on the complexity of computing the volume of a polyhedron we show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix

computation of minimum volume covering ellipsoids we present a practical algorithm for computing the minimum volume n dimensional ellipsoid that must contain m given points a1 am n this convex constrained problem arises in a variety of applied computational settings particularly in data mining and robust statistics its structure makes it particularly amenable to solution by interior point methods and it has been the subject of much theoretical complexity analysis here we focus on computation we present a combined interior point and active set method for solving this problem our computational results demonstrate that our method solves very large problem instances m 30 000 and n 30 to a high degree of accuracy in under 30 seconds on a personal computer

multiple solutions for nonlinear systems with gyroscopic terms abstract in this paper we investigate the existence of multiple solutions to some nonlinear systems with gyroscopic terms via variational methods some new results are obtained and some results from the literature are improved

optimal sampling pattern for free final time linear quadratic regulator the scalar case the optimal sampling problem is the selection of the optimal sampling instants together with the optimal control actions such that a given cost function is minimised in this article we solve the

projective transformations for interior point algorithms and a superlinearly convergent algorithm for the w center problem the purpose of this study is to broaden the scope of projective transformation methods in mathematical programming both in terms of theory and algorithms we start by generalizing the concept of the analytic center of a polyhedral system of constraints to the w center of a polyhedral system which stands for weighted center where there is a positive weight on the logarithmic barrier term for each inequality constraint defining the polyhedronx we prove basic results regarding contained and containing ellipsoids centered at the w center of the systemx we next shift attention to projective transformations and we exhibit an elementary projective transformation that transforms the polyhedronx to another polyhedronz and that transforms the current interior point to the w center of the transformed polyhedronz we work throughout with a polyhedral system of the most general form namely both inequality and equality costraints

threshold solutions and sharp transitions for nonautonomous parabolic equations on mathbb r n this paper is devoted to a class of nonautonomous parabolic equations of the form ut u f t u on mathbb r n we consider a monotone one parameter family of initial data with compact support such that for small values of the parameter the corresponding solutions decay to zero whereas for large values they exhibit a different behavior either blowup in finite time or locally uniform convergence to a positive constant steady state we are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs we refer to such values as threshold values and to the corresponding solutions as threshold solutions we prove that the transition from decay to the other behavior is sharp there is just one threshold value we also describe the behavior of the threshold solution it is global bounded and asymptotically symmetric in the sense that all its limit profiles as t are radially symmetric about the same center our proofs rely on parabolic liouville theorems asymptotic symmetry results for nonlinear parabolic equations and theorems on exponential separation and principal floquet bundles for linear parabolic equations

on a class of damped vibration problems with super quadratic potentials abstract the main purpose of this paper is to study the following damped vibration problems 1 u t q t i n n b u t f t u t a t u t 1 2 q t b u t a e t 0 t u 0 u t u t u t 0 where t 0 q l 1 0 t r with 0 t q t d t 0 a t a i j t is a symmetric n n matrix valued function defined in 0 t with a i j l 0 t for all i j 1 2 n b b i j is an antisymmetry n n constant matrix by establishing a proper variational set one existence result and two multiplicity results of non trivial periodic solutions of 1 are obtained

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

suboptimal receding horizon estimation via noise blocking abstract for discrete time linear systems we propose a suboptimal approach to constrained estimation so that the associated computation burden is reduced this is achieved by enforcing a move blocking mb structure in the estimated process noise sequence pns we show that full information estimation fie and receding horizon estimation rhe with mb are both stable in the sense of an observer the techniques in proving stability are inspired by those that have been proposed for standard rhe to be specific stability results are mainly achieved by i carefully embellishing the general assumptions for standard rhe to accommodate the mb requirement ii exploiting the principle of optimality as well as convexity of the quadratic programs qps associated with fie and rhe iii relying on the fact that the kalman filter is the best linear estimator in the least squares sense a crucial requirement in achieving stability for mb rhe is that the segment structure ss of the pns of mb fie for the optimization steps within the receding horizon i e steps between t n and t 1 has to be enforced in the mb rhe optimization as a result the mb rhe strategy becomes a dynamic estimator with a periodically varying computational complexity the theoretical results have been illustrated with examples

existence of many positive solutions of semilinear elliptic equations on annulus abstract we study the existence of many nonradial positive solutions in an annulus of rn it is an improvement of results in c v coffman a nonlinear boundary value problem with many positive solutions j differential equations 54 1984 429 437 an annulus is invariant under many group actions and these group actions act naturaly on the functional and leave the functional invariant therefore when we look for critical points of the functional we can restrict the functional to the manifold consisting of all fixed points under the group action we are going to use different group actions to obtain critical points and furthermore we can distinguish them

reducing variables in model predictive control of linear system with disturbances using singular value decomposition abstract arising from the need to reduce online computations of model predictive controller this paper proposes an approach for a linear system with bounded additive disturbance using fewer variables than the standard the new variables are chosen so that they transfer the maximal energy to the control inputs several other features are introduced these include an auxiliary state to ensure recursive feasibility an initialization procedure that recovers a substantial portion of the original domain of attraction arising from the use of fewer variables a comparison of the domains of attraction associated with the new variables is also discussed run time computational advantage of more than an order of magnitude compared with standard approach is demonstrated using several numerical examples although a more expensive initialization is needed

duality of ellipsoidal approximations via semi infinite programming in this paper we develop duality of the minimum volume circumscribed ellipsoid and the maximum volume inscribed ellipsoid problems we present a unified treatment of both problems using convex semi infinite programming we establish the known duality relationship between the minimum volume circumscribed ellipsoid problem and the optimal experimental design problem in statistics the duality results are obtained using convex duality for semi infinite programming developed in a functional analysis setting

periodic solutions for some second order systems abstract in this paper we use the minimax theorem to investigate periodic solutions of some second order systems and give some sufficient conditions for the existence for periodic solutions the results generalize and improve some known results

projective re normalization for improving the behavior of a homogeneous conic linear system in this paper we study the homogeneous conic system f ax 0 x in c setminus 0 we choose a point bar s in rm int c that serves as a normalizer and consider computational properties of the normalized system f bar s ax 0 bar s t x 1 x in c we show that the computational complexity of solving f via an interior point method depends only on the complexity value vartheta of the barrier for c and on the symmetry of the origin in the image set h bar s ax bar s tx 1 x in c where the symmetry of 0 in h bar s is rm sym 0 h bar s rm max alpha y in h bar s rightarrow alpha y in h bar s interior point iterations in order to improve the theoretical and practical computation of a solution of f we next present a general theory for projective re normalization of the feasible region f bar s and the image set h bar s and prove the existence of a normalizer bar s such that rm sym 0 h bar s ge 1 m provided that f has an interior solution we develop a methodology for constructing a normalizer bar s such that rm sym 0 h bar s ge 1 m with high probability based on sampling on a geometric random walk with associated probabilistic complexity analysis while such a normalizer is not itself computable in strongly polynomial time the normalizer will yield a conic system that is solvable in o sqrt vartheta rm ln m vartheta iterations which is strongly polynomial time finally we implement this methodology on randomly generated homogeneous linear programming feasibility problems constructed to be poorly behaved our computational results indicate that the projective re normalization methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems for instance we observe a 46 decrease in average ipm iterations for 100 randomly generated poorly behaved problem instances of dimension 1 000 5 000

ellipsoidal approximations of convex sets based on the volumetric barrier let c rn be a convex set we assume that x 1 for all x c and that c contains a ball of radius 1 r for x rn r r and b and n n symmetric positive definite matrix let ex b r y y xtby x r2 a rounding of c is an ellipsoid ex b r such that ex b r c ex b r in the case that c is characterized by a separation oracle it is well known that an on3 2 rounding of c can be obtained using the shallow cut ellipsoid method in on3 lnnr oracle calls we show that a modification of the volumetric cutting plane method obtains an on3 2 rounding of c in on2 lnnr oracle calls we also consider the problem of obtaining an on rounding of c when c has an explicit polyhedral description our analysis uses a new characterization of circumscribing ellipsoids centered at or near the volumetric center of a polyhedral set

spreading vanishing dichotomy in a diffusive logistic model with a free boundary ii abstract we study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for simplicity we assume that the environment and the solution are radially symmetric in the special case of one space dimension and homogeneous environment this free boundary problem was investigated in du and lin 2010 10 we prove that the spreading vanishing dichotomy established in du and lin 2010 10 still holds in the more general and ecologically realistic setting considered here moreover when spreading occurs we obtain best possible upper and lower bounds for the spreading speed of the expanding front when the environment is asymptotically homogeneous at infinity these two bounds coincide our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension

on vaidya s volumetric cutting plane method for convex programming we describe a simplified and strengthened version of vaidya s volumetric cutting plane method for finding a point in a convex set x1d49e rn at each step the algorithm has a system of linear inequality constraints which defines a polyhedron x1d4ab x1d49e and an interior point x x1d4ab the algorithm then either drops one constraint or calls an oracle to check if x x1d49e and if not obtain a new constraint that separates x from x1d49e following the addition or deletion of a constraint the algorithm takes a small number of newton steps for the volumetric barrier v progress of the algorithm is measured in terms of changes in v the algorithm is terminated when either it is discovered that x x1d49e or v becomes large enough to demonstrate that the volume of x1d49e must be below some prescribed amount the complexity of the algorithm compares favorably with that of the ellipsoid method especially in terms of the number of calls to the separation oracle compared to vaidya s original analysis we decrease the total number

brief paper least restrictive move blocking model predictive control move blocking lowers the computational complexity of model predictive control mpc problems by reducing the number of optimization variables however this may render states close to constraints infeasible thus move blocking generally results in control laws that are restrictive the controller domains may be unacceptably and unnecessarily small furthermore different move blocking strategies may result in controller domains of different sizes all other factors being equal in this paper an approach is proposed to design move blocking mpc control laws that are least restrictive i e the controller domain is equal to the maximum controlled invariant set the domains of different move blocking controllers are then by design equal to each other this allows comparison of differing move blocking strategies based on cost performance only without needing to consider domain size also thus this paper is a step towards being able to derive optimal move blocking mpc control laws

a new algorithm for minimizing convex functions over convex sets let s subseteq mathbb r n be a convex set for which there is an oracle with the following property given any pointz n the oracle returns a yes ifz s whereas ifz s then the oracle returns a no together with a hyperplane that separatesz froms the feasibility problem is the problem of finding a point ins the convex optimization problem is the problem of minimizing a convex function overs we present a new algorithm for the feasibility problem the notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm the algorithm for the feasibility problem easily adapts to the convex optimization problem

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

solving convex programs by random walks minimizing a convex function over a convex set in n dimensional space is a basic general problem with many interesting special cases here we present a simple new algorithm for convex optimization based on sampling by a random walk it extends naturally to minimizing quasi convex functions and to other generalizations

random walks on polytopes and an affine interior point method for linear programming let k be a polytope in rn defined by m linear inequalities we give a new markov chain algorithm to draw a nearly uniform sample from k the underlying markov chain is the first to have a mixing time that is strongly polynomial when started from a central point we use this result to design an affine interior point algorithm that does a single random walk to solve linear programs approximately

robust variable horizon mpc with move blocking abstract this paper introduces a new formulation of variable horizon model predictive control vh mpc that utilises move blocking for reducing computational complexity various results pertaining to move blocking are derived following which a generalised blocked vh mpc controller is formulated for linear discrete time systems robustness to bounded disturbances is ensured through the use of tightened constraints the resulting time varying control scheme is shown to guarantee robust recursive feasibility and finite time completion an example is then presented for a particular choice of blocking regime as would be applicable to vehicle man uvring problems simulations demonstrate the efficacy of the formulation

blow up and asymptotic behavior of solutions for reaction diffusion equations with free boundaries abstract we study the blow up phenomena and the asymptotic behavior of time global solutions for reaction diffusion equations u t u x x a u u p a r and p 1 with free boundaries and initial data x we give a sharp threshold value a p 0 such that the solution blows up in finite time when vanishes i e u 0 as t when and when it converges as t to 0 or to an evenly decreasing positive stationary solution depending on whether a 0 or a 0 moreover when blow up happens we show that the blow up set is compact in the occupying domain of initial data and the free boundaries keep bounded

improved complexity for maximum volume inscribed ellipsoids let cal p x vert ax le b where a is an m n matrix we assume that cal p contains a ball of radius one centered at the origin and is itself contained in a ball of radius r centered at the origin we consider the problem of approximating the maximum volume ellipsoid inscribed in cal p such ellipsoids have a number of interesting applications including the inscribed ellipsoid method for convex optimization we describe an optimization algorithm that obtains an ellipsoid whose volume is at least a factor e epsilon of the maximum possible in o m 3 5 ln mr epsilon operations our result provides an alternative to a saddlepoint based approach with the same complexity developed by nemirovskii we also show that a further reduction in complexity can be obtained by first computing an approximation of the analytic center of cal p

analysis on a coupled parabolic system with free boundary abstract the purpose of this paper is to investigate a parabolic problem with coupled reaction terms and free boundary including the existence uniqueness regularity and long time behavior of positive solutions firstly by the contraction mapping theorem we establish the local existence and uniqueness of positive solutions then we prove the solution blows up in finite time with large initial data by comparison principle and give more details for these large initial data by exhibiting an energy condition the simultaneous blow up result of the maximum of u and v is obtained and the blow up set of blow up solutions is a compact subset of 0 h 0 furthermore there is a global and fast solution which decays uniformly at an exponential rate if the initial datum is small while there is a global and slow solution provided that the initial datum is suitably large finally for initial data x x we obtain a trichotomy conclusion by considering the size of parameter

on the complexity of approximating the maximal inscribed ellipsoid for a polytope we give a new polynomial bound on the complexity of approximating the maximal inscribed ellipsoid for a polytope

optimization over state feedback policies for robust control with constraints this paper is concerned with the optimal control of linear discrete time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input it is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence this implies that a broad class of constrained finite horizon robust and optimal control problems where the optimization is over affine state feedback policies can be solved in a computationally efficient fashion using convex optimization methods this equivalence result is used to design a robust receding horizon control rhc state feedback policy such that the closed loop system is input to state stable iss and the constraints are satisfied for all time and all allowable disturbance sequences the cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance free state and input sequences the value of the receding horizon control law can be calculated at each sample instant using a single tractable and convex quadratic program qp if the disturbance set is polytopic or a tractable second order cone program socp if the disturbance set is given by a 2 norm bound

simulated annealing for convex optimization we apply the method known as simulated annealing to the following problem in convex optimization minimize a linear function over an arbitrary convex set where the convex set is specified only by a membership oracle using distributions from the boltzmann gibbs family leads to an algorithm that needs only o n phases for instances in rn this gives an optimization algorithm that makes o n4 5 calls to the membership oracle in the worst case compared to the previous best guarantee of o n5 the benefits of using annealing here are surprising because such problems have no local minima that are not also global minima hence we conclude that one of the advantages of simulated annealing in addition to avoiding poor local minima is that in these problems it converges faster to the minima that it finds we also give a proof that under certain general conditions the boltzmann gibbs distributions are optimal for annealing on these convex problems

on numerical solution of the maximum volume ellipsoid problem in this paper we study practical solution methods for finding the maximum volume ellipsoid inscribing a given full dimensional polytope in re n defined by a finite set of linear inequalities our goal is to design a general purpose algorithmic framework that is reliable and efficient in practice to evaluate the merit of a practical algorithm we consider two key factors the computational cost per iteration and the typical number of iterations required for convergence in addition numerical stability is an important factor we investigate some new formulations upon which we build primal dual type interior point algorithms and we provide theoretical justifications for the proposed formulations and algorithmic framework extensive numerical experiments have shown that one of the new algorithms is the method of choice among those tested

rounding of polytopes in the real number model of computation let a be a set of m points in rn we show that the problem of 1 en rounding of a i e the problem of computing an ellipsoid e rn such that 1 en 1e conv hulla e can be solved in omn2e 1 ln n ln ln m arithmetic operations and comparisons this result implies that the problem of approximating the minimum volume ellipsoid circumscribed about a can be solved in om3 5 lnme 1 operations to a relative accuracy of e in the volume the latter bound also applies to the 1 en rounding problem our bounds hold for the real number model of computation

periodic solutions for second order hamiltonian systems with a change sign potential an existence theorem is obtained for periodic solutions of nonautonomous second order hamiltonian systems with a change sign potential by the minimax methods in critical point theory

the explicit linear quadratic regulator for constrained systems we present a technique to compute the explicit state feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints we show that this closed form solution is piecewise linear and continuous as a practical consequence of the result constrained linear quadratic regulation becomes attractive also for systems with high sampling rates as on line quadratic programming solvers are no more required for the implementation

invariant approximations of the minimal robust positively invariant set this note provides results on approximating the minimal robust positively invariant mrpi set also known as the 0 reachable set of an asymptotically stable discrete time linear time invariant system it is assumed that the disturbance is bounded persistent and acts additively on the state and that the constraints on the disturbance are polyhedral results are given that allow for the computation of a robust positively invariant outer approximation of the mrpi set conditions are also given that allow one to a priori specify the accuracy of this approximation

polynomial time algorithms for linear programming based only on primal scaling and projected gradients of a potential function this paper presents extensions and further analytical properties of algorithms for linear programming based only on primal scaling and projected gradients of a potential function the paper contains extensions and analysis of two polynomial time algorithms for linear programming we first present an extension of gonzaga s o nl iteration algorithm that computes dual variables and does not assume a known optimal objective function value this algorithm uses only affine scaling and is based on computing the projected gradient of the potential function q ln x t s sum limits j 1 n ln x j wherex is the vector of primal variables ands is the vector of dual slack variables and q n sqrt n the algorithm takes either a primal step or recomputes dual variables at each iteration we next present an alternate form of ye s o sqrt n l iteration algorithm that is an extension of the first algorithm of the paper but uses the potential function q ln x t s sum limits j 1 n ln x j sum limits j 1 n ln s j where q n sqrt n we use this alternate form of ye s algorithm to show that ye s algorithm is optimal with respect to the choice of the parameterq in the following sense suppose thatq n n t wheret 0 then the algorithm will solve the linear program in o n r l iterations wherer max t 1 t thus the value oft that minimizes the complexity bound ist 1 2 yielding ye s o sqrt n l iteration bound

computational complexity of inexact gradient augmented lagrangian methods application to constrained mpc we study the computational complexity certification of inexact gradient augmented lagrangian methods for solving convex optimization problems with complicated constraints we solve the augmented lagrangian dual problem that arises from the relaxation of complicating constraints with gradient and fast gradient methods based on inexact first order information moreover since the exact solution of the augmented lagrangian primal problem is hard to compute in practice we solve this problem up to some given inner accuracy we derive relations between the inner and the outer accuracy of the primal and dual problems and we give a full convergence rate analysis for both gradient and fast gradient algorithms we provide estimates on the primal and dual suboptimality and on primal feasibility violation of the generated approximate primal and dual solutions our analysis relies on the lipschitz property of the dual function and on inexact dual gradients we also discuss implementation aspects of the proposed algor

optimal move blocking strategies for model predictive control this paper presents a systematic methodology for designing move blocking strategies to reduce the complexity of a model predictive controller for linear systems with explicit optimisation of the blocking structure using mixed integer programming given a move blocked predictive controller with a terminal invariant set constraint for stability combined with an input parameterisation to preserve recursive feasibility two different optimisation problems are formulated for blocking structure selection the first problem calculates the maximum achievable reduction in the number of input decision variables and prediction horizon length subject to the controller s region of attraction containing a specified subset of the state space then for a given fixed horizon length and block count determined by hardware capabilities the second problem seeks to maximise the volume of an inner approximation to the region of attraction numerical examples show that the resulting blocking structures are able to optimally reduce controller complexity and improve region of attraction volume

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

the diffusive competition problem with a free boundary in heterogeneous time periodic environment abstract in this paper we consider the diffusive competition problem with a free boundary and sign changing intrinsic growth rate in heterogeneous time periodic environment consisting of an invasive species with density u and a native species with density v we assume that v undergoes diffusion and growth in r n and u exists initially in a ball b h 0 0 but invades into the environment with spreading front r h t the effect of the dispersal rate d 1 the initial occupying habitat h 0 the initial density u 0 of invasive species u and the parameter see 1 3 on the dynamics of this free boundary problem are studied a spreading vanishing dichotomy is obtained and some sufficient conditions for the invasive species spreading and vanishing are provided moreover when spreading of u happens some rough estimates of the spreading speed are also given

semi explicit mpc based on subspace clustering this paper presents a new strategy of simplifying the online computations in linear model predictive control mpc employing a specific type of state dependent parameterization for the optimization variable in mpc advantages of explicit mpc are combined with those of online optimization based mpc into an efficient mpc scheme the parameterization is computed offline applying a tailored subspace clustering algorithm to training data consisting of states and corresponding solutions to the mpc optimization problem it is then refined to guarantee feasibility of the parameterized optimization during the offline design phase complexity of the parameterization can be adjusted and control performance can be traded off against online computational effort and storage requirements numerical examples evaluate the presented methods and illustrate their benefits

direct theorems in semi infinite convex programming we show that a semi infinite quasi convex program with certain regularity conditions possesses finitely constrained subprograms with the same optimal value this result is applied to various problems

existence and multiplicity of homoclinic solutions for a class of damped vibration problems abstract the main purpose of this paper is to study the following damped vibration problems 1 1 u t b u t a t u t f t u t a e t r u t 0 u t 0 as t where a a i j t c r r n 2 is an n n symmetric matrix valued function b b i j is an antisymmetry n n constant matrix f c 1 r r n r and f t u u f t u by a symmetric mountain pass theorem and a generalized mountain pass theorem an existence result and a multiplicity result of homoclinic solutions of 1 1 are obtained