pp 151 156 north holland amsterdam 1979 and math ann 266 1984 519 537 and the ky fan minimax inequality by introducing a class of the generalized closedness and continuity conditions which are called the transfer closedness and transfer continuities we then apply these results to prove the existence of maximal elements of binary relations under very weak assumptions we also prove the existence of price equilibrium and the complementarity problem without the continuity assumptions thus our results generalize many of the existence theorems in the literature

the divergence of fluctuations for shape in first passage percolation we consider the first passage percolation model on z d for d 2 in this model we assign independently to each edge the value zero with probability p and the value one with probability 1 p we denote by t 0 the passage time from the origin to for r d and

typical cluster size for two dimensional percolation processes the typical cluster size for two dimensional percolation models is discussed it is shown that forw 0 x z 2 0 x lim n 1 n logp p w 0 n 1 p p c asp p c provided thate p w 0 2 e p w 0 p p c asp p c furthermore we introduce a new quantityf s p which may be thought of as the singular part of the free energy and show thatf s p p p c 2v provided that the correlation length p p c v asp p c

distinguishability of particles in glass forming systems abstract the distinguishability of particles has important implications for calculating the partition function in statistical mechanics while there are standard formulations for systems of identical particles that are either fully distinguishable or fully indistinguishable many realistic systems do not fall into either of these limiting cases in particular the glass transition involves a continuous transition from an ergodic liquid system of indistinguishable particles to a nonergodic glassy system where the particles become distinguishable while the question of partial distinguishability of microstates has been treated previously in quantum information theory this issue has not yet been addressed for a system of classical particles in this paper we present a general theoretical formalism for quantifying particle distinguishability in classical systems this formalism is based on a classical definition of relative entropy such as applied in quantum information theory example calculations for a simple glass forming system demonstrate the continuous onset of distinguishability as temperature is lowered we also examine the loss of distinguishability in the limit of long observation time coinciding with the restoration of ergodicity we discuss some of the general implications of our work including the direct connection to topological constraint theory of glass we also discuss qualitative features of distinguishability as they relate to the second and third laws of thermodynamics

a central limit theorem for critical first passage percolation in two dimensions consider independent first passage percolation on the edges of 2 denote the passage time of the edge e in 2 by t e and assume that p t e 0 1 2 p 0 t e c 0 0 for some constant c 0 0 and that e t e 4 denote by b 0 n the passage time from 0 to the halfplane x y x n and by t 0 nu the passage time from 0 to the nearest lattice point to nu for u a unit vector we prove that there exist constants 0 c 1 c 2 and n such that c 1 log n 1 2 n c 2 log n 1 2 and such that n 1 b 0 n eb 0 n and 2 n 1 t 0 nu et 0 nu converge in distribution to a standard normal variable as n u fixed a similar result holds for the site version of first passage percolation on 2 when the common distribution of the passage times t v of the vertices satisfies p t v 0 1 p t v c 0 p c 2 site critical probability of site percolation on 2 and e t u 4

random walks in asymmetric random environments we consider random walks on zd with transition ratesp x y given by a random matrix ifp is a small random perturbation of the simple random walk we show that the walk remains diffusive for almost all environmentsp ifd 2 the result also holds for a continuous time markov process with a random drift the corresponding path space measures converge weakly in the scaling limit to the wiener process for almost everyp

outlets of 2d invasion percolation and multiple armed incipient infinite clusters we study invasion percolation in two dimensions focusing on properties of the outlets of the invasion and their relation to critical percolation and to incipient infinite clusters iic s first we compute the exact decay rate of the distribution of both the weight of the k th outlet and the volume of the k th pond next we prove bounds for all moments of the distribution of the number of outlets in an annulus this result leads to almost sure bounds for the number of outlets in a box b 2 n and for the decay rate of the weight of the k th outlet to pc we then prove existence of multiplearmed iic measures for any number of arms and for any color sequence which is alternating or monochromatic we use these measures to study the invaded region near outlets and near edges in the invasion backbone far from the origin

asymptotics for 2d critical and near critical first passage percolation we study bernoulli first passage percolation fpp on the triangular lattice in which sites have 0 and 1 passage times with probability p and 1 p respectively denote by mathcal c infty the infinite cluster with 0 time sites when p p c where p c 1 2 is the critical probability denote by t 0 mathcal c infty the passage time from the origin 0 to mathcal c infty first we obtain explicit limit theorem for t 0 mathcal c infty as p searrow p c the proof relies on the limit theorem in the critical case the critical exponent for correlation length and kesten s scaling relations next for the usual point to point passage time a 0 n in the critical case we construct subsequences of sites with different growth rate along the axis the main tool involves the large deviation estimates on the nesting of cle 6 loops derived by miller et al ann probab 44 1013 1052 2016 finally we apply the limit theorem for critical bernoulli fpp to a random graph called cluster graph obtaining explicit strong law of large numbers for graph distance

infinite clusters in percolation models the qualitative nature of infinite clusters in percolation models is investigated the results which apply to both independent and correlated percolation in any dimension concern the number and density of infinite clusters the size of their external surface the value of their total surface to volume ratio and the fluctuations in their density in particular it is shown thatn0 the number of distinct infinite clusters is either 0 1 or and the casen0 which might occur in sufficiently high dimension is analyzed

strong duality in nonconvex quadratic optimization with two quadratic constraints we consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints when the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions we then develop a connection between the image of the real and complex spaces under a quadratic mapping which together with the results in the complex case lead to a condition that ensures strong duality in the real setting preliminary numerical simulations suggest that for random instances of the extended trust region subproblem the sufficient condition is satisfied with a high probability furthermore we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems finally we discuss an application of our results to robust least squares problems

an intersection theorem with applications in minimax theory and equilibrium problem in this paper we obtain a very general intersection theorem for the values of a map from this we derive an analytic alternative a new type of minimax inequality and an alternative theorem concerning three abstract equilibrium problems

an efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints in this paper we investigate two approaches to minimizing a quadratic form subject to the intersection of finitely many ellipsoids the first approach is the d c difference of convex functions optimization algorithm abbr dca whose main tools are the proximal point algorithm and or the projection subgradient method in convex minimization the second is a branch and bound scheme using lagrangian duality for bounding and ellipsoidal bisection in branching the dca was first introduced by pham dinh in 1986 for a general d c program and later developed by our various work is a local method but from a good starting point it provides often a global solution this motivates us to combine the dca and our branch and bound algorithm in order to obtain a good initial point for the dca and to prove the globality of the dca in both approaches we attempt to use the ellipsoidal constrained quadratic programs as the main subproblems the idea is based upon the fact that these programs can be efficiently solved by some available polynomial and nonpolynomial time algorithms among them the dca with restarting procedure recently proposed by pham dinh and le thi has been shown to be the most robust and fast for large scale problems several numerical experiments with dimension up to 200 are given which show the effectiveness and the robustness of the dca and the combined dca branch and bound algorithm

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

aging and relaxation dynamics in free energy landscapes with multiple minima abstract we consider the stochastic dynamics of a system thermally relaxing in a free energy landscape with multiple attractors and show that lack of translational homogeneity in this landscape leads to aging effects e g to the dependence of the susceptibilities on the time elapsed from a thermal quench to the imposition of the probing field we then prove an inequality between response and correlation which generalizes the fluctuation dissipation theorem to a situation far from thermodynamical equilibrium as an application and a check we specialize our formalism in a way which we suggest is appropriate for spin glass systems we assume a hierarchical organization of the landscape and find aging behavior in the response curves in good agreement with relevant experimental data we finally conclude with a summary and a brief discussion of different approaches to slow relaxation in complex systems

incipient infinite percolation clusters in 2d we study several kinds of large critical percolation clusters in two dimensions we show that from the microscopic lattice scale perspective these clusters can be described by kesten s incipient infinite cluster iic as was conjectured by aizenman more specifically we establish this for incipient spanning clusters large clusters in a finite box and the inhomogeneous model of chayes chayes and durrett our results prove the equivalence of several natural definitions of the iic we also show that for any k 1 the difference in size between the kth and k 1 st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes to infinity in addition the distribution of the chayes chayes durrett cluster is shown to be singular with respect to the iic

infinite canonical super brownian motion and scaling limits we construct a measure valued markov process which we call infinite canonical super brownian motion and which corresponds to the canonical measure of super brownian motion conditioned on non extinction infinite canonical super brownian motion is a natural candidate for the scaling limit of various random branching objects on mathbb z d when these objects are critical mean field and infinite we prove that icsbm is the scaling limit of the spread out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension we conjecture that it also arises as the scaling limit in various other models above the upper critical dimension such as the incipient infinite lattice tree above 8 dimensions the incipient infinite cluster for unoriented percolation above 6 dimensions uniform spanning trees above 4 dimensions and invasion percolation above 6 dimensions

on the classical necessary second order optimality conditions in the presence of equality and inequality constraints for nonconvex optimization problems in rn a counterexample was recently given by anitescu siam j optim 10 2000 pp 1116 1135 that showed that the mangasarian fromovitz constraint qualification mfcq is not sufficient for the classical necessary second order optimality conditions to hold we prove these optimality conditions with the assumptions that the set of lagrange multipliers is a bounded line segment and a relaxed strict complementary slackness scs condition holds a new constraint qualification is presented in this paper we assume that the mfcq holds and the active constraint rank deficiency is 1 this assumption relaxes the linear independence constraint qualification and enforces the mfcq in addition the set of lagrange multipliers is a bounded line segment

on the solution of a two ball trust region subproblem in this paper we investigate the structure of a two ball trust region subproblem arising frequently in nonlinear parameter identification problems and propose a method for its solution the method decomposes the subproblem and allows the application of efficient well studied methods for the solution of trust region subproblems arising in unconstrained optimization in the discussion of the structure we focus on the case where both constraints are active and on the treatment of the unconstrained problem

invasion percolation and the incipient infinite cluster in 2d we establish two links between two dimensional invasion percolation and kesten s incipient infinite cluster iic we first prove that the kth moment of the number of invaded sites within the box n n n n is of order n2 n k for k 1 where n is the probability that the origin in critical percolation is connected to the boundary of a box of radius n this improves a result of y zhang we show that the size of the invaded region when scaled by n2 n is tight

random graph asymptotics on high dimensional tori we investigate the scaling of the largest critical percolation cluster on a large d dimensional torus for nearest neighbor percolation in sufficiently high dimensions or when d 6 for sufficiently spread out percolation we use a relatively simple coupling argument to show that this largest critical cluster is with high probability bounded above by a large constant times v 2 3 and below by a small constant times v 2 3 log v 4 3 where v is the volume of the torus we also give a simple criterion in terms of the subcritical percolation two point function on mathbb z d under which the lower bound can be improved to small constant times v 2 3 i e we prove random graph asymptotics for the largest critical cluster on the high dimensional torus this establishes a conjecture by 1 apart from logarithmic corrections we discuss implications of these results on the dependence on boundary conditions for high dimensional percolation

a role for a brachistochrone in the broken ergodicity scenario abstract we show that the time evolution of ergodic decomposition is outlined by a variational principle for a rather universal class of complex systems with a quasicontinuous spectrum of intrinsic relaxation timescales in turn this variational approach enables us to single out a logarithmic scaling law for the barriers on the hierarchical level such a logarithmic dependence has already been shown to hold valid for certain complex systems moreover it represents a key factor to explain the experimentally ubiquitous debye kohlrausch relaxation law and plays a central role on different theoretical model descriptions within this field

random walk on the incipient infinite cluster for oriented percolation in high dimensions we consider simple random walk on the incipient infinite cluster for the spread out model of oriented percolation on mathbb z d times mathbb z in dimensions d 6 we obtain bounds on exit times transition probabilities and the range of the random walk which establish that the spectral dimension of the incipient infinite cluster is frac 4 3 and thereby prove a version of the alexander orbach conjecture in this setting the proof divides into two parts one part establishes general estimates for simple random walk on an arbitrary infinite random graph given suitable bounds on volume and effective resistance for the random graph a second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d 6 by extending results about critical oriented percolation obtained previously via the lace expansion

generalized ensemble and tempering simulations a unified view from the underlying master equations we derive one dimensional stochastic processes that describe generalized ensemble simulations as well as tempering simulated and parallel simulations the representations obtained are either in the form of a one dimensional fokker planck equation or a hopping process on a one dimensional chain in particular we discuss the conditions under which these representations are valid approximate markovian descriptions of the random walk in order parameter or control parameter space they allow a unified discussion of the stationary distribution on as well as of the stationary flow across each space we demonstrate that optimizing the flow is equivalent to minimizing the first passage time for crossing the space and discuss the consequences of our results for optimizing simulations finally we point out the limitations of these representations under conditions of broken ergodicity

computing a celis dennis tapia trust region step for equality constrained optimization we study an approach for minimizing a convex quadratic function subject to two quadratic constraints this problem stems from computing a trust region step for an sqp algorithm proposed by celis dennis and tapia 1985 for equality constrained optimization our approach is to reformulate the problem into a univariate nonlinear equation 0 where the function is continuous at least piecewise differentiable and monotone well established methods then can be readily applied we also consider an extension of our approach to a class of non convex quadratic functions and show that our approach is applicable to reduced hessian sqp algorithms numerical results are presented indicating that our algorithm is reliable robust and has the potential to be used as a building block to construct trust region algorithms for small sized problems in constrained optimization

local time on the exceptional set of dynamical percolation and the incipient infinite cluster in dynamical critical site percolation on the triangular lattice or bond percolation on z2 we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster we show that at a typical time with respect to this measure the percolation configuration has the law of kesten s incipient infinite cluster in the most technical result of this paper we show that on the other hand at the first exceptional time the law of the configuration is different we believe that the two laws are mutually singular but do not show this we also study the collapse of the infinite cluster near typical exceptional times and establish a relation between static and dynamic exponents analogous to kesten s near critical relation

on local solutions of the celis dennis tapia subproblem we discuss the distribution of the local solutions of the celis dennis tapia cdt subproblem which appears in some trust region algorithms for nonlinear optimization we also give some examples to show the differences between the cdt subproblem and the single ball constraint subproblem these results show that the complexity of the cdt subproblem does not depend on the complexity of the structure of the dual plane thus they provide the possibility to search for the global minimizer in the dual plane

relations between invasion percolation and critical percolation in two dimensions we study invasion percolation in two dimensions we compare connectivity properties of the origin s invaded region to those of a the critical percolation cluster of the origin and b the incipient infinite cluster to exhibit similarities we show that for any k 1 the k point function of the first socalled pond has the same asymptotic behavior as the probability that k points are in the critical cluster of the origin more prominent though are the differences we show that there are infinitely many ponds that contain many large disjoint pc open clusters further for k 1 we compute the exact decay rate of the distribution of the radius of the kth pond and see that it differs from that of the radius of the critical cluster of the origin we finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular

the supercritical phase of percolation is well behaved we prove a general result concerning the critical probabilities of subsets of a lattice l it is a consequence of this result that the critical probability of a percolation process on l equals the limit of the critical probability of a slice of l as the thickness of the slice tends to infinity this verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation

incorporating nonmonotone strategies into the trust region method for unconstrained optimization this paper concerns a nonmonotone line search technique and its application to the trust region method for unconstrained optimization problems in our line search technique the current nonmonotone term is a convex combination of the previous nonmonotone term and the current objective function value instead of an average of the successive objective function values that was introduced by zhang and hager h zhang w w hager a nonmonotone line search technique and its application to unconstrained optimization siam j optim 14 4 2004 1043 1056 we incorporate this nonmonotone scheme into the traditional trust region method such that the new algorithm possesses nonmonotonicity unlike the traditional trust region method our algorithm performs a nonmonotone line search to find a new iteration point if a trial step is not accepted instead of resolving the subproblem under mild conditions we prove that the algorithm is global and superlinear convergence holds primary numerical results are reported

an ellipsoidal branch and bound algorithm for global optimization a branch and bound algorithm is developed for global optimization branching in the algorithm is accomplished by subdividing the feasible set using ellipses lower bounds are obtained by replacing the concave part of the objective function by an affine underestimate a ball approximation algorithm obtained by generalizing of a scheme of lin and han is used to solve the convex relaxation of the original problem the ball approximation algorithm is compared to sedumi as well as to gradient projection algorithms using randomly generated test problems with a quadratic objective and ellipsoidal constraints

invasion percolation a new form of percolation theory a new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system the model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces but in principle it may be applied to any kind of invasion process which proceeds along a path at least resistance the name invasion percolation is proposed for this new process similarities to and differences from ordinary percolation theory are discussed

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

central limit theorem for first passage percolation time across thin cylinders we prove that first passage percolation times across thin cylinders of the form 0 n hn hn d 1 obey gaussian central limit theorems as long as hn grows slower than n1 d 1 it is an open question as to what is the fastest that hn can grow so that a gaussian clt still holds under the natural but unproven assumption about existence of fluctuation and transversal exponents and strict convexity of the limiting shape in the direction of 1 0 0 we prove that in dimensions 2 and 3 the clt holds all the way up to the height of the unrestricted geodesic we also provide some numerical evidence in support of the conjecture in dimension 2

a trust region method for norm constrained problems a trust region method for the solution of nonlinear optimization problems with norm constraints is analyzed such optimization problems arise in parameter identification and nonlinear eigenvalue problems the characteristics of these applications motivate the formulation and analysis of the trust region method the algorithms studied here allow for inexact gradient information and the use of subspace methods for the approximate solution of subproblems characterizations and the descent properties of trust region steps are given criteria for the existence of successful iterations under inexact gradient information and under the use of subspace methods are established and global convergence of the method is proven

nonmonotone trust region method for solving optimization problems in this paper a trust region tr method with nonmonotone technique for optimization is proposed we construct a new ratio of actual descent and predicted descent which is a simple and natural generalization of the modified armijo line search rule the paper exposes the relationship between the trust region method and line search approach since this method possesses the robust properties of trust region subproblem it is globally convergent although we employ the nonmonotone sequence of function values instead of the monotone sequence in addition the proof of convergence is obviously simpler than one of newton type method with nonmonotone line search finally applications of the nonmonotone tr algorithm to some optimization problems are discussed

second order cone constraints for extended trust region subproblems the classical trust region subproblem trs minimizes a nonconvex quadratic objective over the unit ball in this paper we consider extensions of trs having extra constraints when two parallel cuts are added to trs we show that the resulting nonconvex problem has an exact representation as a semidenite program with additional linear and second order cone constraints for the case where an additional ellipsoidal constraint is added to trs resulting in the two trust region subproblem ttrs we provide a new relaxation including second order cone constraints that strengthens the usual sdp relaxation

scaling limit of the invasion percolation cluster on a regular tree we prove existence of the scaling limit of the invasion percolation cluster ipc on a regular tree the limit is a random real tree with a single end the contour and height functions of the limit are described as certain diusive stochastic processes this convergence allows us to recover and make precise certain asymptotic results for the ipc in particular we relate the limit of the rescaled level sets of the ipc to the local time of the scaled height function

the scaling limit of the minimum spanning tree of the complete graph consider the minimum spanning tree mst of the complete graph with nn vertices when edges are assigned independent random weights endow this tree with the graph distance renormalized by n1 3n1 3 and with the uniform measure on its vertices we show that the resulting space converges in distribution as n n to a random compact measured metric space in the gromov hausdorff prokhorov topology we additionally show that the limit is a random binary rr tree and has minkowski dimension 33 almost surely in particular its law is mutually singular with that of the brownian continuum random tree or any rescaled version thereof our approach relies on a coupling between the mst problem and the erd s renyi random graph we exploit the explicit description of the scaling limit of the erd s renyi random graph in the so called critical window established in probab theory related fields 152 2012 367 406 and provide a similar description of the scaling limit for a critical minimum spanning forest contained within the mst in order to accomplish this we introduce the notion of rr graphs which generalise rr trees and are of independent interest

the incipient infinite cluster for high dimensional unoriented percolation we consider bond percolation on mathbb z d at the critical occupation density p c for d 6 in two different models the first is the nearest neighbor model in dimension d 6 the second model is a spread out model having long range parameterized by l in dimension d 6 in the spread out case we show that the cluster of the origin conditioned to contain the site x weakly converges to an infinite cluster as x when d 6 and l is sufficiently large we also give a general criterion for this convergence to hold which is satisfied in the case d 6 in the nearest neighbor model by work of hara 12 we further give a second construction by taking p 6 the probability for invasion percolation to reach a site x is asymptotic to c x d 4 as x

invasion percolation on regular trees we consider invasion percolation on a rooted regular tree for the infinite cluster invaded from the root we identify the scaling behavior of its r point function for any r 2 and of its volume both at a given height and below a given height we find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation the scaling functions differ thus somewhat surprisingly the two clusters behave differently in fact we prove that their laws are mutually singular in addition we derive scaling estimates for simple random walk on the cluster starting from the root we show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster far above the root the two clusters have the same law locally but not globally a key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster these weights decay toward the critical value for ordinary percolation but only slowly and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster

on some properties of quadratic programs with a convex quadratic constraint in this paper we consider the problem of minimizing a possibly nonconvex quadratic function with a quadratic constraint we point out some new properties of the problem in particular in the first part of the paper we show that i given a kkt point that is not a global minimizer it is easy to find a better feasible point ii strict complementarity holds at the local nonglobal minimizer in the second part of this paper we show that the original constrained problem is equivalent to the unconstrained minimization of a piecewise quartic merit function using the unconstrained formulation we give in the nonconvex case a new second order necessary condition for global minimizers in the third part of this paper algorithmic applications of the preceding results are briefly outlined and some preliminary numerical experiments are reported

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

local minimizers of quadratic functions on euclidean balls and spheres in this paper a characterization of the local nonglobal minimizes of a quadratic function defined on a euclidean ball or a sphere is given it is proven that there exists at most one local nonglobal minimizes and that the lagrange multiplier that corresponds to this minimizes is the largest solution of a nonlinear scalar equation an algorithm is proposed for computing the local nonglobal minimizes

phase space geometry and slow dynamics we describe a non arrhenius mechanism for the slowing down of dynamics that is inherent to the high dimensionality of the phase space we show that such a mechanism is at work both in a family of mean field spin glass models without any domain structure and in the case of ferromagnetic domain growth the marginality of spin glass dynamics as well as the existence of a quasi equilibrium regime can be understood within this scenario we discuss the question of ergodicity in an out of equilibrium situation

bernoulli percolation above threshold an invasion percolation analysis on utilise le processus de percolation d invasion pour demontrer des resultats relatifs a la percolation de bernoulli sur z d

exponential growth of ponds in invasion percolation on regular trees in invasion percolation the edges of successively maximal weight the outlets divide the invasion cluster into a chain of ponds separated by outlets on the regular tree the ponds are shown to grow exponentially with law of large numbers central limit theorem and large deviation results the tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster with a logarithmic correction

solving generalized cdt problems via two parameter eigenvalues we consider solving a nonconvex quadratic minimization problem with two quadratic constraints one of which being convex this problem is a generalization of the celis denis tapia cdt problem and thus we refer to it as gcdt generalized cdt the cdt problem has been widely studied but no polynomial time algorithm was known until bienstock s recent work his algorithm solves the cdt problem in polynomial time with respect to the number of bits in data and log epsilon 1 by admitting an epsilon error in the constraints the algorithm however appears to be difficult to implement in this paper we present another algorithm for gcdt which is guaranteed to find a global solution for almost all gcdt instances and slightly perturbed ones in some exceptionally rare cases in exact arithmetic including eigenvalue computation our algorithm is based on the approach proposed by iwata nakatsukasa and takeda 2015 for computing the signed distance between overlapping ellipsoids our algorithm computes all the lagrange multipliers of gcdt by solving a two parameter linear eigenvalue problem obtains the corresponding kkt points and finds a global solution as the kkt point with the smallest objective value in practice in finite precision arithmetic our algorithm requires o n 6 log log u 1 computational time where n is the number of variables and u is the unit roundoff although we derive our algorithm under the unrealistic assumption that exact eigenvalues can be computed numerical experiments illustrate that our algorithm performs well in finite precision arithmetic

computing the local nonglobal minimizer of a large scale trust region subproblem we propose two algorithms for computing the local nonglobal minimizer of a quadratic function subject to a constraint set that is a euclidean sphere we also discuss the case where the constraint set is a euclidean ball at each iteration of the algorithms we compute the two smallest eigenvalues of a parametric matrix using an arpack subroutine only matrix vector multiplications are required hence we are able to exploit the possible sparsity of the hessian matrix of the quadratic objective making the algorithms suitable for large problems this improves previous approaches based on matrix factorizations we also give a geometric relationship based on extremal ellipsoids between the global and the local nonglobal minimizers of the quadratic function under the given sphere constraint

construction of the incipient infinite cluster for spread out oriented percolation above 4 1 dimensions we construct the incipient infinite cluster measure iic for sufficiently spread out oriented percolation on d for d 1 4 1 we consider two different constructions for the first construction we define n e by taking the probability of the intersection of an event e with the event that the origin is connected to x n d summing this probability over x d and normalising the sum to get a probability measure we let n and prove existence of a limiting measure the iic for the second construction we condition the connected cluster of the origin in critical oriented percolation to survive to time n and let n under the assumption that the critical survival probability is asymptotic to a multiple of n 1 we prove existence of a limiting measure with in addition we study the asymptotic behaviour of the size of the level set of the cluster of the origin and the dimension of the cluster of the origin under our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super brownian motion for d 1 4 1

an approach to identification for robust control given measured data generated by a discrete time linear system we propose a model consisting of a linear time invariant system affected by norm bounded perturbation under mild assumptions the plants belonging to the resulting uncertain family form a convex set the approach depends on two key parameters an a priori given bound of the perturbation and the input used to generate the data it turns out that the size of the uncertain family can be reduced by intersecting the model families obtained by making use of different inputs the model validation problem in this identification scheme is analyzed for a given energy level the invalidation problem yields the family of those models which can never be invalidated for any possible input of fixed energy and any possible perturbation this leads to the intersection of all uncertain families a consequence of the invalidation problem is that for finite length measurements not all models can be invalidated using fixed energy inputs

a note on quadratic forms we extend an interesting theorem of yuan 12 for two quadratic forms to three matrices let c 1 c 2 c 3 be three symmetric matrices in n n if max x t c 1 x x t c 2 x x t c 3 x 0 for all x n it is proved that there exist t i 0 i 1 2 3 such that i 1 3 t i 1 and i 1 3 t i c i has at most one negative eigenvalue

a central limit theorem with applications to percolation epidemics and boolean models suppose x x x x zd is a white noise process and h b defined for finite subsets b of z d is determined in a stationary way by the restriction of x to b using a martingale approach we prove a central limit theorem clt for h as b becomes large subject to h satisfying a stabilization condition the effect of changing x x at a single site needs to be local this clt is then applied to component counts for percolation and boolean models to the size of the big cluster for percolation on a box and to the final size of a spatial epidemic

a constrained least squares regularization method for nonlinear ill posed problems this paper deals with a method for solving ill posed nonlinear hilbert space operator equations f x y regularization is obtained by solving a constrained least squares regularization problem min f x y 2 quad text subject to j x leqq beta 2 beta serves as a regularization parameter and j x is a quadratic penalty functional to robustly and efficiently solve this regularization problem we apply a trust region method at each iteration the quadratic penalty constraint is retained a gauss newton approximation to the objective functional is taken and we add a quadratic trust region constraint the resulting quadratic subproblem is then reformulated as a nonlinear complementarily problem and solved using newton s method this paper applies methods to find approximate solutions to a severely ill posed nonlinear first kind integral equation arising in geophysics the method of generalized cross validation gcv is used to pick the regularization parameter when random er

random walk in random environment a counterexample we describe a family of random walks in random environments which have exponentially decaying correlations nearest neighbor transition probabilities which are bounded away from 0 and yet are subdiffusive in any dimensiond

on the speed of convergence for two dimensional first passage ising percolation consider first passage ising percolation on z2 let denote the reciprocal temperature and let h denote an external magnetic field denote by 8 the critical temperature and for j3 t let h f h sup h 0 3 h 0 where 0 8 h is the probability that the origin is connected by an infinite cluster with these definitions let us consider first passage ising percolation on z2 let ao 7 denote the first passage time from 0 0 to n 0 it follows from a subadditive argument that

retention capacity of random surfaces we introduce a water retention model for liquids captured on a random surface with open boundaries and investigate the model for both continuous and discrete surface heights 0 1 n 1 on a square lattice with a square boundary the model is found to have several intriguing features including a nonmonotonic dependence of the retention on the number of levels for many n the retention is counterintuitively greater than that of an n 1 level system the behavior is explained using percolation theory by mapping it to a 2 level system with variable probability results in one dimension are also found

inhomogeneous percolation problems and incipient infinite clusters the authors consider inhomogeneous percolation models with density pc f x and examine the forms of f x which produce incipient structures taking f x approximately mod x mod lambda and assuming the existence of a correlation length exponent v for the homogeneous percolation model they prove that in d 2 the borderline value of lambda is lambda b 1 v if lambda 1 v then with probability one there is no infinite cluster while if lambda 2 the models studied suggest what sort of incipient objects should be examined in random surface models

an alternative theorem for quadratic forms and extensions abstract this note concerns an alternative theorem for quadratic forms established recently by y yuan we explore some directions in which yuan s result can be extended and we indicate also some important situations where an alternative theorem a la yuan does not hold

mesh independence for nonlinear least squares problems with norm constraints if one solves an infinite dimensional optimization problem by introducing discretizations and applying a solution method to the resulting finite dimensional problem one often observes the very stable behavior of this method with respect to varying discretizations the most striking observation is the constancy of the number of iterations needed to satisfy a given stopping criterion in this paper an analysis of these phenomena is given and the so called mesh independence for non linear least squares problems with norm constraints ncnlls is proved a gauss newton method for the solution of ncnlls is discussed and its convergence properties are analyzed the mesh independence is proven in its sharpest formulation sufficient conditions for the mesh independence to hold are related to conditions guaranteeing convergence of the gauss newton method the results are demonstrated on a two point boundary value problem

coincidence and maximal element theorems and their applications to generalized equilibrium problems and minimax inequalities abstract in this paper we obtain first a very general coincidence theorem from this we derive a new coincidence theorem and two alternative theorems concerning existence of maximal elements applications of these results to generalized equilibrium problems and minimax inequalities are given in the last sections

shape fluctuations are different in different directions we consider the first passage percolation model on z 2 in this model we assign independently to each edge e a passage time t e with a common distribution f let t u v be the passage time from u to v in this paper we show that whenever f 0 0 and f r p c it is known that there exists m such that for all n 2 t 0 0 n n m these results tell us that shape fluctuations not only depend on distribution f but also on direction when showing this result we find the following interesting geometrical property with the special distribution above any long piece with r edges in an optimal path from 0 0 to n 0 has to be very circuitous

a trust region algorithm for equality constrained optimization a trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function under certain conditions global convergence and local superlinear convergence results are proved

a global convergence theory for dennis el alem and maciel s class of trust region algorithms for constrained optimization without assuming regularity this work presents a convergence theory for dennis el alem and maciel s class of trust region based algorithms for solving the smooth nonlinear programming problem with equality constraints the results are proved under very mild conditions on the quasi normal and tangential components of the trial steps the lagrange multiplier estimates and the hessian estimates are assumed to be bounded no regularity assumption is made in particular linear independence of the gradients of the constraints is not assumed the theory proves global convergence for the class in particular it shows that a subsequence of the iteration sequence satisfies one of four types of mayer bliss stationary conditions in the limit

new results on quadratic minimization in this paper we present several new results on minimizing an indefinite quadratic function under quadratic linear constraints the emphasis is placed on the case in which the constraints are two quadratic inequalities this formulation is termed the extended trust region subproblem in this paper to distinguish it from the ordinary trust region subproblem in which the constraint is a single ellipsoid the computational complexity of the extended trust region subproblem in general is still unknown in this paper we consider several interesting cases related to this problem and show that for those cases the corresponding semidefinite programming relaxation admits no gap with the true optimal value and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization for the extended trust region subproblem itself we introduce a parameterized problem and prove the existence of a trajectory that will lead to an optimal solution combining this with a result obtained in the first part of the paper we propose a polynomial time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint

broken ergodicity and the geometry of rugged landscapes we study the nature and mechanisms of broken ergodicity be in specific random walk models corresponding to diffusion on random potential surfaces in both one dimension and high dimension using both rigorous results and nonrigorous methods we confirm several aspects of the standard be picture and show that others apply in one dimension but need to be modified in higher dimensions these latter aspects include the notions that a fixed temperature confining barriers increase logarithmically with time that components are necessarily bounded regions of state space which depend on the observational time scale and that the system continually revisits previously traversed regions of state space we examine our results in the context of several experiments and discuss some implications of our results for the dynamics of disordered and or complex systems

potpourri of conjectures and open questions in nonlinear analysis and optimization we present a collection of fourteen conjectures and open problems in the fields of nonlinear analysis and optimization these problems can be classified into three groups problems of pure mathematical interest problems motivated by scientific computing and applications and problems whose solutions are known but for which we would like to know better proofs for each problem we provide a succinct presentation a list of appropriate references and a view of the state of the art of the subject

the fractal volume of the two dimensional invasion percolation cluster we consider both invasion percolation and standard bernoulli bond percolation on thez2 lattice denote byv andc the invasion cluster and the occupied cluster of the origin respectively let mathcal v n mathcal v cap n n 2 and open image in new window let e 0 be given here we show that with a probability tending to 1 n 2 varepsilon pi n leqq mathcal v n leqq n 2 varepsilon pi n assuming the existence of an exponent 1 for n it can be seen that with probability tending to one n 2 1 rho varepsilon leqq mathcal v n leqq n 2 1 rho varepsilon moreover by den nijs and nienhuis et al s computations n 1 8958389583 varepsilon n 1 tfrac 43 48 varepsilon leqq mathcal v n leqq n 1 tfrac 43 48 varepsilon n 1 8958389583 varepsilon with a probability tending to one the result matches wilkinson and willemsen s numerical computation mathcal v n sim n 1 89 the method allows us also to show the same argument for any planar graph therefore any two planar invasion clusters have the same fractal dimension 2 1 if one believes universality

the incipient infinite cluster in two dimensional percolation letpp be the probability measure on the configurations of occupied and vacant vertices of a two dimensional graphg under which all vertices are independently occupied respectively vacant with probabilityp respectively 1 p letph be the critical probability for this system andw the occupied cluster of some fixed vertexw0 we show that for many graphsg such as mathbb z 2 or its covering graph which corresponds to bond percolation on mathbb z 2 the following two conditional probability measures converge and have the same limit v say

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

limit theorems for 2d invasion percolation we prove limit theorems and variance estimates for quantities related to ponds and outlets for 2d invasion percolation we first exhibit several properties of a sequence o n of outlet variables the nth of which gives the number of outlets in the box centered at the origin of side length 2n the most important of these properties describes the sequence s renewal structure and exponentially fast mixing behavior we use these to prove a central limit theorem and strong law of large numbers for o n we then show consequences of these limit theorems for the pond radii and outlet weights

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems