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

convergence of neural networks for programming problems via a nonsmooth lojasiewicz inequality this paper considers a class of neural networks nns for solving linear programming lp problems convex quadratic programming qp problems and nonconvex qp problems where an indefinite quadratic objective function is subject to a set of affine constraints the nns are characterized by constraint neurons modeled by ideal diodes with vertical segments in their characteristic which enable to implement an exact penalty method a new method is exploited to address convergence of trajectories which is based on a nonsmooth lstrokojasiewicz inequality for the generalized gradient vector field describing the nn dynamics the method permits to prove that each forward trajectory of the nn has finite length and as a consequence it converges toward a singleton furthermore by means of a quantitative evaluation of the lstrokojasiewicz exponent at the equilibrium points the following results on convergence rate of trajectories are established 1 for nonconvex qp problems each trajectory is either exponentially convergent or convergent in finite time toward a singleton belonging to the set of constrained critical points 2 for convex qp problems the same result as in 1 holds moreover the singleton belongs to the set of global minimizers and 3 for lp problems each trajectory converges in finite time to a singleton belonging to the set of global minimizers these results which improve previous results obtained via the lyapunov approach are true independently of the nature of the set of equilibrium points and in particular they hold even when the nn possesses infinitely many nonisolated equilibrium points

a second order multi agent network for bound constrained distributed optimization this technical note presents a second order multi agent network for distributed optimization with a sum of convex objective functions subject to bound constraints in the multi agent network the agents connect each others locally as an undirected graph and know only their own objectives and constraints the multi agent network is proved to be able to reach consensus to the optimal solution under mild assumptions moreover the consensus of the multi agent network is converted to the convergence of a dynamical system which is proved using the lyapunov method compared with existing multi agent networks for optimization the second order multi agent network herein is capable of solving more general constrained distributed optimization problems simulation results on two numerical examples are presented to substantiate the performance and characteristics of the multi agent network

higher traces noncommutative motives and the categorified chern character abstract we propose a categorification of the chern character that refines earlier work of toen and vezzosi and of ganter and kapranov if x is an algebraic stack our categorified chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over x which we introduce to s 1 equivariant perfect complexes on the derived free loop stack l x as an application of the theory we show that toen and vezzosi s secondary chern character factors through secondary k theory our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal n categories which is of independent interest

one layer continuous and discrete time projection neural networks for solving variational inequalities and related optimization problems this paper presents one layer projection neural networks based on projection operators for solving constrained variational inequalities and related optimization problems sufficient conditions for global convergence of the proposed neural networks are provided based on lyapunov stability compared with the existing neural networks for variational inequalities and optimization the proposed neural networks have lower model complexities in addition some improved criteria for global convergence are given compared with our previous work a design parameter has been added in the projection neural network models and it results in some improved performance the simulation results on numerical examples are discussed to demonstrate the effectiveness and characteristics of the proposed neural networks

uniqueness of the multiplicative cyclotomic trace abstract making use of the theory of noncommutative motives we characterize the topological dennis trace map as the unique multiplicative natural transformation from algebraic k theory to topological hochschild homology thh and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology tc moreover we prove that the space of all multiplicative structures on algebraic k theory is contractible we also show that the algebraic k theory functor from small stable categories to spectra is lax symmetric monoidal which in particular implies that e n ring spectra give rise to e n 1 ring algebraic k theory spectra along the way we develop a multiplicative morita theory establishing a symmetric monoidal equivalence between the category of small idempotent complete stable categories and the morita localization of the category of spectral categories

sliding modes in solving convex programming problems sliding modes are used to analyze a class of dynamical systems that solve convex programming problems the analysis is carried out using concepts from the theory of differential equations with discontinuous right hand sides and lyapunov stability theory it is shown that the equilibrium points of the system coincide with the minimizers of the convex programming problem and that irrespective of the initial state of the system the state trajectory converges to the solution set of the problem the dynamic behavior of the systems is illustrated by two numerical examples

automatic integration of systems of second order ode s separated into subsystems this paper continues to lay the foundation for runge kutta type methods for integrating systems of ode s in which some of the variables are more slowly varying than others within the specified error bounds second rather than first order systems are considered because they permit the development of more efficient algorithms it is assumed that the system has been separated into several subsystems requiring different stepsizes after the presentation of general theorems and a stepsize control analysis a particular runge kutta type method is developed and applied to two example problems including a perturbation problem it is shown that a wide class of perturbation problems may be solved in a like manner

neural network for nonsmooth nonconvex constrained minimization via smooth approximation a neural network based on smoothing approximation is presented for a class of nonsmooth nonconvex constrained optimization problems where the objective function is nonsmooth and nonconvex the equality constraint functions are linear and the inequality constraint functions are nonsmooth convex this approach can find a clarke stationary point of the optimization problem by following a continuous path defined by a solution of an ordinary differential equation the global convergence is guaranteed if either the feasible set is bounded or the objective function is level bounded specially the proposed network does not require 1 the initial point to be feasible 2 a prior penalty parameter to be chosen exactly 3 a differential inclusion to be solved numerical experiments and comparisons with some existing algorithms are presented to illustrate the theoretical results and show the efficiency of the proposed network

derived azumaya algebras and generators for twisted derived categories we introduce a notion of derived azumaya algebras over ring and schemes generalizing the notion of azumaya algebras of grothendieck le groupe de brauer i algebres d azumaya et interpretations diverses dix exposes sur la cohomologie des schemas pp 46 66 north holland amsterdam 1968 we prove that any such algebra b on a scheme x provides a class b in h 1 et x mathbb z times h 2 et x mathbb g m we prove that for x a quasi compact and quasi separated scheme defines a bijective correspondence and in particular that any class in h 2 et x mathbb g m torsion or not can be represented by a derived azumaya algebra on x our result is a consequence of a more general theorem about the existence of compact generators in twisted derived categories with coefficients in any local system of reasonable dg categories generalizing the well known existence of compact generators in derived categories of quasi coherent sheaves of bondal and van den bergh mosc math j 3 1 1 36 2003 a huge part of this paper concerns the treatment of twisted derived categories as well as the proof that the existence of compact generator locally for the fppf topology implies the existence of a global compact generator we present explicit examples of derived azumaya algebras that are not represented by classical azumaya algebras as well as applications of our main result to the localization for twisted algebraic k theory and to the stability of saturated dg categories by direct push forwards along smooth and proper maps

solving systems of linear equations via gradient systems with discontinuous righthand sides application to ls svm a gradient system with discontinuous righthand side that solves an underdetermined system of linear equations in the l sub 1 norm is presented an upper bound estimate for finite time convergence to a solution set of the system of linear equations is shown by means of the persidskii form of the gradient system and the corresponding nonsmooth diagonal type lyapunov function this class of systems can be interpreted as a recurrent neural network and an application devoted to solving least squares support vector machines ls svm is used as an example

a projection neural network for constrained quadratic minimax optimization this paper presents a projection neural network described by a dynamic system for solving constrained quadratic minimax programming problems sufficient conditions based on a linear matrix inequality are provided for global convergence of the proposed neural network compared with some of the existing neural networks for quadratic minimax optimization the proposed neural network in this paper is capable of solving more general constrained quadratic minimax optimization problems and the designed neural network does not include any parameter moreover the neural network has lower model complexities the number of state variables of which is equal to that of the dimension of the optimization problems the simulation results on numerical examples are discussed to demonstrate the effectiveness and characteristics of the proposed neural network

a one layer projection neural network for nonsmooth optimization subject to linear equalities and bound constraints this paper presents a one layer projection neural network for solving nonsmooth optimization problems with generalized convex objective functions and subject to linear equalities and bound constraints the proposed neural network is designed based on two projection operators linear equality constraints and bound constraints the objective function in the optimization problem can be any nonsmooth function which is not restricted to be convex but is required to be convex pseudoconvex on a set defined by the constraints compared with existing recurrent neural networks for nonsmooth optimization the proposed model does not have any design parameter which is more convenient for design and implementation it is proved that the output variables of the proposed neural network are globally convergent to the optimal solutions provided that the objective function is at least pseudoconvex simulation results of numerical examples are discussed to demonstrate the effectiveness and characteristics of the proposed neural network

a generalized neural network for solving a class of minimax optimization problems with linear constraints abstract in this paper a generalized neural network was proposed based on projection method and differential inclusions which is contributed to solve a class of minimax optimization problems with linear constraints it is proved that the solution trajectory can converge to the feasible region in the finite time when the initial point is not in the feasible region once the solution trajectory reaches the feasible region it will stay therein thereafter in addition we investigate the global convergence and exponential convergence furthermore three illustrative examples are given to show the efficiency of the proposed neural network

a recurrent neural network for nonlinear optimization with a continuously differentiable objective function and bound constraints this paper presents a continuous time recurrent neural network model for nonlinear optimization with any continuously differentiable objective function and bound constraints quadratic optimization with bound constraints is a special problem which can be solved by the recurrent neural network the proposed recurrent neural network has the following characteristics 1 it is regular in the sense that any optimum of the objective function with bound constraints is also an equilibrium point of the neural network if the objective function to be minimized is convex then the recurrent neural network is complete in the sense that the set of optima of the function with bound constraints coincides with the set of equilibria of the neural network 2 the recurrent neural network is primal and quasiconvergent in the sense that its trajectory cannot escape from the feasible region and will converge to the set of equilibria of the neural network for any initial point in the feasible bound region 3 the recurrent neural network has an attractivity property in the sense that its trajectory will eventually converge to the feasible region for any initial states even at outside of the bounded feasible region 4 for minimizing any strictly convex quadratic objective function subject to bound constraints the recurrent neural network is globally exponentially stable for almost any positive network parameters simulation results are given to demonstrate the convergence and performance of the proposed recurrent neural network for nonlinear optimization with bound constraints

formulas for modular integration of systems of ode s abstract formulas are presented for modular integration of systems of first or second order ordinary differential equations which have been separated into subsystems using these formulas each subsystem or module may be integrated essentially independently using an appropriate processor or specialized integration technique included is a general mathematical procedure for devising formulas for modular integration depending on the order of accuracy required and other considerations

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

a neural network for a class of convex quadratic minimax problems with constraints in this paper we propose a neural network for solving a class of convex quadratic minimax problems with constraints four sufficient conditions are provided to ensure the asymptotic stability of the proposed network furthermore the exponential stability of the proposing network is also proved under certain conditions the results obtained here can be further extended to the globally projected dynamical system in addition some new stability conditions for the system are also obtained since our stability conditions can be easily checked in practice these results becomes more attractive in real applications

deriving sufficient conditions for global asymptotic stability of delayed neural networks via nonsmooth analysis ii in this paper we obtain new sufficient conditions ensuring existence uniqueness and global asymptotic stability gas of the equilibrium point for a general class of delayed neural networks dnns via nonsmooth analysis which makes full use of the lipschitz property of functions defining dnns based on this new tool of nonsmooth analysis we first obtain a couple of general results concerning the existence and uniqueness of the equilibrium point then those results are applied to show that existence assumptions on the equilibrium point in some existing sufficient conditions ensuring gas are actually unnecessary and some strong assumptions such as the boundedness of activation functions in some other existing sufficient conditions can be actually dropped finally we derive some new sufficient conditions which are easy to check comparison with some related existing results is conducted and advantages are illustrated with examples throughout our paper spectral properties of the matrix a a sup spl tau play an important role which is a distinguished feature from previous studies here a and a sup spl tau are respectively the feedback and the delayed feedback matrix defining the neural network under consideration

a runge kutta method with stepsize control for separated systems of first order odes abstract this paper is concerned with a method developed earlier that may be used to solve a system of first order odes separated into two subsystems each system may be integrated essentially independently of the other one a runge kutta technique with automatic stepsize control is developed for the method taking both truncation and separation errors into account the paper also provides a proof of the existence of a solution to the separated problem

convergence analyses on sparse feedforward neural networks via group lasso regularization in this paper a new variant of feedforward neural networks has been proposed for a class of nonsmooth optimization problems the penalty term of the presented neural networks stems from the group lasso method which selects hidden variables in a grouped manner to deal with the non differentiability of the original penalty term l 1 l 2 norm and avoid oscillations smoothing techniques have been used to approximate the objective function it is assumed that the training samples are supplied to the networks in a specific incremental way during training that is in each cycle samples are supplied in a fixed order then under suitable assumptions on learning rate penalization coefficients and smoothing parameters the weak and strong convergence of the training process for the smoothing neural networks have been proved the convergence analysis shows that the gradient of the smoothing error function approaches zero and the weight sequence converges to a fixed point respectively we demonstrate how the smoothing approximation parameter can be updated in the training procedure so as to guarantee the convergence of the procedure to a clarke stationary point of the original optimization problem in addition we have proved that the original nonsmoothing algorithm with l 1 l 2 norm penalty converges consistently to the same optimum solution with the corresponding smoothed algorithm numerical simulations demonstrate the convergence and effectiveness of the proposed training algorithm

universal homotopy theories abstract begin with a small category c the goal of this short note is to point out that there is such a thing as a universal model category built from c we describe applications of this to the study of homotopy colimits the dwyer kan theory of framings sheaf theory and the homotopy theory of schemes

a two time scale neurodynamic approach to constrained minimax optimization this paper presents a two time scale neurodynamic approach to constrained minimax optimization using two coupled neural networks one of the recurrent neural networks is used for minimizing the objective function and another is used for maximization it is shown that the coupled neurodynamic systems operating in two different time scales work well for minimax optimization the effectiveness and characteristics of the proposed approach are illustrated using several examples furthermore the proposed approach is applied for h infty model predictive control

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

robust pole assignment for synthesizing feedback control systems using recurrent neural networks this paper presents a neurodynamic optimization approach to robust pole assignment for synthesizing linear control systems via state and output feedback the problem is formulated as a pseudoconvex optimization problem with robustness measure i e the spectral condition number as the objective function and linear matrix equality constraints for exact pole assignment two coupled recurrent neural networks are applied for solving the formulated problem in real time in contrast to existing approaches the exponential convergence of the proposed neurodynamics to global optimal solutions can be guaranteed even with lower model complexity in terms of the number of variables simulation results of the proposed neurodynamic approach for 11 benchmark problems are reported to demonstrate its superiority

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

an analysis of a class of neural networks for solving linear programming problems a class of neural networks that solve linear programming problems is analyzed the neural networks considered are modeled by dynamic gradient systems that are constructed using a parametric family of exact nondifferentiable penalty functions it is proved that for a given linear programming problem and sufficiently large penalty parameters any trajectory of the neural network converges in finite time to its solution set for the analysis lyapunov type theorems are developed for finite time convergence of nonsmooth sliding mode dynamic systems to invariant sets the results are illustrated via numerical simulation examples

k theory and the bridge from motives to noncommutative motives abstract in this work we present a new approach to the theory of noncommutative motives and use it to explain the different flavors of algebraic k theory of schemes and dg categories the work is divided into three main parts in the first part we use the techniques of higher algebra developed in 63 to provide a universal characterization for the symmetric monoidal 1 category underlying the motivic stable a 1 homotopy theory of morel voevodsky 107 67 more precisely given a symmetric monoidal model category v together with an object x v we characterize the underlying symmetric monoidal 1 category of the symmetric monoidal model category sp v x introduced by hovey in 43 by means of a universal property amongst symmetric monoidal 1 categories this characterization trivializes the problem of finding motivic monoidal realizations in the second part we introduce a new approach to the theory of noncommutative motives by constructing a stable motivic homotopy theory for the noncommutative spaces of kontsevich 56 55 54 the key ingredient is a notion of nisnevich topology in the noncommutative setting compatible with the classical notion this compatibility together with the universal property proved in the first part ensures the existence of a canonical monoidal map from the stable motivic theory of morel voevodsky towards these new noncommutative motives that allow us to compare the two theories in the last part of this paper we explain how this bridge can be used to explain the various flavors of algebraic k theory of dg categories more precisely we prove that the non connective k theory of dg categories introduced by schlichting 82 is the non commutative nisnevich sheafification of connective algebraic k theory then we prove that its further non commutative a 1 localization is a tensor unit in our noncommutative motives as a corollary we obtain a precise proof for an original conjecture of kontsevich claiming that k theory gives the correct mapping spaces in noncommutative motives our major application is the discovery of a canonical factorization of our motivic bridge through the 1 category of modules over the commutative algebra object representing homotopy invariant algebraic k theory of schemes the results in 77 imply that this bridge is fully faithful over a field k with resolutions of singularities so that at the motivic level no information below k theory is lost by passing to the noncommutative world

recurrent neural network for non smooth convex optimization problems with application to the identification of genetic regulatory networks a recurrent neural network is proposed for solving the non smooth convex optimization problem with the convex inequality and linear equality constraints since the objective function and inequality constraints may not be smooth the clarke s generalized gradients of the objective function and inequality constraints are employed to describe the dynamics of the proposed neural network it is proved that the equilibrium point set of the proposed neural network is equivalent to the optimal solution of the original optimization problem by using the lagrangian saddle point theorem under weak conditions the proposed neural network is proved to be stable and the state of the neural network is convergent to one of its equilibrium points compared with the existing neural network models for non smooth optimization problems the proposed neural network can deal with a larger class of constraints and is not based on the penalty method finally the proposed neural network is used to solve the identification problem of genetic regulatory networks which can be transformed into a non smooth convex optimization problem the simulation results show the satisfactory identification accuracy which demonstrates the effectiveness and efficiency of the proposed approach

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem

smoothing neural network for constrained non lipschitz optimization with applications in this paper a smoothing neural network snn is proposed for a class of constrained non lipschitz optimization problems where the objective function is the sum of a nonsmooth nonconvex function and a non lipschitz function and the feasible set is a closed convex subset of using the smoothing approximate techniques the proposed neural network is modeled by a differential equation which can be implemented easily under the level bounded condition on the objective function in the feasible set we prove the global existence and uniform boundedness of the solutions of the snn with any initial point in the feasible set the uniqueness of the solution of the snn is provided under the lipschitz property of smoothing functions we show that any accumulation point of the solutions of the snn is a stationary point of the optimization problem numerical results including image restoration blind source separation variable selection and minimizing condition number are presented to illustrate the theoretical results and show the efficiency of the snn comparisons with some existing algorithms show the advantages of the snn

twisted fourier mukai partners of enriques surfaces bridgeland and maciocia showed that a complex enriques surface x has no fourier mukai partners apart from itself that is if d b x cong d b y then x cong y we extend this to twisted fourier mukai partners if alpha is the non trivial element of mathrm br x mathbb z 2 and d b x alpha cong d b y beta then x cong y and beta is non trivial our main tools are twisted topological k theory and twisted mukai lattices

dynamical behaviors of delayed neural network systems with discontinuous activation functions in this letter without assuming the boundedness of the activation functions we discuss the dynamics of a class of delayed neural networks with discontinuous activation functions a relaxed set of sufficient conditions is derived guaranteeing the existence uniqueness and global stability of the equilibrium point convergence behaviors for both state and output are discussed the constraints imposed on the feedback matrix are independent of the delay parameter and can be validated by the linear matrix inequality technique we also prove that the solution of delayed neural networks with discontinuous activation functions can be regarded as a limit of the solutions of delayed neural networks with high slope continuous activation functions

derived azumaya algebras and twisted k theory abstract we construct a relative version of topological k theory in the sense of 11 over an arbitrary quasi compact quasi separated c scheme x this has as input a perf x linear stable category and output a sheaf of spectra on x c the space of complex points of x we then characterize the values of this functor on inputs of the form perf x a for a a derived azumaya algebra over x in such cases we show that this coincides with the twisted topological k theory of x c for some appropriately defined twist of k theory we use this to provide a topological analogue of a classical result of quillen s on the algebraic k theory of severi brauer varieties