a new neural network for solving linear and quadratic programming problems a new neural network for solving linear and quadratic programming problems is presented and is shown to be globally convergent the new neural network improves existing neural networks for solving these problems it avoids the parameter turning problem it is capable of achieving the exact solutions and it uses only simple hardware in which no analog multipliers for variables are required furthermore the network solves both the primal problems and their dual problems simultaneously

existence uniqueness and l q estimates for the stokes problem in an exterior domain as is well known the study of the properties of solutions to the stokes system plays an essential role in the mathematical theory of viscous fluid flows governed by the navier stokes equations it is not art overstatement to say that existence uniqueness and a priori estimates in sobolev spaces for the stokes system are fundamental in every basic question related to the navier stokes equations such as existence regularity asymptotic behavior in time etc see e g ladyzhenskaya 1969 solonni ov 1977 and te am 1977 so far however the stokes system has been fully analyzed only for bounded domains see sobolevskn 1960 ladvznenskaya 1959 solonnu ov 1960 vorovlch yot oowcr 1961 cattabrma 1961 in particular in the paper of cattabriga a complete treatment is given in the sobotev spaces w l 1 1 q c for space dimension n 3 cattabriga s results essentially state that if 2 is a bounded domain in r a of class c m m max l 2 2 l 1 then given f e wt q y2 3 gc wt l q o q e wl 2 1 q q 8o a 1 q there exists one and only one distributional solution v c w1 z q q 3 p e wt l q o to the stokes system

a novel neural network for nonlinear convex programming in this paper we present a neural network for solving the nonlinear convex programming problem in real time by means of the projection method the main idea is to convert the convex programming problem into a variational inequality problem then a dynamical system and a convex energy function are constructed for resulting variational inequality problem it is shown that the proposed neural network is stable in the sense of lyapunov and can converge to an exact optimal solution of the original problem compared with the existing neural networks for solving the nonlinear convex programming problem the proposed neural network has no lipschitz condition no adjustable parameter and its structure is simple the validity and transient behavior of the proposed neural network are demonstrated by some simulation results

on solving constrained optimization problems with neural networks a penalty method approach deals with the use of neural networks to solve linear and nonlinear programming problems the dynamics of these networks are analyzed in particular the dynamics of the canonical nonlinear programming circuit are analyzed the circuit is shown to be a gradient system that seeks to minimize an unconstrained energy function that can be viewed as a penalty method approximation of the original problem next the implementations that correspond to the dynamical canonical nonlinear programming circuit are examined it is shown that the energy function that the system seeks to minimize is different than that of the canonical circuit due to the saturation limits of op amps in the circuit it is also noted that this difference can cause the circuit to converge to a different state than the dynamical canonical circuit to remedy this problem a new circuit implementation is proposed

for the vorticity velocity pressure form of the navier stokes equations on a bounded plane domain with corners abstract we show existence and regularity for the boundary value problems of the navier stokes equations with non standard bcs on a bounded plane domain with non convex corners we assign the vorticity value 0 and the velocity normal component u n u 0 n over the non convex corner the dynamic pressure value p u 2 2 p 0 over inflow and outflow boundaries and so on we construct the corner singularity functions for the stokes operator with zero vorticity and velocity normal component bcs subtract its leading singularity from the solution by defining the coefficient of the singularity and show increased regularity for the remainder the solution is determined by the smoother part and the coefficients of the singularities it is seen from the singularity that the dynamic pressure has a transition layer that changes the sign at 2 in the domain the obtained results can be applied to general polygonal domains and the cavity flows

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

boundary value problems for the laplacian in convex and semiconvex domains abstract we study the fully inhomogeneous dirichlet problem for the laplacian in bounded convex domains in r n when the size smoothness of both the data and the solution are measured on scales of besov and triebel lizorkin spaces as a preamble we deal with the dirichlet and regularity problems for harmonic functions in convex domains with optimal nontangential maximal function estimates as a corollary sharp estimates for the green potential are obtained in a variety of contexts including local hardy spaces a substantial part of this analysis applies to bounded semiconvex domains i e lipschitz domains satisfying a uniform exterior ball condition

a general methodology for designing globally convergent optimization neural networks we present a general methodology for designing optimization neural networks we prove that the neural networks constructed by using the proposed method are guaranteed to be globally convergent to solutions of problems with bounded or unbounded solution sets in contrast with the gradient methods whose convergence is not guaranteed we show that the proposed method contains both the gradient methods and nongradient methods employed in existing optimization neural networks as special cases based on the theoretical results of the proposed method we study the convergence and stability of general gradient models in the case of unisolated solutions using the proposed method we derive some new neural network models for a very large class of optimization problems in which the equilibrium points correspond to exact solutions and there is no variable parameter finally some numerical examples show the effectiveness of the method

resolvent estimates in l p for the stokes operator in lipschitz domains we establish the l p resolvent estimates for the stokes operator in lipschitz domains in mathbb r d d geqq 3 for frac 1 p frac 1 2 frac 1 2d varepsilon the result implies that the stokes operator in a three dimensional lipschitz domain generates a bounded analytic semigroup in l p for 3 2 e p 3 e this gives an affirmative answer to a conjecture of m taylor progr nonlinear differential equations appl vol 42 pp 320 334

second order neural nets for constrained optimization analog neural nets for constrained optimization are proposed as an analogue of newton s algorithm in numerical analysis the neural model is globally stable and can converge to the constrained stationary points nonlinear neurons are introduced into the net making it possible to solve optimization problems where the variables take discrete values i e combinatorial optimization

nonsmooth neural network for convex time dependent constraint satisfaction problems this paper introduces a nonsmooth ns neural network that is able to operate in a time dependent td context and is potentially useful for solving some classes of ns td problems the proposed network is named nonsmooth time dependent network ntn and is an extension to a td setting of a previous ns neural network for programming problems suppose c t t ge 0 is a nonempty td convex feasibility set defined by td inequality constraints the constraints are in general ns nondifferentiable functions of the state variables and time ntn is described by the subdifferential with respect to the state variables of an ns td barrier function and a vector field corresponding to the unconstrained dynamics this paper shows that for suitable values of the penalty parameter the ntn dynamics displays two main phases in the first phase any solution of ntn not starting in c 0 at t 0 is able to reach the moving set c cdot in finite time t h whereas in the second phase the solution tracks the moving set i e it stays within c t for all subsequent times t ge t h ntn is thus able to find an exact feasible solution in finite time and also to provide an exact feasible solution for subsequent times this new and peculiar dynamics displayed by ntn is potentially useful for addressing some significant td signal processing tasks as an illustration this paper discusses a number of examples where ntn is applied to the solution of ns td convex feasibility problems

a posteriori error bounds for the linearly constrained varitional inequality problem in this paper we derive a posteriori error bound of an approximate solution to a linearly constrained variational inequality problem three measures of error are used with each depending on a particular characterization of an exact solution and related to a certain class of iterative methods

equivalence of variational inequality problems to unconstrained minimization in this paper we propose a class of merit functions for variational inequality problems vi through these merit functions the variational inequality problem is cast as unconstrained minimization problem we estimate the growth rate of these merit functions and give conditions under which the stationary points of these functions are the solutions of vi

on solving systems of linear inequalities with artificial neural networks the implementation of the relaxation projection algorithm by artificial neural networks to solve sets of linear inequalities is examined the different versions of this algorithm are described and theoretical convergence results are given the best known analog optimization solvers are shown to use the simultaneous projection version of it neural networks that implement each version are described the results of tests made with simulated realizations of these networks are reported these tests consisted in having all networks solve some sample problems the results obtained help determine good values for the step size parameters and point out the relative merits of the different networks

global exponential stability of recurrent neural networks for solving optimization and related problems global exponential stability is a desirable property for dynamic systems the paper studies the global exponential stability of several existing recurrent neural networks for solving linear programming problems convex programming problems with interval constraints convex programming problems with nonlinear constraints and monotone variational inequalities in contrast to the existing results on global exponential stability the present results do not require additional conditions on the weight matrices of recurrent neural networks and improve some existing conditions for global exponential stability therefore the stability results in the paper further demonstrate the superior convergence properties of the existing neural networks for optimization

a novel neural network for generally constrained variational inequalities this paper presents a novel neural network for solving generally constrained variational inequality problems by constructing a system of double projection equations by defining proper convex energy functions the proposed neural network is proved to be stable in the sense of lyapunov and converges to an exact solution of the original problem for any starting point under the weaker cocoercivity condition or the monotonicity condition of the gradient mapping on the linear equation set furthermore two sufficient conditions are provided to ensure the stability of the proposed neural network for a special case the proposed model overcomes some shortcomings of existing continuous time neural networks for constrained variational inequality and its stability only requires some monotonicity conditions of the underlying mapping and the concavity of nonlinear inequality constraints on the equation set the validity and transient behavior of the proposed neural network are demonstrated by some simulation results

vorticity velocity pressure formulation for stokes problem we propose a three field formulation for efficiently solving a twodimensional stokes problem in the case of nonstandard boundary conditions more specifically we consider the case where the pressure and either normal or tangential components of the velocity are prescribed at some given parts of the boundary the proposed computational methodology consists in reformulating the considered boundary value problem via a mixed type formulation where the pressure and the vorticity are the principal unknowns while the velocity is the lagrange multiplier the obtained formulation is then discretized and a convergence analysis is performed a priori error estimates are established and some numerical results are presented to highlight the perfomance of the proposed computational methodology

linear and quadratic programming neural network analysis neural networks for linear and quadratic programming are analyzed the network proposed by m p kennedy and l o chua ieee trans circuits syst vol 35 pp 554 562 may 1988 is justified from the viewpoint of optimization theory and the technique is extended to solve optimization problems such as the least squares problem for quadratic programming the network converges either to an equilibrium or to an exact solution depending on whether the problem has constraints or not the results also suggest an analytical approach to solve the linear system bx b without calculating the matrix inverse the results are directly applicable to optimization problems with c sup 2 convex objective functions and linear constraints the dynamics and applicability of the networks are demonstrated by simulation the distance between the equilibria of the networks and the problem solutions can be controlled by the appropriate choice of a network parameter

one problem of the navier type for the stokes system in planar domains we study the problem u f u f u g u g in u gu g h h on for a bounded simply connected lipschitz domain in the plane for f 0f 0 g 0g 0 g lp g lp h lq h lq we study a solution in the sense of a nontangential limit for f ws 1 q r2 f ws 1 q r2 g ws q g ws q g wt 1 p p g wt 1 p p h ws 1 q q h ws 1 q q with t s 1t s 1 we prove the existence of a unique solution u wt p r2 ws q u wt p r2 ws q for f bs 1q r r2 g bsq r g bt 1 pp h bs 1 qq r with t s 1t s 1 we prove the existence of a unique solution u btp r2 bsq r for f ck 1 r2 g ck h ck h ck g ck 1 g ck 1 we prove the existence of a unique solution u ck 1 r2 ck

navier stokes equations on lipschitz domains in riemannian manifolds the navier stokes equations are a system of nonlinear evolution equations modeling the flow of a viscous incompressible fluid one ingredient in the analysis of this system is the stationary linear system known as the stokes system a boundary value problem bvp that will be described in detail in the next section layer potential methods in smoothly bounded domains in euclidean space have proven useful in the analysis of the stokes system starting with work of odqvist and lichtenstein and including work of solonnikov and many others see the discussion in chapter iii of 10 and in 17 for the case of flow in regions with smooth boundary a treatment based on the modern language of pseudodifferential operators can be found in 18 in 1988 e fabes c kenig and g verchota 6 extended this classical layer potential approach to cover lipschitz domains in euclidean space in 6 the main result concerning the constant coefficient stokes system on lipschitz domains with connected boundary in euclidean space is the treatment of the l dirichlet boundary value problem and its regular version to achieve this the authors solve certain auxiliary neumann type problems and then exploit the duality between these and the original bvp s at the level of boundary integral operators p deuring and w von wahl 4 made use of the analysis in 6 to demonstrate the short time existence of solutions to the navier stokes equations in bounded lipschitz domains in threedimensional euclidean space it was necessary in 4 to include the hypothesis that the boundary be connected the hypothesis that the boundary be connected pervaded much work on the application of layer potentials to analysis on lipschitz domains it was certainly natural to speculate that this restriction was an artifact of the methods used and not partly supported by nsf grant dms 9870018 partially supported by nsf grant dms 9877077 1991mathematics subject classification primary 35q30 76d05 35j25 secondary 42b20 45e05

weighted estimates in l 2 for laplace s equation on lipschitz domains let c r d d 3 be a bounded lipschitz domain for laplace s equation au 0 in q we study the dirichlet and neumann problems with boundary data in the weighted space l 2 leader where q q q 0 q 0 is a fixed point on and da denotes the surface measure on we prove that there exists e e 0 2 such that the dirichlet problem is uniquely solvable if 1 d d 3 e and the neumann problem is uniquely solvable if 3 d e a d 1 if q is a c 1 domain one may take e 2 the regularity for the dirichlet problem with data in the weighted sobolev space l 2 1 d is also considered finally we establish the weighted l 2 estimates with general ap weights for the dirichlet and regularity problems

the neumann problem and helmholtz decomposition in convex domains we show that the neumann problem for laplace s equation in a convex domain with boundary data in lp is uniquely solvable for 1 p as a consequence we obtain the helmholtz decomposition of vector fields in lp rd

neural data fusion algorithms based on a linearly constrained least square method two novel neural data fusion algorithms based on a linearly constrained least square lcls method are proposed while the lcls method is used to minimize the energy of the linearly fused information two neural network algorithms are developed to overcome the ill conditioned and singular problems of the sample covariance matrix arisen in the lcls method the proposed neural fusion algorithms are samples for implementation using both software and hardware compared with the existing fusion methods the proposed neural data fusion method has an unbiased statistical property and does not require any a priori knowledge about the noise covariance it is shown that the proposed neural fusion algorithms converge globally to the optimal fusion solution when the sample covariance matrix is singular and converge globally with exponential rate when the sample covariance matrix is nonsingular we apply the proposed neural fusion method to image and signal fusion and it is shown that the quality of the solution can be greatly enhanced by the proposed technique

neural network for quadratic optimization with bound constraints a recurrent neural network is presented which performs quadratic optimization subject to bound constraints on each of the optimization variables the network is shown to be globally convergent and conditions on the quadratic problem and the network parameters are established under which exponential asymptotic stability is achieved through suitable choice of the network parameters the system of differential equations governing the network activations is preconditioned in order to reduce its sensitivity to noise and to roundoff errors the optimization method employed by the neural network is shown to fall into the general class of gradient methods for constrained nonlinear optimization and in contrast with penalty function methods is guaranteed to yield only feasible solutions

a neural network approach for solving nonlinear bilevel programming problem a neural network model is presented for solving nonlinear bilevel programming problem which is a np hard problem the proposed neural network is proved to be lyapunov stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem the asymptotic properties of the neural network are analyzed and the condition for asymptotic stability solution feasibility and solution optimality are derived the transient behavior of the neural network is simulated and the validity of the network is verified with numerical examples

lp theory for stokes and navier stokes equations with navier boundary condition abstract this work was intended as an attempt at studying stationary stokes and navier stokes problem with navier boundary conditions 1 3 we wish to investigate some results of existence uniqueness and regularity of solutions in hilbert case and in l p theory

a delayed neural network for solving linear projection equations and its analysis in this paper we present a delayed neural network approach to solve linear projection equations the lyapunov krasovskii theory for functional differential equations and the linear matrix inequality lmi approach are employed to analyze the global asymptotic stability and global exponential stability of the delayed neural network compared with the existing linear projection neural network theoretical results and illustrative examples show that the delayed neural network can effectively solve a class of linear projection equations and some quadratic programming problems

the lp boundary value problems on lipschitz domains abstract let be a bounded lipschitz domain in r n we develop a new approach to the invertibility on l p of the layer potentials associated with elliptic equations and systems in as a consequence for n 4 and 2 n 1 n 1 e p 2 where e 0 depends on we obtain the solvability of the l p neumann type boundary value problems for second order elliptic systems the analogous results for the biharmonic equation are also established

normal maps inducted by linear transformations we study a certain piecewise linear manifold which we call the normal manifold associated with a polyhedral convex set and a family of continuous functions called normal maps that are induced on this manifold by continuous functions from rn to rn these normal maps occur frequently in optimization and equilibrium problems and the subclass of normal maps induced by linear transformations plays a key role our main result is that the normal map induced by a linear transformation is a lipschitzian homeomorphism if and only if the determinant of the map in each n cell of the normal manifold has the same nonzero sign

neurodynamics based robust pole assignment for high order descriptor systems in this paper a neurodynamic optimization approach is proposed for synthesizing high order descriptor linear systems with state feedback control via robust pole assignment with a new robustness measure serving as the objective function the robust eigenstructure assignment problem is formulated as a pseudoconvex optimization problem a neurodynamic optimization approach is applied and shown to be capable of maximizing the robust stability margin for high order singular systems with guaranteed optimality and exact pole assignment two numerical examples and vehicle vibration control application are discussed to substantiate the efficacy of the proposed approach

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

on the navier problem for the stationary navier stokes equations the navier problem is to find a solution of the steady state navier stokes equations such that the normal component of the velocity and a linear combination of the tangential components of the velocity and the traction assume prescribed value a and s at the boundary if is exterior it is required that the velocity converges to an assigned constant vector u0 at infinity we prove that a solution exists in a bounded domain provided a l2 is less than a computable positive constant and is unique if a w1 2 2 s l2 is suitably small as far as exterior domains are concerned we show that a solution exists if a l2 a u0 n l2 is small

neural network for solving extended linear programming problems a neural network for solving extended linear programming problems is presented and is shown to be globally convergent to exact solutions the proposed neural network only uses simple hardware in which no analog multiplier for variables is required and has no parameter tuning problem finally an application of the neural network to the l sub 1 norm minimization problem is given

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

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

a new regression estimator with neural network realization a new regression estimator viewed as the solution of a strictly convex quadratic programming problem is introduced in this paper two recurrent neural networks in continuous time and discrete time respectively are proposed to solve the quadratic programming problem in real time the continuous time neural network is shown to have a global stability including the global asymptotic and exponential stability the discrete time neural network is shown to have a global convergence with a fixed step length this fixed step length can be independent of the regression problem size by scaling a design parameter since the sizes of the proposed neural networks depend only on the constraints of the optimization problems the proposed new regression estimator can obtained by two novel neural networks with lower implementation costs than the conventional methods our simulation results confirm that the proposed neural networks are effective in solving various kind of regression problems

a high performance feedback neural network for solving convex nonlinear programming problems based on a new idea of successive approximation this paper proposes a high performance feedback neural network model for solving convex nonlinear programming problems differing from existing neural network optimization models no dual variables penalty parameters or lagrange multipliers are involved in the proposed network it has the least number of state variables and is very simple in structure in particular the proposed network has better asymptotic stability for an arbitrarily given initial point the trajectory of the network converges to an optimal solution of the convex nonlinear programming problem under no more than the standard assumptions in addition the network can also solve linear programming and convex quadratic programming problems and the new idea of a feedback network may be used to solve other optimization problems feasibility and efficiency are also substantiated by simulation examples

the neumann problem in lp on lipschitz and convex domains abstract given p 2 and a bounded lipschitz domain we obtain a necessary and sufficient condition for the solvability of the neumann problem for laplace s equation u 0 in with boundary data in l p as a result we show that the l p neumann problem on convex domains in r d is solvable for 1 p if d 2 for 1 p 4 if d 3 and for 1 p 3 e if d 4

a high performance neural network for solving linear and quadratic programming problems two classes of high performance neural networks for solving linear and quadratic programming problems are given we prove that the new system converges globally to the solutions of the linear and quadratic programming problems in a neural network network parameters are usually not specified the proposed models can overcome numerical difficulty caused by neural networks with network parameters and obtain desired approximate solutions of the linear and quadratic programming problems

potential theory on lipschitz domains in riemannian manifolds the case of dini metric tensors we study the applicability of the method of layer potentials in the treatment of boundary value problems for the laplace beltrami operator on lipschitz sub domains of riemannian manifolds in the case when the metric tensor g jk dx j desk has low regularity under the assumption that g jk x g jk y c x y where the modulus of continuity satisfies a dini type condition we prove the well posedness of the classical dirichlet and neumann problems with l p boundary data for sharp ranges of p s and with optimal nontangential maximal function estimates

hopfield neural networks in large scale linear optimization problems abstract hopfield neural networks and affine scaling interior point methods are combined in a hybrid approach for solving linear optimization problems the hopfield networks perform the early stages of the optimization procedures providing enhanced feasible starting points for both primal and dual affine scaling interior point methods thus facilitating the steps towards optimality the hybrid approach is applied to a set of real world linear programming problems the results show the potential of the integrated approach indicating that the combination of neural networks and affine scaling interior point methods can be a good alternative to obtain solutions for large scale optimization problems

solving convex programs by means of ordinary differential equations within a hilbert space we consider nonsmooth convex programs with sharp constraints examples include all problem instances that are bounded and strictly feasible to solve such programs we pursue an absolutely continuous trajectory generated by a differential inclusion of subgradient type whenever this inclusion offers some freedom of choice we select a steepest descent direction it is shown that the proposed algorithm converges to an optimal solution in finite time

hardy spaces and the neumann problem in l p for laplace s equation in lipschitz domains on donne des resultats optimaux pour la resolubilite du probleme de neumann dans des domaines de lipschitz a donnees dans l p on obtient des resultats du point extreme correspondant pour des espaces de hardy

neural network for solving linear programming problems with bounded variables a new neural network for solving linear programming problems with bounded variables is presented the network is shown to be completely stable and globally convergent to the solutions to the linear programming problems the proposed new network is capable of achieving the exact solutions in contrast to existing optimization neural networks which need a suitable choice of the network parameters and thus can obtain only approximate solutions furthermore both the primal problems and their dual problems are solved simultaneously by the new network

two metric projection methods for constrained optimization this paper is concerned with the problem min f x mid x in x where x is a convex subset of a linear space h and f is a smooth real valued function on h we propose the class of methods x k 1 p x k alpha k g k where p denotes projection on x with respect to a hilbert space norm cdot g k denotes the frechet derivative of f at x k with respect to another hilbert space norm cdot k on h and alpha k is a positive scalar stepsize we thus remove an important restriction in the original proposal of goldstein 1 and levitin and poljak 2 where the norms cdot and cdot k must be the same it is therefore possible to match the norm cdot with the structure of x so that the projection operation is simplified while at the same time reserving the option to choose cdot k on the basis of approximations to the hessian of f so as to attain a typically superlinear rate of convergence the resulting methods are particularly attrac