an o n log log n time algorithm for triangulating a simple polygon given a simple n vertex polygon the triangulation problem is to partition the interior of the polygon into n 2 triangles by adding n 3 nonintersecting diagonals we propose an o n log log n time algorithm for this problem improving on the previously best bound of o n log n and showing that triangulation is not as hard as sorting improved algorithms for several other computational geometry problems including testing whether a polygon is simple follow from our result

multivariate product shot noise cox point process models we introduce a new multivariate product shot noise cox process which is useful for modeling multi species spatial point patterns with clustering intra specific interactions and neutral negative or positive inter specific interactions the auto and cross pair correlation functions of the process can be obtained in closed analytical forms and approximate simulation of the process is straightforward we use the proposed process to model interactions within and among five tree species in the barro colorado island plot

using sampling and simplex derivatives in pattern search methods in this paper we introduce ways of making a pattern search more efficient by reusing previous evaluations of the objective function based on the computation of simplex derivatives e g simplex gradients at each iteration one can attempt to compute an accurate simplex gradient by identifying a sampling set of previously evaluated points with good geometrical properties this can be done using only past successful iterates or by considering all past function evaluations the simplex gradient can then be used to reorder the evaluations of the objective function associated with the directions used in the poll step or to update the mesh size parameter according to a sufficient decrease criterion neither of which requires new function evaluations we present these procedures in detail and apply them to a set of problems from the cuter collection numerical results show that these procedures can enhance significantly the practical performance of pattern search methods

convergence of the restricted nelder mead algorithm in two dimensions the nelder mead algorithm a longstanding direct search method for unconstrained optimization published in 1965 is designed to minimize a scalar valued function f of n real variables using only function values without any derivative information each nelder mead iteration is asso ciated with a nondegenerate simplex defined by n 1 vertices and their function values a typical iteration produces a new simplex by replacing the worst vertex by a new point despite the method s widespread use theoretical results have been limited for strictly convex objective functions of one variable with bounded level sets the algorithm always converges to the minimizer for such func tions of two variables the diameter of the simplex converges to zero but examples constructed by mckinnon show that the algorithm may converge to a nonminimizing point this paper considers the restricted nelder mead algorithm a variant that does not allow expansion steps in two dimen sions we show that for any nondegenerate starting simplex and any twice continuously differentiable function with positive definite hessian and bounded level sets the algorithm always converges to the minimizer the proof is based on treating the method as a discrete dynamical system and relies on several techniques that are nonstandard in convergence proofs for unconstrained optimization

numerical analysis of mathcal v mathcal u decomposition mathcal u gradient and mathcal u hessian approximations advances in bundle methods for nonsmooth optimization have lead to the development of mathcal v mathcal u decompositions the mathcal u gradient and the mathcal u hessian these variational analysis constructs have proven extremely valuable and lead to the development of the superlinearly convergent mathcal v mathcal u algorithm for nonsmooth optimization in this paper we examine these constructs from the viewpoint of derivative free optimization we show that given a finite max function f x max i 0 1 ldots m f i x and a black box which returns function values for each f i it is possible to construct approximations of the mathcal v mathcal u decompositions mathcal u gradient and mathcal u hessian the approximations do not require excessive black box calls and the accuracy of the approximations is directly related to the accuracy of the approximate gradient and hessians for each f i

reducing the number of function evaluations in mesh adaptive direct search algorithms the mesh adaptive direct search mads class of algorithms is designed for nonsmooth optimization where the objective function and constraints are typically computed by launching a time consuming computer simulation each iteration of a mads algorithm attempts to improve the current best known solution by launching the simulation at a finite number of trial points common implementations of mads generate 2n trial points at each iteration where n is the number of variables in the optimization problem the objective of the present work is to dynamically reduce that number we present an algorithmic framework that reduces the number of simulations to exactly n 1 without impacting the theoretical guarantees from the convergence analysis numerical experiments are conducted for several different contexts the results suggest that these strategies allow the new algorithms to reach a better solution with fewer function evaluations

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 the convergence of the multidirectional search algorithm this paper presents the convergence analysis for the multidirectional search algorithm a direct search method for unconstrained minimization the analysis follows the classic lines of proofs of convergence for gradient related methods the novelty of the argument lies in the fact that explicit calculation of the gradient is unnecessary although it is assumed that the function is continuously differentiable over some subset of the domain the proof can be extended to treat most nonsmooth cases of interest the argument breaks down only at points where the derivative exists but is not continuous finally it is shown how a general convergence theory can be developed for an entire class of direct search methods which includes such methods as the factorial design algorithm and the pattern search algorithm that share a key feature of the multidirectional search algorithm

pattern search algorithms for bound constrained minimization we present a convergence theory for pattern search methods for solving bound constrained nonlinear programs the analysis relies on the abstract structure of pattern search methods and an understanding of how the pattern interacts with the bound constraints this analysis makes it possible to develop pattern search methods for bound constrained problems while only slightly restricting the flexibility present in pattern search methods for unconstrained problems we prove global convergence despite the fact that pattern search methods do not have explicit information concerning the gradient and its projection onto the feasible region and consequently are unable to enforce explicitly a notion of sufficient feasible decrease

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

frames and grids in unconstrained and linearly constrained optimization a nonsmooth approach this paper describes a class of frame based direct search methods for unconstrained and linearly constrained optimization a template is described and analyzed using clarke s break nonsmooth calculus this provides a unified and simple approach to earlier results for grid and frame based methods and also provides partial convergence results when the objective function is not smooth undefined in some places or both the template also covers many new methods which combine elements of previous ideas using frames and grids these new methods include grid based simple descent algorithms which allow moving to points off the grid at every iteration and can automatically control the grid size provided function values are available the concept of a grid is also generalized to that of an admissible set which allows sets for example with circular symmetries the method is applied to linearly constrained problems using a simple barrier approach

controller design via nonsmooth multidirectional search we propose an algorithm which combines multidirectional search mds with nonsmooth optimization techniques to solve difficult problems in automatic control applications include static and fixed order output feedback controller design simultaneous stabilization h 2 h infty synthesis and much else we show how to combine direct search techniques with nonsmooth descent steps in order to obtain convergence certificates in the presence of nonsmoothness our technique is efficient when small and medium size controllers for plants with large state dimension are sought our numerical testing includes several benchmark examples for instance our algorithm needs 0 41 s to compute a static output feedback stabilizing controller for the boeing 767 flutter benchmark problem e e j davison ifac technical committee reports pergamon press oxford 1990 a system with 55 states the first static controller without performance specifications for this system was obtained in j burke a lewis and m overton siam j optim 15 2003 pp 751 779

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

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

on the convergence of pattern search algorithms we introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems our definition unifies an important collection of optimization methods that neither compute nor explicitly approximate derivatives we exploit our characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of sufficient decrease our analysis is possible because the iterates of a pattern search method lie on a scaled translated integer lattice this allows us to relax the classical requirements on the acceptance of the step at the expense of stronger conditions on the form of the step and still guarantee global convergence

parallel space decomposition of the mesh adaptive direct search algorithm this paper describes a parallel space decomposition psd technique for the mesh adaptive direct search mads algorithm mads extends a generalized pattern search for constrained nonsmooth optimization problems the objective of the present work is to obtain good solutions to larger problems than the ones typically solved by mads the new method psd mads is an asynchronous parallel algorithm in which the processes solve problems over subsets of variables the convergence analysis based on the clarke calculus is essentially the same as for the mads algorithm a practical implementation is described and some numerical results on problems with up to 500 variables illustrate the advantages and limitations of psd mads

on the convergence of grid based methods for unconstrained optimization the convergence of direct search methods for unconstrained minimization is examined in the case where the underlying method can be interpreted as a grid or pattern search over successively refined meshes an important aspect of the main convergence result is that translation rotation scaling and shearing of the successive grids are allowed

a non model based approach to bandwidth selection for kernel estimators of spatial intensity functions we propose a new bandwidth selection method for kernel estimators of spatial point process intensity functions the method is based on an optimality criterion motivated by the campbell formula applied to the reciprocal intensity function the new method is fully nonparametric does not require knowl edge of higher order moments and is not restricted to a specific class of point process our approach is computationally straightforward and does not require numerical approximation of integrals

an optimal algorithm for detecting weak visibility of a polygon notation and a theorem are presented which using a result of b chazelle and l j guibas 1985 enable the authors to design an o n log n algorithm for reporting all visibility edges of a given n vertex polygon improving on this bound to o n is presently focused upon this problem is solved for polygons with at least one given visibility edge it is assumed that both endpoints of this edge are convex vertices subsequently it is shown how to drop this restriction the general case of detecting weak edge visibility of an arbitrary simple polygon is dealt with

large sample properties of simulations using latin hypercube sampling latin hypercube sampling mckay conover and beckman 1979 is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables the asymptotic variance of such an estimate is obtained the estimate is also shown to be asymptotically normal asymptotically the variance is less than that obtained using simple random sampling with the degree of variance reduction depending on the degree of additivity in the function being integrated a method for producing latin hypercube samples when the components of the input variables are statistically dependent is also described these techniques are applied to a simulation of the performance of a printer actuator

pattern search methods for linearly constrained minimization we extend pattern search methods to linearly constrained minimization we develop a general class of feasible point pattern search algorithms and prove global convergence to a karush kuhn tucker point as in the case of unconstrained minimization pattern search methods for linearly constrained problems accomplish this without explicit recourse to the gradient or the directional derivative of the objective key to the analysis of the algorithms is the way in which the local search patterns conform to the geometry of the boundary of the feasible region

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

unconstrained derivative free optimization by successive approximation we present an algorithmic framework for unconstrained derivative free optimization based on dividing the search space in regions partitions every partition is assigned a representative point the representative points form a grid a piecewise constant approximation to the function subject to optimization is constructed using a partitioning and its corresponding grid the convergence of the framework to a stationary point of a continuously differentiable function is guaranteed under mild assumptions the proposed framework is appropriate for upgrading heuristics that lack mathematical analysis into algorithms that guarantee convergence to a local minimizer a convergent variant of the nelder mead algorithm that conforms to the given framework is constructed the algorithm is compared to two previously published convergent variants of the nm algorithm the comparison is conducted on the more garbow hillstrom set of test problems and on four variably dimensional functions with dimension up to 100 the results of the comparison show that the proposed algorithm outperforms both previously published algorithms

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

on a convex hull algorithm for polygons and its application to triangulation problems abstract a frequently used algorithm for finding the convex hull of a simple polygon in linear running time has been recently shown to fail in some cases due to its simplicity the algorithm is nevertheless attractive in this paper it is shown that the algorithm does in fact work for a family of simple polygons known as weakly externally visible polygons some application areas where such polygons occur are briefly discussed in addition it is shown that with a trivial modification the algorithm can be used to internally and externally triangulate certain classes of polygons in 0 n time

a pattern search filter method for nonlinear programming without derivatives this paper formulates and analyzes a pattern search method for general constrained optimization based on filter methods for step acceptance roughly a filter method accepts a step that improves either the objective function value or the value of some function that measures the constraint violation the new algorithm does not compute or approximate any derivatives penalty constants or lagrange multipliers a key feature of the new algorithm is that it preserves the division into search and local poll steps which allows the explicit use of inexpensive surrogates or random search heuristics in the search step it is shown here that the algorithm identifies limit points at which optimality conditions depend on local smoothness of the functions and to a greater extent on the choice of a certain set of directions stronger optimality conditions are guaranteed for smoother functions and in the constrained case for a fortunate choice of the directions on which the algorithm depends these directional conditions generalize those given previously for linear constraints but they do not require a feasible starting point in the absence of general constraints the proposed algorithm and its convergence analysis generalize previous work on unconstrained bound constrained and linearly constrained generalized pattern search the algorithm is illustrated on some test examples and on an industrial wing planform engineering design application

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

second order analysis of inhomogeneous spatial point processes with proportional intensity functions we consider pairs of spatial point processes with intensity functions sharing a common multiplicative term we introduce two novel approaches to estimating the pair correlation function for one of the point processes by treating the other as a baseline so as to account for the unspecified part of the intensity functions the first approach is based on nonparametric kernel smoothing whereas the second approach uses a conditional likelihood estimation approach to fit a parametric model for the pair correlation function a great advantage of the proposed methods is that they do not require the often difficult estimation of the unspecified part of the intensity functions we establish the consistency of the resulting estimators and discuss how the parametric estimator can be applied in model diagnostics and inference on regression parameters for the intensity functions we apply the proposed procedures to two spatial point patterns regarding the spatial distributions of birds in the u k s peak district in 1

generalized pattern searches with derivative information a common question asked by users of direct search algorithms is how to use derivative information at iterates where it is available this paper addresses that question with respect to generalized pattern search gps methods for unconstrained and linearly constrained optimization specifically this paper concentrates on the gps pollstep polling is done to certify the need to refine the current mesh and it requires o n function evaluations in the worst case we show that the use of derivative information significantly reduces the maximum number of function evaluations necessary for pollsteps even to a worst case of a single function evaluation with certain algorithmic choices given here furthermore we show that rather rough approximations to the gradient are sufficient to reduce the pollstep to a single function evaluation we prove that using these less expensive pollsteps does not weaken the known convergence properties of the method all of which depend only on the pollstep

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

a heuristic iterated subspace minimization method with pattern search for unconstrained optimization recently an increasing attention was paid on different procedures for an unconstrained optimization problem when the information of the first derivatives is unavailable or unreliable in this paper we consider a heuristic iterated subspace minimization method with pattern search for solving such unconstrained optimization problems the proposed method is designed to reduce the total number of function evaluations for the implementation of high dimensional problems meanwhile it keeps the advantages of general pattern search algorithm i e the information of the derivatives is not needed at each major iteration of such a method a low dimensional manifold the iterated subspace is constructed and an approximate minimizer of the objective function in this manifold is then determined by a pattern search method numerical results on some classic test examples are given to show the efficiency of the proposed method in comparison with a conventional pattern search method and a derivative free method

optimization by direct search new perspectives on some classical and modern methods direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis nonetheless users remained loyal to these methods most of which were easy to program some of which were reliable in the past fifteen years these methods have seen a revival due in part to the appearance of mathematical analysis as well as to interest in parallel and distributed com puting this review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results the underlying principles allow general ization to handle bound constraints and linear constraints we also discuss extensions to problems with nonlinear constraints

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

direct search methods on parallel machines direct search methods are methods designed to solve unconstrained minimization problems of the form mathop min limits x in mathbb r n f x where f n these methods are distinguished by the fact that they neither use nor require explicit derivative information the search for a local minimizer is driven solely by function information popular methods in this class include the factorial design algorithm of box 1 the pattern search algorithm of hooke and jeeves 4 and the simplex method of neider and mead 5

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

cfd shape optimization using an incomplete gradient adjoint formulation a design methodology based on the adjoint approach for flow problems governed by the incompressible euler equations is presented the main feature of the algorithm is that it avoids solving the adjoint equations which saves an important amount of cpu time furthermore the methodology is general in the sense it does not depend on the geometry representation all the grid points on the surface to be optimized can be chosen as design parameters in addition the methodology can be applied to any type of mesh the partial derivatives of the flow equations with respect to the design parameters are computed by finite differences in this way this computation is independent of the numerical scheme employed to obtain the flow solution once the design parameters have been updated the new solid surface is obtained with a pseudo shell approach in such a way that local singularities which can degrade or inhibit the convergence to the optimal solution are avoided some 2d and 3d numerical examples are shown to demonstrate the proposed methodology copyright 2001 john wiley sons ltd

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

finding optimal algorithmic parameters using derivative free optimization the objectives of this paper are twofold we devise a general framework for identifying locally optimal algorithmic parameters algorithmic parameters are treated as decision variables in a problem for which no derivative knowledge or existence is assumed a derivative free method for optimization seeks to minimize some measure of performance of the algorithm being fine tuned this measure is treated as a black box and may be chosen by the user examples are given in the text the second objective is to illustrate this framework by specializing it to the identification of locally optimal trust region parameters in unconstrained optimization the derivative free method chosen to guide the process is the mesh adaptive direct search a generalization of pattern search methods we illustrate the flexibility of the latter and in particular make provision for surrogate objectives locally optimal parameters with respect to overall computational time on a set of test problems are identified each function call may take several hours and may not always return a predictable result a tailored surrogate function is used to guide the search towards a local solution the parameters thus identified differ from traditionally used values and allow one to solve a problem that remained otherwise unsolved in a reasonable time using traditional values

solving unconstrained optimization problem with a filter based nonmonotone pattern search algorithm abstract this paper proposes a new frame based pattern search algorithm for unconstrained optimization the implementation of filter leads the sequence of function values at iterates x k nonmonotonically decreasing and thus motivates us to consider nonmonotone technique furthermore the use of these two techniques can improve the efficiency of general pattern search algorithms the numerical results show that the new algorithm is practical and efficient

a new quasi newton pattern search method based on symmetric rank one update for unconstrained optimization this paper proposes a new robust and quickly convergent pattern search method based on an implementation of ocssr1 optimal conditioning based self scaling symmetric rank one algorithm m r osborne l p sun a new approach to symmetric rank one updating ima journal of numerical analysis 19 1999 497 507 for unconstrained optimization this method utilizes the factorization of approximating hessian matrices to provide a series of convergent positive bases needed in pattern search process numerical experiments on some famous optimization test problems show that the new method performs well and is competitive in comparison with some other derivative free methods

analysis of direct searches for discontinuous functions it is known that the clarke generalized directional derivative is nonnegative along the limit directions generated by directional direct search methods at a limit point of certain subsequences of unsuccessful iterates if the function being minimized is lipschitz continuous near the limit point in this paper we generalize this result for discontinuous functions using rockafellar generalized directional derivatives upper subderivatives we show that rockafellar derivatives are also nonnegative along the limit directions of those subsequences of unsuccessful iterates when the function values converge to the function value at the limit point this result is obtained assuming that the function is directionally lipschitz with respect to the limit direction it is also possible under appropriate conditions to establish more insightful results by showing that the sequence of points generated by these methods eventually approaches the limit point along the locally best branch or step function when the number of steps is equal to two the results of this paper are presented for constrained optimization and illustrated numerically

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

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

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

a quasi newton based pattern search algorithm for unconstrained optimization this study proposes a new robust quasi newton algorithm for unconstrained optimization problem the factorization of approximating hessian matrices is investigated to provide a series of positive bases for pattern search experiments on some well known optimization test problems are presented to show the efficiency and robustness of the proposed algorithm it is found that the proposed algorithm is competitive and outperforms some other derivative free algorithms

semi parametric multinomial logistic regression for multivariate point pattern data we propose a new method for analysis of multivariate point pattern data observed in a heterogeneous environment and with complex intensity functions we suggest semi parametric models for the inten

a derivative free multidimensional optimal search method using lucas number we consider a unimodal continuous function of n variables over an n dimensional rectangular parallelopiped and explain a search method to find the domain of uncertainty which contains the point that maximizes the function our search method does not require differentiability assumption the function is projected along each dimension an algorithm is developed to eliminate the subset of the domain at each step which does not contain the solution lucas numbers are used in the elimination process

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

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

on the convergence of asynchronous parallel pattern search in this paper we prove global convergence for asynchronous parallel pattern search in standard pattern search decisions regarding the update of the iterate and the step length control parameter are synchronized implicitly across all search directions we lose this feature in asynchronous parallel pattern search since the search along each direction proceeds semiautonomously by bounding the value of the step length control parameter after any step that produces decrease along a single search direction we can prove that all the processes share a common accumulation point and if the function is continuously differentiable that such a point is a stationary point of the standard nonlinear unconstrained optimization problem

analysis of generalized pattern searches this paper contains a new convergence analysis for the lewis and torczon generalized pattern search gps class of methods for unconstrained and linearly constrained optimization this analysis is motivated by a desire to understand the successful behavior of the algorithm under hypotheses that are satisfied by many practical problems specifically even if the objective function is discontinuous or extended valued the methods find a limit point with some minimizing properties simple examples show that the strength of the optimality conditions at a limit point depends not only on the algorithm but also on the directions it uses and on the smoothness of the objective at the limit point in question the contribution of this paper is to provide a simple convergence analysis that supplies detail about the relation of optimality conditions to objective smoothness properties and to the defining directions for the algorithm and it gives previous results as corollaries

pattern search algorithms for mixed variable programming many engineering optimization problems involve a special kind of discrete variable that can be represented by a number but this representation has no significance such variables arise when a decision involves some situation like a choice from an unordered list of categories this has two implications the standard approach of solving problems with continuous relaxations of discrete variables is not available and the notion of local optimality must be defined through a user specified set of neighboring points we present a class of direct search algorithms to provide limit points that satisfy some appropriate necessary conditions for local optimality for such problems we give a more expensive version of the algorithm that guarantees additional necessary optimality conditions a small example illustrates the differences between the two versions a real thermal insulation system design problem illustrates the efficacy of the user controls for this class of algorithms

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

second order behavior of pattern search previous analyses of pattern search algorithms for unconstrained and linearly constrained minimization have focused on proving convergence of a subsequence of iterates to a limit point satisfying either directional or first order necessary conditions for optimality depending on the smoothness of the objective function in a neighborhood of the limit point even though pattern search methods require no derivative information we are able to prove some limited directional second order results although not as strong as classical second order necessary conditions these results are stronger than the first order conditions that many gradient based methods satisfy under fairly mild conditions we can eliminate from consideration all strict local maximizers and an entire class of saddle points

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

stochastic quasi likelihood for case control point pattern data we propose a novel stochastic quasi likelihood estimation procedure for case control point processes quasi likelihood for point processes depends on a certain optimal weight function and for the new method the weight function is stochastic since it depends on the control point pattern the new procedure also provides a computationally efficient implementation of quasi likelihood for univariate point processes in which case a synthetic control point process is simulated by the user under mild conditions the proposed approach yields consistent and asymptotically normal parameter estimators we further show that the estimators are optimal in the sense that the associated godambe information is maximal within a wide class of estimating functions for case control point processes the effectiveness of the proposed method is further illustrated using extensive simulation studies and two data examples

a nearly optimal sensor placement algorithm for boundary coverage locating visual sensors in 2d can be often modeled as an art gallery problem tasks such as surveillance require observing or covering the interior of a polygon with a minimum number of sensors or guards for other tasks such as inspection and image based rendering observing the boundaries of the environment is sufficient as interior covering ic also edge covering ec is np hard and no finite algorithm is known for its exact solution approximate ec solutions are provided by many heuristic algorithms but their performances with respect to optimality minimum number of sensors is unknown in this paper we propose a new ec sensors location technique the algorithm is incremental and converges toward the optimal solution it refines an initial approximation provided by an integer covering algorithm iec where each edge is observed entirely by at least one sensor a lower bound for the number of sensors specific of the polygon considered is used at each step for evaluating the quality of the current solution and a set of rules are provided for performing a local refinement to reduce the computational burden the algorithm has been implemented and tests over hundreds of random polygons show that it supplies solutions very close to and often coincident with the lower bound and then suboptimal or optimal in addition the approximate starting solutions provided by the iec algorithms are on the average close to optimum the tight lower bound can also be used for testing other ec sensor location algorithms running times allow dealing with polygons with up to a few hundreds of edges which appears adequate for many practical cases an enhanced version of the algorithm also taking into account range and incidence constraints has also been implemented and tested

a globally convergent augmented lagrangian pattern search algorithm for optimization with general constraints and simple bounds we give a pattern search method for nonlinearly constrained optimization that is an adaption of a bound constrained augmented lagrangian method first proposed by conn gould and toint siam j numer anal 28 1991 pp 545 572 in the pattern search adaptation we solve the bound constrained subproblem approximately using a pattern search method the stopping criterion proposed by conn gould and toint for the solution of the subproblem requires explicit knowledge of derivatives such information is presumed absent in pattern search methods however we show how we can replace this with a stopping criterion based on the pattern size in a way that preserves the convergence properties of the original algorithm in this way we proceed by successive inexact bound constrained minimization without knowing exactly how inexact the minimization is as far as we know this is the first provably convergent direct search method for general nonlinear programming

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

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

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

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