spherical prediction variance properties of central composite and box behnken designs for quadratic regression on the hypercube a single number criterion such as a g efficiency that is based on the prediction variance is often included as one of the criteria when selecting a response surface design as an alternative to the single number criterion approach the variance dispersion graph presented by giovannitti jensen and myers is a graphical technique for evaluating prediction variance properties throughout the experimental region three properties of interest are the maximum minimum and average spherical prediction variances given the spherical radius as an alternative to the computer based approach requiring an optimization algorithm to evaluate these properties the maximum minimum and spherical prediction variances for central composite and box behnken designs can be determined analytically and are functions only of the radius and the design parameters these functions yield the exact values of the spherical prediction variance properties of central composite and box behnke

a large sample confidence region useful in characterizing the stationary point of a quadratic response surface a second order response surface model is often used to approximate the relationship between a response variable and a set of explanatory variables the nature of the stationary point of the surface can be determined by considering the eigenvalues of the matrix of the model s second order terms since the elements of this matrix are estimated from the data however it follows that the eigenvalues are random variables hence the sampling properties of these eigenvalues should be considered in characterizing the nature of the stationary point in this article it is shown how a confidence region around these eigenvalues can be used to aid in the characterization of the stationary point and in an improved ridge analysis of the response surface the delta method is used to construct an approximate confidence region for these eigenvalues box and draper 1987 gave a result that simplifies the calculation of such a confidence region for rotatable or nearly rotatable designs a simulation study was performed to

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

majorization for a optimal designs in a regression setting the greatest lower bound for the trace of the covariance matrix of the simple least squares estimator is given based on the idea of majorization the problem raised by n n chan schur convexity for a optimal designs j math anal appl 122 1987 1 6 thus possesses a concise solution

a measure of rotatability for response surface designs a measure that quantifies the amount of rotatability in a given response surface design is introduced in this article the measure which is expressible as a percentage takes the value 100 if and only if the design is rotatable one of the main advantages of this measure is that it can be used to repair a nonrotatable design by the addition of experimental runs that maximize the percent rotatability over a spherical region of interest four numerical examples are given to illustrate the applications of this measure

second order characterizations of tilt stability with applications to nonlinear programming the paper is devoted to the study of tilt stable local minimizers of general optimization problems in finite dimensional spaces and its applications to classical nonlinear programs with twice continuously differentiable data the importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization and this notion has been extensively studied in the literature based on advanced tools of second order variational analysis and generalized differentiation we develop a new approach to tilt stability which allows us to derive not only qualitative but also quantitative characterizations of tilt stable minimizers with calculating the corresponding moduli the implementation of this approach and general results in the classical framework of nonlinear programming provides complete characterizations of tilt stable minimizers under new second order qualification and optimality conditions

on the relation between u hessians and second order epi derivatives abstract we consider second order objects for a convex function defined as the maximum of a finite number of c2 functions variational analysis yields explicit formulae for the second order epi derivatives of such max functions another second order object can be defined by means of a space decomposition that allows us to identify a subspace on which a lagrangian related to a max function is smooth this decomposition yields an explicit expression for the so called u hessian of the function defined as the hessian of the related lagrangian we show that the second order epi derivative relative to the u subspace and the u hessian are equivalent second order objects thus the u lagrangian of a max function captures the function s second order epi differential behavior with ordinary second derivatives

i optimal design of mixture experiments abstractin mixture experiments the factors under study are proportions of the ingredients of a mixture the special nature of the factors necessitates specific types of regression models and specific types of experimental designs although mixture experiments usually are intended to predict the response s for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients little research has been done concerning their i optimal design this is surprising given that i optimal designs minimize the average variance of prediction and therefore seem more appropriate for mixture experiments than the commonly used d optimal designs which focus on a precise model estimation rather than precise predictions in this article we provide the first detailed overview of the literature on the i optimal design of mixture experiments and identify several contradictions for the second order and the special cubic model we present continuous i optimal designs and contrast the

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

tilt stability uniform quadratic growth and strong metric regularity of the subdifferential we prove that uniform second order growth tilt stability and strong metric regularity of the subdifferential three notions that have appeared in entirely different settings are all essentially equivalent for any lower semicontinuous extended real valued function

active sets nonsmoothness and sensitivity nonsmoothness pervades optimization but the way it typically arises is highly structured nonsmooth behavior of an objective function is usually associated locally with an active manifold on this manifold the function is smooth whereas in normal directions it is vee shaped active set ideas in optimization depend heavily on this structure important examples of such functions include the pointwise maximum of some smooth functions and the maximum eigenvalue of a parametrized symmetric matrix among possible foundations for practical nonsmooth optimization this broad class of partly smooth functions seems a promising candidate enjoying a powerful calculus and sensitivity theory in particular we show under a natural regularity condition that critical points of partly smooth functions are stable small perturbations to the function cause small movements of the critical point on the active manifold

graphical tools for assessing the sensitivity of response surface designs to model misspecification when comparing different designs for an experiment optimality criteria and other measures often depend on the correctness of the assumed model in this article we develop and illustrate an approach for comparing designs given the potential effect of bias due to an underspecified model we illustrate this approach using graphical summaries of the expected mean squared error emse that allow assessment of the robustness of designs to model misspecification for response surface designs in cuboidal regions the excellent performance of the central composite designs when the quadratic model is correct is tempered when bias is present the box behnken designs demonstrate superiority in the presence of missing cubic terms for a screening design scenario the amount of bias present leads to different conclusions as to which design is best

some sufficient optimality conditions in nonsmooth analysis in this paper we study some second order conditions that may be added to the first order nessessary optimality condition 0 in partial p f bar x with partial p f denoting the proximal subdifferential in order to obtain a sufficient condition for a strict local minimum for extended real valued nonsmooth functions three different types of second order conditions are investigated all based on a different second order subdifferential namely the subhessian the graphical derivative and the contingent coderivative to the proximal subdifferential

partial smoothness tilt stability and generalized hessians we compare two recent variational analytic approaches to second order conditions and sensitivity analysis for nonsmooth optimization we describe a broad setting where computing the generalized hessian of mordukhovich is easy in this setting the idea of tilt stability introduced by poliquin and rockafellar is equivalent to a classical smooth second order condition

identifiable surfaces in constrained optimization the concept of a class c p identifiable surface of a convex set in euclidean space is introduced the paper shows how the smoothness of these surfaces is related to the smoothness of the projection operator and presents finite identification results for certain algorithms for minimization of a function over this set the work uses a partially geometric view of constrained optimization to generalize previous finite identification results

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a optimality for regression designs abstract use is made of a result of horn amer j math 76 1954 620 630 on the existence of a symmetric matrix with prescribed diagonal elements and eigenvalues a necessary and sufficient condition is then given for the existence of an a optimal design for a regression experiment in the dorogovcev selected transl math statist probab 10 1971 35 41 setting

comment on classification of lie point symmetries for quadratic lienard type equation ẍ f x ẋ2 g x 0 j math phys 54 053506 2013

primal dual gradient structured functions second order results links to epi derivatives and partly smooth functions we give second order expansions for quite general nonsmooth functions from the cal v cal u space decomposition point of view the results depend on primal dual gradient structure which we relate to general concepts of second order epi derivatives and partly smooth functions expressions for the associated second order objects are given in terms of cal u subspace hessians

graphical assessment of the prediction capability of response surface designs measures of the quality of prediction at locations on the surface of a hyperspherc are presented these measures are used to form a graphical method of assessing the overall prediction capability of an experimental design throughout the region of interest a plot of the spherical variance and the maximum and minimum prediction variances for locations on a sphere against the radius of the sphere a vuriance dispersion graph is used to give a comprehensive picture of the behavior of the prediction variances throughout a region and hence of the quality of the predicted responses obtained with a particular design such plots are used to investigate and compare the prediction capabilities of certain response surface designs currently available to the researcher

comparison of design optimality criteria of reduced models for response surface designs in the hypercube results from the comparison of d g and a efficiencies and the scaled average prediction variance iv criterion are presented for the central composite small composite notz hoke box draper and computer generated designs these design optimality criteria are evaluated over the cuboidal design region for three four and five design variables

limiting subhessians limiting subjets and their calculus we study calculus rules for limiting subjets of order two these subjets are obtained as limits of sequences of subjets a subjet of a function f at some point x being the taylor expansion of a twice differentiable function which minorizes f and coincides with f at x these calculus rules are deduced from approximate or fuzzy calculus rules for subjets of order two in turn these rules are consequences of delicate results of crandall ishii lions we point out the similarities and the differences with the case of first order limiting subdifferentials

response to comment on classification of lie point symmetries for quadratic lienard type equation ẍ f x ẋ2 g x 0 j math phys 61 044101 2020 we respond to the comment on classification of lie point symmetries for quadratic lienard type equation f x 2 g x 0 j math phys 61 044101 2013 by iacono regarding linearizability and isochronicity we assert here that the condition for linearization of the equation f x 2 g x 0 given by us in our paper is correct with the condition g1 02 0 we present the explicit form of local and nonlocal transformations that transform the quadratic lienard equation x f 11 xx2 x 1 x 1 dx 0 into the harmonic oscillator equation for the four cases mentioned in the comment and confirm the statements given in our paper are all valid

a graphical approach for evaluating response surface designs in terms of the mean squared error of prediction one of the fundamental purposes of response surface methodology is to develop a relatively simple linear model to explain a more complex phenomenon since the proposed model is only an approximation analysts almost always must address the potential of bias due to model misspecification this article proposes a graphical approach for evaluating response surface designs in terms of the mean squared error of prediction in particular this article develops measures of the maximum and minimum mean squared error of prediction over the surface of spheres through a region of interest these measures are then plotted against radius the resulting plots reveal at a glance how a response surface design performs in the presence of model misspecification these plots are used to investigate the performances of several common response surface designs

unusual lienard type nonlinear oscillator a lienard type nonlinear oscillator of the form x kxx k2 9 x3 lambda1x 0 which may also be considered as a generalized emden type equation is shown to possess unusual nonlinear dynamical properties it is shown to admit explicit nonisolated periodic orbits of conservative hamiltonian type for lambda1 0 these periodic orbits exhibit the unexpected property that the frequency of oscillations is completely independent of amplitude and continues to remain as that of the linear harmonic oscillator this is completely contrary to the standard characteristic property of nonlinear oscillators interestingly the system though appears deceptively a dissipative type for lambda1 or 0 does admit a conserved hamiltonian description where the characteristic decay time is also independent of the amplitude the results also show that the criterion for conservative hamiltonian system in terms of divergence of flow function needs to be generalized

newton methods for nonsmooth convex minimization connections among lagrangian riemannian newton and sqp methods this paper studies newton type methods for minimization of partly smooth convex functions sequential newton methods are provided using local parameterizations obtained from lagrangian theory and from riemannian geometry the hessian based on the lagrangian depends on the selection of a dual parameter g by revealing the connection to riemannian geometry a natural choice of g emerges for which the two newton directions coincide this choice of g is also shown to be related to the least squares multiplier estimate from a sequential quadratic programming sqp approach and with this multiplier sqp gives the same search direction as the newton methods

center points in second order response surface designs previous approaches for selecting the number of center points in a second order response surface design are explained and an integrated variance criterion is offered calculations extending results in the literature are made for various composite and box behnken designs for 2 k 8 the results are compared and discussed and indicate that fewer center points than initially recommended are generally appropriate

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

diagonal elements and eigenvalues of a real symmetric matrix abstract a simple and constructive proof is given for the existence of a real symmetric matrix with prescribed diagonal elements and eigenvalues numerically implementable algorithms for constructing such a matrix are discussed

an index of rotatability an index of rotatability is suggested that will enable the experimenter to obtain an immediate appreciation of the overall shape of specified variance contours for symmetrical second order designs values of the index are tabulated for the central composite designs for two to eight factors three designs of roquemore 1976 are assessed via the index comparisons are made with an entirely different index suggested by khuri 1988 it is concluded that both indexes are useful and sensibly consistent

the quadratic sub lagrangian of a prox regular function given a finite valued convex function f the u lagrangian as defined by lemarechal oustry and sagastizabal provides an envelope of f that maintains these properties by selecting the appropriate uv decomposition on one can also ensure that the u lagrangian is not only continuous at the origin but differentiable there this thesis extends these properties to proxregular functions by creating a new envelope which we call the quadratic sublagrangian in further exploration of this envelope it will be demonstrated how to use a quadratic expansion of the quadratic sub lagrangian to create a quadratic expansion for the original function finally some properties that guarantee such an expansion for the quadratic sub lagrangian will be developed acknowledgements and dedication to dr h bell for making me think it would be possible to dr r kerman for making me think it would be interesting and to dr r poliquin for making both these things true i would also like to express my deep gratitude to the natural sciences and engineering research council of canada and the university of alberta for the financial support they have given me throughout my studies finally i would like to thank dr tuszynski and dr van roessel who dedicated much of their time and energy to the improvement of this thesis

book review implicit functions and solution mappings a view from variational analysis second edition by a l dontchev and r t rockafellar springer new york 2014

a study of tilt stable optimality and sufficient conditions abstract recent results by eberhard et al 2006 4 and eberhard and wenczel 2009 3 on the interaction of single and double envelope operations of nonsmooth functions and their interaction with second order derivations have been used to study tilt stability of local minima this continues the study begun by poliquin and rockafellar 1998 1 but now armed with new tools we are able to make some new observations we observe that tilt stability entails a local density within the graph of the proximal subderivative of strict local minima order two of the tilted function indeed it also entails the strict local minimality order two of the tilt stable local minimum itself for prox regular subdifferentially continuous functions this density property characterises tilt stability

conjugation and second order properties of convex functions abstract we single out some second order properties of convex functions that are well behaved with respect to the conjugacy operator as an application we prove that if a convex lower semicontinuous function f r n r has a second order taylor expansion at the origin f x 1 2 ax x o x 2 as x 0 and the matrix a is symmetric and positive definite then the conjugate function f has a second order taylor expansion at the origin too given by f y 1 2 a 1 y y o y 2 as y 0

mixed response surface models with heterogeneous within block error variances the experimental error variance in a response surface model with a block effect has traditionally been assumed to be constant in some experimental situations however this variance may be different for the different blocks that compose the associated design this article discusses the analysis of a response surface model with a random block effect and heterogeneous error variances among the blocks the model includes interactions between the fixed polynomial effects and the block effect procedures for estimating and testing the fixed effects and the random effects in the model are described in particular anova based procedures for testing equality of the error variances and testing the random effects are introduced

necessary and sufficient conditions for isolated local minima of nonsmooth functions we consider the mathematical programming problem find inf f x mid x in c where f is an arbitrary extended real valued function and c a subset of a finite dimensional space we give necessary and sufficient optimality conditions for this problem generalizing previous results of a auslendex this journal 22 1984 pp 239 254

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