general balance and treatment permutations abstract many equi replicate designs whether ordinary block designs or multi stratum designs have a structure which is left unchanged by a group g of permutations of the treatments this group g is often used in the construction of the design a survey of the properties of such designs is given and useful necessary conditions on g are found for such a design to be generally balanced if g is abelian then the necessary conditions are satisfied in this case explicit formulas for the efficiency factors are given

practical aspects of the moreau yosida regularization theoretical preliminaries when computing the infimal convolution of a convex function f with the squared norm the so called moreau yosida regularization of f is obtained among other things this function has a lipschitzian gradient we investigate some more of its properties relevant for optimization the most important part of our study concerns second order differentiability existence of a second order development of f implies that its regularization has a hessian for the converse we disclose the importance of the decomposition of bbb r n along cal u the subspace where f is smooth and cal v the subspace parallel to the subdifferential of f

changeover designs and factorial structure summary existing changeover designs given in the literature are not necessarily suitable when the treatments come from a factorial experiment the concept of factorial structure in changeover designs is defined and shown to be a desirable requirement for these types of designs different classes of designs are then examined in the light of this concept in particular it is shown that all generalized cyclic gc n changeover designs have factorial structure thereby providing a flexible class of useful designs

identifying structure of nonsmooth convex functions by the bundle technique we consider the problem of minimizing nonsmooth convex functions defined piecewise by a finite number of functions each of which is either convex quadratic or twice continuously differentiable with positive definite hessian on the set of interest this is a particular case of functions with primal dual gradient structure a notion closely related to the so called cal vu space decomposition at a given point nonsmoothness is locally restricted to the directions of the subspace cal v while along the subspace cal u the behavior of the function is twice differentiable constructive identification of the two subspaces is important because it opens the way to devising fast algorithms for nonsmooth optimization by following iteratively the manifold of smoothness on which superlinear cal u newton steps can be computed in this work we show that for the class of functions in consideration the information needed for this identification can be obtained from the output of a standard bundle method for computing proximal points provided a minimizer satisfies the nondegeneracy and strong transversality conditions

a combined smoothing and regularization method for monotone second order cone complementarity problems the second order cone complementarity problem soccp is a wide class of problems containing the nonlinear complementarity problem ncp and the second order cone programming problem socp recently fukushima luo and tseng siam j optim 12 2001 pp 436 460 extended some merit functions and their smoothing functions for ncp to soccp moreover they derived computable formulas for the jacobians of the smoothing functions and gave conditions for the jacobians to be invertible in this paper we propose a globally and quadratically convergent algorithm which is based on smoothing and regularization methods for solving monotone soccp in particular we study strong semismoothness and jacobian consistency which play an important role in establishing quadratic convergence of the algorithm furthermore we examine the effectiveness of the algorithm by means of numerical experiments

differentiability of relations and differential stability of perturbed optimization problems several new concepts of differentiability of multifunctions are introduced special attention is paid to relations given by perturbed inequalities applications are given to the study of the value or marginal function of a perturbed nonlinear program

on eigenvalue optimization in this paper we study optimization problems involving eigenvalues of symmetric matrices one of the difficulties with numerical analysis of such problems is that the eigenvalues considered as functions of a symmetric matrix are not differentiable at those points where they coalesce we present a general framework for a smooth differentiable approach to such problems it is based on the concept of transversality borrowed from differential geometry in that framework we discuss first and second order optimality conditions and rates of convergence of the corresponding second order algorithms finally we present some results on the sensitivity analysis of such problems

composite proximal bundle method we consider minimization of nonsmooth functions which can be represented as the composition of a positively homogeneous convex function and a smooth mapping this is a sufficiently rich class that includes max functions largest eigenvalue functions and norm 1 regularized functions the bundle method uses an oracle that is able to compute separately the function and subgradient information for the convex function and the function and derivatives for the smooth mapping with this information it is possible to solve approximately certain proximal linearized subproblems in which the smooth mapping is replaced by its taylor series linearization around the current serious step our numerical results show the good performance of the composite bundle method for a large class of problems

necessary and sufficient optimality conditions for a class of nonsmooth minimization problems the purpose of this paper is to derive in a unified way second order necessary and sufficient optimality criteria for four types of nonsmooth minimization problems thediscrete minimax problem thediscrete l1 approximation the minimization of theexact penalty function and the minimization of theclassical exterior penalty function our results correct and supplement conditions obtained by various authors in recent papers

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

the u lagrangian of the maximum eigenvalue function in this paper we apply the u lagrangian theory to the maximum eigenvalue function la and to its precomposition with affine matrix valued mappings we first give geometrical interpretations of the u objects that we introduce we also show that the u lagrangian of la has a hessian which can be explicitly computed the second order development of the u lagrangian provides a second order development of la along a characteristic smooth manifold the set of symmetric matrices whose maximal eigenvalues have a fixed multiplicity the same results can be obtained when we precompose la with an affine matrix valued mapping a provided that this mapping satisfies a regularity condition em transversality condition we show that the hessian of the u lagrangian of la circ a coincides with the reduced hessian encountered in sequential quadratic programming finally we use the u lagrangian to derive second order algorithms for minimizing la circ a

a proximal method for composite minimization we consider minimization of functions that are compositions of convex or prox regular functions possibly extended valued with smooth vector functions a wide variety of important optimization problems fall into this framework we describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem preliminary computational results on both convex and nonconvex examples are promising

two factor designs balanced for the neighbour effect of one factor summary we consider the situation when treatments are the combinations of levels of two factors one of which exerts competition effects experimental units consist of one or several rows of plots with border plots classes of balanced designs which are optimal for the estimation of direct treatment effects and competition effects are introduced and methods of construction are given for various numbers of levels of each factor

the supremum of gaussian processes with a constant variance we obtain an estimate of the distribution of the large values of the supremum of a sample bounded gaussian process having a constant variance this estimate uses the entropy function of the parameter space endowed as usual with the pseudo metric induced by the l2 norm of the increments of the process

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

on a class of nonsmooth composite functions we discuss in this paper a class of nonsmooth functions which can be represented in a neighborhood of a considered point as a composition of a positively homogeneous convex function and a smooth mapping which maps the considered point into the null vector we argue that this is a sufficiently rich class of functions and that such functions have various properties useful for purposes of optimization

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

minima of h valued gaussian processes we study low local extremes of gaussian random fields with values in a separable hilbert space and constant variance our results are sharp for certain stationary processes on the line and for these processes we also prove global limits

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

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs