feasible and noninterior path following in constrained minimization with low multiplier regularity primal dual path following methods for constrained minimization problems in function space with low multiplier regularity are introduced and analyzed regularity properties of the path are proved the path structure allows us to define approximating models which are used for controlling the path parameter in an iterative process for computing a solution of the original problem the moreau yosida regularized subproblems of the new path following technique are solved efficiently by semismooth newton methods the overall algorithmic concept is provided and numerical tests including a comparison with primal dual path following interior point methods for state constrained optimal control problems show the efficiency of the new concept

finite element discretization of state constrained elliptic optimal control problems with semilinear state equation we study a class of semilinear elliptic optimal control problems with pointwise state constraints the purpose of this paper is twofold first we present convergence results for the finite element discretization of this problem class similarly to known results with finite dimensional control space thus extending results that are for control functions only available for linear quadratic convex problems we rely on a quadratic growth condition for the continuous problem that follows from second order sufficient conditions second we show that the second order sufficient conditions for the continuous problem transfer to its discretized version this is of interest for example when considering questions of local uniqueness of solutions or the convergence of solution algorithms such as the sqp method

preconditioning of active set newton methods for pde constrained optimal control problems we address the problem of preconditioning a sequence of saddle point linear systems arising in the solution of pde constrained optimal control problems via active set newton methods with control and regularized state constraints we present two new preconditioners based on a full block matrix factorization of the schur complement of the jacobian matrices where the active set blocks are merged into the constraint blocks we discuss the robustness of the new preconditioners with respect to the parameters of the continuous and discrete problems numerical experiments on three dimensional problems are presented including comparisons with existing approaches based on preconditioned conjugate gradients in a nonstandard inner product

second order and stability analysis for state constrained elliptic optimal control problems with sparse controls an optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state the main novelty of the paper is the presence of the l 1 norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions second order sufficient optimality conditions are analyzed they are applied to show the convergence of optimal solutions for vanishing l 2 regularization parameter for the control the associated convergence rate is estimated

l2 estimates for the finite element method for the dirichlet problem with singular data in this paper we consider the approximation by the finite element method of second order elliptic problems on convex domains and homogeneous dirichlet condition on the boundary in these problems the data are borel measures using a quasiuniform mesh of finite elements and polynomials of degree 1 we prove that in two dimensions the convergence is of orderh inl 2 and in three dimensions of orderh 1 2

c0 interior penalty methods for an elliptic state constrained optimal control problem with neumann boundary condition abstract we study c 0 interior penalty methods for an elliptic optimal control problem with pointwise state constraints on two dimensional convex polygonal domains the approximation of the optimal state is obtained by solving a fourth order variational inequality and the approximation of the optimal control is computed by a post processing procedure we prove the convergence of numerical solutions with rates in the h 2 like energy error by using the complementarity form of the variational inequality furthermore we develop an a posteriori analysis for a residual based error estimator and introduce an adaptive algorithm numerical experiments are provided to gauge the performance of the proposed methods

barrier methods for optimal control problems with state constraints we study barrier methods for state constrained optimal control problems with pdes the focus of our analysis is the path of minimizers of the barrier subproblems with the aim to provide a solid theoretical basis for function space oriented path following algorithms we establish results on existence continuity and convergence of this path moreover we consider the structure of barrier subdifferentials which play the role of dual variables

force traction microscopy an inverse problem with pointwise observations abstract force traction microscopy is an inversion method that allows one to obtain the stress field applied by a living cell on the environment on the basis of a pointwise knowledge of the displacement produced by the cell itself this classical biophysical problem usually addressed in terms of green functions can be alternatively tackled using a variational framework and then a finite elements discretization in such a case a variation of the error functional under suitable regularization is operated in view of its minimization this setting naturally suggests the introduction of a new equation based on the adjoint operator of the elasticity problem in this paper we illustrate the rigorous theory of the two dimensional and three dimensional problems involving in the former case a distributed control and in the latter case a surface control the pointwise observations require one to exploit the theory of elasticity extended to forcing terms that are borel measures

non commutative discretize then optimize algorithms for elliptic pde constrained optimal control problems abstract in this paper we analyze the convergence of several optimize then discretize and discretize then optimize algorithms based on either a second order or a fourth order finite difference discretization for solving elliptic pde constrained optimal control problems to ensure the convergence of a discretize then optimize algorithm one well accepted criterion is to design the discretization scheme such that the resulting discretize then optimize algorithm commutes with the corresponding optimize then discretize algorithm in other words both algorithms should give rise to exactly the same discrete optimality system however such a restrictive criterion is not trivial to fulfill by investigating a distributed control problem governed by an elliptic equation we first show that enforcing such a stringent condition of commutative property is only sufficient but not necessary for achieving the desired convergence we then introduce some suitable h 1 semi norm penalty regularization terms to recover the lost convergence due to the inconsistency caused by the loss of commutativity numerical experiments are carried out to verify our theoretical analysis and also validate the effectiveness of our proposed regularization techniques

optimal control problems with partially polyhedric constraints this paper discusses a class of state constrained optimal control problems for which it is possible to formulate second order necessary or sufficient conditions for local optimality or quadratic growth that do not involve all curvature terms for the constraints this kind of result is classical in the case of polyhedric control constraints our theory of optimization problems with partially polyhedric constraints allows to extend these results to the case when the control constraints are polyhedric in the presence of state constraints satisfying some specific hypotheses the analysis is based on the assumption that some strict semilinearized qualification condition is satisfied we apply the theory to some optimal control problems of elliptic equations with state and control constraints

composite rough sets for dynamic data mining as a soft computing tool rough set theory has become a popular mathematical framework for pattern recognition data mining and knowledge discovery it can only deal with attributes of a specific type in the information system by using a specific binary relation however there may be attributes of multiple different types in information systems in real life applications such information systems are called as composite information systems in this paper a composite relation is proposed to process attributes of multiple different types simultaneously in composite information systems then an extended rough set model called as composite rough sets is presented we also redefine lower and upper approximations positive boundary and negative regions in composite rough sets through introducing the concepts of the relation matrix the decision matrix and the basic matrix we propose matrix based methods for computing the approximations positive boundary and negative regions in composite information systems which is crucial for feature selection and knowledge discovery moreover combined with the incremental learning technique a novel matrix based method for fast updating approximations is proposed in dynamic composite information systems extensive experiments on different data sets from uci and user defined data sets show that the proposed incremental method can process large data sets efficiently

an interior point algorithm with inexact step computation in function space for state constrained optimal control we consider an interior point method in function space for pde constrained optimal control problems with state constraints our emphasis is on the construction and analysis of an algorithm that integrates a newton path following method with adaptive grid refinement this is done in the framework of inexact newton methods in function space where the discretization error of each newton step is controlled by adaptive grid refinement in the innermost loop this allows to perform most of the required newton steps on coarse grids such that the overall computational time is dominated by the last few steps for this purpose we propose an a posteriori error estimator for a problem suited norm

error estimates for spectral approximation of elliptic control problems with integral state and control constraints abstract the goal of this paper is to investigate the legendre galerkin spectral approximation of elliptic optimal control problems with integral state and control constraints thanks to the appropriate base functions of the discrete spaces the discrete system is with sparse coefficient matrices we first present the optimality conditions of the control system then a priori and a posteriori error estimates both in h 1 and l 2 norms are derived some numerical tests indicate that the spectral accuracy can be achieved and the proposed method is competitive for solving control problems

state constrained optimal control of the three dimensional stationary navier stokes equations abstract in this paper an optimal control problem for the stationary navier stokes equations in the presence of state constraints is investigated existence of optimal solutions is proved and first order necessary conditions are derived the regularity of the adjoint state and the state constraint multiplier is also studied lipschitz stability of the optimal control state and adjoint variables with respect to perturbations is proved and a second order sufficient optimality condition for the case of pointwise state constraints is stated

finite element approximations of an optimal control problem with integral state constraint an integral state constrained optimal control problem governed by an elliptic partial differential equation and its finite element approximation are considered the finite element approximation is constructed on multimeshes an l 2 norm a priori error estimate of the finite element approximation is obtained further some superconvergence results are proved based on these superconvergence results almost optimal l infty norm error estimates are derived some recovery algorithms are then proposed to produce a posteriori error estimators of gradient type to solve the finite element system a simple and yet efficient iterative gradient projection algorithm is proposed and its convergence rate is proved some numerical examples are performed to confirm theoretical analysis

moreau yosida regularization in state constrained elliptic control problems error estimates and parameter adjustment an adjustment scheme for the regularization parameter of a moreau yosida based regularization or relaxation approach to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced the method utilizes error estimates of an associated finite element discretization of the regularized problems for the optimal selection of the regularization parameter in dependence on the mesh size of discretization and error estimates for the approximation error due to regularization the theoretical results are verified numerically

optimal control of the stationary navier stokes equations with mixed control state constraints in this paper we consider the distributed optimal control of the navier stokes equations in the presence of pointwise mixed control state constraints after deriving a first order necessary condition the regularity of the mixed constraint multiplier is investigated second order sufficient optimality conditions are studied as well in the last part of the paper a semismooth newton method is applied for the numerical solution of the control problem the convergence of the method is proved and numerical experiments are carried out

a priori error analysis for discretization of sparse elliptic optimal control problems in measure space in this paper an optimal control problem is considered where the control variable lies in a measure space and the state variable fulfills an elliptic equation this formulation leads to a sparse structure of the optimal control in this setting we prove a new regularity result for the optimal state and the optimal control moreover a finite element discretization based on e casas c clason and k kunisch siam j control optim 50 2012 pp 1735 1752 is discussed and a priori error estimates are derived which significantly improve the estimates from that paper numerical examples for problems in two and three space dimensions illustrate our results

regularized state constrained boundary optimal control of the navier stokes equations abstract the numerical solution of the dirichlet boundary optimal control problem of the navier stokes equations in presence of pointwise state constraints is investigated two different regularization techniques are considered first a moreau yosida regularization of the problem is studied optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved a source representation of the control combined with a lavrentiev type regularization strategy is also presented the analysis concerning optimality conditions and convergence of the regularized solutions is carried out in the last part of the paper numerical experiments are presented for the numerical solution of each regularized problem a semi smooth newton method is applied

constrained lqr problems in elliptic distributed control systems with point observations on convergence rates in this paper we continue to study the bound constrained linear quadratic regulator problem in distributed boundary control systems governed by the elliptic equation with point observations due to the appearance of singularities in the problem the traditional galerkin variational method that leads to an adjoint system is not desirable and the classical lagrangian multiplier method is not reliable for providing numerical results a characterization formula of the optimal control and its singularity decomposition formula are derived in z ding l ji and j zhou siam j control optim 34 1996 pp 264 294 z ding and j zhou appl math optim to appear by using the boundary integral equation and potential theory coupled with a variational inequality in a banach space setting based on the characterization formula a conditioned gradient projection method cgpm has been proposed in z ding and j zhou appl math optim to appear numerical experiment has shown that cgpm is efficient and also insensitive to the partition number of the boundary in this paper we estimate the rate of convergence for cgpm first it is proved that for cal n 2 cgpm converges exponentially in the l2 norm and for cal n 3 cgpm converges subexponentially in the l p norm then under a reasonable condition it is proved that for cal n 2 cgpm converges uniformly exponentially and for cal n 3 cgpm converges uniformly subexponentially

optimal a priori error estimates for an elliptic problem with dirac right hand side it is well known that finite element solutions for elliptic pdes with dirac measures as source terms converge due to the fact that the solution is not in h 1 suboptimal in classical norms a standard remedy is to use graded meshes then quasioptimality i e optimal up to a log factor for low order finite elements can be recovered e g in the l 2 norm here we show for the lowest order case quasioptimality and for higher order finite elements optimal order a priori estimates on a family of quasi uniform meshes in an l 2 seminorm the seminorm is defined as an l 2 norm on a fixed subdomain which excludes the locations of the delta source terms our motivation in the use of such a norm results from the observation that in many applications the error at the singularity is dominated by the model error e g in dimension reduced settings or is not the quantity of interest e g in optimal control problems the quasi optimal and optimal order a priori bounds are obtained recursively by using aub

a priori l 2 discretization error estimates for the state in elliptic optimization problems with pointwise inequality state constraints in this paper an elliptic optimization problem with pointwise inequality constraints on the state is considered the main contributions of this paper are a priori l 2 error estimates for the discretization error in the optimal states due to the non separability of the space for the lagrange multipliers for the inequality constraints the problem is tackled by separation of the discretization error into two components first the state constraints are discretized second with discretized inequality constraints a duality argument for the error due to the discretization of the pde is employed for the second stage an a priori error estimate is derived with constants depending on the regularity of the dual problem finally we discuss two cases in which these constants can be bounded in a favorable way leading to higher order estimates than those induced by the known l 2 error in the control variable more precisely we consider a given fixed number of pointwise inequality constraints and a case of infinitely many but only weakly active constraints

spike controls for elliptic and parabolic pdes abstract we analyze the use of measures of the minimal norm to control elliptic and parabolic equations we prove the sparsity of the optimal control in the parabolic case we prove that the solution of the optimization problem is a borel measure supported in a set of lebesgue measure zero in both cases the approximate controllability can be achieved efficiently by means of controls that are activated in some finite number of pointwise locations we also analyze the corresponding dual problem

a rough set approach to attribute generalization in data mining abstract this paper presents a method for updating approximations of a concept incrementally the results can be used to implement a quasi incremental algorithm for learning classification rules from very large data bases generalized by dynamic conceptual hierarchies provided by users in general the process of attribute generalization may introduce inconsistency into a generalized relation this issue is resolved by using the inductive learning algorithm lers based on rough set theory

a self concordant interior point approach for optimal control with state constraints we propose an infeasible interior point method for pointwise state constrained optimal control problems with linear elliptic pdes a smoothed constraint violation functional is used to develop a self concordant barrier approach in an infinite dimensional setting we provide a detailed convergence analysis in function space for this approach the quality of the smoothing is described by a parameter by fixing this parameter we obtain a perturbed version of the original problem we establish complexity estimates and convergence rates for the methods that we propose to solve a given perturbed problem we also estimate the distance between the optimal solution of the perturbed problem and the optimal solution of the original problem moreover our approach yields a rigorous measure for the proximity of the actual iterate to the minimizer of the perturbed and the original problems we report on numerical experiments to illustrate efficiency and mesh independence

mathematical programs with complementarity constraints in function space c and strong stationarity and a path following algorithm an optimal control problem governed by an elliptic variational inequality is studied the feasible set of the problem is relaxed and a path following type method is used to regularize the constraint on the state variable first order optimality conditions for the relaxed regularized subproblems are derived and convergence of stationary points with respect to the relaxation and regularization parameters is shown in particular c and strong stationarity as well as variants thereof are studied the subproblems are solved by using semismooth newton methods the overall algorithmic concept is provided and its performance is discussed by means of examples including problems with bilateral constraints and a nonsymmetric operator

state constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions we consider a control and state constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions these conditions occur during the modeling of diffuse gray conductive radiative heat transfer the nonlocal radiation interface condition and the pointwise state constraints represent the particular features of this problem to deal with the state constraints continuity of the state is shown which allows us to derive first order necessary conditions afterwards we establish second order sufficient conditions that account for strongly active sets and ensure local optimality in an l 2 neighborhood

state constrained optimal control problems with states of low regularity we consider first order optimality conditions for state constrained optimal control problems in particular we study the case where the state equation does not have enough regularity to admit existence of a slater point in function space we overcome this difficulty by a special transformation under a density condition we show existence of lagrange multipliers which have a representation via measures and additional regularity properties

regularization for state constrained optimal control problems by half spaces based decoupling abstract in this paper we study an abstract constrained optimization problem which subsumes a common model for the optimal control of linear partial differential equations our emphasis is the case when the ordering cone for the optimization problem has an empty interior to circumvent this difficulty we propose a new regularization approach which is based on the decoupling of the ordering cone by using certain half spaces the approach results in a family of regular optimization problems having a simpler structure than the original optimization problem existence theorems convergence analysis and optimality conditions are given for the regularized problems we present two numerical examples and the results are quite encouraging showing the potential of the proposed approach

first and second order optimality conditions for a class of optimal control problems with quasilinear elliptic equations a class of optimal control problems for quasilinear elliptic equations is considered where the coefficients of the elliptic differential operator depend on the state function first and second order optimality conditions are discussed for an associated control constrained optimal control problem main emphasis is laid on second order sufficient optimality conditions to this aim the regularity of the solutions to the state equation and its linearization is studied in detail and the pontryagin maximum principle is derived one of the main difficulties is the nonmonotone character of the state equation

a priori mesh grading for an elliptic problem with dirac right hand side the green function of the poisson equation in two dimensions is not contained in the sobolev space h 1 omega such that finite element error estimates for the discretization of a problem with the dirac measure on the right hand side are nonstandard and quasi uniform meshes are inappropriate by using graded meshes l 2 error estimates of almost optimal order are shown as a byproduct we show for the poisson equation with a right hand side in l 2 that appropriate mesh refinement near some interior point diminishes the error at this point by nearly one order

sufficient second order optimality conditions for an elliptic optimal control problem with pointwise control state constraints an optimal control problem for a semilinear elliptic equation is investigated where pointwise constraints are given on control and state the state constraints are of mixed bottleneck type where associated lagrange multipliers can be chosen as bounded and measurable functions based on this property a second order sufficient optimality condition is established that takes into account strongly active constraints

on the structure of lagrange multipliers for state constrained optimal control problems abstract state constrained optimal control problems represent severe analytical and numerical challenges it is shown that under certain conditions on the active set the measure representing the lagrange multiplier associated to the state constraint can be decomposed in a distributed part in l2 with support on the active set and a boundary measure concentrated on the interface between active and inactive sets

the multiplier penalty method for generalized nash equilibrium problems in banach spaces this paper deals with generalized nash equilibrium problems gneps in banach spaces we give an existence result for normalized equilibria of jointly convex gneps and then propose an augmented lag

fast iterative solution of reaction diffusion control problems arising from chemical processes pde constrained optimization problems and the development of preconditioned iterative methods for the efficient solution of the arising matrix systems is a field of numerical analysis that has recently been attracting much attention in this paper we analyze and develop preconditioners for matrix systems that arise from the optimal control of reaction diffusion equations which themselves result from chemical processes important aspects of our solvers are saddle point theory mass matrix representation and effective schur complement approximation as well as the incorporation of control constraints and application of the outer newton iteration to take into account the nonlinearity of the underlying pdes

a high accuracy nfem for constrained optimal control problems governed by elliptic equations in this paper a new nonconforming finite element method nfem is proposed for the constrained optimal control problems ocps governed by elliptic equations the state and co state are approximated by the nonconforming eq 1 rot element and the control is approximated by the piecewise constant element respectively the superclose properties for the above three variables are obtained moreover for the state and co state the superconvergence results are achieved in a broken energy norm by using the post processing technique lastly some numerical examples are provided to verify the theoretical analysis

pontryagin s principle for state constrained control problems governed by parabolic equations with unbounded controls this paper deals with optimal control problems governed by semilinear parabolic equations with pointwise state constraints and unbounded controls under some strong stability assumption we obtain necessary optimality conditions in the form of a pontryagin s minimum principle in qualified form a pontryagin s principle in nonqualified form is also proved without any stability condition

control of an elliptic problem with pointwise state constraints this paper deals with a quadratic control problem for elliptic equations with pointwise state constraints existence and uniqueness of the solution is proved optimality conditions are given and regularity of the optimal solution is investigated

local error analysis for the stokes equations with a punctual source term the solution of the stokes problem with a punctual force in source term is not h 1 times mathbb l 2 and therefore the approximation by a finite element method is suboptimal in the case of poisson problem with a dirac mass in the right hand side an optimal convergence for the lagrange finite elements has been shown on a subdomain which excludes the singularity in mathbb l 2 norm by koppl and wohlmuth here we show a quasi optimal local convergence in h 1 times mathbb l 2 norm for a mathbb p k mathbb p k 1 finite element method k geqslant 2 and for the mathbb p 1 mathrm b mathbb p 1 the error is still analysed on a subdomain which does not contain the singularity the proof is based on local arnold and liu error estimates a weak version of aubin nitsche duality lemma applied to the stokes problem and discrete inf sup conditions these theoretical results are generalized to a wide class of finite element methods before being illustrated by numerical simulations

existence of regular lagrange multipliers for a nonlinear elliptic optimal control problem with pointwise control state constraints a class of optimal control problems for semilinear elliptic equations with mixed control state constraints is considered the existence of bounded and measurable lagrange multipliers is proven as a particular application the lavrentiev type regularization of pointwise state constraints is discussed here the existence of associated regular multipliers is shown too

an extension of pontryagin s principle for state constrained optimal control of semilinear elliptic equations and variational inequalities this paper deals with state constrained optimal control problems governed by semilinear elliptic equations or variational inequalities by using ekeland s principle a minimum principle of pontryagin s type under some stability conditions of the optimal cost with respect to the state constraints is derived

set valued information systems set valued information systems are generalized models of single valued information systems incomplete information systems can be viewed as disjunctively interpreted set valued information systems since some objects in set valued information systems may have more than one value for an attribute so we define tolerance relation and use the maximal tolerance classes to classify the universe of discourse in order to derive optimal decision rules from set valued decision information systems we propose the concept of relative reduct of maximal tolerance classes and define a kind of discernibility function to compute the relative reduct by boolean reasoning techniques finally we define three kinds of relative reducts for set valued information systems and used them to evaluate the significance of attributes

penalty techniques for state constrained optimal control problems with the wave equation we consider a system that is governed by a pde of hyperbolic type namely the wave equation and is controlled by a dirichlet boundary control for the penalization of terminal constraints we compare the convergence of three penalization techniques a differentiable penalty method an exact penalization and a smoothed exact penalization the error of the solution of the differentiably penalized problem is at most of the order 1 sqrt gamma where gamma is the penalty parameter the exact penalization yields the exact solution if gamma is sufficiently large if gamma is sufficiently large for the smoothed exact penalization we obtain an error bound of the order 1 sqrt beta where beta is the additional smoothing parameter this method yields differentiable objective functions that remain uniformly strongly convex since gamma need not tend to infinity to obtain convergence we also consider the penalization of distributed inequality state constraints that prescribe an upper bound for the l infty norm of the state we prove the convergence with respect to the l 2 norm for a penalization of the constraint violation measured in the l 1 norm we use this penalization technique to prove the existence of optimal controls for the inequality state constrained problem

essential prime divisors and projectively equivalent ideals let z be an ideal in a noetherian ring r we concern ourselves with the essential prime divisors of z an interesting subset of ass r p for all large n we first take z br with b a regular element of r we show that there is a ring t with r c te r such that t is a finite r module and the essential primes of bt are exactly the prime divisors of bt we next consider an arbitrary ideal z and apply our principal arguments to the element u in the kees ring of z we thereby deduce that there is an idcal j projectively equivalent to z such that the set of essential primes of z equals the set u ass r j over ii 1 2 3 notation let z be an ideal in a noetherian ring r we will use w or

a bracket power characterization of analytic spread one ideals the main theorem characterizes in terms of bracket powers analytic spread one ideals in local rings specifically let bl bg x be r egular nonunits in a local noetherian ring r m and assume that i c xr a the integral closure of xr where i b1 bg x r then the main result shows that for all but finitely many units u1 ug in r that are non congruent modulo m and for all large integers n and k it holds that jin ii for j 1 i k and j not divisible by char r m where i i is the j th bracket power bi uix i bg ugx i xi r of i bi ulx bg ugx x r and conversely if there exist positive integers g n and k n g such that i has a basis 31 39 x such that ilkn 3i k 3gk xk n r then i has analytic spread one

mesh independence of semismooth newton methods for lavrentiev regularized state constrained nonlinear optimal control problems a class of nonlinear elliptic optimal control problems with mixed control state constraints arising e g in lavrentiev type regularized state constrained optimal control is considered based on its first order necessary optimality conditions a semismooth newton method is proposed and its fast local convergence in function space as well as a mesh independence principle for appropriate discretizations are proved the paper ends by a numerical verification of the theoretical results including a study of the algorithm in the case of vanishing lavrentiev parameter the latter process is realized numerically by a combination of a nested iteration concept and an extrapolation technique for the state with respect to the lavrentiev parameter

approximation of elliptic control problems in measure spaces with sparse solutions optimal control problems in measure spaces governed by elliptic equations are considered for distributed and neumann boundary control which are known to promote sparse solutions optimality conditions are derived and some of the structural properties of their solutions in particular sparsity are discussed a framework for their approximation is proposed which is efficient for numerical computations and for which we prove convergence and provide error estimates

an interior point method in function space for the efficient solution of state constrained optimal control problems we propose and analyze an interior point path following method in function space for state constrained optimal control our emphasis is on proving convergence in function space and on constructing a practical path following algorithm in particular the introduction of a pointwise damping step leads to a very efficient method as verified by numerical experiments

solving elliptic control problems with interior point and sqp methods control and state constraints abstract we study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints both boundary control and distributed control problems are considered with boundary conditions of dirichlet or neumann type by introducing suitable discretization schemes the control problem is transcribed into a nonlinear programming problem necessary conditions of optimality are discussed both for the continuous and the discretized control problem it is shown that the recently developed interior point method loqo of 35 is capable of solving these problems even for high discretizations four numerical examples with dirichlet and neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang bang controls

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

an a posteriori error analysis of an elliptic optimal control problem in measure space abstract we propose an a posteriori error estimator for an optimal control problem with sparsity promoting cost the control variable lies in the space of regular borel measures we consider a solution technique that relies on the discretization of the control variable as a linear combination of dirac measures the proposed a posteriori error estimator can be decomposed into the sum of two contributions an error estimator in the maximum norm for the discretization of the adjoint equation and an estimator in the l 2 norm that accounts for the approximation of the state equation we prove that the designed error estimator is locally efficient and we explore its reliability properties the analysis is valid for two and three dimensional domains we illustrate the theory with numerical examples

on the use of state constraints in optimal control of singular pdes abstract we consider optimal control of nonlinear partial differential equations involving potentially singular solution dependent terms singularity can be prevented by either restricting controls to a closed admissible set for which well posedness of the equation can be guaranteed or by explicitly enforcing pointwise bounds on the state by means of an elliptic model problem we contrast the requirements for deriving the existence of solutions and first order optimality conditions for both the control constrained and the state constrained formulation our analysis as well as numerical tests illustrate that control constraints lead to severe restrictions on the attainable states which is not the case for state constraints

strong stationarity for optimal control of the obstacle problem with control constraints we consider the distributed optimal control of the obstacle problem with control constraints since mignot proved in 1976 the necessity of a system which is equivalent to strong stationarity it has been an open problem whether such a system is still necessary in the presence of control constraints using moderate regularity of the optimal control and an assumption on the control bounds which is implied by u a 0 le u b quasi everywhere in omega in the case of an upper obstacle y le psi we can answer this question in the affirmative we also present counterexamples showing that strong stationarity may not hold if u a 0 or 0 le u b are violated an erratum is attached

optimal control of a fractional diffusion equation with state constraints this paper is concerned with the state constrained optimal control problems of a fractional diffusion equation in a bounded domain the fractional time derivative is considered in the riemann liouville sense under a slater type condition we prove the existence a lagrange multiplier and a decoupled optimality system

a level set approach for the solution of a state constrained optimal control problem state constrained optimal control problems for linear elliptic partial differential equations are considered the corresponding first order optimality conditions in primal dual form are analyzed and linked to a free boundary problem resulting in a novel algorithmic approach with the boundary interface between the active and inactive sets as optimization variable the new algorithm is based on the level set methodology the speed function involved in the level set equation for propagating the interface is computed by utilizing techniques from shape optimization encouraging numerical results attained by the new algorithm are reported on

hp spectral element approximation for integral state constrained optimal control problems governed by harmonic equations abstract in this paper an integral state constrained optimal control problem governed by first bi harmonic equation and its h p spectral element approximation are considered we first present the optimality conditions of the continuous and discrete optimal control problems respectively then a priori error estimates of h p spectral element approximation for the optimal control problem are derived rigorously moreover a posteriori error estimates are established for the coupled state and control approximations in detail such estimators can be used to construct reliable adaptive methods for optimal control problems

immersed finite elements for optimal control problems of elliptic pdes with interfaces this paper presents a numerical method and analysis based on the variational discretization concept for optimal control problems governed by elliptic pdes with interfaces the method uses a simple uniform mesh which is independent of the interface due to the jump of the coefficient across the interface the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used therefore the immersed finite element method ifem developed in li et al 20 is used to discretize the state equation required in the variational discretization approach optimal error estimates for the control state and adjoint state are derived numerical examples are provided to confirm the theoretical results

optimal control for an elliptic system with polygonal state constraints this work is devoted to stationary optimal control problems with polygonal constraints on the components of the state existence of lagrange multipliers of different regularities is verified for the cases with and without slater condition holding for the numerical realization a semismooth newton method is proposed for an appropriately chosen family of regularized problems the asymptotic behavior of the regularized problem class is studied and numerical feasibility of the method is shown

analysis on inexact block diagonal preconditioners for elliptic pde constrained optimization problems abstract by using the galerkin finite element method the distributed control problems can be discretized into a saddle point problem with a coefficient matrix of block three by three which can be reduced to a linear system with lower order we first introduce a class of inexact block diagonal preconditioners and estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively based on the cholesky decomposition of the known matrices we also analyze a lower triangular preconditioner to accelerate the minimal residual method for the reduced linear system and discuss its real and complex eigenvalues respectively moreover these bounds do not rely on the regularization parameter and the eigenvalues of the matrices in the discrete system numerical experiments are also presented to demonstrate the effectiveness and robustness of the two new preconditioners

error estimates of the space time spectral method for parabolic control problems abstract this work is devoted to investigate the spectral approximation of optimal control of parabolic problems the space time method is used to boost high order accuracy by applying dual petrov galerkin spectral scheme in time and spectral method in space the optimality conditions are derived and the a priori error estimates indicate the convergence of the proposed method numerical tests confirm the theoretical results and show the efficiency of the method

optimal control of a system governed by an elliptic partial differential equation abstract in this paper we consider the existence and uniqueness of a state constrained boundary control problem governed by an elliptic partial differential equation with twice differentiable conditions on the state constraint a linear reformulation of the minimax control problem has advantages which are enhanced by considering the implications of the continuous and discrete maximum principles the continuous optimality conditions of the linear reformulation of the control problem are developed and stated in an algebraic setting

an efficient nfem for optimal control problems governed by a bilinear state equation abstract in this paper a nonconforming finite element method nfem is proposed for the constrained optimal control problems ocps governed by a bilinear state equation the state and adjoint state are approximated by the nonconforming e q 1 r o t element and the control is approximated by the orthogonal projection through the state and adjoint state some superclose and superconvergence properties are obtained by full use of the distinguish characters of this e q 1 r o t element such as the interpolation operator equals the ritz projection and the consistency error is one order higher than its interpolation error in the broken energy norm finally some numerical results are provided to verify the theoretical analysis

a posteriori error estimates for optimal control problems with state and control constraints we discuss the full discretization of an elliptic optimal control problem with pointwise control and state constraints we provide the first reliable a posteriori error estimator that contains only computable quantities for this class of problems moreover we show that the error estimator converges to zero if one has convergence of the discrete solutions to the solution of the original problem the theory is illustrated by numerical tests

a stable nonconforming mixed finite element scheme for elliptic optimal control problems a new stable nonconforming mixed finite element mfe scheme is proposed for the constrained optimal control problem ocp governed by elliptic equations the nonconforming fes are employed to approximate the co state the piecewise constant element for the state and control respectively the corresponding optimal order error estimates are derived by full use of the distinguish properties of the nonconforming element i e a the divergence of the function in the fe space is a constant b the consistency error is one order higher than its interpolation error in the broken energy norm finally some numerical results are carried out to verify the theoretical analysis

triangular powers of integers from determinants of binomial coefficient matrices abstract formulas are obtained for the determinants of certain matrices whose entries are zero and either binomial coefficients or their negatives a consequence is that for all integers n 2 and k 2 there exists an n 1 k 1 n 1 k 1 matrix m n k whose entries are the alternating binomial coefficients 1 j 1 jn and zeros such that det m n k ktn 1 where tn 1 is the n l th triangular number further if we form the infinite matrix download download full size image whose kth row is 0k 1k 2k then each of the above mentioned determinants is up to sign the determinant of an n n submatrix a of download download full size image obtained by selecting the initial n columns and some choice of n rows of download download full size image the matrices m n k and others that we will consider also have the unexpected property that det m n k det m n k where m denotes the matrix obtained from m by replacing each entry with its absolute value