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Showing papers in "Journal of Optimization Theory and Applications in 2016"


Journal ArticleDOI
TL;DR: In this article, the alternating directions method of multipliers is used to solve the homogeneous self-dual embedding, an equivalent feasibility problem involving finding a nonzero point in the intersection of a subspace and a cone.
Abstract: We introduce a first-order method for solving very large convex cone programs. The method uses an operator splitting method, the alternating directions method of multipliers, to solve the homogeneous self-dual embedding, an equivalent feasibility problem involving finding a nonzero point in the intersection of a subspace and a cone. This approach has several favorable properties. Compared to interior-point methods, first-order methods scale to very large problems, at the cost of requiring more time to reach very high accuracy. Compared to other first-order methods for cone programs, our approach finds both primal and dual solutions when available or a certificate of infeasibility or unboundedness otherwise, is parameter free, and the per-iteration cost of the method is the same as applying a splitting method to the primal or dual alone. We discuss efficient implementation of the method in detail, including direct and indirect methods for computing projection onto the subspace, scaling the original problem data, and stopping criteria. We describe an open-source implementation, which handles the usual (symmetric) nonnegative, second-order, and semidefinite cones as well as the (non-self-dual) exponential and power cones and their duals. We report numerical results that show speedups over interior-point cone solvers for large problems, and scaling to very large general cone programs.

597 citations


Journal ArticleDOI
TL;DR: In this article, the relation between the optimal transport problem and the Schrodinger bridge problem from a stochastic control perspective was investigated and connections between the two problems were made.
Abstract: We take a new look at the relation between the optimal transport problem and the Schrodinger bridge problem from a stochastic control perspective. Our aim is to highlight new connections between the two that are richer and deeper than those previously described in the literature. We begin with an elementary derivation of the Benamou---Brenier fluid dynamic version of the optimal transport problem and provide, in parallel, a new fluid dynamic version of the Schrodinger bridge problem. We observe that the latter establishes an important connection with optimal transport without zero-noise limits and solves a question posed by Eric Carlen in 2006. Indeed, the two variational problems differ by a Fisher information functional. We motivate and consider a generalization of optimal mass transport in the form of a (fluid dynamic) problem of optimal transport with prior. This can be seen as the zero-noise limit of Schrodinger bridges when the prior is any Markovian evolution. We finally specialize to the Gaussian case and derive an explicit computational theory based on matrix Riccati differential equations. A numerical example involving Brownian particles is also provided.

230 citations


Journal ArticleDOI
TL;DR: A comprehensive study of a general class of linear-quadratic mean field games, and establishes a neat and instructive sufficient condition, which is apparently absent in the literature and only depends on coefficients, for the unique existence of the solution for a class of nonsymmetric Riccati equations.
Abstract: We provide a comprehensive study of a general class of linear-quadratic mean field games. We adopt the adjoint equation approach to investigate the unique existence of their equilibrium strategies. Due to the linearity of the adjoint equations, the optimal mean field term satisfies a forward---backward ordinary differential equation. For the one-dimensional case, we establish the unique existence of the equilibrium strategy. For a dimension greater than one, by applying the Banach fixed point theorem under a suitable norm, a sufficient condition for the unique existence of the equilibrium strategy is provided, which is independent of the coefficients of controls in the underlying dynamics and is always satisfied whenever the coefficients of the mean field term are vanished, and hence, our theories include the classical linear-quadratic stochastic control problems as special cases. As a by-product, we also establish a neat and instructive sufficient condition, which is apparently absent in the literature and only depends on coefficients, for the unique existence of the solution for a class of nonsymmetric Riccati equations. Numerical examples of nonexistence of the equilibrium strategy will also be illustrated. Finally, a similar approach has been adopted to study the linear-quadratic mean field type stochastic control problems and their comparisons with mean field games.

208 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that a real tensor is strictly semi-positive if and only if the corresponding tensor complementarity problem has a unique solution for any nonnegative vector.
Abstract: In this paper, we prove that a real tensor is strictly semi-positive if and only if the corresponding tensor complementarity problem has a unique solution for any nonnegative vector and that a real tensor is semi-positive if and only if the corresponding tensor complementarity problem has a unique solution for any positive vector. It is shown that a real symmetric tensor is a (strictly) semi-positive tensor if and only if it is (strictly) copositive.

137 citations


Journal ArticleDOI
TL;DR: A class of related structured tensors is introduced and it is shown that the corresponding tensor complementarity problem has the property of global uniqueness and solvability.
Abstract: Recently, the tensor complementarity problem has been investigated in the literature. An important question involving the property of global uniqueness and solvability for a class of tensor complementarity problems was proposed by Song and Qi (J Optim Theory Appl, 165:854---873, 2015). In the present paper, we give an answer to this question by constructing two counterexamples. We also show that the solution set of this class of tensor complementarity problems is nonempty and compact. In particular, we introduce a class of related structured tensors and show that the corresponding tensor complementarity problem has the property of global uniqueness and solvability.

123 citations


Journal ArticleDOI
TL;DR: It is proved that a unique solution of the NCP exists under the condition of diagonalizable tensors.
Abstract: The main purpose of this paper was to investigate some kinds of nonlinear complementarity problems (NCP). For the structured tensors, such as symmetric positive-definite tensors and copositive tensors, we derive the existence theorems on a solution of these kinds of nonlinear complementarity problems. We prove that a unique solution of the NCP exists under the condition of diagonalizable tensors.

119 citations


Journal ArticleDOI
TL;DR: The numerical results show that this modified Hestenes–Stiefel conjugate gradient algorithm can be used to solve large-scale nonsmooth problems with convex and nonconvex properties and the global convergence of the presented algorithm is established.
Abstract: It is well known that nonlinear conjugate gradient methods are very effective for large-scale smooth optimization problems. However, their efficiency has not been widely investigated for large-scale nonsmooth problems, which are often found in practice. This paper proposes a modified Hestenes---Stiefel conjugate gradient algorithm for nonsmooth convex optimization problems. The search direction of the proposed method not only possesses the sufficient descent property but also belongs to a trust region. Under suitable conditions, the global convergence of the presented algorithm is established. The numerical results show that this method can successfully be used to solve large-scale nonsmooth problems with convex and nonconvex properties (with a maximum dimension of 60,000). Furthermore, we study the modified Hestenes---Stiefel method as a solution method for large-scale nonlinear equations and establish its global convergence. Finally, the numerical results for nonlinear equations are verified, with a maximum dimension of 100,000.

90 citations


Journal ArticleDOI
TL;DR: In this article, a new subclass of tensors is introduced and it is proved that this class of new tensors can be defined by the feasible region of the corresponding tensor complementarity problem.
Abstract: In this paper, a new subclass of tensors is introduced and it is proved that this class of new tensors can be defined by the feasible region of the corresponding tensor complementarity problem. Furthermore, the boundedness of solution set of the tensor complementarity problem is equivalent to the uniqueness of solution for such a problem with zero vector. For the tensor complementarity problem with a strictly semi-positive tensor, we proved the global upper bounds of its solution set. In particular, such upper bounds are closely associated with the smallest Pareto eigenvalue of such a tensor.

80 citations


Journal ArticleDOI
TL;DR: In this article, the convergence of a forward-backward-forward proximal-type algorithm with inertial and memory effects when minimizing the sum of a nonsmooth function with a smooth one in the absence of convexity is investigated.
Abstract: We investigate the convergence of a forward---backward---forward proximal-type algorithm with inertial and memory effects when minimizing the sum of a nonsmooth function with a smooth one in the absence of convexity. The convergence is obtained provided an appropriate regularization of the objective satisfies the Kurdyka---źojasiewicz inequality, which is for instance fulfilled for semi-algebraic functions.

80 citations


Journal ArticleDOI
TL;DR: This paper revisits the numerical approach to some classical variational inequalities, with monotone and Lipschitz continuous mapping A, by means of a projected reflected gradient-type method and establishes the convergence of the method in a more general setting that allows to use varying step-sizes without any requirement of additional projections.
Abstract: In this paper, we revisit the numerical approach to some classical variational inequalities, with monotone and Lipschitz continuous mapping A, by means of a projected reflected gradient-type method. A main feature of the method is that it formally requires only one projection step onto the feasible set and one evaluation of the involved mapping per iteration. Contrary to what was done so far, we establish the convergence of the method in a more general setting that allows us to use varying step-sizes without any requirement of additional projections. A linear convergence rate is obtained, when A is assumed to be strongly monotone. Preliminary numerical experiments are also performed.

79 citations


Journal ArticleDOI
TL;DR: The first method is an extension of the Intermediate Gradient Method proposed by Devolder, Glineur and Nesterov for problems with deterministic inexact oracle and can be applied to problems with composite objective function, both deterministic and stochastic inexactness of the oracle, and allows using a non-Euclidean setup.
Abstract: In this paper, we introduce new methods for convex optimization problems with stochastic inexact oracle. Our first method is an extension of the Intermediate Gradient Method proposed by Devolder, Glineur and Nesterov for problems with deterministic inexact oracle. Our method can be applied to problems with composite objective function, both deterministic and stochastic inexactness of the oracle, and allows using a non-Euclidean setup. We estimate the rate of convergence in terms of the expectation of the non-optimality gap and provide a way to control the probability of large deviations from this rate. Also we introduce two modifications of this method for strongly convex problems. For the first modification, we estimate the rate of convergence for the non-optimality gap expectation and, for the second, we provide a bound for the probability of large deviations from the rate of convergence in terms of the expectation of the non-optimality gap. All the rates lead to the complexity estimates for the proposed methods, which up to a multiplicative constant coincide with the lower complexity bound for the considered class of convex composite optimization problems with stochastic inexact oracle.

Journal ArticleDOI
TL;DR: This work allows for the subproblem to be solved inexactly, leading to an inexact block coordinate descent method, which incorporates the best known results for exact updates as a special case.
Abstract: One of the key steps at each iteration of a randomized block coordinate descent method consists in determining the update to a block of variables. Existing algorithms assume that in order to compute the update, a particular subproblem is solved exactly. In this work, we relax this requirement and allow for the subproblem to be solved inexactly, leading to an inexact block coordinate descent method. Our approach incorporates the best known results for exact updates as a special case. Moreover, these theoretical guarantees are complemented by practical considerations: the use of iterative techniques to determine the update and the use of preconditioning for further acceleration.

Journal ArticleDOI
TL;DR: The simulation results indicate that the proposed optimal tuned cascade control is effective and efficient and clearly demonstrate that applied techniques exhibit a significant performance improvement over classical tuning methods.
Abstract: This paper presents the optimal tuning of cascade load force controllers for a parallel robot platform. A parameter search for the proposed cascade controller is difficult because there is no methodology to set the parameters and the search space is broad. The proposed parameter search scheme is based on a bat algorithm, which attracts a lot of attention in the evolutionary computation area due to the empirical evidence of its superiority in solving various nonconvex problems. The control design problem is formulated as an optimization problem under constraints. Typical constraints, such as mechanical limits on positions and maximal velocities of hydraulic actuators as well as on servo-valve positions, are included in the proposed algorithm. The simulation results indicate that the proposed optimal tuned cascade control is effective and efficient. These results clearly demonstrate that applied techniques exhibit a significant performance improvement over classical tuning methods.

Journal ArticleDOI
TL;DR: In this paper, a consensus-based dual decomposition is proposed to remove the need for such a master node and still enable the computing nodes to generate an approximate dual solution for the underlying convex optimization problem.
Abstract: Dual decomposition has been successfully employed in a variety of distributed convex optimization problems solved by a network of computing and communicating nodes. Often, when the cost function is separable but the constraints are coupled, the dual decomposition scheme involves local parallel subgradient calculations and a global subgradient update performed by a master node. In this paper, we propose a consensus-based dual decomposition to remove the need for such a master node and still enable the computing nodes to generate an approximate dual solution for the underlying convex optimization problem. In addition, we provide a primal recovery mechanism to allow the nodes to have access to approximate near-optimal primal solutions. Our scheme is based on a constant stepsize choice, and the dual and primal objective convergence are achieved up to a bounded error floor dependent on the stepsize and on the number of consensus steps among the nodes.

Journal ArticleDOI
TL;DR: In this article, a local convergence rate is established for an orthogonal collocation method based on Gauss quadrature applied to an unconstrained optimal control problem, where the continuous problem has a smooth solution and the Hamiltonian satisfies a strong convexity condition, and the discrete problem possesses a local minimizer in a neighborhood of the continuous solution.
Abstract: A local convergence rate is established for an orthogonal collocation method based on Gauss quadrature applied to an unconstrained optimal control problem. If the continuous problem has a smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as the number of collocation points increases, the discrete solution converges to the continuous solution at the collocation points, exponentially fast in the sup-norm. Numerical examples illustrating the convergence theory are provided.

Journal ArticleDOI
TL;DR: In this article, the convergence of a stochastic extension of the forward-backward method for monotone inclusions is analyzed. Butler et al. provide a non-asymptotic error analysis in expectation for the strongly-monotone case, and almost sure convergence under weaker assumptions.
Abstract: We propose and analyze the convergence of a novel stochastic algorithm for monotone inclusions that are sum of a maximal monotone operator and a single-valued cocoercive operator. The algorithm we propose is a natural stochastic extension of the classical forward---backward method. We provide a non-asymptotic error analysis in expectation for the strongly monotone case, as well as almost sure convergence under weaker assumptions. For minimization problems, we recover rates matching those obtained by stochastic extensions of the so-called accelerated methods. Stochastic quasi-Fejer's sequences are a key technical tool to prove almost sure convergence.

Journal ArticleDOI
TL;DR: Without requiring any restrictive assumptions about the sign of the sectional curvature of the manifold, full convergence is obtained for any bounded sequence generated by the proximal point method, in the case that the objective function satisfies the Kurdyka–Lojasiewicz inequality.
Abstract: In this paper, we present a new approach to the proximal point method in the Riemannian context. In particular, without requiring any restrictive assumptions about the sign of the sectional curvature of the manifold, we obtain full convergence for any bounded sequence generated by the proximal point method, in the case that the objective function satisfies the Kurdyka---Lojasiewicz inequality. In our approach, we extend the applicability of the proximal point method to be able to solve any problem that can be formulated as the minimizing of a definable function, such as one that is analytic, restricted to a compact manifold, on which the sign of the sectional curvature is not necessarily constant.

Journal ArticleDOI
TL;DR: Geometric and information frameworks for constructing global optimization algorithms are considered, and several new ideas to speed up the search are proposed, and a smart mixture of new and traditional computational steps leads to 22 different global optimization methods.
Abstract: Geometric and information frameworks for constructing global optimization algorithms are considered, and several new ideas to speed up the search are proposed. The accelerated global optimization methods automatically realize a local behavior in the promising subregions without the necessity to stop the global optimization procedure. Moreover, all the trials executed during the local phases are used also in the course of the global ones. The resulting geometric and information global optimization methods have a similar structure, and a smart mixture of new and traditional computational steps leads to 22 different global optimization algorithms. All of them are studied and numerically compared on three test sets including 120 benchmark functions and 4 applied problems.

Journal ArticleDOI
TL;DR: The results show that directly applying two-block alternating direction method of multipliers with a large step length of the golden ratio to the linearly constrained convex optimization problem with a quadratically coupled objective function is convergent under mild conditions.
Abstract: In this paper, we establish the convergence properties for a majorized alternating direction method of multipliers for linearly constrained convex optimization problems, whose objectives contain coupled functions. Our convergence analysis relies on the generalized Mean-Value Theorem, which plays an important role to properly control the cross terms due to the presence of coupled objective functions. Our results, in particular, show that directly applying two-block alternating direction method of multipliers with a large step length of the golden ratio to the linearly constrained convex optimization problem with a quadratically coupled objective function is convergent under mild conditions. We also provide several iteration complexity results for the algorithm.

Journal ArticleDOI
TL;DR: From the perspective of some special matrices and iteration forms, some new and useful results for the unique solvability of the absolute value equation are obtained.
Abstract: In this paper, the unique solvability of the absolute value equation is further discussed. From the perspective of some special matrices and iteration forms, some new and useful results for the unique solvability of the absolute value equation are obtained.

Journal ArticleDOI
TL;DR: A new computational approach is proposed, which combines the control parameterization technique with a hybrid time-scaling strategy, for solving a class of nonlinear time-delay optimal control problems with canonical equality and inequality constraints.
Abstract: In this paper, we consider a class of nonlinear time-delay optimal control problems with canonical equality and inequality constraints. We propose a new computational approach, which combines the control parameterization technique with a hybrid time-scaling strategy, for solving this class of optimal control problems. The proposed approach involves approximating the control variables by piecewise constant functions, whose heights and switching times are decision variables to be optimized. Then, the resulting problem with varying switching times is transformed, via a new hybrid time-scaling strategy, into an equivalent problem with fixed switching times, which is much preferred for numerical computation. Our new time-scaling strategy is hybrid in the sense that it is related to two coupled time-delay systems--one defined on the original time scale, in which the switching times are variable, the other defined on the new time scale, in which the switching times are fixed. This is different from the conventional time-scaling transformation widely used in the literature, which is not applicable to systems with time-delays. To demonstrate the effectiveness of the proposed approach, we solve four numerical examples. The results show that the costs obtained by our new approach are lower, when compared with those obtained by existing optimal control methods.

Journal ArticleDOI
TL;DR: A Hamilton–Jacobi–Bellman fixed-point algorithm is compared to a steepest descent method issued from calculus of variations and an extended Bellman’s principle is derived by a different argument.
Abstract: We investigate a model problem for optimal resource management. The problem is a stochastic control problem of mean-field type. We compare a Hamilton---Jacobi---Bellman fixed-point algorithm to a steepest descent method issued from calculus of variations. For mean-field type control problems, stochastic dynamic programming requires adaptation. The problem is reformulated as a distributed control problem by using the Fokker---Planck equation for the probability distribution of the stochastic process; then, an extended Bellman's principle is derived by a different argument than the one used by P. L. Lions. Both algorithms are compared numerically.

Journal ArticleDOI
TL;DR: This paper presents how to apply second-order cone programming, a subclass of convex optimization, to rapidly solve a highly nonlinear optimal control problem arisen from smooth entry trajectory planning of hypersonic glide vehicles with high lift/drag ratios.
Abstract: This paper presents how to apply second-order cone programming, a subclass of convex optimization, to rapidly solve a highly nonlinear optimal control problem arisen from smooth entry trajectory planning of hypersonic glide vehicles with high lift/drag ratios. The common phugoid oscillations are eliminated by designing a smooth flight path angle profile. The nonconvexity terms of the optimal control problem, which include the nonlinear dynamics and nonconvex control constraints, are convexified via techniques of successive linearization, successive approximation, and relaxation. Lossless relaxation is also proved using optimal control theory. After discretization, the original nonconvex optimal control problem is converted into a sequence of second-order cone programming problems each of which can be solved in polynomial time using existing primal---dual interior-point algorithms whenever a feasible solution exists. Numerical examples are provided to show that rather smooth entry trajectory can be obtained in about 1 s on a desktop computer.

Journal ArticleDOI
TL;DR: A second-order image decomposition model is dealt with to perform denoising and texture extraction and gives qualitative properties of solutions using the dual problem and inf-convolution formulation.
Abstract: We deal with a second-order image decomposition model to perform denoising and texture extraction that was previously presented. We look for the decomposition of an image as the summation of three different order terms. For highly textured images, the model gives a two-scale texture decomposition: The first-order term can be viewed as a macro-texture (larger scale) which oscillations are not too large, and the zero-order term is the micro-texture (very oscillating) that contains the noise. Here, we perform mathematical analysis of the model and give qualitative properties of solutions using the dual problem and inf-convolution formulation.

Journal ArticleDOI
TL;DR: This paper addresses the problem of modifying the vertex weights of a block graph at minimum total cost so that a prespecified vertex becomes a 1-median of the perturbed graph and develops an O(Mlog M) algorithm that solves the problem on block graphs with M vertices.
Abstract: This paper addresses the problem of modifying the vertex weights of a block graph at minimum total cost so that a prespecified vertex becomes a 1-median of the perturbed graph. We call this problem the inverse 1-median problem on block graphs with variable vertex weights. For block graphs with equal edge lengths in each block, we can formulate the problem as a univariate optimization problem. By the convexity of the objective function, the local optimizer is also the global one. Therefore, we use the convexity to develop an $$O(M\log M)$$O(MlogM) algorithm that solves the problem on block graphs with M vertices.

Journal ArticleDOI
TL;DR: Under validity of a constraint qualification, it is shown that the stationary points of the regularized problem converge to a stationary point of the relaxed reformulation and under additional condition it is even a stationary Point of the original problem.
Abstract: We deal with chance constrained problems with differentiable nonlinear random functions and discrete distribution. We allow nonconvex functions both in the constraints and in the objective. We reformulate the problem as a mixed-integer nonlinear program and relax the integer variables into continuous ones. We approach the relaxed problem as a mathematical problem with complementarity constraints and regularize it by enlarging the set of feasible solutions. For all considered problems, we derive necessary optimality conditions based on Frechet objects corresponding to strong stationarity. We discuss relations between stationary points and minima. We propose two iterative algorithms for finding a stationary point of the original problem. The first is based on the relaxed reformulation, while the second one employs its regularized version. Under validity of a constraint qualification, we show that the stationary points of the regularized problem converge to a stationary point of the relaxed reformulation and under additional condition it is even a stationary point of the original problem. We conclude the paper by a numerical example.

Journal ArticleDOI
TL;DR: Directional differentiability is proved both for finite-dimensional problems and for problems in function spaces, under suitable assumptions on the active set, and a trust-region algorithm is proposed for the solution of the optimization problems.
Abstract: We investigate optimality conditions for optimization problems constrained by a class of variational inequalities of the second kind. Based on a nonsmooth primal---dual reformulation of the governing inequality, the differentiability of the solution map is studied. Directional differentiability is proved both for finite-dimensional problems and for problems in function spaces, under suitable assumptions on the active set. A characterization of Bouligand and strong stationary points is obtained thereafter. Finally, based on the obtained first-order information, a trust-region algorithm is proposed for the solution of the optimization problems.

Journal ArticleDOI
TL;DR: It is shown for the first time that the augmented Munk's minimum induced drag theorem is also applicable to closed systems, joined wings and generic biwings, and the nonuniqueness of the optimal circulation for a closed wing system is rigorously addressed.
Abstract: An analytical formulation for the induced drag minimization of closed wing systems is presented. The method is based on a variational approach, which leads to the Euler---Lagrange integral equation in the unknown circulation distribution. It is shown for the first time that the augmented Munk's minimum induced drag theorem, formulated in the past for open single-wing systems, is also applicable to closed systems, joined wings and generic biwings. The quasi-closed C-wing minimum induced drag conjecture discussed in the literature is addressed. Using the variational procedure presented in this work, it is also shown that in a general biwing, under optimal conditions, the aerodynamic efficiency of each wing is equal to the aerodynamic efficiency of the entire wing system (biwing's minimum induced drag theorem). This theorem holds even if the two wings are not identical and present different shapes and wingspans; an interesting direct consequence of the theorem is discussed. It is then verified (but yet not demonstrated) that in a closed path, the minimum induced drag of the biwing is identical to the optimal induced drag of the corresponding closed system (closed system's biwing limit theorem). Finally, the nonuniqueness of the optimal circulation for a closed wing system is rigorously addressed, and direct implications in the design of joined wings are discussed.

Journal ArticleDOI
TL;DR: This work studies in detail the synthesis of the avoidance strategies with a set-point stabilization control law and proves that the agents converge to the desired configurations while avoiding collisions and deadlocks (i.e., unwanted local minima).
Abstract: A set of cooperative and noncooperative collision avoidance strategies for a pair of interacting agents with acceleration constraints, bounded sensing uncertainties, and limited sensing ranges is presented. We explicitly consider the case in which position information from the other agent is unreliable, and develop bounded control inputs using Lyapunov-based analysis, that guarantee collision-free trajectories for both agents. The proposed avoidance control strategies can be appended to any other stable control law (i.e., main control objective) and are active only when the agents are close to each other. As an application, we study in detail the synthesis of the avoidance strategies with a set-point stabilization control law and prove that the agents converge to the desired configurations while avoiding collisions and deadlocks (i.e., unwanted local minima). Simulation results are presented to validate the proposed control formulation.

Journal ArticleDOI
TL;DR: An existence and uniqueness result of a class of second-order sweeping processes, with velocity in the moving set under perturbation in infinite-dimensional Hilbert spaces, is studied by using an implicit discretization scheme.
Abstract: In this paper, an existence and uniqueness result of a class of second-order sweeping processes, with velocity in the moving set under perturbation in infinite-dimensional Hilbert spaces, is studied by using an implicit discretization scheme. It is assumed that the moving set depends on the time, the state and is possibly unbounded. The assumptions on the Lipschitz continuity and the compactness of the moving set, and the linear growth boundedness of the perturbation force are weaker than the ones used in previous papers.