problem method classification in optimization and control abstract there are many classifications in optimization and control continuous discrete integer problems convex or concave minimization steepest descent branch and bound path following space filling random search methods smooth or non smooth problems etc the aim of this short note is not to propose yet another classification but to clarify existing ambiguities and imprecisions in problem formulations and to emphasize certain fundamental distinctions in solution methods currently used

least squares splines with free knots global optimization approach splines with free knots have been extensively studied in regard to calculating the optimal knot positions the dependence of the accuracy of approximation on the knot distribution is highly nonlinear and optimisation techniques face a difficult problem of multiple local minima the domain of the problem is a simplex which adds to the complexity we have applied a recently developed cutting angle method of deterministic global optimisation which allows one to solve a wide class of optimisation problems on a simplex the results of the cutting angle method are subsequently improved by local discrete gradient method the resulting algorithm is sufficiently fast and guarantees that the global minimum has been reached the results of numerical experiments are presented

onset of cooperative entrainment in limit cycle oscillators with uniform all to all interactions bifurcation of the order function abstract the concept of order function was previously proposed as the key to a general theory of macroscopic mutual or cooperative entrainment in large populations of limit cycle oscillators with weak interactions such that every element is linked to all the other as well as with weakly dispersed intrinsic frequencies that is limit cycle oscillators that can be modeled by globally coupled phase oscillators with distributed frequencies following previous work a bifurcation theory of the order function is developed on the basis of its self consistent functional equation to elucidate in particular generic scaling behavior of such systems at the onset of cooperative entrainment among other results when the onset is not abrupt the critical exponent of fundamental order parameters turns out to generically differ from the conventional value 1 2 taken by the well studied sinusoidal coupling model as well as by typical mean fields models of thermodynamic phase transitions to which coupled oscillator models investigated here are analogous the theory also reveals what happens in nongeneric cases moreover a criterion is found of whether the bifurcation is normal or inverted all these analytical results and predictions are verified not only by numerically solving the equation of the order function but also by numerical simulations although this paper is mainly concerned with the critical behaviors noncritical regimes are also explored to demonstrate overall power of the order function theory by reproducing simulation results such as average frequency spectra the theory however keeps some room to be further generalized a finding which suggests this is put forth

multidimensional bisection a dual viewpoint abstract this paper provides an alternative viewpoint of multidimensional bisection global optimization methods of wood a dual coordinate representation of convex bodies is introduced which leads to an easy implementation and eliminates the need to see the geometry of intersecting simplexes although developed in the context of global optimization the techniques deal more generally with regions represented as the union of convex bodies with this dual framework the algorithm can be implemented efficiently using any multiattribute index data structure that allows for quick range queries a c version using a multi key double linked skip list based on pugh s skip list has been implemented

multiple solutions for some quasilinear equation of n laplacian type and containing a gradient term abstract in this paper a new variational tool is used to establish a multiplicity result for some quasilinear equations involving a gradient term and a nonlinear term having a critical exponential growth

slowly coupled oscillators phase dynamics and synchronization in this paper we extendthe results of frankel andkiemel siam j appl math 5 3 1993 pp 1436 1446 to a network of slowly coupledoscillators first we use malkin s theorem to derive a canonical phase model that describes synchronization properties of a slowly coupled network then we illustrate the result using slowly coupledoscillators 1 near and ronov hopf bifurcations 2 near saddle node on invariant circle bifurcations and 3 near relaxation oscillations we compare andcontrast synchronization properties of slowly andweakly coupledoscillators

functional identification of spike processing neural circuits we introduce a novel approach for a complete functional identification of biophysical spike processing neural circuits the circuits considered accept multidimensional spike trains as their input and comprise a multitude of temporal receptive fields and conductance based models of action potential generation each temporal receptive field describes the spatiotemporal contribution of all synapses between any two neurons and incorporates the passive processing carried out by the dendritic tree the aggregate dendritic current produced by a multitude of temporal receptive fields is encoded into a sequence of action potentials by a spike generator modeled as a nonlinear dynamical system our approach builds on the observation that during any experiment an entire neural circuit including its receptive fields and biophysical spike generators is projected onto the space of stimuli used to identify the circuit employing the reproducing kernel hilbert space rkhs of trigonometric polynomials to describe input stimuli we quantitatively describe the relationship between underlying circuit parameters and their projections we also derive experimental conditions under which these projections converge to the true parameters in doing so we achieve the mathematical tractability needed to characterize the biophysical spike generator and identify the multitude of receptive fields the algorithms obviate the need to repeat experiments in order to compute the neurons rate of response rendering our methodology of interest to both experimental and theoretical neuroscientists

a new filled function for unconstrained global optimization abstract seeking global optima of an unconstrained and multi modal global optimization problem min x f x by constructing a filled function is concerned in this paper on the basis of analyzing filled functions presented before a new filled function is proposed and it is proved to meet the properties of a filled function moreover solutions of numerical experiments show that the function is quite effective

mesh adaptive direct search algorithms for constrained optimization this paper addresses the problem of minimization of a nonsmooth function under general nonsmooth constraints when no derivatives of the objective or constraint functions are available we introduce the mesh adaptive direct search mads class of algorithms which extends the generalized pattern search gps class by allowing local exploration called polling in an asymptotically dense set of directions in the space of optimization variables this means that under certain hypotheses including a weak constraint qualification due to rockafellar mads can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained the main gps convergence result is to identify limit points hat x where the clarke generalized derivatives are nonnegative in a finite set of directions called refining directions although in the unconstrained case nonnegative combinations of these directions span the whole space the fact that there can only be finitely many gps refining directions limits rigorous justification of the barrier approach to finitely many linear constraints for gps the main result of this paper is that the general mads framework is flexible enough to allow the generation of an asymptotically dense set of refining directions along which the clarke derivatives are nonnegative we propose an instance of mads for which the refining directions are dense in the hypertangent cone at hat x with probability 1 whenever the iterates associated with the refining directions converge to a single hat x the instance of mads is compared to versions of gps on some test problems we also illustrate the limitation of our results with examples

on ground states of superlinear p laplacian equations in rn abstract we consider superlinear p laplacian equations in r n with a potential which is periodic or has a bounded potential well without assuming the ambrosetti rabinowitz type condition and the monotonicity of the function t f x t t p 1 we prove the existence of ground states even for the case p 2 our results extend the recent results of li wang and zeng ann inst h poincare anal non lineaire 23 2006 829 837

singularly perturbed critical choquard equations abstract in this paper we study the semiclassical limit for the singularly perturbed choquard equation e 2 u v x u e 3 r 3 q y g u y x y d y q x g u in r 3 where 0 3 e is a positive parameter v q are two continuous real function on r 3 and g is the primitive of g which is of critical growth due to the hardy littlewood sobolev inequality under suitable assumptions on g we first establish the existence of ground states for the critical choquard equation with constant coefficients next we establish existence and multiplicity of semi classical solutions and characterize the concentration behavior by variational methods

global optimization of univariate lipschitz functions ii new algorithms and computational comparison we consider the following global optimization problems for a lipschitz functionf implicitly defined on an interval a b problem p find a globallye optimal value off and a corresponding point problem q find a set of disjoint subintervals of a b containing only points with a globallye optimal value and the union of which contains all globally optimal points a two phase algorithm is proposed for problem p in phase i this algorithm obtains rapidly a solution which is often globallye optimal moreover a sufficient condition onf for this to be the case is given in phase ii the algorithm proves thee optimality of the solution obtained in phase i or finds a sequence of points of increasing value containing one with a globallye optimal value the new algorithm is empirically compared on twenty problems from the literature with a best possible algorithm for which the optimal value is assumed to be known with a passive algorithm and with the algorithms of evtushenko galperin shen and zhu piyavskii timonov and schoen for smalle the new algorithm requires only a few percent more function evaluations than the best possible one an extended version of piyavskii s algorithm is proposed for problem q a sufficient condition onf is given for the globally optimal points to be in one to one correspondance with the obtained intervals this result is achieved for all twenty test problems

cluster based population initialization for differential evolution frameworks this article proposes a procedure to perform an intelligent initialization for population based algorithms the proposed pre processing procedure namely cluster based population initialization cbpi consists of three consecutive stages at the first stage the individuals belonging to a randomly sampled population undergo two subsequent local search algorithms i e a simple local search that performs moves along the axes and rosenbrock algorithm at the second stage the solutions processed by the two local searches undergo the k means clustering algorithm and are grouped into sets on the basis of their euclidean distance at the third stage the best individuals belonging to each cluster are saved into the initial population of a generic optimization algorithm if the population has not been yet filled the other individuals of the population are sampled within the clusters by using a fitness based probabilistic criterion this three stage procedure implicitly performs an initial screening of the problem features in order to roughly estimate the most interesting regions of the decision space the proposed cbpi has been tested on multiple classical and modern differential evolution variants on a wide array of test problems and dimensionality values as well as on a real world problem the proposed intelligent sampling appears to have a significant impact on the algorithmic functioning as it consistently enhances the performance of the algorithms with which it is integrated

two alternatives for the cubic algorithm abstract two alternatives are proved that allow application of the cubic algorithm for finding approximate solutions of the general non convex and non robust mathematical programming problem

a deterministic algorithm for global optimization we present an algorithm for finding the global maximum of a multimodal multivariate function for which derivatives are available the algorithm assumes a bound on the second derivatives of the function and uses this to construct an upper envelope successive function evaluations lower this envelope until the value of the global maximum is known to the required degree of accuracy the algorithm has been implemented in ratfor and execution times for standard test functions are presented at the end of the paper

fast global oscillations in networks of integrate and fire neurons with low firing rates we study analytically the dynamics of a network of sparsely connected inhibitory integrate and fire neurons in a regime where individual neurons emit spikes irregularly and at a low rate in the limit when the number of neurons n 1 the network exhibits a sharp transition between a stationary and an oscillatory global activity regime where neurons are weakly synchronized the activity becomes oscillatory when the inhibitory feedback is strong enough the period of the global oscillation is found to be mainly controlled by synaptic times but depends also on the characteristics of the external input in large but finite networks the analysis shows that global oscillations of finite coherence time generically exist both above and below the critical inhibition threshold their characteristics are determined as functions of systems parameters in these two different regimes the results are found to be in good agreement with numerical simulations

type i membranes phase resetting curves and synchrony type i membrane oscillators such as the connor model connor et al 1977 and the morris lecar model morris and lecar 1981 admit very low frequency oscillations near the critical applied current hansel et al 1995 have numerically shown that synchrony is difficult to achieve with these models and that the phase resetting curve is strictly positive we use singular perturbation methods and averaging to show that this is a general property of type i membrane models we show in a limited sense that so called type ii resetting occurs with models that obtain rhythmicity via a hopf bifurcation we also show the differences between synapses that act rapidly and those that act slowly and derive a canonical form for the phase interactions

convergence and restart in branch and bound algorithms for global optimization application to concave minimization and d c optimization problems a general branch and bound conceptual scheme for global optimization is presented that includes along with previous branch and bound approaches also grid search techniques the corresponding convergence theory as well as the question of restart capability for branch and bound algorithms used in decomposition or outer approximation schemes are discussed as an illustration of this conceptual scheme a finite branch and bound algorithm for concave minimization is described and a convergent branch and bound algorithm based on the previous one is developed for the minimization of a difference of two convex functions

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

on perturbations of a class of a periodic m laplacian equation with critical growth 1 after the well known results of brezis nirenberg 10 problems involving ellipticequationswithcriticalgrowthinboundeddomainshavebeenstudiedbyseveralauthors see for example 7 15 11 13 17 and the references therein the special features of this class of problems considered in this paper is that it isdened in r

a naive multi scale search algorithm for global optimization problems this paper proposes a multi scale search algorithm for solving global optimization problems given a finite number of function evaluations we refer to this algorithm as the naive multi scale search optimization nmso nmso looks for the optimal solution by optimistically partitioning the search space over multiple scales in a hierarchical fashion based on a weak assumption about the function smoothness we present a theoretical analysis on its finite time and asymptotic convergence an empirical assessment of the algorithm has been conducted on the noiseless black box optimization benchmarking bbob testbed and compared with the state of the art optimistic as well as stochastic algorithms moreover the efficacy of nmso has been validated on the black box optimization competition within the gecco 15 conference where it has secured the third place out of twenty eight participating algorithms overall nmso is suitable for problems with limited function evaluations low dimensionality search space and objective functions that are separable or multi modal otherwise it is comparable with the top performing algorithms

lipschitz minorants of brownian motion and levy processes for alpha 0 the alpha lipschitz minorant of a function f mathbb r rightarrow mathbb r is the greatest function m mathbb r rightarrow mathbb r such that m le f and m s m t le alpha s t for all s t in mathbb r should such a function exist if x x t t in mathbb r is a real valued levy process that is not pure linear drift with slope pm alpha then the sample paths of x have an alpha lipschitz minorant almost surely if and only if mathbb e x 1 alpha denoting the minorant by m we investigate properties of the random closed set fancyscript z t in mathbb r m t x t wedge x t which since it is regenerative and stationary has the distribution of the closed range of some subordinator made stationary in a suitable sense we give conditions for the contact set fancyscript z to be countable or to have zero lebesgue measure and we obtain formulas that characterize the levy measure of the associated subordinator we study the limit of fancyscript z as alpha rightarrow infty and find for the so called abrupt levy processes introduced by vigon that this limit is the set of local infima of x when x is a brownian motion with drift beta such that beta alpha we calculate explicitly the densities of various random variables related to the minorant

knapsack problems with sigmoid utilities approximation algorithms via hybrid optimization we study a class of non convex optimization problems involving sigmoid functions we show that sigmoid functions impart a combinatorial element to the optimization variables and make the global optimization computationally hard we formulate versions of the knapsack problem the generalized assignment problem and the bin packing problem with sigmoid utilities we merge approximation algorithms from discrete optimization with algorithms from continuous optimization to develop approximation algorithms for these np hard problems with sigmoid utilities

convergence rates of a global optimization algorithm this paper presents a best and worst case analysis of convergence rates for a deterministic global optimization algorithm superlinear convergence is proved for lipschitz functions which are convex in the direction of the global maximum concave in the direction of the global minimum computer results are given which confirm the theoretical convergence rates

sequential and parallel algorithms for global minimizing functions with lipschitzian derivatives abstract in this paper sequential and parallel algorithms using derivatives for solving unconstrained one dimensional global optimization problems are described sufficient conditions of convergence to all global minimizers are established for both methods parallel algorithm conditions which guarantee significant speed up in comparison to the sequential version of the method are presented the sequential method is numerically compared with the algorithms of breiman and cutler pijavskii and strongin on a set of 20 test functions taken from literature we also present results of numerical experiments illustrating the performance of the parallel method all experiments have been executed on the parallel computer alliant fx 80

variation free iterative method for global optimal control abstract a variation free method for global optimal control is proposed the method is based on a global parameter optimization algorithm and does not employ variational principles or variation based techniques

on the number of iterations of piyavskii s global optimization algorithm piyavskii s algorithm maximizes a univariate function f satisfying a lipschitz condition we compare the numbers of iterations needed by piyavskir s algorithm np to obtain a bounding function whose maximum value is within e of the unknown globally optimal value with that required by the best possible algorithm nb the main result is that np 2nb 1 and that this bound is sharp we also show that the number of iterations needed by piyavskii s algorithm to obtain a globally e optimal value together with a corresponding point npy satisfies npy 4nb 1 and that this bound is again sharp lower and upper bounds on nb are obtained as functions of f x e l0 and l1 where l0 is the smallest valid lipschitz constant and l1 the lipschitz constant used several properties of nb and of the distribution of evaluation points are deduced from these bounds

inclusion functions and global optimization 11 inclusion functions combined with special subdivision strategies are an effective means of solving the global unconstrained optimization problem although these techniques were determined and numerically tested about ten years ago they are nearly unknown and scarcely used in order to make the role of inclusion functions and subdivision strategies more widespread and transparent we will discuss a related simplified basic algorithm it computes approximations of the global minimum and at the same time bounds the absolute approximation error we will show that the algorithm works and converges under more general assumptions than it has been known hitherto that is only appropriate inclusion functions are expected to exist the number of minimal points finite or infinite is not of importance lipschitz conditions or continuity are not assumed as shown in the appendix the required inclusion functions can be constructed and programmed without difficulty in a natural way using interval analysis

on the phase reduction and response dynamics of neural oscillator populations weundertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli recalling and extending results from the literature we compute phase response curves prcs valid near bifurcations to periodic firing for hindmarsh rose hodgkin huxley fitzhugh nagumo and morris lecar models encompassing the four generic codimension one bifurcations phase density equations are then used to analyze the role of the bifurcation and the resulting prc in responses to stimuli in particular we explore the interplay among stimulus duration baseline firing frequency and population level response patterns we interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions

optimal search for the global maximum of functions with bounded seminorm this paper deals with methods seeking the global maximum of a real valued function defined over a real interval a b with prescribed seminorm i e the integral of the square of the derivative an optimal one step search algorithm is devised which provides the minimum guaranteed value of the lebesgue measure of the set of possible values of the global maximum after evaluating a given information operator at the next point at each step of the algorithm the value of the function at a new point x n and the se minorm taken over the interval a x n are evaluated in order to obtain the optimal algorithm the interpolatory envelope of a certain class of functions is constructed utilizing the minimum seminorm properties of the first order spline functions a simple search method which is near optimal is also described

the cubic algorithm for global games with application to pursuit evasion games abstract new algorithms for solution of nonconvex global games defined over a cube are presented application to differential games and to pursuit evasion games is considered and extension is made for games over general compact and robust sets defined by constraints that may depend on time and on state variables the methods are deterministic and monotonically set convergent to deliver in the limit the unique exact full globally optimal minimax and maximin solutions for both players or the entire global saddle set if it exists a stopping rule is provided for determining approximate solutions with a given precision in a finite number of iterations a modification is considered to play on mistakes of the opponent on line replacement of cost functionals is possible in the course of iterations according to changing interests of the players no separability nor convexity concavity assumptions are imposed on the cost functional and variational methods are not used the ideas are illustrated by examples and application is made to determining the globally optimal closed loop strategies for the ship torpedo collision avoidance differential game with manoeuvrability constraints

global search based on efficient diagonal partitions and a set of lipschitz constants in the paper the global optimization problem of a multidimensional black box function satisfying the lipschitz condition over a hyperinterval with an unknown lipschitz constant is considered a new efficient algorithm for solving this problem is presented at each iteration of the method a number of possible lipschitz constants are chosen from a set of values varying from zero to infinity this idea is unified with an efficient diagonal partition strategy a novel technique balancing usage of local and global information during partitioning is proposed a new procedure for finding lower bounds of the objective function over hyperintervals is also considered it is demonstrated by extensive numerical experiments performed on more than 1600 multidimensional test functions that the new algorithm shows a very promising performance

the bisection method in higher dimensions is the familiar bisection method part of some larger scheme the aim of this paper is to present a natural and useful generalisation of the bisection method to higher dimensions the algorithm preserves the salient features of the bisection method it is simple convergence is assured and linear and it proceeds via a sequence of brackets whose infinite intersection is the set of points desired brackets are unions of similar simplexes an immediate application is to the global minimisation of a lipschitz continuous function defined on a compact subset of euclidean space

a hybrid global optimization method the multi dimensional case we extend the hybrid global optimization method proposed by xu j comput appl math 147 2002 301 314 for the one dimensional case to the multi dimensional case the method consists of two basic components local optimizers and feasible point finders local optimizers guarantee efficiency and speed of producing a local optimal solution in the neighbourhood of a feasible point feasible point finders provide the theoretical guarantee for the new method to always produce the global optimal solution s correctly if a nonlinear nonconvex inverse problem has multiple global optimal solutions our algorithm is capable of finding all of them correctly three synthetic examples which have failed simulated annealing and genetic algorithms are used to demonstrate the proposed method

multidimensional bisection applied to global optimisation abstract a family of deterministic algorithms is introduced designed to solve the global optimisation problem for lipschitz continuous functions of many variables all the algorithms can be considered as generalisations of the bisection method they proceed via a sequence of brackets whose infinite intersection is the set of global optima brackets are unions of similar simplexes acceleration methods convergence properties and optimality questions are considered

local tuning and partition strategies for diagonal go methods in this paper global optimization go lipschitz problems are considered where the multi dimensional multiextremal objective function is determined over a hyperinterval an efficient one dimensional go method using local tuning on the behavior of the objective function is generalized to the multi dimensional case by the diagonal approach using two partition strategies global convergence conditions are established for the obtained diagonal geometric methods results of a wide numerical comparison show a strong acceleration reached by the new methods working with estimates of the local lipschitz constants over different subregions of the search domain in comparison with the traditional approach

elliptic equations and systems with nonstandard growth conditions existence uniqueness and localization properties of solutions abstract we study the dirichlet problem for the elliptic equations i d i a i x u d i u p i x 2 d i u c x u u x 2 u f x in a bounded domain r n and the class of elliptic systems j d j a i j x u f i x u i 1 n u u 1 u n satisfying the growth condition x s v r n 2 i j a i j x v v i j a 0 i j v i j p i j x a 0 const 0 the exponents p i j x p i x x are known functions these equations are usually referred to as elliptic equations with nonstandard growth conditions we prove first the theorems of existence of bounded weak solutions and establish sufficient conditions of uniqueness of a weak solution our main purpose is the study of the localization properties of weak solutions we show that the weak solution may identically vanish on a set of nonzero measure in a dead core and derive estimates on the size and location of these dead cores in terms of the problem data the study of the localization properties is performed via the method of local energy estimates

global optimization of univariate lipschitz functions i survey and properties we consider the following global optimization problems for a univariate lipschitz functionf defined on an interval a b problem p find a globally optimal value off and a corresponding point problem p find a globally optimal value off and a corresponding point problem q localize all globally optimal points problem q find a set of disjoint subintervals of small length whose union contains all globally optimal points problem q find a set of disjoint subintervals containing only points with a globally optimal value and whose union contains all globally optimal points we present necessary conditions onf for finite convergence in problem p and problem q recall the concepts necessary for a worst case and an empirical study of algorithms i e those ofpassive and ofbest possible algorithms summarize and discuss algorithms of evtushenko piyavskii shubert timonov schoen galperin shen and zhu presenting them in a simplified and uniform way in a high level computer language we address in particular the problems of using an approximation for the lipschitz constant reducing as much as possible the expected length of the region of indeterminacy which contains all globally optimal points and avoiding remaining subintervals without points with a globally optimal value new algorithms for problems p and q and an extensive computational comparison of algorithms are presented in a companion paper

changes of firing rate induced by changes of phase response curve in bifurcation transitions we study dynamical mechanisms responsible for changes of the firing rate during four different bifurcation transitions in the two dimensional hindmarsh rose 2dhr neuron model the saddle node on an invariant circle snic bifurcation to the supercritical andronov hopf ah one the snic bifurcation to the saddle separatrix loop ssl one the ah bifurcation to the subcritical ah sah one and the ssl bifurcation to the ah one for this purpose we study slopes of the firing rate curve with respect to not only an external input current but also temperature that can be interpreted as a timescale in the 2dhr neuron model these slopes are mathematically formulated with phase response curves prcs expanding the firing rate with perturbations of the temperature and external input current on the one dimensional space of the phase in the 2dhr oscillator by analyzing the two different slopes of the firing rate curve with respect to the temperature and external input current we find that during changes of the firing rate in all of the bifurcation transitions the calculated slope with respect to the temperature also changes this is largely dependent on changes in the prc size that is also related to the slope with respect to the external input current furthermore we find phase transition like switches of the firing rate with a possible increase of the temperature during the ssl to ah bifurcation transition

a filled function method with one parameter for box constrained global optimization in this paper a new auxiliary function with one parameter on box constrained for escaping the current local minimizer of global optimization problem is proposed first a new definition of the filled function for box constrained minimization problem is given and under mild assumptions this new auxiliary function is really a filled function then a new solution algorithm is proposed according to the theoretical analysis and some numerical results demonstrate the efficiency of this method for box constrained global optimization

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

the beta algorithm abstract a set monotonic non gradient algorithm is proposed for finding global minima of general non convex mathematical programming problems the algorithm is based on the cubic algorithm e a galperin j math anal appl m 112 1985 635 640 equipped with a semi certain distinction operator and the marginal comparison constant generator

integral global optimization method for differential games with application to pursuit evasion games abstract a variation free method for global solution of differential games is proposed the method is based on the integral global optimization algorithm and it does not employ variation based techniques nor the notion of convexity the existence of a saddle point is not assumed a priori instead the method delivers the global minimax and maximin solutions for both players a necessary and sufficient global minimax condition is presented an application is made to the solution of nonlinear pursuit evasion games the ideas are illustrated by examples

correlation transfer in stochastically driven neural oscillators over long and short time scales in the absence of synaptic coupling two or more neural oscillators may become synchronized by virtue of the statistical correlations in their noisy input streams recent work has shown that the degree of correlation transfer from input currents to output spikes depends not only on intrinsic oscillator dynamics but also on the length of the observation window over which the correlation is calculated in this paper we use stochastic phase reduction and regular perturbations to derive the correlation of the total phase elapsed over long time scales a quantity that provides a convenient proxy for the spike count correlation over short time scales we derive the spike count correlation directly using straightforward probabilistic reasoning applied to the density of the phase difference our approximations show that output correlation scales with the autocorrelation of the phase resetting curve over long time scales we also find a concise expression for the influence of the shape of the phase resetting curve on the initial slope of the output correlation over short time scales these analytic results together with numerical simulations provide new intuitions for the recent counterintuitive finding that type i oscillators transfer correlations more faithfully than do type ii over long time scales while the reverse holds true for the better understood case of short time scales

an implicit enumeration method for global optimization problems abstract this paper presents a new method for the global minimization of a function subject to linear constraints the feasible set is first covered by a union of simpler sets then some of them are eliminated using some global analytic information about the objective function e g lipschitz constants properties of the derivatives etc the remaining part of the cover is then refined and the process is repeated estimates for the computational complexity are given for some classes of problems numerical experience is also reported

qualitative changes in phase response curve and synchronization at the saddle node loop bifurcation prominent changes in neuronal dynamics have previously been attributed to a specific switch in onset bifurcation the bogdanov takens bt point this study unveils another relevant and so far underestimated transition point the saddle node loop bifurcation which can be reached by several parameters including capacitance leak conductance and temperature this bifurcation turns out to induce even more drastic changes in synchronization than the bt transition this result arises from a direct effect of the saddle node loop bifurcation on the limit cycle and hence spike dynamics in contrast the bt bifurcation exerts its immediate influence upon the subthreshold dynamics and hence only indirectly relates to spiking we specifically demonstrate that the saddle node loop bifurcation i ubiquitously occurs in planar neuron models with a saddle node on invariant cycle onset bifurcation and ii results in a symmetry breaking of the system s phase response curve the latter entails an increase in synchronization range in pulse coupled oscillators such as neurons the derived bifurcation structure is of interest in any system for which a relaxation limit is admissible such as josephson junctions and chemical oscillators

a critical point theorem via the ekeland variational principle abstract the aim of this paper is to establish the existence of a local minimum for a continuously g teaux differentiable function possibly unbounded from below without requiring any weak continuity assumption several special cases are also emphasized moreover a novel definition of palais smale condition which is more general than the usual one is presented and a mountain pass theorem is pointed out as a consequence multiple critical points theorems are then established finally as an example of applications an elliptic dirichlet problem with critical exponent is investigated

population encoding with hodgkin huxley neurons the recovery of weak stimuli encoded with a population of hodgkin huxley neurons is investigated in the absence of a stimulus the hodgkin huxley neurons are assumed to be tonically spiking the methodology employed calls for 1 finding an input output i o equivalent description of the hodgkin huxley neuron and 2 devising a recovery algorithm for stimuli encoded with the i o equivalent neuron s a hodgkin huxley neuron with multiplicative coupling is i o equivalent with an integrate and fire neuron with a variable threshold sequence for bandlimited stimuli a perfect recovery of the stimulus can be achieved provided that a nyquist type rate condition is satisfied a hodgkin huxley neuron with additive coupling and deterministic conductances is first order i o equivalent with a project integrate and fire neuron that integrates a projection of the stimulus on the phase response curve the stimulus recovery is formulated as a spline interpolation problem in the space of finite length bounded energy signals a hodgkin huxley neuron with additive coupling and stochastic conductances is shown to be first order i o equivalent with a project integrate and fire neuron with random thresholds for stimuli modeled as elements of sobolev spaces the reconstruction algorithm minimizes a regularized quadratic optimality criterion finally all previous recovery results of stimuli encoded with hodgkin huxley neurons with multiplicative and additive coupling and deterministic and stochastic conductances are extended to stimuli encoded with a population of hodgkin huxley neurons

multiplicity of solutions for a class of quasilinear equations with exponential critical growth abstract this paper deals with some results of existence and multiplicity of solutions for quasilinear problems involving the n laplacian operator when the nonlinearity has exponential critical growth to obtain our results we utilize ekeland variational principle mountain pass theorem and inequalities of the type trudinger moser bounded and unbounded domains are considered

convergence qualification of adaptive partition algorithms in global optimization following the presentation of a general partition algorithm scheme for seeking the globally best solution in multiextremal optimization problems necessary and sufficient convergence conditions are formulated in terms of respectively implied or postulated properties of the partition operator the convergence results obtained are pertinent to a number of deterministic algorithms in global optimization permitting their diverse modifications and generalizations

an algorithm for finding the global maximum of a multimodal multivariate function this algorithm for global optimization uses an arbitrary starting point requires no derivatives uses comparatively few function evaluations and is not side tracked by nearby relative optima the algorithm builds a gradually closer piecewise differentiable approximation to the objective function the computer program exhibits a theoretically expected strong tendency to cluster around relative optima close to the global results of testing with several standard functions are given

global one dimensional optimization using smooth auxiliary functions in this paper new global optimization algorithms are proposed for solving problems where the objective function is univariate and has lipschitzean first derivatives to solve this problem smooth auxiliary functions which are adaptively improved during the course of the search are constructed three new algorithms are introduced the first used the exact a priori known lipschitz constant for derivatives the second when this constant is unknown estimates it during the course of the search and finally the last method uses neither the exact global lipschitz constant nor its estimate but instead adaptively estimates the local lipschitz constants in different sectors of the search region during the course of optimization convergence conditions of the methods are investigated from a general viewpoint and some numerical results are also given 1998 the mathematical programming society inc published by elsevier science b v

a non gradient approach to global extremum seeking an adaptation of the shubert algorithm the main purpose of this paper is to adapt the so called shubert algorithm for extremum seeking control of general dynamic plants this algorithm is a good representative of the sampling optimization methods that achieve global extremum seeking on compact sets in the presence of local extrema the algorithm applies to lipschitz mappings the model of the system is assumed unknown but the knowledge of its lipschitz constant is assumed the controller depends on a design parameter the waiting time and tuning guidelines that relate the design parameter and the region of convergence and accuracy of the algorithm are presented the analysis shows that semi global practical convergence in the initial states to the global extremum can be achieved in presence of local extrema if compact sets of inputs are considered numerical simulations for global optimization in the presence of local extrema are provided to demonstrate the proposed approach

the integer cubic algorithm abstract the integer and mixed cubic algorithms are proposed for full global solution of integer and mixed optimization problems the paper presents further development of the ideas in 1 4

a semilinear elliptic pde not in divergence form via variational methods in this paper we consider a semilinear equation driven by an operator not in divergence form precisely the principal part of the operator is in divergence form but it has also a lower order term depending on du while the right hand side of the equation satisfies superlinear and subcritical growth conditions at zero and at infinity the problem has not a variational structure but despite that we use variational techniques in order to prove an existence and regularity result for the equation

on multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in rn abstract in this work we study the existence multiplicity and concentration of positive solutions for the following class of quasilinear problems n n u v x u n 2 u f u in r n u x 0 x r n where n u is the n laplacian operator is a positive parameter n 2 v r n r is a continuous function and f r r is a c 1 function having critical exponential growth

existence and multiplicity of solutions to equations of n laplacian type with critical exponential growth in rn in this paper we deal with the existence of solutions to the nonuniformly elliptic equation of the form 0 1 div a x u v x u n 2u f x u x eh x in rn when f rn r r behaves like exp u n n 1 when u and satisfies the ambrosetti rabinowitz condition in particular in the case of n laplacian i e a x u u n 2 u we obtain multiplicity of weak solutions of 0 1 moreover we can get the nontriviality of the solution in this case when e 0 finally we show that the main results remain true if one replaces the ambrosetti rabinowitz condition on the nonlinearity by weaker assumptions and thus we establish the existence and multiplicity results for a wider class of nonlinearity see section 7 for more details

the alpha algorithm and the application of the cubic algorithm in case of unknown lipschitz constant abstract a new algorithm for full global optimization of a lipschitzian function over an arbitrary bounded set in r n is presented a procedure is proposed to handle functions with unknown lip constants application to the cubic algorithm is given and illustrative examples are presented

global optimization to prescribed accuracy abstract programmable methods are outlined for rigorous global optimization to prescribed accuracy with references to enough detail to enable the implementation of the techniques on any computer some directions for future research are indicated

iterative methods for the localization of the global maximum methods for the search of a maximum utilizing a sequence of majorants to generate a sequence of localizations are analyzed necessary and sufficient conditions for the convergence of this sequence are given the case of lipschitz continuous functions is studied in detail and an analogy with iterative and relaxation methods for the solution of a system of linear equations is made methods of jacobi gauss seidel and chaotic type are proposed

global optimization in problems with uncertainties the gamma algorithm abstract problems with uncertainties can be viewed and formalized making use of multifunctions or general set valued functions a new concept of global optimality is proposed which allows us to solve global optimization problems with uncertainties in natural setting without imposing artificial constraints on uncertainties nor introducing a kind of partial ordering in order to apply conventional optimality concepts and optimization techniques nor considering solution in probability with the new concept deterministic optimization requires two optimization procedures a study of the subject is presented with many illustrative examples then a monotonic iterative algorithm is developed which renders approximate solutions with precision specified in advance a notion of piecewise continuous function of several variables is proposed and the method is then generalized for uncertain functions defined by a closed set valued function with piecewise continuous upper and lower boundaries the max2 f reduction precision and decomposition lemmas are proved to facilitate practical applications deferred deletions of sets with discontinuities are introduced and convergence theorem is proved for the modified algorithm

several filled functions with mitigators several filled functions were reported to seek the global minimum of multimodal functions of multiple variables this paper proposes an alternative formulation that may reduce the negative definite effect of the hessian of a filled function proposed before furthermore a class of mitigators is defined and applied to improve the computational characteristics of filled functions results of numerical experiments on typical testing functions are also reported

quasilinear equations with dependence on the gradient via mountain pass techniques in rn abstract we consider the quasilinear problem p u u p 2 u f u u p 2 u in r n where 1 p n and f is a nonlinearity depending also on the gradient of the solution the existence of a positive solution is stated through an iterative method based on mountain pass techniques