branch and bound search for experimental designs based on d optimality and other criteria this article presents a branch and bound algorithm that constructs a catalog of all d optimal n point designs for specified design region linear model and number of observations n while the primary design criterion is d optimality the algorithm may also be used to find designs performing well by other secondary criteria if a small sacrifice in ellkiency as measured by d optimahty is accepted finally some designs are supplied for a quadratic response surface model

optimal impulse control for a multidimensional cash management system with generalized cost functions abstract we consider the optimal control of a multidimensional cash management system where the cash balances fluctuate as a homogeneous diffusion process in r n we formulate the model as an impulse control problem on an unbounded domain with unbounded cost functions under general assumptions we characterize the value function as a weak solution of a quasi variational inequality in a weighted sobolev space and we show the existence of an optimal policy moreover we prove the local uniform convergence of a finite element scheme to compute numerically the value function and the optimal cost we compute the solution of the model in two dimensions with linear and distance cost functions showing what are the shapes of the optimal policies in these two simple cases finally our third numerical experiment computes the solution in the realistic case of the cash concentration of two bank accounts made by a centralized treasury

pointwise error estimates for a class of elliptic quasi variational inequalities with nonlinear source terms in this paper we derive l error estimates of a piecewise linear approximation method for a class of elliptic quasi variational inequalities qvis with nonlinear source terms our approach stands on a discrete l stability property with respect to the right hand side in linear elliptic qvis

representability and specht problem for g graded algebras abstract let w be an associative pi algebra over a field f of characteristic zero graded by a finite group g let id g w denote the t ideal of g graded identities of w we prove 1 g graded pi equivalence there exists a field extension k of f and a finite dimensional z 2 z g graded algebra a over k such that id g w id g a where a is the grassmann envelope of a 2 g graded specht problem the t ideal id g w is finitely generated as a t ideal 3 g graded pi equivalence for affine algebras let w be a g graded affine algebra over f then there exists a field extension k of f and a finite dimensional algebra a over k such that id g w id g a

graded identities of matrix algebras and the universal graded algebra in the last decade group gradings and graded identities of finite dimensional central simple algebras have been an active area of research we refer the reader to bahturin et al 6 and 7 there are two basic kinds of group grading elementary and fine it was proved by bahturin and zaicev 7 that any group grading of mn c is given by a certain composition of an elementary grading and a fine grading in this paper we are concerned with fine gradings on mn c and their corresponding graded identities let r be a simple algebra finite dimensional over its center k and g a finite group we say that r is fine graded by g if r g grg is a grading and dimk rg 1 thus any component is either 0 or isomorphic to k as a k vector space it is easy to show that supp r the subset of elements of g for which rg is not 0 is a subgroup of g moreover

l asymptotic behavior for a finite element approximation in parabolic quasi variational inequalities related to impulse control problem abstract in this paper the parabolic quasi variational inequalities are transformed into a noncoercive elliptic quasi variational inequalities a new iterative discrete algorithm is proposed to show the existence and uniqueness and a simple proof to asymptotic behavior in uniform norm is also given using the theta time scheme combined with a finite element spatial approximation the proposed approach stands on a discrete l stability property with respect to the right hand side and obstacle defined as an impulse control problem

the finite element approximation of variational inequalities related to ergodic control problems abstract in this paper we study the numerical approximation of variational inequalities vi where the discount factor i e the zero order term is set to 0 optimal error estimates are proved in the l norm using the concept of subsolutions

constrained maximin designs for computer experiments finding a design scheme for expensive computer experiments in arbitrary design regions is often a difficult task a well known criterion that is often used for computer experiments is the minimal euclidean distance between any pair of design sites which should be maximal in this article we describe methods to create such a constrained maximin simulation scheme we give some numerical results for both theoretical and practical examples

the construction of exact d optimum experimental designs with application to blocking response surface designs summary the properties of optimum designs are investigated when there are both qualitative and quantitative factors algorithms for the construction of exact designs are described which are of general applicability the case when the qualitative factors represent blocking variables is given special treatment a modified algorithm provides the opportunity to divide the experimental trials into blocks of specified sizes examples are given of second order models in the quantitative factors divided into two or three blocks of a variety of sizes use of the adjustment algorithm to improve such designs generated from a list of candidate points is demonstrated

discrete minimum and maximum principles for finite element approximations of non monotone elliptic equations uniform lower and upper bounds for positive finite element approximations to semilinear elliptic equations in several space dimensions subject to mixed dirichlet neumann boundary conditions are derived the main feature is that the non linearity may be non monotone and unbounded the discrete minimum principle provides a positivity preserving approximation if the discretization parameter is small enough and if some structure conditions on the non linearity and the triangulation are assumed the discrete maximum principle also holds for degenerate diffusion coefficients the proofs are based on stampacchia s truncation technique and on a variational formulation both methods are settled on careful estimates on the truncation operator

an optimum design for estimating the first derivative an optimum design of experiment for a class of estimates of the first derivative at 0 used in stochastic approximation and density estimation is shown to be equivalent to the problem of finding a point of minimum of the function gamma defined by gamma x det lbrack 1 x 3 ldots x 2m 1 rbrack det lbrack x x 3 ldots x 2m 1 rbrack on the set of all m dimensional vectors with components satisfying 0 x 1 x 2 cdots 1 m 1 x m and pi x i 1 in the determinants 1 is the column vector with all components 1 and x i has components of x raised to the i th power the minimum of gamma is shown to be m and the point at which the minimum is attained is characterized by chebyshev polynomials of the second kind

l error estimate for a system of elliptic quasi variational inequalities with noncoercive operators this paper deals with the standard finite element approximation of a noncoercive system of quasi variational inequalities qvis arising in stochastic control problems we improve a result obtained in boulbrachene comput math appl 45 983 989 2003 and establish the optimal l convergence order making use of the concepts of subsolutions and discrete regularity

computing exact d optimal designs by mixed integer second order cone programming i 1 wiaia t i where ai i 1 s are known matrices with m rows in this paper we show that the criterion of d optimality is secondorder cone representable as a result the method of second order cone programming can be used to compute an approximate d optimal design with any system of linear constraints on the vector of weights more importantly the proposed characterization allows us to compute an exact d optimal design which is possible thanks to highquality branch and cut solvers specialized to solve mixed integer second order cone programming problems our results extend to the case of the criterion of dk optimality which measures the quality of w for the estimation of a linear parameter subsystem defined by a full rank coefficient matrixk we prove that some other widely used criteria are also secondorder cone representable for instance the criteria of a ak gand i optimality we present several numerical examples demonstrating the efficiency and general applicability of the proposed method we show that in many cases the mixed integer second order cone programming approach allows us to find a provably optimal exact design while the standard heuristics systematically miss the optimum

experimental designs for model discrimination abstract we present designs that perform well for several objectives simultaneously three different approaches are discussed to augment a given design in an optimal way to evaluate a mixture of the various criteria and to optimize one objective subject to achieving a prescribed efficiency level for the others our sample designs are for the situation of discriminating between a second and third degree polynomial fit under the d criterion and geometric mixtures of d criteria

the application of the annealing algorithm to the construction of exact optimal designs for linear regression models this article reports the application of the annealing algorithm to the construction of exact d i and g optimal designs for polynomial regression of degree 5 on the interval 1 l and for the second order model in two factors on the design space 1 l 1 1 details of the perturbation scheme and the annealing schedules used are given and the method of implementation is illustrated by means of a simple example the algorithm is assessed by comparing its performance in terms of computer time and effkiency with the modified fedorov procedure and it is shown to be particularly effective in finding g optimal designs the salient features of the exact designs constructed in this study are also summarized

on finite element approximation in the l norm of variational inequalities we are interested in the approximation in thel norm of variational inequalities with non linear operators and somewhat irregular obstacles we show that the order of convergence will be the same as that of the equation associated with the non linear operator if the discrete maximum principle is verified

identities and isomorphisms of graded simple algebras abstract let g be an arbitrary abelian group and let a and b be two finite dimensional g graded simple algebras over an algebraically closed field f such that the orders of all finite subgroups of g are invertible in f we prove that a and b are isomorphic if and only if they satisfy the same g graded identities we also describe all isomorphism classes of finite dimensional g graded simple algebras

multidimensional designs for two factor experiments abstract a class of multidimensional designs is presented for factorial experiments with two treatment factors and two or more orthogonal blocking factors we assume that interactions between all factors except for the two treatment factors are negligible a table of efficient three dimensional designs is given together with tables of component designs that can be used to construct two three or higher dimensional designs all the multidimensional designs considered in this article have d 2 crossed orthogonal blocking factors and exactly one treatment combination observed at each combination of levels of the blocking factors they are constructed by amalgamating one dimensional generalized cyclic designs with known properties the purpose of this article is to increase the availability of multidimensional designs three dimensional designs in particular our main objective is to achieve efficient estimation of the factorial effects via least squares estimators that are uncorrelated after adjustment

sensor placement by maximal projection on minimum eigenspace for linear inverse problems this paper presents two new greedy sensor placement algorithms named minimum nonzero eigenvalue pursuit mnep and maximal projection on minimum eigenspace mpme for linear inverse problems with greater emphasis on the mpme algorithm for performance comparison with existing approaches in both mnep and mpme we select the sensing locations one by one in this way the least number of required sensor nodes can be determined by checking whether the estimation accuracy is satisfied after each sensing location is determined for the mpme algorithm the minimum eigenspace is defined as the eigenspace associated with the minimum eigenvalue of the dual observation matrix for each sensing location the projection of its observation vector onto the minimum eigenspace is shown to be monotonically decreasing w r t the worst case error variance wcev of the estimated parameters we select the sensing location whose observation vector has the maximum projection onto the minimum eigenspace of the current dual observation matrix the proposed mpme is shown to be one of the most computationally efficient algorithms our monte carlo simulations showed that mpme outperforms the convex relaxation method the sparsense method and the framesense method in terms of wcev and the mean square error mse of the estimated parameters especially when the number of available sensor nodes is very limited

p i algebras with hopf algebra actions abstract ifais a p i algebra in characteristic zero with action from a finite dimensional semisimple hopf algebrah thenahas a nilpotenth idealnsuch thata nwill beh verbally semiprime everyh verbally semiprime algebra ish p i equivalent to a direct sum ofh verbally prime algebras in the case of a finite group action or a grading by an abelian group we show that the sum can be taken to be finite in the case of an action by a finite cyclic groupg we classify allg p i algebras up to equivalence this paper generalizes the work of a r kerner 1985 math ussr izv 25

general weighted optimality of designed experiments the standard approach to finding optimal experimental designs employs conventional measures of design efficacy such as the a e and d criterion that assume equal interest in all estimable functions of model parameters this paper develops a general theory for weighted optimality allowing precise design selection according to expressed relative interest in different functions in the estimation space the approach employs a very general class of matrix specified weighting schemes that produce easily interpretable weighted optimality criteria in particular for any set of estimable functions and any selected corresponding weights analogs of standard optimality criteria are found that guide design selection according to the weighted variances of estimators of those particular functions the results are applied to solve the a optimal design problem for baseline factorial effects in unblocked experiments

the usefulness of optimum experimental designs optimum experimental designs were originally developed by kiefer mainly for response surface models this survey of recent developments emphasizes potential or actual usefulness for linear models the construction of exact designs particularly over irregular design regions is stressed as is the blocking of response surface designs other important areas include systematic designs that are robust against trend and designs for mixtures with irregular design regions several industrial examples are mentioned both d and c optimum designs are found for a non linear model of the economic response of cereal production to fertilizer level the c optimum design being for the conditions of maximum economic return locally optimum and bayesian designs are both described similar results for generalized linear models lead to designs for the ld 95 in a logistic model in which male and female subjects respond differently designs with structure in the variance suggest alternatives to the potentially wasteful product designs of taguchi designs for sequential clinical trials to include random balance are presented the last section outlines some applications including life testing and models in which time is a factor

optimal additions to and deletions from two level orthogonal arrays summary consider the problem of selecting a two level factorial design it is well known that two level orthogonal arrays of strength 4 or more with e extra runs have various optimality properties including generalized cheng type 1 optimality when e 1 restricted cheng type 1 optimality when e 2 and e optimality when 3e5 more general schur optimality results are derived for more general values of e within the more restricted class of augmented two level orthogonal arrays similar results are derived for the class of orthogonal arrays with deletions examples are used to illustrate the results and in many cases the designs are confirmed to be optimal across all two level designs

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

optimum continuous block designs the incidence matrix of a block design is replaced by a normalized version n in which the entries are non negative numbers whose sum is unity the so called c matrix the information matrix for estimation of treatment contrasts is similarly replaced by the normalized analogue c n we study the set of ordered eigenvalues of all c n and give a complete specification for three treatments rows for any number of treatments we characterize the eigenvalues of an important subclass of designs for which the non zero entries in any given block are equal it is suggested that the natural ordering between designs is upper weak majorization of the eigenvalues using this we show how to improve a given n matrix and this leads to several optimality statements

finite element error estimates in non energy norms for the two dimensional scalar signorini problem this paper is concerned with error estimates for the piecewise linear finite element approximation of the two dimensional scalar signorini problem on a convex polygonal domain varomega using a cea type lemma a supercloseness result and a non standard duality argument we prove w 1 p varomega l infty varomega w 1 infty varomega and h 1 2 partial varomega error estimates under reasonable assumptions on the regularity of the exact solution and l p varomega error estimates under comparatively mild assumptions on the involved contact sets the obtained orders of convergence turn out to be optimal for problems with essentially bounded right hand sides our results are accompanied by numerical experiments which confirm the theoretical findings

verbally prime t ideals and graded division algebras abstract let f be an algebraically closed field of characteristic zero and let g be a finite group we consider graded verbally prime t ideals in the free g graded algebra it turns out that equivalent definitions in the ordinary case i e ungraded extend to nonequivalent definitions in the graded case namely verbally prime g graded t ideals and strongly verbally prime t ideals at first following kemer s ideas we classify g graded verbally prime t ideals the main bulk of the paper is devoted to the stronger notion we classify g graded strongly verbally prime t ideals which are t ideal of affine g graded algebras or equivalently g graded t ideals that contain a capelli polynomial it turns out that these are precisely the t ideal of g graded identities of finite dimensional g graded central over f i e z a e f which admit a g graded division algebra twisted form over a field k which contains f or equivalently over a field k which contains enough roots of unity e g a primitive n root of unity where n o r d g

bayes d optimal and e optimal block designs summary bayes designs are considered for a two way analysis of variance model with a normal prior for treatments and blocks one of the treatments acting as a control the existence of an optimal allocation is proved for each bayesian optimality criterion i e each criterion defined on the posterior covariance matrix for the treatments under a special assumption on the treatment prior the structure of the optimal design is specified for the case of only one test treatment such a design is universally optimal bayes d and e optimal designs are found as special cases and results are compared with classical ones two examples are included

optimal row column designs in high throughput screening experiments saturated row column designs are studied to eliminate row and column effects in primary high throughput screening experiments all paired comparisons of treatments in the designs recommended are estimable within each microplate in spite of the existence of row and column effects the m s criterion is used to select optimal and eliminate inefficient designs it turns out that all m s optimal designs are binary that is no treatments appear twice in any row or column optimal designs are not unique with respect to design isomorphism a series of m s optimal designs is constructed and all paired comparisons of treatments in the constructed designs are estimable regardless of the two way heterogeneity an m s optimal design for 8 12 microplates is provided and optimal designs of other dimensions can be constructed systematically

another look at first order saturated designs the p efficient designs the problem of constructing first order saturated designs that are optimal in some sense has received a great deal of attention in the literature since these saturated designs are frequently used in screening situations the focus will be on the potential projective models rather than the full model this article discusses some practical concerns in choosing a design and presents some first order saturated designs having two desirable properties near equal occurrence and near orthogonality these saturated designs are shown to be reasonably efficient for estimating the parameters of projective submodels and thus are called p efficient designs comparisons with the efficiency of d optimal designs are given for designs for all n from 3 to 30

multiobjective sampling design for parameter estimation and model discrimination in groundwater solute transport sampling design for site characterization studies of solute transport in porous media is formulated as a multiobjective problem optimal design of a sampling network is a sequential process in which the next phase of sampling is designed on the basis of all available physical knowledge of the system three objectives are considered model discrimination parameter estimation and cost minimization for the first two objectives physically based measures of the value of information obtained from a set of observations are specified in model discrimination value of information of an observation point is measured in terms of the difference in solute concentration predicted by hypothesized models of transport points of greatest difference in predictions can contribute the most information to the discriminatory power of a sampling design sensitivity of solute concentration to a change in a parameter contributes information on the relative variance of a parameter estimate inclusion of points in a sampling design with high sensitivities to parameters tends to reduce variance in parameter estimates cost minimization accounts for both the capital cost of well installation and the operating costs of collection and analysis of field samples sensitivities discrimination information and well installation and sampling costs are used to form coefficients in the multiobjective problem in which the decision variables are binary zero one each corresponding to the selection of an observation point in time and space the solution to the multiobjective problem is a noninferior set of designs to gain insight into effective design strategies a one dimensional solute transport problem is hypothesized then an approximation of the noninferior set is found by enumerating 120 designs and evaluating objective functions for each of the designs trade offs between pairs of objectives are demonstrated among the models the value of an objective function for a given design is shown to correspond to the ability of a design to actually meet an objective

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

approximation of obstacle problems by continuation methods nonlinear variational inequalities associated with obstacle problems are considered which may exhibit multiple solutions and turning points a predictor corrector continuation method for following a solution branch is presented essentially it is shown that regular or simple singular solutions of the nonlinear eigenvalue problem are regular for an augmented problem the main difficulty here is due to the fact that the equations are not differentiable in the frechet sense the numerical schemes that are used in the implementation of the method are developed for the predictor step a deflated block algorithm is used and for the corrector step algorithms are set up using projected sor projected newton and projected preconditioned conjugate gradients the efficiency of the various algorithms is compared by applying the continuation method to a finite element discretization of a two dimensional problem numerical results are presented

the coordinate exchange algorithm for constructing exact optimal experimental designs we describe the cyclic coordinate exchange algorithm for constructing d optimal and linear optimal experimental designs the algorithm uses a variant of the gauss southwell cyclic coordinate descent algorithm within the k exchange algorithm to achieve substantive reductions in required computing among its advantages are the following candidate sets which grow exponentially in the number of factors need not be explicitly constructed or enumerated convex design spaces or mixed convex by discrete design spaces are handled directly without the need for sophisticated nonlinear programming routines or candidate set adjustment for design problems having 10 or more factors the reductions in execution time are typically two or more orders of magnitude when compared to standard candidateset based procedures such as k exchange yet the designs produced exhibit no loss of efficiency

compound d and d s optimum designs for determining the order of a chemical reaction estimation of the order of a chemical reaction is often at least as important as estimation of the rate of the reaction locally optimum experimental designs are found for the order and the rate separately and together compound d optimum designs provide a method for designing experiments with specified efficiency for rate or order determination a relationship between the compound designs and d optimum designs for both rate and order aids interpretation of the plots of design efficiencies bayesian designs incorporating prior uncertainty are exemplified monte carlo sampling of the prior is used to design an experiment for the esterification of acetic anhydride

model robust response surface designs scaling two level factorials summary response surface methods use simple graduating functions to study the relationship between an experimental response variable and a set of continuous explanatory variables in designing a response surface study an experimenter must decide how far apart to set the levels of each factor i e how to scale the design a good choice should be sensitive to the fact that the graduating function is only an approximation to the true response function a bayesian model is proposed that makes explicit assumptions about the inadequacy of an assumed model and a design criterion based on the model leads to reasonable choices of scale for two level factorial designs the choice of scale is found to be insensitive to the prior distributions in the model

efficient computational algorithm for optimal continuous experimental designs abstract a simple yet efficient computational algorithm for computing the continuous optimal experimental design for linear models is proposed an alternative proof of the monotonic convergence for d optimal criterion on continuous design spaces is provided we further show that the proposed algorithm converges to the d optimal design we also provide an algorithm for the a optimality and conjecture that the algorithm converges monotonically on continuous design spaces different numerical examples are used to demonstrate the usefulness and performance of the proposed algorithms

a fixed point approximation method for some noncoercive variational problems abstract this paper deals with the finite element approximation of some variational problems namely linear elliptic boundary value problems variational inequalities and quasi variational inequalities with noncoercive operators to prove optimal l error estimates we introduce a simple and direct argument combining continuous piecewise linear finite elements with the banach fixedpoint theorem

construction and optimality of affine resolvable designs summary affine resolvable designs are constructed from orthogonal arrays and shown to be optimal among resolvable designs with respect to the usual criteria tables and text show how to construct such designs in up to seven replicates whenever the number of treatments properly divides the square of the block size data from experiments which use affine resolvable designs can be simply analysed by using pseudofactors in many experiments where an incomplete block design is used a resolvable design that is a design in which the blocks can be partitioned into sets containing each treatment once is desirable sometimes the blocks are naturally grouped into larger blocks contain ing each treatment exactly once an example from microbiology is an experiment in which jars of treated bacteria are placed in incubators so that the bacteria may multiply blocks are the shelves within the incubators and large blocks are the incubators use of a resolv able design ensures that the large blocks are orthogonal to treatments in other cases particularly in agricultural trials neighbouring blocks are grouped into large blocks for management purposes then a resolvable design not only keeps management effects ortho gonal to treatments it also gives some protection against the loss of a whole large block for all treatments are disadvantaged equally by such a loss the large blocks are often called replicates although somewhat ambiguously because their existence is independent of the treatment allocation suppose that a resolvable design is needed for v treatments in blocks of size k where v sk for some integer s greater than 1 and that r replicates are required for r 2 williams patterson john 1976 showed that an optimal resolvable design is equivalent to an optimal incomplete block design for s treatments in s blocks of size k thus optimality results about nonresolvable designs may be used to find the most efficient resolvable

the noncoercive quasi variational inequalities related to impulse control problems abstract this paper deals with the numerical analysis of noncoercive quasi variational inequalities of impulse control problems optimally l error estimates are derived using qualitative properties of both the continuous and finite element approximation solutions and the notion of subsolutions

an adjustment algorithm for the construction of exact d optimum experimental designs existing algorithms for the construction of exact d optimum designs customarily search a list of candidate points this article describes a computationally simple method of improving such designs tables are given of designs for second order models with as many as five factors found by search over candidate points these are compared with designs already in the literature and used as the starting point for the adjustment algorithm

on group gradings on pi algebras abstract we show that there exists a constant k such that for any pi algebra w and any nondegenerate g grading on w where g is any group possibly infinite there exists an abelian subgroup u of g with g u exp w k a g grading w g g w g is said to be nondegenerate if w g 1 w g 2 w g r 0 for any r 1 and any r tuple g 1 g 2 g r in g r

optimal sequential designs for on line item estimation replenishing item pools for on line ability testing requires innovative and efficient data collection designs by generating locald optimal designs for selecting individual examinees and consistently estimating item parameters in the presence of error in the design points sequential procedures are efficient for on line item calibration the estimating error in the on line ability values is accounted for with an item parameter estimate studied by stefanski and carroll locallyd optimaln point designs are derived using the branch and bound algorithm of welch in simulations the overall sequential designs appear to be considerably more efficient than random seeding of items

finite element approximation of the volume matching problem optimal orderh 1 andl error bounds are obtained for a continuous piecewise linear finite element approximation of the volume matching problem this problem consists of minimising v 1 2 overv h 1 subject to the inequality constraintv 0 and a number of linear equality constraints the presence of the equality constraints leads to lagrange multipliers which in turn lead to complications with the standard error analysis for variational inequalities finally we consider an algorithm for solving the resulting algebraic problem