subgradient based neural networks for nonsmooth nonconvex optimization problems this paper presents a subgradient based neural network to solve a nonsmooth nonconvex optimization problem with a nonsmooth nonconvex objective function a class of affine equality constraints and a class of nonsmooth convex inequality constraints the proposed neural network is modeled with a differential inclusion under a suitable assumption on the constraint set and a proper assumption on the objective function it is proved that for a sufficiently large penalty parameter there exists a unique global solution to the neural network and the trajectory of the network can reach the feasible region in finite time and stay there thereafter it is proved that the trajectory of the neural network converges to the set which consists of the equilibrium points of the neural network and coincides with the set which consists of the critical points of the objective function in the feasible region a condition is given to ensure the convergence to the equilibrium point set in finite time moreover under suitable assumptions the coincidence between the solution to the differential inclusion and the ldquoslow solutionrdquo of it is also proved furthermore three typical examples are given to present the effectiveness of the theoretic results obtained in this paper and the good performance of the proposed neural network

fractal first order partial differential equations the present paper is concerned with semi linear partial differential equations involving a particular pseudo differential operator it investigates both fractal conservation laws and non local hamilton jacobi equations the idea is to combine an integral representation of the operator and duhamel s formula to prove on the one hand the key a priori estimates for the scalar conservation law and the hamilton jacobi equation and on the other hand the smoothing effect of the operator as far as hamilton jacobi equations are concerned a non local vanishing viscosity method is used to construct a viscosity solution when existence of regular solutions fails and a rate of convergence is provided turning to conservation laws global in time existence and uniqueness are established we also show that our formula allows us to obtain entropy inequalities for the non local conservation law and thus to prove the convergence of the solution as the non local term vanishes toward the entropy solution of the pure conservation law

pedestrian detection in images via cascaded l1 norm minimization learning method a new cascaded l1 norm minimization learning clml method for pedestrian detection in images is proposed in this paper the proposed clml method which is designed from the perspective of vapnic s theory in the statistical learning integrates feature selection with classifier construction via solving meaningful optimization models the method incorporates three stages weak classifier learning strong classifier learning and the cascaded classifier construction in the weak classifier learning the l1 norm minimization learning lml and min max penalty function model are presented in the strong classifier learning an integer programming optimization model is built equaling the reformulation of lml in the integer space finally a cascade of lml classifiers is constructed to promote detection speed during the classifier learning and pedestrian detection histograms of oriented gradients of variable sized blocks v hog are used as feature descriptors experimental results on the inria and sdl human datasets show that the proposed method achieves a higher performance and speed than the state of the art methods

fractional nonlinear degenerate diffusion equations on bounded domains part i existence uniqueness and upper bounds abstract we investigate quantitative properties of nonnegative solutions u t x 0 to the nonlinear fractional diffusion equation t u l f u 0 posed in a bounded domain x r n with appropriate homogeneous dirichlet boundary conditions as l we can use a quite general class of linear operators that includes the two most common versions of the fractional laplacian s 0 s 1 in a bounded domain with zero dirichlet boundary conditions but it also includes many other examples since our theory only needs some basic properties that are typical of linear heat semigroups the nonlinearity f is assumed to be increasing and is allowed to be degenerate the prototype is the power case f u u m 1 u with m 1 in this paper we propose a suitable class of solutions of the equation and cover the basic theory we prove existence uniqueness of such solutions and we establish upper bounds of two forms absolute bounds and smoothing effects as well as weighted l 1 estimates the class of solutions is very well suited for that work the standard laplacian case s 1 is included and the linear case m 1 can be recovered in the limit in a companion paper bonforte and vazquez in preparation we will complete the study with more advanced estimates like the upper and lower boundary behaviour and harnack inequalities for which the results of this paper are needed

nonlocal filtration equations with rough kernels abstract we study the nonlinear and nonlocal cauchy problem t u l u 0 in r n r u 0 u 0 where l is a levy type nonlocal operator with a kernel having a singularity at the origin as that of the fractional laplacian the nonlinearity is nondecreasing and continuous and the initial datum u 0 is assumed to be in l 1 r n we prove existence and uniqueness of weak solutions for a wide class of nonlinearities including the porous media case u u m 1 u m 1 these solutions turn out to be bounded and holder continuous for t 0 we also describe the large time behaviour when the nonlinearity resembles a power for u 0 and the kernel associated to l is close at infinity to that of the fractional laplacian

a one layer recurrent neural network for pseudoconvex optimization subject to linear equality constraints in this paper a one layer recurrent neural network is presented for solving pseudoconvex optimization problems subject to linear equality constraints the global convergence of the neural network can be guaranteed even though the objective function is pseudoconvex the finite time state convergence to the feasible region defined by the equality constraints is also proved in addition global exponential convergence is proved when the objective function is strongly pseudoconvex on the feasible region simulation results on illustrative examples and application on chemical process data reconciliation are provided to demonstrate the effectiveness and characteristics of the neural network

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

a novel recurrent neural network with finite time convergence for linear programming in this letter a novel recurrent neural network based on the gradient method is proposed for solving linear programming problems finite time convergence of the proposed neural network is proved by using the lyapunov method compared with the existing neural networks for linear programming the proposed neural network is globally convergent to exact optimal solutions in finite time which is remarkable and rare in the literature of neural networks for optimization some numerical examples are given to show the effectiveness and excellent performance of the new recurrent neural network

a dual neural network for redundancy resolution of kinematically redundant manipulators subject to joint limits and joint velocity limits in this paper a recurrent neural network called the dual neural network is proposed for online redundancy resolution of kinematically redundant manipulators physical constraints such as joint limits and joint velocity limits together with the drift free criterion as a secondary task are incorporated into the problem formulation of redundancy resolution compared to other recurrent neural networks the dual neural network is piecewise linear and has much simpler architecture with only one layer of neurons the dual neural network is shown to be globally exponentially convergent to optimal solutions the dual neural network is simulated to control the pa10 robot manipulator with effectiveness demonstrated

finite time convergent recurrent neural network with a hard limiting activation function for constrained optimization with piecewise linear objective functions this paper presents a one layer recurrent neural network for solving a class of constrained nonsmooth optimization problems with piecewise linear objective functions the proposed neural network is guaranteed to be globally convergent in finite time to the optimal solutions under a mild condition on a derived lower bound of a single gain parameter in the model the number of neurons in the neural network is the same as the number of decision variables of the optimization problem compared with existing neural networks for optimization the proposed neural network has a couple of salient features such as finite time convergence and a low model complexity specific models for two important special cases namely linear programming and nonsmooth optimization are also presented in addition applications to the shortest path problem and constrained least absolute deviation problem are discussed with simulation results to demonstrate the effectiveness and characteristics of the proposed neural network

a general fractional porous medium equation we develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion u t 2 u m 1 u 0 x r n t 0 u x 0 f x x r n we consider data f l 1 r n and all exponents 0 0 existence and uniqueness of a weak solution is established for m m n n giving rise to an l 1 contraction semigroup in addition we obtain the main qualitative properties of these solutions in the lower range 0 m m existence and uniqueness of solutions with good properties happen under some restrictions and the properties are different from the case above m we also study the dependence of solutions on f m and moreover we consider the above questions for the problem posed in a bounded domain

on numerical methods and error estimates for degenerate fractional convection diffusion equations first we introduce and analyze a convergent numerical method for a large class of nonlinear nonlocal possibly degenerate convection diffusion equations secondly we develop a new kuznetsov type theory and obtain general and possibly optimal error estimates for our numerical methods even when the principal derivatives have any fractional order between 1 and 2 the class of equations we consider includes equations with nonlinear and possibly degenerate fractional or general levy diffusion special cases are conservation laws fractional conservation laws certain fractional porous medium equations and new strongly degenerate equations

graph cuts via ell_1 norm minimization graph cuts have become an increasingly important tool for solving a number of energy minimization problems in computer vision and other fields in this paper the graph cut problem is reformulated as an unconstrained l1 norm minimization that can be solved effectively using interior point methods this reformulation exposes connections between graph cuts and other related continuous optimization problems eventually the problem is reduced to solving a sequence of sparse linear systems involving the laplacian of the underlying graph the proposed procedure exploits the structure of these linear systems in a manner that is easily amenable to parallel implementations experimental results obtained by applying the procedure to graphs derived from image processing problems are provided

a two layer recurrent neural network for nonsmooth convex optimization problems in this paper a two layer recurrent neural network is proposed to solve the nonsmooth convex optimization problem subject to convex inequality and linear equality constraints compared with existing neural network models the proposed neural network has a low model complexity and avoids penalty parameters it is proved that from any initial point the state of the proposed neural network reaches the equality feasible region in finite time and stays there thereafter moreover the state is unique if the initial point lies in the equality feasible region the equilibrium point set of the proposed neural network is proved to be equivalent to the karush kuhn tucker optimality set of the original optimization problem it is further proved that the equilibrium point of the proposed neural network is stable in the sense of lyapunov moreover from any initial point the state is proved to be convergent to an equilibrium point of the proposed neural network finally as applications the proposed neural network is used to solve nonlinear convex programming with linear constraints and l 1 norm minimization problems

uniqueness and properties of distributional solutions of nonlocal equations of porous medium type abstract we study the uniqueness existence and properties of bounded distributional solutions of the initial value problem for the anomalous diffusion equation t u l u 0 here l can be any nonlocal symmetric degenerate elliptic operator including the fractional laplacian and numerical discretizations of this operator the function r r is only assumed to be continuous and nondecreasing the class of equations include nonlocal generalized porous medium equations fast diffusion equations and stefan problems in addition to very general uniqueness and existence results we obtain stability l 1 contraction and a priori estimates we also study local limits continuous dependence and properties and convergence of a numerical approximation of our equations

the nonlocal porous medium equation barenblatt profiles and other weak solutions a degenerate nonlinear nonlocal evolution equation is considered it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal we show the existence of sign changing weak solutions to the corresponding cauchy problem moreover we construct explicit compactly supported self similar solutions which generalize barenblatt profiles the well known solutions of the classical porous medium equation

a one layer recurrent neural network with a discontinuous hard limiting activation function for quadratic programming in this paper a one layer recurrent neural network with a discontinuous hard limiting activation function is proposed for quadratic programming this neural network is capable of solving a large class of quadratic programming problems the state variables of the neural network are proven to be globally stable and the output variables are proven to be convergent to optimal solutions as long as the objective function is strictly convex on a set defined by the equality constraints in addition a sequential quadratic programming approach based on the proposed recurrent neural network is developed for general nonlinear programming simulation results on numerical examples and support vector machine svm learning show the effectiveness and performance of the neural network

a novel recurrent neural network for solving nonlinear optimization problems with inequality constraints this paper presents a novel recurrent neural network for solving nonlinear optimization problems with inequality constraints under the condition that the hessian matrix of the associated lagrangian function is positive semidefinite it is shown that the proposed neural network is stable at a karush kuhn tucker point in the sense of lyapunov and its output trajectory is globally convergent to a minimum solution compared with variety of the existing projection neural networks including their extensions and modification for solving such nonlinearly constrained optimization problems it is shown that the proposed neural network can solve constrained convex optimization problems and a class of constrained nonconvex optimization problems and there is no restriction on the initial point simulation results show the effectiveness of the proposed neural network in solving nonlinearly constrained optimization problems

the fisher kpp problem with doubly nonlinear diffusion abstract the famous fisher kpp reaction diffusion model combines linear diffusion with the typical kpp reaction term and appears in a number of relevant applications in biology and chemistry it is remarkable as a mathematical model since it possesses a family of travelling waves that describe the asymptotic behaviour of a large class solutions 0 u x t 1 of the problem posed in the real line the existence of propagation waves with finite speed has been confirmed in some related models and disproved in others we investigate here the corresponding theory when the linear diffusion is replaced by the slow doubly nonlinear diffusion and we find travelling waves that represent the wave propagation of more general solutions even when we extend the study to several space dimensions a similar study is performed in the critical case that we call pseudo linear i e when the operator is still nonlinear but has homogeneity one with respect to the classical model and the pseudo linear case the slow travelling waves exhibit free boundaries

on nonlocal quasilinear equations and their local limits we introduce a new class of quasilinear nonlocal operators and study equations involving these operators the operators are degenerate elliptic and may have arbitrary growth in the gradient included are new nonlocal versions of p laplace laplace mean curvature of graph and even strongly degenerate operators in addition to some nonlocal quasilinear operators appearing in the existing literature our main results are comparison uniqueness and existence results for viscosity solutions of linear and fully nonlinear equations involving these operators because of the structure of our operators especially the existence proof is highly non trivial and non standard we also identify the conditions under which the nonlocal operators converge to local quasilinear operators and show that the solutions of the corresponding nonlocal equations converge to the solutions of the local limit equations finally we give a formal stochastic representation formula for the solutions and provide many examples

uniqueness of very weak solutions for a fractional filtration equation abstract we prove existence and uniqueness of distributional bounded solutions to a fractional filtration equation in r d with regards to uniqueness it was shown even for more general equations in 22 that if two bounded solutions u w of 1 1 satisfy u w l 1 r d 0 t then u w we obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions for nonnegative initial data we first show that a minimal solution exists and then that any other solution must coincide with it a similar procedure is carried out for sign changing solutions as a consequence distributional solutions have locally finite energy

a widder s type theorem for the heat equation with nonlocal diffusion the main goal of this work is to prove that every non negative strong solutionu x t to the problem u t delta alpha 2 u 0 rm for x t in mathbb r n times 0 t 0 alpha 2 can be written as u x t int mathbb r n p t x y u y 0 dy where p t x frac 1 t n alpha p left frac x t 1 alpha right and p x int mathbb r n e i x cdot xi xi alpha d xi this result shows uniqueness in the setting of non negative solutions and extends some classical results for the heat equation by widder in 15 to the nonlocal diffusion framework

a recurrent neural network for solving nonlinear convex programs subject to linear constraints in this paper we propose a recurrent neural network for solving nonlinear convex programming problems with linear constraints the proposed neural network has a simpler structure and a lower complexity for implementation than the existing neural networks for solving such problems it is shown here that the proposed neural network is stable in the sense of lyapunov and globally convergent to an optimal solution within a finite time under the condition that the objective function is strictly convex compared with the existing convergence results the present results do not require lipschitz continuity condition on the objective function finally examples are provided to show the applicability of the proposed neural network

a fractional porous medium equation abstract we develop a theory of existence uniqueness and regularity for the following porous medium equation with fractional diffusion u t 1 2 u m 1 u 0 x r n t 0 u x 0 f x x r n with m m n 1 n n 1 and f l 1 r n an l 1 contraction semigroup is constructed and the continuous dependence on data and exponent is established nonnegative solutions are proved to be continuous and strictly positive for all x r n t 0

stochastic processes slaves and supersymmetry we extend the work of t nase nicola and kurchan on the structure of diffusion processes and the associated supersymmetry algebra by examining the responses of a simple statistical system to external disturbances of various kinds we consider both the stochastic differential equations sdes for the process and the associated diffusion equation the influence of the disturbances can be understood by augmenting the original sde with an equation for slave variables the evolution of the slave variables describes the behaviour of line elements carried along in the stochastic flow these line elements together with the associated surface and volume elements constructed from them provide the basis of the supersymmetry properties of the theory for ease of visualization and in order to emphasize a helpful electromagnetic analogy we work in three dimensions the results are all generalizable to higher dimensions and can be specialized to one and two dimensions the electromagnetic analogy is a useful starting point for calculating asymptotic results at low temperature that can be compared with direct numerical evaluations we also examine the problems that arise in a direct numerical simulation of the stochastic equation together with the slave equations we pay special attention to the dependence of the slave variable statistics on temperature we identify in specific models the critical temperature below which the slave variable distribution ceases to have a variance and consider the effect on estimates of susceptibilities

a one layer recurrent neural network with a discontinuous activation function for linear programming a one layer recurrent neural network with a discontinuous activation function is proposed for linear programming the number of neurons in the neural network is equal to that of decision variables in the linear programming problem it is proven that the neural network with a sufficiently high gain is globally convergent to the optimal solution its application to linear assignment is discussed to demonstrate the utility of the neural network several simulation examples are given to show the effectiveness and characteristics of the neural network

worst case complexity of smoothing quadratic regularization methods for non lipschitzian optimization in this paper we propose a smoothing quadratic regularization sqr algorithm for solving a class of nonsmooth nonconvex perhaps even non lipschitzian minimization problems which has wide applications in statistics and sparse reconstruction the proposed sqr algorithm is a first order method at each iteration the sqr algorithm solves a strongly convex quadratic minimization problem with a diagonal hessian matrix which has a simple closed form solution that is inexpensive to calculate we show that the worst case complexity of reaching an epsilon scaled stationary point is o epsilon 2 moreover if the objective function is locally lipschitz continuous the sqr algorithm with a slightly modified updating scheme for the smoothing parameter and iterate can obtain an epsilon clarke stationary point in at most o epsilon 3 iterations

dynamical analysis of neural networks of subgradient system in this technical note we consider a class of neural network which is a generalization of neural network models considered in the optimization context under some mild assumptions this neural network can be translated into a negative subgradient dynamical system at first we study the existence and uniqueness of solution of this neural network then by nonsmooth ojasiewicz inequality we prove the convergence of the trajectories of this neural network in the end a constrained minimization problem is studied which can be associated with this neural network it is proved that the local constrained minimum of the cost function coincides with the stable equilibria point of this neural network

uniqueness of entropy solutions to fractional conservation laws with fully infinite speed of propagation our goal is to study the uniqueness of bounded entropy solutions for a multidimensional conservation law including a non lipschitz convection term and a diffusion term of nonlocal porous medium type the nonlocality is given by a fractional power of the laplace operator for a wide class of nonlinearities the l 1 contraction principle is established despite the fact that the finite infinite speed of propagation alibaud jee 2007 cannot be exploited in our framework existence is deduced with perturbation arguments the method of proof adapted from andreianov maliki nodea 2010 requires a careful analysis of the action of the fractional laplacian on truncations of radial powers

entropy solutions for nonlinear degenerate problems we consider a class of elliptic hyperbolic degenerate equations g u delta b u divg phi u f with dirichlet homogeneous boundary conditions and a class of elliptic parabolic hyperbolic degenerate equations g u t delta b u divg phi u f with homogeneous dirichlet conditions and initial conditions existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function satisfying rather general conditions comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for continuous

the cauchy problem for ut δum when 0 m 1 on considere le probleme de cauchy pour l equation de diffusion non lineaire u t u u m 1 0 sur 0 r n u 0 u 0 quand 0 m 1 on montre qu il existe une solution temporelle globale pour une fonction localement integrable u 0 on montre l unicite dans cette classe

quantitative local and global a priori estimates for fractional nonlinear diffusion equations abstract we establish quantitative estimates for solutions u t x to the fractional nonlinear diffusion equation t u s u m 0 in the whole range of exponents m 0 0 s 1 the equation is posed in the whole space x r d we first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data in the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions depending only on the norm of the initial data in a certain ball the estimates take a different form in three exponent ranges slow diffusion good range of fast diffusion and very fast diffusion finally we show existence and uniqueness of initial traces

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems