the convergence speed of interval methods for global optimization abstract three particular algorithms from a class of interval subdivision methods for global optimization are studied the theoretical upper bound on the convergence speed of hansen s method is given the three methods by hansen moore skelboe and a new one with a random actual box selection rule are compared numerically

iterative methods for large convex quadratic programs a survey in this paper we give a state of the art review of many iterative methods for solving large convex quadratic programs we attempt to classify several of the more basic methods in two categories within each of which a unified iterative scheme will be introduced and its convergence analyzed hybrid iterative methods such as the proximal point algorithm and a diagonalization scheme that make use of the more basic schemes will also be described the results of an extensive computer experimentation which is aimed at comparing the relative performance of the various methods will be reported and discussed finally several important topics which require future research will be highlighted

iteration complexity of block decomposition algorithms and the alternating direction method of multipliers in this paper we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a point to set maximal monotone operator with a separable two block structure and introduce a framework of block decomposition prox type algorithms for solving it which allows for each one of the single block proximal subproblems to be solved in an approximate sense moreover by showing that any method in this framework is also a special instance of the hybrid proximal extragradient hpe method introduced by solodov and svaiter we derive corresponding convergence rate results we also describe some instances of the framework based on specific and inexpensive schemes for solving the single block proximal subproblems finally we consider some applications of our methodology to establish for the first time i the iteration complexity of an algorithm for finding a zero of the sum of two arbitrary maximal monotone operators and as a consequence the ergodic iteration complexity of the douglas

solving a huff like competitive location and design model for profit maximization in the plane abstract a chain wants to set up a single new facility in a planar market where similar facilities of competitors and possibly of its own chain are already present fixed demand points split their demand probabilistically over all facilities in the market proportionally with their attraction to each facility determined by the different perceived qualities of the facilities and the distances to them through a gravitational or logit type model both the location and the quality design of the new facility are to be found so as to maximise the profit obtained for the chain several types of constraints and costs are considered two solution methods are developed and tested the first is a repeated local optimisation heuristic extending earlier proposals to the supplementary design question and the presence of locational constraints the second is an exact global optimisation technique based on reliable computing using interval analysis incorporating several novel features an example and comparative computational results demonstrate that this difficult and very multi modal problem can be solved by such techniques the local optimisation method turns out not to be very robust in its results even after numerous repetitions whereas the global optimisation method yields very useful and complete information on guaranteed near to optimal solutions after an important but still quite acceptable computational effort

reduced forms and reduced witt rings of higher level the present paper and its successor s deal with an extension of the socalled reduced theory of quadratic forms over a formally real field k 5 7 lo this theory deals with the reduced witt ring wred k w k nil w k and its natural embedding into c x z the ring of integer valued continuous functions on the space x x k of orderings of k recently it has been found that there is a far reaching generalization of the notion of ordering namely the orderings of higher level 11 the orderings of level s where s divides a given natural number n are related to sums of 2nth powers in the same way as the usual orderings are to sums of squares their intersection is the set of all sums of 2nth powers in k 11 in 6 orderings of levels were replaced by signatures of this level these are those characters x i s of order 2s whose kernels are additively closed where k k o it is exactly this change in point of view which leads to a natural definition of reduced witt rings of higher level let x x k denote the set of all signatures of k whose order divides 2n so that x k x k x being a closed subspace of the compact character group of x the set x k is also a compact topological space each element a e i induces a continuous function a x c in c x c defined by x x a where c is given the discrete topology by analogy with the natural representation of the ordinary reduced witt ring as a subring of 477 0021 8693 85 3 00

extended admm and bcd for nonseparable convex minimization models with quadratic coupling terms convergence analysis and insights in this paper we establish the convergence of the proximal alternating direction method of multipliers admm and block coordinate descent bcd method for nonseparable minimization models with quadratic coupling terms the novel convergence results presented in this paper answer several open questions that have been the subject of considerable discussion we firstly extend the 2 block proximal admm to linearly constrained convex optimization with a coupled quadratic objective function an area where theoretical understanding is currently lacking and prove that the sequence generated by the proximal admm converges in point wise manner to a primal dual solution pair moreover we apply randomly permuted admm rpadmm to nonseparable multi block convex optimization and prove its expected convergence for a class of nonseparable quadratic programming problems when the linear constraint vanishes the 2 block proximal admm and rpadmm reduce to the 2 block cyclic proximal bcd method and randomly permuted bcd rpbcd our study provides the first iterate convergence result for 2 block cyclic proximal bcd without assuming the boundedness of the iterates we also theoretically establish the expected iterate convergence result concerning multi block rpbcd for convex quadratic optimization in addition we demonstrate that rpbcd may have a worse convergence rate than cyclic proximal bcd for 2 block convex quadratic minimization problems although the results on rpadmm and rpbcd are restricted to quadratic minimization models they provide some interesting insights 1 random permutation makes admm and bcd more robust for multi block convex minimization problems 2 cyclic bcd may outperform rpbcd for nice problems and rpbcd should be applied with caution when solving general convex optimization problems especially with a few blocks

subdivision direction selection in interval methods for global optimization the role of the interval subdivision selection rule is investigated in branch and bound algorithms for global optimization the class of rules that allows convergence for the model algorithm is characterized and it is shown that the four rules investigated satisfy the conditions of convergence a numerical study with a wide spectrum of test problems indicates that there are substantial differences between the rules in terms of the required cpu time the number of function and derivative evaluations and space complexity and two rules can provide substantial improvements in efficiency

smoothing and first order methods a unified framework we propose a unifying framework that combines smoothing approximation with fast first order algorithms for solving nonsmooth convex minimization problems we prove that independently of the structure of the convex nonsmooth function involved and of the given fast first order iterative scheme it is always possible to improve the complexity rate and reach an o varepsilon 1 efficiency estimate by solving an adequately smoothed approximation counterpart our approach relies on the combination of the notion of smoothable functions that we introduce with a natural extension of the moreau infimal convolution technique along with its connection to the smoothing mechanism via asymptotic functions this allows for clarification and unification of several issues on the design analysis and potential applications of smoothing methods when combined with fast first order algorithms

a modified weiszfeld algorithm for the fermat weber location problem this paper gives a new simple monotonically convergent algorithm for the fermat weber location problem with extensions covering more general cost functions

convergence rates in forward backward splitting forward backward splitting methods provide a range of approaches to solving large scale optimization problems and variational inequalities in which structures conducive to decomposition can be utilized apart from special cases where the forward step is absent and a version of the proximal point algorithm comes out efforts at evaluating the convergence potential of such methods have so far relied on lipschitz properties and strong monotonicity or inverse strong monotonicity of the mapping involved in the forward step the perspective mainly being that of projection algorithms here convergence is analyzed by a technique that allows properties of the mapping in the backward step to be brought in as well for the first time in such a general setting global and local contraction rates are derived moreover they are derived in a form which makes it possible to determine the optimal step size relative to certain constants associated with the given problem insights are thereby gained into the effects of shifting strong monotonicity between the forward and backward mappings when a splitting is selected

total variation projection with first order schemes this article proposes a new algorithm to compute the projection on the set of images whose total variation is bounded by a constant the projection is computed through a dual formulation that is solved by first order non smooth optimization methods this yields an iterative algorithm that applies iterative soft thresholding to the dual vector field and for which we establish convergence rate on the primal iterates this projection algorithm can then be used as a building block in a variety of applications such as solving inverse problems under a total variation constraint or for texture synthesis numerical results are reported to illustrate the usefulness and potential applicability of our tv projection algorithm on various examples including denoising texture synthesis inpainting deconvolution and tomography problems we also show that our projection algorithm competes favorably with state of the art tv projection methods in terms of convergence speed

open questions concerning weiszfeld s algorithm for the fermat weber location problem the fermat weber location problem is to find a point in n that minimizes the sum of the weighted euclidean distances fromm given points in n a popular iterative solution method for this problem was first introduced by weiszfeld in 1937 in 1973 kuhn claimed that if them given points are not collinear then for all but a denumerable number of starting points the sequence of iterates generated by weiszfeld s scheme converges to the unique optimal solution we demonstrate that kuhn s convergence theorem is not always correct we then conjecture that if this algorithm is initiated at the affine subspace spanned by them given points the convergence is ensured for all but a denumerable number of starting points

centers of higher degree forms let be a symmetric d linear form on a vector space v of dimension n over a field k its center cent is the analog of the space of symmetric matrices for a bilinear form if d 2 the center is a commutative subalgebra of end v it seems difficult to determine which subalgebras can be realized as cent for some some d linear form as a first step we conjecture that the center has dimension at most n the conjecture is proved for n 5

a class of dantzig wolfe type decomposition methods for variational inequality problems we consider a class of decomposition methods for variational inequalities which is related to the classical dantzig wolfe decomposition of linear programs our approach is rather general in that it can be used with certain types of set valued or nonmonotone operators as well as with various kinds of approximations in the subproblems of the functions and derivatives in the single valued case also subproblems may be solved approximately convergence is established under reasonable assumptions we also report numerical experiments for computing variational equilibria of the game theoretic models of electricity markets our numerical results illustrate that the decomposition approach allows to solve large scale problem instances otherwise intractable if the widely used path solver is applied directly without decomposition

trace form preservers abstract if x m n k the set of n n matrices over a field define td x tr xd a preserver of td is a linear map f m n k m n k with tr f x d tr xd for every x a transformation is standard if it lies in the group generated by maps of the types 1 conjugation by something in gln k 2 transpose and 3 multiplication by a scaler dth root of unity if d 2 and d 0 in k o ryan shapiro m o ryan d b shapiro linear alg appl 246 1996 313 333 proved that every preserver of td is standard guralnick r m guralnick linear alg appl 212 213 1994 249 257 using methods of algebraic group theory extended this to infinite fields of characteristic p 0 when d 1 pm he found a fourth type of td preserver 4 f x x tr x where k satisfies n d d 1 0 call a transformation standard if it is the group generated by the standard maps together with these f s if they exist

on proving existence of feasible points in equality constrained optimization problems various algorithms can compute approximate feasible points or approximate solutions to equality and bound constrained optimization problems in exhaustive search algorithms for global optimizers and other contexts it is of interest to construct bounds around such approximate feasible points then to verify computationally but rigorously that an actual feasible point exists within these bounds hansen and others have proposed techniques for proving the existence of feasible points within given bounds but practical implementations have not to our knowledge previously been described various alternatives are possible in such an implementation and details must be carefully considered also in addition to hansen s technique for handling the underdetermined case it is important to handle the overdetermined case when the approximate feasible point corresponds to a point with many active bound constraints the basic ideas along with experimental results from an actual implementation are summarized here

primal and dual predicted decrease approximation methods we introduce the notion of predicted decrease approximation pda for constrained convex optimization a flexible framework which includes as special cases known algorithms such as generalized conditional gradient proximal gradient greedy coordinate descent for separable constraints and working set methods for linear equality constraints with bounds the new scheme allows the development of a unified convergence analysis for these methods we further consider a partially strongly convex nonsmooth model and show that dual application of pda based methods yields new sublinear convergence rate estimates in terms of both primal and dual objectives as an example of an application we provide an explicit working set selection rule for smo type methods for training the support vector machine with an improved primal convergence analysis

n real valuations and the higher level version of the krull baer theorem abstract in comm algebra 29 2001 193 201 cimpric gave examples of division rings containing an ordering of level 2m but not of level m for m n his examples were quite complicated we give substantially simplified examples in section 2 in sections 3 and 4 we investigate this phenomenon using valuation theory we define almost real and n real valuations and study liftings of orderings from the residue division ring to the original division ring such liftings are not always possible as is the case in the commutative setting but we give a necessary and sufficient condition for a lifting to exist we also prove a suitable generalization of the baer krull theorem finally in the last section we use our examples and the theory developed to answer a question given by marshall and zhang j algebra 212 1 1999 190 207

the cyclic block conditional gradient method for convex optimization problems in this paper we study the convex problem of optimizing the sum of a smooth function and a compactly supported nonsmooth term with a specific separable form we analyze the block version of the generalized conditional gradient method when the blocks are chosen in a cyclic order a global sublinear rate of convergence is established for two different stepsize strategies commonly used in this class of methods numerical comparisons of the proposed method to both the classical conditional gradient algorithm and its random block version demonstrate the effectiveness of the cyclic block update rule

applications of splitting algorithm to decomposition in convex programming and variational inequalities recently han and lou proposed a highly parallelizable decomposition algorithm for minimizing a strongly convex cost over the intersection of closed convex sets it is shown that their algorithm is in fact a special case of a splitting algorithm analyzed by gabay for finding a zero of the sum of two maximal monotone operators gabay s convergence analysis for the splitting algorithm is sharpened and new applications of this algorithm to variational inequalities convex programming and the solution of linear complementarily problems are proposed for convex programs with a certain separable structure a multiplier method that is closely related to the alternating direction method of multipliers of gabay mercier and of glowinski marrocco but which uses both ordinary and augmented lagrangians is obtained

stability and performance verification of optimization based controllers this paper presents a method to verify closed loop properties of optimization based controllers for deterministic and stochastic constrained polynomial discrete time dynamical systems the closed loop properties amenable to the proposed technique include global and local stability performance with respect to a given cost function both in a deterministic and stochastic setting and the mathcal l 2 gain the method applies to a wide range of practical control problems for instance a dynamical controller e g a pid plus input saturation model predictive control with state estimation inexact model and soft constraints or a general optimization based controller where the underlying problem is solved with a fixed number of iterations of a first order method are all amenable to the proposed approach the approach is based on the observation that the control input generated by an optimization based controller satisfies the associated karush kuhn tucker kkt conditions which provided all data is polynomial are a system of polynomial equalities and inequalities the closed loop properties can then be analyzed using sum of squares sos programming

chain signatures and higher level witt rings in many respects becker s theory of orderings of higher level is very much reminiscent of the familiar artin schreier theory of ordered fields a few examples of similarities are fields with higher level orderings have real closures 1 3 orderings of higher level can be used to study sums of powers in fields 3 41 there is a theory of witt rings for fields with higher level preorderings 7 81 however among all these similarities one very obvious difference stands out a field with a higher level ordering has nonisomorphic real closures becker showed that in order to describe the isomorphism classes of real closures a whole sequence of higher level orderings is required 1 p 162 theorem 121 subsequently this has been relined and reformulated through the introduction of chains of higher level orderings 121 and chain signatures 27 see alo 10 11 i in view of this situation one should expect that if results from the usual theory of ordered fields level 1 carry over to the case of orderings of higher level then quite different results will also be brought in by the chains of higher level orderings the purpose of this paper is to look for such results in the theory of higher level witt rings developed recently by becker and rosenberg 7 81 to keep things from getting too complicated this will be done only for the case where all levels are powers of 2 thus throughout the term higher level will be synonymous with level 2 for some n iv w k denotes the witt ring of the field k with respect to the higher level preordering k c k2 nonzero sums of 2 th powers of 9 just as the use of signatures s leads to homomorphisms w 7 chain signatures lead to homomorphisms

convergence rate analysis for the alternating direction method of multipliers with a substitution procedure for separable convex programming recently in he et al he bs tao m yuan xm 2012 alternating direction method with gaussian back substitution for separable convex programming siam j optim 22 2 313 340 we have showed the first possibility of combining the douglas rachford alternating direction method of multipliers admm with a gaussian back substitution procedure for solving a convex minimization model with a general separable structure this paper is a further study on this theme we first derive a general algorithmic framework to combine admm with either a forward or backward substitution procedure then we show that convergence of this framework can be easily proved from the contraction perspective and its local linear convergence rate is provable if certain error bound condition is assumed without such an error bound assumption we can estimate its worst case convergence rate measured by the iteration complexity

the impact of accelerating tools on the interval subdivision algorithm for global optimization abstract the interval subdivision method of moore and skelboe is considered for global optimization the new implementation allows the use of a long list in the procedure which is limited only by the available computer memory standard global optimization test problems are used to measure the efficiency of different versions with or without the monotonicity and the cut off tests of the algorithm the new implementation and the inclusion of the monotonicity test made it possible for the first time for all standard test problems to be solved completely by an interval method moreover in some problems the efficiency of the algorithm was better than that of the best traditional techniques which do not give guaranteed reliability results

holomorphy rings and higher level orders on skew fields the intimate connections which exist between higher level orders sums of 2n th powers valuation theory and real valued places on a field have been explored extensively craven initiated the study of similar notions for skew fields in 4 in 12 we pursued these ideas but the best results required a 2 primary assumption this paper extends the work of 12 in particular we prove an artin schreier theorem for preordered skew fields namely that a preorder t in a skew field is the intersection of the kernels of all the signatures that contain t this allows us to obtain the results of 12 without the 2 primary assumption in particular we have that preordered skew fields give rise to spaces of signatures mulcahy s abstract setting for the theory of higher level forms and reduced witt rings our main tools are valuation theory and the kadison dubois representation theorem for partially ordered rings as in the field case we make use of the ring a associated to a preorder t which is the intersection of certain valuation rings along the way to our main theorem we prove some facts about noncommutative priifer rings and we show that a is a priifer ring while our study of preordered skew fields follows along lines which are familiar from the field context there are significant differences between the two theories for example there are skew fields which have no orders but admit valuation rings with ordered residue fields

on the o 1 n convergence rate of the douglas rachford alternating direction method alternating direction methods adms have been well studied in the literature and they have found many efficient applications in various fields in this note we focus on the douglas rachford adm scheme proposed by glowinski and marrocco and we aim at providing a simple approach to estimating its convergence rate in terms of the iteration number the linearized version of this adm scheme which is known as the split inexact uzawa method in the image processing literature is also discussed

indecomposable forms of higher degree all nondegenerate indecomposable forms of higher degree over a perfect field k can be realized as traces of nondegenerate absolutely indecomposable forms of higher degree over a suitable algebraic field extension of k with the help of trace forms of certain nonassociative algebras we construct classes of indecomposable forms of degree d 3

on the convergence of alternating minimization for convex programming with applications to iteratively reweighted least squares and decomposition schemes this paper is concerned with the alternating minimization am method for solving convex minimization problems where the decision variables vector is split into two blocks the objective function is a sum of a differentiable convex function and a separable possibly nonsmooth extended real valued convex function and consequently constraints can be incorporated we analyze the convergence rate of the method and establish a nonasymptotic sublinear rate of convergence where the multiplicative constant depends on the minimal block lipschitz constant we then analyze the iteratively reweighted least squares irls method for solving convex problems involving sums of norms based on the results derived for the am method we establish a nonasymptotic sublinear rate of convergence of the irls method in addition we show an asymptotic rate of convergence whose efficiency estimate does not depend on the data of the problem finally we study the convergence properties of a decomposition based approach designed t

operator splittings bregman methods and frame shrinkage in image processing we examine the underlying structure of popular algorithms for variational methods used in image processing we focus here on operator splittings and bregman methods based on a unified approach via fixed point iterations and averaged operators in particular the recently proposed alternating split bregman method can be interpreted from different points of view as a bregman as an augmented lagrangian and as a douglas rachford splitting algorithm which is a classical operator splitting method we also study similarities between this method and the forward backward splitting method when applied to two frequently used models for image denoising which employ a besov norm and a total variation regularization term respectively in the first setting we show that for a discretization based on parseval frames the gradient descent reprojection and the alternating split bregman algorithm are equivalent and turn out to be a frame shrinkage method for the total variation regularizer we also present a numerical comparison with multistep methods

waveform design for radar stap in signal dependent interference waveform design is a pivotal component of the fully adaptive radar construct in this paper we consider waveform design for radar space time adaptive processing stap accounting for the waveform dependence of the clutter correlation matrix due to this dependence in general the joint problem of receiver filter optimization and radar waveform design becomes an intractable nonconvex optimization problem nevertheless it is however shown to be individually convex either in the filter or in the waveform variables we derive constrained versions of a the alternating minimization algorithm b proximal alternating minimization and c the constant modulus alternating minimization which at each step iteratively optimizes either the stap filter or the waveform independently a fast and slow time model permits waveform design in radar stap but the primary bottleneck is the computational complexity of the algorithms

rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization this paper presents two classes of decomposition algorithms based on the proximal method of multipliers pmm introduced in the mid 1970s by rockafellar for convex minimization we first show that

on waring s problem for cubic forms let w be a cubic form in n variables over the complex field and r be the smallest integer such that w can be expressed as a sum of cubes of r linear forms some theorems are proved that connect r with the ranks of certain matrices associated with w for arbitrary n and for n 5 based on these theorems an algorithm to express a cubic form in five variables as a sum of cubes of linear forms is proposed the paper is concluded with two examples of indecomposable cubic forms for the first form it is shown that r 7 and that there exists a one parameter set of representations of the form as a sum of cubes of seven linear forms while for the second form it is shown that r 8

valuations and higher level orders in commutative rings valuation theory is one of the main tools for studying higher level orders and the reduced theory of forms over fields see for example br in mw the theory of higher level orders and reduced forms was generalized to rings with many units and many of the results for fields carried over to this setting while it seems desirable to extend these results further the techniques used for rings with many units will not work for general commutative rings at the same time there is a general theory of valuations in commutative rings see lm m and g which in ma was used to study orders and the reduced theory of quadratic forms over general commutative rings thus it seems natural to ask if the connections between valuations and higher level orders in fields exist in commutative rings in this paper we use valuation theory to study the space of orders and the reduced witt ring relative to a higher level preorder in a commutative ring as in ma we first localize our ring at a multiplicative set without changing the space of orders in order to make the valuation theory work better this is a standard idea from real algebraic geometry remarkably many of the notions methods and results for fields carry over to this new setting we define compatiblity between valuations and orders and preorders and the ring a t associated to a preorder t which turns out to be prufer ring as in the field case we define the relation of dependency on the set of valuations associated to a preorder and we use this to prove a decomposition theorem for the space of orders we can then apply this to show that under a certain finiteness condition the space of orders is equivalent to the space of orders of a preordered field

new interval methods for constrained global optimization interval analysis is a powerful tool which allows to design branch and bound algorithms able to solve many global optimization problems in this paper we present new adaptive multisection rules which enable the algorithm to choose the proper multisection type depending on simple heuristic decision rules moreover for the selection of the next box to be subdivided we investigate new criteria both the adaptive multisection and the subinterval selection rules seem to be specially suitable for being used in inequality constrained global optimization problems the usefulness of these new techniques is shown by computational studies

solving constrained optimization problems using a novel genetic algorithm a novel genetic algorithm is described in this paper for the problem of constrained optimization the algorithm incorporates modified genetic operators that preserve the feasibility of the trial solutions encoded in the chromosomes the stochastic application of a local search procedure and a stopping rule which is based on asymptotic considerations the algorithm is tested on a series of well known test problems and a comparison is made against the algorithms c somga and donlp2

determinants of r fold symmetric multilinear forms in hi harrison has constructed and investigated higher degree forms and the witt grothendieck ring of forms we speculate that the signature theory of ordered fields may be extended to the signature theory of partially ordered fields which will be needed for higher degree forms cf br k it seems that higher degree forms and the witt grothendieck ring have important meaning for it in this paper we introduce a notion of determinant as an extension of ordinary determinant and an invariant of forms we shall use the terminologies and notations in h let r be a commutative ring with identity 1 for an r module e let 8 e x e x x e r be an r fold symmetric multilinear form that is 0 x x2 x 0 x x x for every permutation p of 1 2 r and 8 ax byl x2 x ao x

on trace forms of higher degree let ti be a central simple algebra over a field k and let tr denote its reduced trace then qd x tr x is the trace form of degree d on a similarity of w vd is a linear map f d d for which there exists a e k such that

automorphisms of certain forms of higher degree over ordered fields abstract we are interested in forms of even degree d 4 over ordered fields which are linear combinations with positive coefficients of d th powers of linear forms usually such a representation is not unique however certain forms have so called unique good representation linear preservers of such forms are studied since id v is a linear preserver of any form of even degree orthogonal groups of considered forms have a central involution we show that any finite group with a central involution is isomorphic with the orthogonal group of such forms of even degree

a parallel algorithm for a class of convex programs a parallel algorithm is proposed in this paper for solving the problem min q x x in c 1 cap cdots cap c m where q is an uniformly convex function and c i are closed convex sets in r n in each iteration of the method we solve in parallel m independent subproblems each minimizing a definite quadratic function over an individual set c i the method has attractive convergence properties and can be implemented as parallel algorithms for tackling definite quadratic programs linear programs systems of linear equations and systems of generalized nonlinear inequalities

on the similarity group of forms of higher degree abstract let x x 1 x n k x 1 x n be a non singular form of degree d 3 over the field k where char k 0 or char k d the similarity group of s k is the subgroup of gl n k consisting of all matrices a such that ax a x for some a k let aut k denote the automorphism group of identifying k with the group of scalar matrices in s k we show using kummer s theory of fields that s k aut k k is a finite abelian group of exponent d we apply this result to give a complete characterization of the similarity group of rational binaries of odd degree

convergence analysis of generalized iteratively reweighted least squares algorithms on convex function spaces the computation of robust regression estimates often relies on minimization of a convex functional on a convex set in this paper we discuss a general technique for a large class of convex functionals to compute the minimizers iteratively which is closely related to majorization minimization algorithms our approach is based on a quadratic approximation of the functional to be minimized and includes the iteratively reweighted least squares algorithm as a special case we prove convergence on convex function spaces for general coercive and convex functionals f and derive geometric convergence in certain unconstrained settings the algorithm is applied to total variation tv penalized quantile regression and is compared with a step size corrected newton raphson algorithm it is found that typically in the first steps the iteratively reweighted least squares algorithm performs significantly better whereas the newton type method outpaces the former only after many iterations finally in the setting of bivariate regression with unimodality constraints we illustrate how this algorithm allows one to utilize highly efficient algorithms for special quadratic programs in more complex settings

a representation theorem for higher level reduced witt rings in 6 becker and rosenberg defined the reduced witt ring wak of a formally real field k with respect to a higher level preorder tin k wak is a subring of the ring c x of continuous c valued functions on the space x of signatures of k which are trivial on t and is built up from higher level t forms over k this higher level reduced theory is a natural generalization of the reduced theory of quadratic forms over fields spaces of signatures were introduced in 18 to provide an abstract framework for the higher level reduced theory over fields along the lines of marshall s spaces of orderings 13 h treatment of the classical reduced theory of quadratic forms there are many advantages to this axiomatic approach it allows for a unified study of reduced witt rings over fields and skew fields powers has shown that becker and rosenberg s study of higher level reduced witt rings can also be carried out over skew fields 22 the absence of technical considerations such as valuation theory not only yields simpler proofs of known results but also new results about the theory over fields 183 the space of orderings version 151 of the result which we prove here has found application in real algebraic geometry and we hope that our abstract higher level result might also prove useful in this context the question which concerns us here is the so called representation problem given fe c x how can we tell if f f for some f w x this has been solved for the higher level reduced witt ring of a field by becker and rosenberg 6 see s also and in the case of spaces of orderings by marshall 15 both solutions being generalizations of the original representation theorems of becker and brlicker 4 5 31 and brown and marshall 9 4 21 in the context of the reduced quadratic form theory over fields under a 2 power assumption we simultaneously extend all of these representation theorems to our abstract higher level setting this entails studying fans and more importantly a generalization called quasifans we not only obtain a common proof of the representation 105 0021 8693 88 3 00