generalized bundle methods we study a class of generalized bundle methods for which the stabilizing term can be any closed convex function satisfying certain properties this setting covers several algorithms from the literature that have been so far regarded as distinct under a different hypothesis on the stabilizing term and or the function to be minimized we prove finite termination asymptotic convergence and finite convergence to an optimal point with or without limits on the number of serious steps and or requiring the proximal parameter to go to infinity the convergence proofs leave a high degree of freedom in the crucial implementative features of the algorithm i e the management of the bundle of subgradients beta strategy and of the proximal parameter t strategy we extensively exploit a dual view of bundle methods which are shown to be a dual ascent approach to one nonlinear problem in an appropriate dual space where nonlinear subproblems are approximately solved at each step with an inner linearization approach this allows us to precisely characterize the changes in the subproblems during the serious steps since the dual problem is not tied to the local concept of varepsilon subdifferential for some of the proofs a generalization of inf compactness called compactness is required this concept is related to that of asymptotically well behaved functions

bundle methods for sum functions with easy components applications to multicommodity network design we propose a version of the bundle scheme for convex nondifferentiable optimization suitable for the case of a sum function where some of the components are easy that is they are lagrangian functions of explicitly known compact convex programs this corresponds to a stabilized partial dantzig wolfe decomposition where suitably modified representations of the easy convex subproblems are inserted in the master problem as an alternative to iteratively inner approximating them by extreme points thus providing the algorithm with exact information about a part of the dual objective function the resulting master problems are potentially larger and less well structured than the standard ones ruling out the available specialized techniques and requiring the use of general purpose solvers for their solution this strongly favors piecewise linear stabilizing terms as opposed to the more usual quadratic ones which in turn may have an adverse effect on the convergence speed of the algorithm so that the overall performance may depend on appropriate tuning of all these aspects yet very good computational results are obtained in at least one relevant application the computation of tight lower bounds for fixed charge multicommodity min cost flow problems

a proximal projection bundle method for lagrangian relaxation including semidefinite programming we give a proximal bundle method for minimizing a convex function f over a convex set c it requires evaluating f and its subgradients with a fixed but possibly unknown accuracy epsilon 0 each iteration involves solving an unconstrained proximal subproblem and projecting a certain point onto c the method asymptotically finds points that are epsilon optimal in lagrangian relaxation of convex programs it allows for e accurate solutions of lagrangian subproblems and finds e optimal primal solutions for semidefinite programming problems it extends the highly successful spectral bundle method to the case of inexact eigenvalue computations

a doubly stabilized bundle method for nonsmooth convex optimization we propose a bundle method for minimizing nonsmooth convex functions that combines both the level and the proximal stabilizations most bundle algorithms use a cutting plane model of the objective function to formulate a subproblem whose solution gives the next iterate proximal bundle methods employ the model in the objective function of the subproblem while level methods put the model in the subproblem s constraints the proposed algorithm defines new iterates by solving a subproblem that employs the model in both the objective function and in the constraints one advantage when compared to the proximal approach is that the level set constraint provides a certain lagrange multiplier which is used to update the proximal parameter in a novel manner we also show that in the case of inexact function and subgradient evaluations no additional procedure needs to be performed by our variant to deal with inexactness as opposed to the proximal bundle methods that require special modifications numerical experiments on almost 1 000 instances of different types of problems are presented our experiments show that the doubly stabilized bundle method inherits useful features of the level and the proximal versions and compares favorably to both of them

quadratic approximations in convex nondifferentiable optimization an implementable descent method for the unconstrained minimization of convex nonsmooth functions of several variables is described the algorithm is characterized by the use of a set of quadratic approximations of the objective function in order to compute the search direction the resulting direction finding subproblem is shown to be equivalent to a structured parametric quadratic programming problem the convergence of the algorithm to the minimum is proved and numerical experience is reported

a note on second order optimality conditions abstract in this note we compare second order sufficient conditions for optimality presented by v jeyakumar and d t luc and by d bedna ik and the author we show that these results can be jointly generalized using some facts from convex analysis

second order directional derivatives of spectral functions a spectral function of a symmetric matrix x is a function which depends only on the eigenvalues of x l 1 x l 2 x l n x and may be written as l 1 x l 2 x l n x for some symmetric function in this paper we assume that is a c 1 1 function and discuss second order directional derivatives of such a spectral function we obtain an explicit expression of second order directional derivative for the spectral function

second order optimality conditions with arbitrary nondifferentiable function in scalar and vector optimization abstract in this paper we introduce a new second order directional derivative and a second order subdifferential of hadamard type for an arbitrary nondifferentiable function we derive second order necessary and sufficient optimality conditions for a local and a global minimum and an isolated local minimum of second order in unconstrained optimization in particular we obtain two results with strongly pseudoconvex functions we also compare our conditions with the results of the recently published paper bednarik and pastor 2008 and a lot of other works published in high level journals and prove that they are particular cases of our necessary and sufficient ones we prove that the necessary optimality conditions concern more functions than the conditions in terms of lower dini directional derivative even the optimality conditions with the last derivative can be applied to a function which does not belong to some special class at last we apply our optimality criteria for scalar problems to derive necessary and sufficient optimality conditions in the cone constrained vector optimization

on scalar and vector l stable functions abstract the notion of a scalar function that is l stable at a point is introduced in d bedna ik k pastor on second order conditions in unconstrained optimization math program ser a 113 2008 283 298 in the present paper a characterization of the l stable functions is obtained further the notion of an l stable function is generalized from scalar to vector functions in an application optimality conditions for constrained vector problems with l stable data are established

the volume algorithm revisited relation with bundle methods we revise the volume algorithm va for linear programming and relate it to bundle methods when first introduced va was presented as a subgradient like method for solving the original problem in its dual form in a way similar to the serious null steps philosophy of bundle methods va produces green yellow or red steps in order to give convergence results we introduce in va a precise measure for the improvement needed to declare a green or serious step this addition yields a revised formulation rva that is halfway between va and a specific bundle method that we call bva we analyze the convergence properties of both rva and bva finally we compare the performance of the modified algorithms versus va on a set of rectilinear steiner problems of various sizes and increasing complexity derived from real world vlsi design instances

analytical study of spatiotemporal chaos control by applying local injections spatiotemporal chaos control by applying local feedback injections is investigated analytically the influence of gradient force on the controllability is investigated it is shown that as the gradient force of the system is larger than a critical value local control can reach very high efficiency to drive the turbulent system of infinite size to a regular target state by using a single control signal the complex ginzburg landau equation is used as a model to confirm the above analysis and a four wave mixing mode is revealed to determine the dynamical behavior of the controlled system at the onset of instability

a technical note concerning one example abstract our aim is to correct one mistake that appeared in nonlinear anal theory methods appl 26 1996 55 example 2 1 and example 5 1 and nonlinear anal theory methods appl 36 1999 595 example 2 2 in calculation of certain cominetti correa generalized second order derivative introduced in siam j control optim 28 1990 789

new variants of bundle methods in this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization convex concave games and variational inequalities we outline the ideas underlying these methods and present rate of convergence estimates

second order moderate derivatives f x f x it is the purpose of the present paper to introduce a notion of second order generalized derivative for arbitrary finite functions which retains the preceding properties in particular which coincides with the usual second order derivative when it exists and enjoys useful calculus rules our proposal is close to the proposal of 3 cf also 4 the variant we use is similar in spirit to the first order variant of the clarke circa derivatives introduced in l 21 here however the concept of fan plays a key role as in 3 51 let us note that the absence of built in upper semicontinuity is intimately linked with the accuracy of the concept under added assumptions of semicontinuity one recovers the notions of 3 51 our results can be found in 6

a algorithm for convex minimization for convex minimization we introduce an algorithm based on space decomposition the method uses a bundle subroutine to generate a sequence of approximate proximal points when a primal dual track leading to a solution and zero subgradient pair exists these points approximate the primal track points and give the algorithm s or corrector steps the subroutine also approximates dual track points that are gradients needed for the method s newton predictor steps with the inclusion of a simple line search the resulting algorithm is proved to be globally convergent the convergence is superlinear if the primal dual track points and the objective s hessian are approximated well enough

a proximal cutting plane method using chebychev center for nonsmooth convex optimization an algorithm is developed for minimizing nonsmooth convex functions this algorithm extends elzinga moore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones thus removing compactness assumption our method is to elzinga moore s algorithm what a proximal bundle method is to kelley s algorithm instead of lower approximations used in proximal bundle methods the present approach is based on some objects regularizing translated functions of the objective function we propose some variants and using some academic test problems we conduct a numerical comparative study with elzinga moore algorithm and two other well known nonsmooth methods

consensus in a heterogeneous influence network we consider a dynamical network model in which a number of agents all move on the plane with the same constant absolute velocity at each time step each agent s direction is updated as the average of its direction plus the directions of other agents who can influence it the influencing capability of each agent is represented by its influencing radius which is randomly chosen according to a power law distribution with a scaling exponent between 2 and ensuremath infty as the value of the scaling exponent decreases the radius distribution becomes more heterogeneous and the network becomes much easier to achieve direction consensus among agents due to the leading roles played by a few hub agents furthermore almost all agents will finally move in the same desired direction in a strong heterogeneous influence network if and only if a small fraction of hub agents can be controlled to move in the desired direction these results also reflect the robust yet fragile feature of a heterogeneous influence network

on second order optimality conditions in constrained multiobjective optimization abstract in this paper we study a multiobjective optimization problem with inequality constraints on finite dimensional spaces a second order necessary condition for local weak efficiency is proved under strict differentiability assumptions we also establish a second order sufficient condition for local firm efficiency of order 2 under l stability assumptions in this way we generalize some corresponding results obtained by p q khanh and n d tuan and by the authors

on first and second order conditions linebreak for error bounds we establish first order and second order sufficient conditions ensuring that a proper lower semicontinuous function f on a banach space x has an error bound we also consider similar problems with constraint namely that f is replaced by its restriction to a subset of x these results are employed to identify exactly when a quadratic function on x has an error bound

collocation methods for solving linear differential algebraic boundary value problems we consider boundary value problems for linear differential algebraic equations with variable coefficients with no restriction on the index a well known regularisation procedure yields an equivalent index one problem with d differential and a n d algebraic equations collocation methods based on the regularised bvp approximate the solution x by a continuous piecewise polynomial of degree k and deliver in particular consistent approximations at mesh points by using the radau schemes under weak assumptions the collocation problems are uniquely and stably solvable and if the unique solution x is sufficiently smooth convergence of order min k 1 2k 1 and superconvergence at mesh points of order 2k 1 is shown finally some numerical experiments illustrating these results are presented

lagrange multipliers and calmness conditions of order p in this paper by assuming that a non lipschitz penalty function is exact new conditions for the existence of lagrange multipliers are established for an inequality and equality constrained continuously differentiable optimization problem this is done by virtue of a first order necessary optimality condition of the penalty problem which is obtained by estimating dini upper directional derivatives of the penalty function in terms of taylor expansions and a farkas lemma relations among the obtained results and some well known constraint qualifications are discussed

regular solutions of nonlinear differential algebraic equations and their numerical determination for a general class of nonlinear possibly higher index differential algebraic equations we show existence and uniqueness of solutions these solutions are regular in the sense that newton s method will converge locally and quadratically on the basis of the presented theoretical results numerical methods for the determination of consistent initial values and for the computation of regular solutions are developed several numerical examples are included

differentiability properties of functions that are l stable at a point abstract the class of the l stable functions is introduced in previous papers of the authors as a generalization of the class of c 1 1 functions the importance of this class is that minimization problems with l stable data admit second order optimality conditions in terms of suitable directional derivatives similar to those for ones with c 1 1 data the present paper studies differentiability properties of l stable functions in normed spaces and generalizes to infinite dimensional spaces some results of the authors

symmetric collocation for unstructured nonlinear differential algebraic equations of arbitrary index we examine a class of symmetric collocation schemes for the solution of nonlinear boundary value problems for unstructured nonlinear systems of differential algebraic equations with arbitrary index we show that these schemes converge with the same orders as one would expect for ordinary differential equations in particular we show superconvergence for a special choice of the collocation points we demonstrate the efficiency of the new approach with some numerical examples

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

convergence of mesh adaptive direct search to second order stationary points a previous analysis of second order behavior of generalized pattern search algorithms for unconstrained and linearly constrained minimization is extended to the more general class of mesh adaptive direct search mads algorithms for general constrained optimization because of the ability of mads to generate an asymptotically dense set of search directions we are able to establish reasonable conditions under which a subsequence of mads iterates converges to a limit point satisfying second order necessary or sufficient optimality conditions for general set constrained optimization problems

second order subdifferentials and convexity of real valued functions abstract in this paper we show that the positive semi definiteness psd of the frechet and or mordukhovich second order subdifferentials can recognize the convexity of c 1 functions however the psd is insufficient for ensuring the convexity of a locally lipschitz function in general a complete characterization of strong convexity via the second order subdifferentials is also given

a proximal bundle method with approximate subgradient linearizations we give a proximal bundle method for minimizing a convex function f over a closed convex set it only requires evaluating f and its subgradients with an accuracy epsilon 0 which is fixed but possibly unknown it asymptotically finds points that are epsilon optimal when applied to lagrangian relaxation it allows for epsilon accurate solutions of lagrangian subproblems and finds epsilon optimal solutions of convex programs

enhancing the convergence efficiency of a self propelled agent system via a weighted model we investigate a weighted self propelled agent system wherein each agent s direction is determined by its spatial neighbors directions with exponential weights according to the neighbor numbers in order to describe the fact that some agents with more neighbors might have larger influence on its neighbors we introduce a scaling exponent of the neighbor number between 0 and ensuremath infty when the exponent is equal to 1 the convergence efficiency is enhanced in our simulation furthermore as the exponent increases i e the effect of weight becomes stronger the network of agents becomes easier to achieve direction consensus

structure identification of uncertain general complex dynamical networks with time delay it is well known that many real world complex networks have various uncertain information such as unknown or uncertain topological structure and node dynamics the structure identification problem has theoretical and practical importance for uncertain complex dynamical networks at the same time time delay often appears in the state variables or coupling coefficients of various practical complex networks this paper initiates a novel approach for simultaneously identifying the topological structure and unknown parameters of uncertain general complex networks with time delay in particular this method is also effective for uncertain delayed complex dynamical networks with different node dynamics moreover the proposed method can be easily extended to monitor the on line evolution of network topological structure finally three representative examples are then given to verify the effectiveness of the proposed approach

a version of the bundle idea for minimizing a nonsmooth function conceptual idea convergence analysis numerical results during recent years various proposals for the minimization of a nonsmooth functional have been made amongst these the bundle concept turned out to be an especially fruitful idea based on this concept a number of authors have developed codes that can successfully deal with nonsmooth problems the aim of the paper is to show that by adding some features of the trust region philosophy to the bundle concept the end result is a distinguished member of the bundle family with a more stable behaviour than some other bundle versions the reliability and efficiency of this code is demonstrated on the standard academic test examples and on some real life problems

some applications of variational inequalities in nonsmooth analysis abstract in this paper we apply some results on variational inequalities to the clarke generalized gradient and the generalized second order directional derivative f x u v and obtain some variational inequalities of them

pinning control of scale free dynamical networks abstract recently it has been demonstrated that many large complex networks display a scale free feature that is their connectivity distributions have the power law form in the present work control of a scale free dynamical network by applying local feedback injections to a fraction of network nodes is investigated the specifically and randomly pinning schemes are considered the specifically pinning of the most highly connected nodes is shown to require a significantly smaller number of local controllers as compared to the randomly pinning scheme the method is applied to an array of chua s oscillators as an example

on second order conditions in unconstrained optimization the main purpose of this paper is to establish the second order nonsmooth sufficient unconstrained optimality condition for so called l stable at some point functions and in this way to generalize some previous results in this direction we provide the comparisons with other results by examples

promoting collective motion of self propelled agents by discarding short range interactions abstract we study the collective motion of self propelled agents with the restricted view the field of view of each agent is an annulus bounded by the outer radius r and inner radius r where is a tunable parameter we find that there exists an optimal value of leading to the highest degree of direction consensus this phenomenon indicates that there exists superfluous communication in the collective motion of self propelled agents and short range interactions hinder the direction consensus of the system the value of optimal decreases as the absolute velocity increases while it increases as the outer radius r and the system size increase for a fixed value of direction consensus is enhanced when the absolute velocity is small the outer radius or the system size is large

second order global optimality conditions for convex composite optimization in recent years second order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established in this paper second order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition this result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification in particular second order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained 1998 the mathematical programming society inc published by elsevier science b v

on characterization of convexity for regularly locally lipschitz functions we use a second order upper dini directional derivative in order to characterize convexity for regularly locally lipschitz functions this improves in a certain sense the previously obtained results of cominetti correa and yang

partial extensions of attouch s theorem with applications to proto derivatives of subgradient mappings attouch s theorem which gives on a reflexive banach space the equivalence between the mosco epi convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients has many important applications in convex optimization in particular generalized derivatives have been defined in terms of the epi convergence or graph convergence of certain difference quotient mappings and attouch s theorem has been used to relate these various generalized derivatives these relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems we prove in a hilbert space several partial extensions of attouch s theorem to functions more general than convex these functions are called primal lower nice furthermore we use our extensions to derive a relationship between the second order epi derivatives of primal lower nice functions and the proto derivative of their associated subgradient mappings

an inexact bundle variant suited to column generation we give a bundle method for constrained convex optimization instead of using penalty functions it shifts iterates towards feasibility by way of a slater point assumed to be known besides the method accepts an oracle delivering function and subgradient values with unknown accuracy our approach is motivated by a number of applications in column generation in which constraints are positively homogeneous so that zero is a natural slater point and an exact oracle may be time consuming finally our convergence analysis employs arguments which have been little used so far in the bundle community the method is illustrated on a number of cutting stock problems

piecewise quadratic approximations in convex numerical optimization we present a bundle method for convex nondifferentiable minimization where the model is a piecewise quadratic convex approximation of the objective function unlike standard bundle approaches the model only needs to support the objective function from below at a properly chosen small subset of points as opposed to everywhere we provide the convergence analysis for the algorithm with a general form of master problem which combines features of trust region stabilization and proximal stabilization taking care of all the important practical aspects such as proper handling of the proximity parameters and the bundle of information numerical results are also reported

factors that predict better synchronizability on complex networks while shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network the suppressing tendency of the heterogeneity of the degree distribution even for shorter characteristic path length has also been reported to see this we investigate the effects of various factors such as the degree characteristic path length heterogeneity and betweenness centrality on synchronization and find a consistent trend between the synchronization and the betweenness centrality the betweenness centrality is thus proposed as a good indicator for synchronizability

decentralized control of multi agent systems for swarming with a given geometric pattern this paper addresses a problem of cooperative formation control of a network of self deployed autonomous agents we propose a decentralized motion coordination control for the agents so that they collectively move in a desired geometric pattern from any initial position there are no predefined leaders in the group and only local information is required for the control the control algorithm is developed using the ideas of information consensus and its effectiveness is illustrated via numerical simulations

flocks herds and schools a quantitative theory of flocking we present a quantitative continuum theory of flocking the collective coherent motion of large numbers of self propelled organisms in agreement with everyday experience our model predicts the existence of an ordered phase of flocks in which all members of even an arbitrarily large flock move together with the same mean velocity stackrel ensuremath rightarrow v ensuremath ne 0 this coherent motion of the flock is an example of spontaneously broken symmetry no preferred direction for the motion is picked out a priori in the model rather each flock is allowed to and does spontaneously pick out some completely arbitrary direction to move in by analyzing our model we can make detailed quantitative predictions for the long distance long time behavior of this broken symmetry state the goldstone modes associated with this spontaneously broken rotational symmetry are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole these goldstone modes mix with modes associated with conservation of bird number to produce propagating sound modes these sound modes lead to enormous fluctuations of the density of the flock far larger at long wavelengths than those in e g an equilibrium gas our model is similar in many ways to the navier stokes equations for a simple compressible fluid in other ways it resembles a relaxational time dependent ginsburg landau theory for an n d component isotropic ferromagnet in spatial dimensions dg4 the long distance behavior is correctly described by a linearized theory and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems for dl4 nonlinear fluctuation effects radically alter the long distance behavior making it different from that of any known equilibrium model in particular we find that in d 2 where we can calculate the scaling exponents exactly flocks exhibit a true long range ordered spontaneously broken symmetry state in contrast to equilibrium systems which cannot spontaneously break a continuous symmetry in d 2 the mermin wagner theorem we make detailed predictions for various correlation functions that could be measured either in simulations or by quantitative imaging of real flocks we also consider an anisotropic model in which the birds move preferentially in an easy e g horizontal plane and make analogous but quantitatively different predictions for that model as well for this anisotropic model we obtain exact scaling exponents for all spatial dimensions including the physically relevant case d 3

piecewise linear approximations in nonconvex nonsmooth optimization we present a bundle type method for minimizing nonconvex nondifferentiable functions of several variables the algorithm is based on the construction of both a lower and an upper polyhedral approximation of the objective function in particular at each iteration a search direction is computed by solving a quadratic program aiming at maximizing the difference between the lower and the upper model a proximal approach is used to guarantee convergence to a stationary point under the hypothesis of weak semismoothness

an incremental method for solving convex finite min max problems we introduce a new approach to minimizing a function defined as the pointwise maximum over finitely many convex real functions next referred to as the component functions with the aim of working on the basis of incomplete knowledge of the objective function a descent algorithm is proposed which need not require at the current point the evaluation of the actual value of the objective function namely of all the component functions thus extending to min max problems the philosophy of the incremental approaches widely adopted in the nonlinear least squares literature given the nonsmooth nature of the problem we resort to the well established machinery of bundle methods we provide global convergence analysis of our method and in addition we study a subgradient aggregation scheme aimed at simplifying the problem of finding a tentative step this paper is completed by the numerical results obtained on a set of standard test problems

a generalized second order derivative in nonsmooth optimization in this work a new notion of generalized second order directional derivative and generalized hessian for nonsmooth real valued functions is studied the general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation two applications of these derivatives in optimization theory are considered first to obtaining necessary and sufficient second order optimality conditions for problems with or without constraints and second to extending the newton method for the minimization of a mathcal c 1 1 function

minimizing nonconvex nonsmooth functions via cutting planes and proximity control we describe an extension of the classical cutting plane algorithm to tackle the unconstrained minimization of a nonconvex not necessarily differentiable function of several variables the method is based on the construction of both a lower and an upper polyhedral approximation to the objective function and is related to the use of the concept of proximal trajectory convergence to a stationary point is proved for weakly semismooth functions

dissipative control system for the stochastic nonlinear h problems abstract we characterize functions satisfying a dissipative inequality associated with a stochastic control problem such a characterization is provided in terms of an upper generalized gaussian solution or a viscosity supersolution to a partial differential equation called hamilton jacobi equation h j links between upper generalized gaussian solutions and viscosity supersolutions to hamilton jacobi equation are studied finally it shows that generalized gaussian solutions is identical to viscosity solutions to hamilton jacobi equation

exploiting invariants in the numerical solution of multipoint boundary value problems for dae this paper presents a new approach to the numerical solution of boundary value problems for higher index differential algebraic equations daes invariants known for the original dae as well as invariants of the reduced index 1 formulation are exploited to stabilize initial value problem ivp integration derivative generation and nonlinear and linear systems solution of an enhanced multiple shooting method extensions to collocation are given applications are presented for two important problem classes parameter estimation in multibody systems given in descriptor form and singular and state constrained optimal control problems in particular generalizations of the internal numerical differentiation technique to dae with invariants and a new multistage least squares decomposition technique for dae boundary value problems are developed which are implemented in the multiple shooting code parfit and in the collocation code colfit further a method is described which minimizes the number of necessary directional derivatives in the presence of multipoint conditions and invariants as numerical applications a parameter identification problem for a slider crank mechanism and a periodic cruise optimal control problem for a motor glider aircraft are treated

consensus analysis of multiagent networks via aggregated and pinning approaches in this paper the consensus problem of multiagent nonlinear directed networks mndns is discussed in the case that a mndn does not have a spanning tree to reach the consensus of all nodes by using the lie algebra theory a linear node and node pinning method is proposed to achieve a consensus of a mndn for all nonlinear functions satisfying a given set of conditions based on some optimal algorithms large size networks are aggregated to small size ones then by applying the principle minor theory to the small size networks a sufficient condition is given to reduce the number of controlled nodes finally simulation results are given to illustrate the effectiveness of the developed criteria

multiple shooting for unstructured nonlinear differential algebraic equations of arbitrary index we study multiple shooting methods for the numerical solution of nonlinear boundary value problems for unstructured nonlinear systems of differential algebraic equations with arbitrary index we give a convergence analysis and demonstrate the results with some numerical examples

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

new approaches for optimizing over the semimetric polytope the semimetric polytope is an important polyhedral structure lying at the heart of several hard combinatorial problems therefore linear optimization over the semimetric polytope is crucial for a number of relevant applications building on some recent polyhedral and algorithmic results about a related polyhedron the rooted semimetric polytope we develop and test several approaches based over lagrangian relaxation and application of non differentiable optimization algorithms for linear optimization over the semimetric polytope we show that some of these approaches can obtain very accurate primal and dual solutions in a small fraction of the time required for the same task by state of the art general purpose linear programming technology in some cases good estimates of the dual optimal solution but not of the primal solution can be obtained even quicker

enhancing consensus in weighted networks with coupling time delay abstract we study the consensus problem in a weighted and directed network composed of self propelled agents the weight which quantifies the relationship between the agents is adjusted according to their state incoherence to suppress the heterogeneity in state the proposed consensus protocol enhances the convergence efficiency of consensus greatly and has better performance than that of two other protocols the convergence efficiency can be further improved by adjusting the parameter of this protocol furthermore the robustness of the system against coupling time delay is significantly increased unlike most consensus acceleration methods no topological information is needed in our model networks with different stuctures are investigated to demonstrate the generality of the method

approximate jacobian matrices for nonsmooth continuous maps and c 1 optimization the notion of approximate jacobian matrices is introduced for a continuous vector valued map it is shown for instance that the clarke generalized jacobian is an approximate jacobian for a locally lipschitz map the approach is based on the idea of convexificators of real valued functions mean value conditions for continuous vector valued maps and taylor s expansions for continuously g teaux differentiable functions i e c1 functions are presented in terms of approximate jacobians and approximate hessians respectively second order necessary and sufficient conditions for optimality and convexity of c1 functions are also given

symmetric collocation methods for linear differential algebraic boundary value problems we present symmetric collocation methods for linear differential algebraic boundary value problems without restrictions on the index or the structure of the differential algebraic equation in particular we do not require a separation into differential and algebraic solution components instead we use the splitting into differential and algebraic equations which arises naturally by index reduction techniques and apply gaus type for the differential part and lobatto type for the algebraic part collocation schemes to obtain a symmetric method which guarantees consistent approximations at the mesh points under standard assumptions we show solvability and stability of the discrete problem and determine its order of convergence moreover we show superconvergence when using the combination of gaus and lobatto schemes and discuss the application of interpolation to reduce the number of function evaluations finally we present some numerical comparisons to show the reliability and efficiency of the new methods

on mathcalvu theory for functions with primal dual gradient structure we consider a general class of convex functions having what we call primal dual gradient structure it includes finitely determined max functions and maximum eigenvalue functions as well as other infinitely defined max functions for a function in this class we discuss a space decomposition that allows us to identify a subspace on which the function appears to be smooth moreover using the special structure of such a function we compute smooth trajectories along which certain second order expansions can be obtained we also give an explicit expression for the hessian of a related lagrangian

a new software package for linear differential algebraic equations we describe the new software package gelda for the numerical solution of linear differential algebraic equations with variable coefficients the implementation is based on the new discretization scheme introduced in p kunkel and v mehrmann siam j numer anal 33 1996 pp 1941 1961 it can deal with systems of arbitrary index and with systems that do not have unique solutions or inconsistencies in the initial values or the inhomogeneity the package includes a computation of all the local invariants of the system a regularization procedure and an index reduction scheme and it can be combined with every solution method for standard index 1 systems nonuniqueness and inconsistencies are treated in a least square sense we give a brief survey of the theoretical analysis of linear differential algebraic equations with variable coefficients and discuss the algorithms used in gelda furthermore we include a series of numerical examples as well as comparisons with results from other codes as far as this is possible

leapfrog for optimal control the leapfrog algorithm so called because of its geometric nature for solving a class of optimal control problems is proposed initially a feasible trajectory is given and subdivided into smaller pieces in each subdivision with the assumption that local optimal controls can easily be calculated a piecewise optimal trajectory is obtained then the junctions of these smaller pieces of optimal control trajectories are updated through a scheme of midpoint maps under some broad assumptions the sequence of trajectories is shown to converge to a trajectory that satisfies the maximum principle the main advantages of the leapfrog algorithm are that i it does not need an initial guess for the costates and ii the piecewise optimal trajectory generated in each iteration is feasible these are illustrated through a numerical implementation of leapfrog on a problem involving the van der pol system

a modified vicsek model for self propelled agents with exponential neighbor weight and restricted visual field abstract large flocks of wild beings can have coordinated motion with neither leading center nor global information the vicsek model and its new versions explained such collective behavior of self propelled agents with the mechanism of un weighted influence of neighbors and global visual field gvf which may not always fit the reality in the present work we take both exponential neighbor weight enw and restricted visual field rvf into account and investigate the combinatory effect of them based on this mechanism in most cases of our simulation stationary direction consensus v a exhibits a maximum or decreases with v f and a minimum convergence time t m i n appears in rvf which demonstrates that gvf is not necessary for coordinated motion while rvf matched enw would be reasonable moreover we simulated various recipes for optimizing v a and t in unfavorable cases and found critical points separating inverse dependent variations of t the present work may be useful in improving the efficiency of direction consensus of large flocks of artificial individuals besides the understanding of biological collective motion in nature

a bundle newton method for nonsmooth unconstrained minimization an algorithm based on a combination of the polyhedral and quadratic approximation is given for finding stationary points for unconstrained minimization problems with locally lips chitz problem functions that are not necessarily convex or differentiable global convergence of the algorithm is established under additional assumptions it is shown that the algorithm generates newton iterations and that the convergence is superlinear some encouraging numerical experience is reported

proximity control in bundle methods for convex proximal bundle methods for minimizing a convex functionf generate a sequence x k by takingxk 1 to be the minimizer of hat f k x u k x x k 2 2 where hat f k is a sufficiently accurate polyhedral approximation tof andu k 0 the usual choice ofu k 1 may yield very slow convergence a technique is given for choosing u k adaptively that eliminates sensitivity to objective scaling some encouraging numerical experience is reported

quasi newton bundle type methods for nondifferentiable convex optimization in this paper we provide implementable methods for solving nondifferentiable convex optimization problems a typical method minimizes an approximate moreau yosida regularization using a quasi newton technique with inexact function and gradient values which are generated by a finite inner bundle algorithm for a bfgs bundle type method global and superlinear convergence results for the outer iteration sequence are obtained

error bounds for lower semicontinuous functions in normed spaces without the convexity or analyticity assumption we study error bounds for an inequality system defined by a general lower semicontinuous function and establish sufficient necessary conditions on the existence of error bounds in infinite dimensional normed spaces some characterizations for a convex inequality system to possess an error bound in a reflexive banach space are also given as applications in dealing with the hoffman error bound result in normed spaces we give a computable lipschitz bound constant which is better than previous lipschitz bound constants in some examples we also consider error bounds for quadratic functions on rn

l stable functions are continuous abstract it has been shown that l stability of the function at some point implies continuity near the point then the previous second order optimality condition introduced by the authors can be stated under weak assumptions

on lower bounds of the second order directional derivatives of ben tal zowe and chaney let f be a regular locally lipschitz real valued function defined on an open convex subset of a normed space we show that at any unit direction u the upper second order derivative d 2f u 0 in the sense of dem yanov and pevnyi dem yanov v f a b pevnyi 1974 expansion with respect to a parameter of the extremal values of game problems ussr computational math and math phys 14 33 45 ben tal and zowe ben tal a j zowe 1982 necessary and sufficient optimality conditions for a class of nonsmooth minimization problems math programming24 70 91 has the same lower bounds as the lower second order derivatives d 2f u 0 consequently one can characterize the convexity of f in terms of these derivatives we also obtain the corresponding results in terms of chaney s second order directional derivatives

support functions of the clarke generalized jacobian and of its plenary hull this paper studies two important mathematical objects which are useful in tackling the first order behaviour of vector valued locally lipschitz functions in a finite dimensional setting the clarke generalized jacobian and its plenary hull we aim at giving analytical expressions of the support functions of these compact convex sets of matrices our study was motivated by earlier works by j b hiriart urruty and recent papers by zs pales and v zeidan the expressions of the support functions are applied for instance to provide a new proof of a chain rule on generalized jacobians of composed locally lipschitz functions without further assumptions applications of our results to the second order behaviour of c 1 functions with locally lipschitz gradients are considered

optimality conditions via exact penalty functions in this paper we study kkt optimality conditions for constrained nonlinear programming problems and strong and mordukhovich stationarities for mathematical programs with complementarity constraints using l p penalty functions with 0 leq p leq1 we introduce some optimality indication sets by using contingent derivatives of penalty function terms some characterizations of optimality indication sets are obtained by virtue of the original problem data we show that the kkt optimality condition holds at a feasible point if this point is a local minimizer of some l p penalty function with p belonging to the optimality indication set our result on constrained nonlinear programming includes some existing results from the literature as special cases

generalized hessian properties of regularized nonsmooth functions we take up the question of second order expansions for a class of functions of importance in optimization namely moreau envelope regularizations of nonsmooth functions f it is shown that when f is prox regular which includes convex functions and the extended real valued functions representing problems of nonlinear programming the many second order properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself this clarifies the circumstances conducive to developing computational methods based on envelope functions such as second order approximations in nonsmooth optimization and variants of the proximal point algorithm the results establish that generalized second order expansions of moreau envelopes at least can be counted on in most situations of interest in finite dimensional optimization

generalized hessian for c 1 1 functions in infinite dimensional normed spaces the subject of this paper is the systematic study of second order notions concerning differentiable functions with lipschitz derivative the results and notions are motivated by recent papers of cominetti correa and hiriart urruty the first goal of this paper is the comparison of several known second order directional derivatives the second goal is the introduction of a generalized hessian which is a set of certain symmetric bilinear forms the relation of this generalized hessian to other existing second order derivatives is also described

on partial smoothness tilt stability and the mathcal vu decomposition under the assumption of prox regularity and the presence of a tilt stable local minimum we are able to show that a mathcal vu like decomposition gives rise to the existence of a smooth manifold on which the function in question coincides locally with a smooth function