Bounded perturbation resilience of extragradient-type methods and their applications

In this paper we study the bounded perturbation resilience of the extragradient and the subgradient extragradient methods for solving a variational inequality (VI) problem in real Hilbert spaces. This is an important property of algorithms which guarantees the convergence of the scheme under summable errors, meaning that an inexact version of the methods can also be considered. Moreover, once an algorithm is proved to be bounded perturbation resilience, superiorization can be used, and this allows flexibility in choosing the bounded perturbations in order to obtain a superior solution, as well explained in the paper. We also discuss some inertial extragradient methods. Under mild and standard assumptions of monotonicity and Lipschitz continuity of the VI’s associated mapping, convergence of the perturbed extragradient and subgradient extragradient methods is proved. In addition we show that the perturbed algorithms converge at the rate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(1/t)$\end{document}O(1/t). Numerical illustrations are given to demonstrate the performances of the algorithms.


Introduction
In this paper we are concerned with the variational inequality (VI) problem of finding a point x * such that where C ⊆ H is a nonempty, closed and convex set in a real Hilbert space H, ·, · denotes the inner product in H, and F : H → H is a given mapping. This problem is a fundamental problem in optimization theory, and it captures various applications such as partial differential equations, optimal control and mathematical programming; for the theory and application of VIs and related problems, the reader is referred, for example, to the works of Ceng et al. Many algorithms for solving VI (.) are projection algorithms that employ projections onto the feasible set C of VI (.), or onto some related set, in order to reach iteratively a solution. Korpelevich [] and Antipin [] proposed an algorithm for solving (.), known as the extragradient method, see also Facchinei and Pang [, Chapter ]. In each iteration of the algorithm, in order to get the next iterate x k+ , two orthogonal projections onto C are calculated according to the following iterative step. Given the current iterate x k , calculate where γ k ∈ (, /L), and L is the Lipschitz constant of F, or γ k is updated by the following adaptive procedure: In the extragradient method there is the need to calculate twice the orthogonal projection onto C in each iteration. In case that the set C is simple enough so that projections onto it can be easily computed, this method is particularly useful; but if C is a general closed and convex set, a minimal distance problem has to be solved (twice) in order to obtain the next iterate. This might seriously affect the efficiency of the extragradient method. Hence, Censor et al. in [-] presented a method called the subgradient extragradient method, in which the second projection (.) onto C is replaced by a specific subgradient projection which can be easily calculated. The iterative step has the following form: where T k is the set defined as and γ ∈ (, /L). In this manuscript we prove that the above methods, the extragradient and the subgradient extragradient methods, are bounded perturbation resilient, and the perturbed methods have the convergence rate of O(/t). This means that that will show that an inexact version of the algorithms allows incorporating summable errors that also converge to a solution of VI (.) and, moreover, their superiorized version can be introduced by choosing the perturbations. In order to obtain a superior solution with respect to some new objective function, for example, by choosing the norm, we can obtain a solution to VI (.) which is closer to the origin.
Our paper is organized as follows. In Section  we present the preliminaries. In Section  we study the convergence of the extragradient method with outer perturbations. Later, in Section , the bounded perturbation resilience of the extragradient method as well as the construction of the inertial extragradient methods are presented.
In the spirit of the previous sections, in Section  we study the convergence of the subgradient extragradient method with outer perturbations, show its bounded perturbation resilience and the construction of the inertial subgradient extragradient methods. Finally, in Section  we present numerical examples in signal processing which demonstrate the performances of the perturbed algorithms.

Preliminaries
Let H be a real Hilbert space with the inner product ·, · and the induced norm · , and let D be a nonempty, closed and convex subset of H. We write x k x to indicate that the sequence {x k } ∞ k= converges weakly to x and x k → x to indicate that the sequence {x k } ∞ k= converges strongly to x. Given a sequence {x k } ∞ k= , denote by ω w (x k ) its weak ωlimit set, that is, any which converges weakly to x. For each point x ∈ H, there exists a unique nearest point in D denoted by P D (x). That is, The mapping P D : H → D is called the metric projection of H onto D. It is well known that P D is a nonexpansive mapping of H onto D, i.e., and even firmly nonexpansive mapping. This is captured in the next lemma.
Lemma . For any x, y ∈ H and z ∈ D, it holds The characterization of the metric projection P D [, Section ] is given by the following two properties in this lemma.

Lemma . Given x ∈ H and z ∈ D. Then z = P D (x) if and only if
is not properly contained in the graph of any other monotone operator.
Based on Rockafellar [, Theorem ], a monotone mapping B is maximal if and only if, for any (x, u) ∈ H × H, if uv, xy ≥  for all (v, y) ∈ G(B), then it follows that u ∈ B(x). Definition . The subdifferential set of a convex function c at a point x is defined as For z ∈ H, take any ξ ∈ ∂c(z) and define This is a half-space, the bounding hyperplane of which separates the set D from the point where the nonnegative sequences {γ k } ∞ k= and {δ k } ∞ k= satisfy ∞ k= γ k < +∞ and ∞ k= δ k < +∞, respectively. Then lim k→∞ a k exists.

The extragradient method with outer perturbations
In order to discuss the convergence of the extragradient method with outer perturbations, we make the following assumptions.
Condition . The solution set of (.), denoted by SOL(C, F), is nonempty.
Observe that while Censor et al. in [, Theorem .] showed the weak convergence of the extragradient method (.) in Hilbert spaces for a fixed step size γ k = γ ∈ (, /L), this can be easily improved in case that the adaptive rule (.) is used. The next theorem shows this, and its proof can easily be derived by following similar lines of the proof of [, Theorem .]. Now we consider the extragradient method with outer perturbations.
Algorithm . The extragradient method with outer perturbations: Step : Select a starting point x  ∈ C and set k = .
Step : Given the current iterate x k , compute where γ k = σρ m k , σ > , ρ ∈ (, ) and m k is the smallest nonnegative integer such that (see []) Calculate the next iterate Step : If x k = y k , then stop. Otherwise, set k ← (k + ) and return to Step .

Convergence analysis
Theorem . Assume that Conditions .-. hold. Then the sequence {x k } ∞ k= generated by Algorithm . converges weakly to a solution of the variational inequality (.).
Proof Take x * ∈ SOL(C, F). From (.) and Lemma .(ii), we have From the Cauchy-Schwarz inequality and the mean value inequality, it follows Using x * ∈ SOL(C, F) and the monotone property of F, we have y kx * , F(y k ) ≥  and, consequently, we get where the equality comes from Using x k+ ∈ C, the definition of y k and Lemma ., we have Therefor, we assume e k  ∈ [, μν) and e k  ∈ [, /), k ≥ , where ν ∈ (, μ). So, using (.), we get Combining (.)-(.) and (.), we obtain where e k := e k  + e k  .
(  .   ) From (.), it follows (  .   ) So, from (.), we have Using (.) and Lemma ., we get the existence of lim k→∞ x kx *  and then the boundedness of {x k } ∞ k= . From (.), it follows which means that We will show thatx is a solution of the variational inequality (.). Let On the other hand, by the definition of y k and Lemma ., it follows that and consequently, Hence we have Taking the limit as i → ∞ in the above inequality and using Lemma ., we obtain Since lim k→∞ x kx * exists and ω w (x k ) ⊆ SOL(C, F), using Lemma ., we conclude that {x k } ∞ k= weakly converges to a solution of the variational inequality (.). This completes the proof. Nemirovski

Convergence rate
Proof Take arbitrarily x ∈ C. From Conditions . and ., we have By (.) and Lemma ., we get Identifying x * with x in (.) and (.), and combining (.) and (.), we get Thus, we have Using the notations of ϒ t and y t in the above inequality, we derive The proof is complete.
thus Algorithm . has O(/t) convergence rate. In fact, for any bounded subset D ⊂ C and given accuracy > , our algorithm achieves

The bounded perturbation resilience of the extragradient method
In this section, we prove the bounded perturbation resilience (BPR) of the extragradient method. This property is fundamental for the application of the superiorization methodology (SM) to them. The superiorization methodology of [-], which originates in the papers by Butnariu, Reich and Zaslavski [-], is intended for constrained minimization (CM) problems of the form where φ : H → R is an objective function and ⊆ H is the solution set of another problem. Here, we assume = ∅ throughout this paper. Assume that the set is a closed convex subset of a Hilbert space H, the minimization problem (.) becomes a standard CM problem. Here, we are interested in the case wherein is the solution set of another CM of the form i.e., we wish to look at provided that is nonempty. If f is differentiable, and let F = ∇f , then CM (.) equals the following variational inequality: to find a point x * ∈ C such that The superiorization methodology (SM) strives not to solve (.) but rather the task is to find a point in which is superior, i.e., has a lower, but not necessarily minimal, value of the objective function φ. This is done in the SM by first investigating the bounded perturbation resilience of an algorithm designed to solve (.) and then proactively using such permitted perturbations in order to steer the iterates of such an algorithm toward lower values of the φ objective function while not loosing the overall convergence to a point in .
In this paper, we do not investigate superiorization of the extragradient method. We prepare for such an application by proving the bounded perturbation resilience that is needed in order to do superiorization.

Algorithm . The basic algorithm:
Initialization: x  ∈ is arbitrary; Iterative step: Given the current iterate vector x k , calculate the next iterate x k+ via The bounded perturbation resilience (henceforth abbreviated to BPR) of such a basic algorithm is defined next. Definition . An algorithmic operator A : H → is said to be bounded perturbations resilient if the following is true. If Algorithm . generates sequences {x k } ∞ k= with x  ∈ that converge to points in , then any sequence {y k } ∞ k= starting from any y  ∈ , generated by Treating the extragradient method as the Basic Algorithm A , our strategy is to first prove convergence of the iterative step (.) with bounded perturbations. We show next how the convergence of this yields BPR according to Definition ..
A superiorized version of any Basic Algorithm employs the perturbed version of the Basic Algorithm as in (.). A certificate to do so in the superiorization method, see [], is gained by showing that the Basic Algorithm is BPR. Therefore, proving the BPR of an algorithm is the first step toward superiorizing it. This is done for the extragradient method in the next subsection.

The BPR of the extragradient method
In this subsection, we investigate the bounded perturbation resilience of the extragradient method whose iterative step is given by (.).
To this end, we treat the right-hand side of (.) as the algorithmic operator A of Definition ., namely, we define, for all k ≥ , and identify the solution set with the solution set of the variational inequality (.) and identify the additional set with C. According to Definition ., we need to show the convergence of the sequence {x k } ∞ k= that, starting from any x  ∈ C, is generated by which can be rewritten as where γ k = σρ m k , σ > , ρ ∈ (, ) and m k is the smallest nonnegative integer such that The sequences {v k } ∞ k= and {λ k } ∞ k= obey conditions (i) and (ii) in Definition ., respectively, and also (iii) in Definition . is satisfied.
The next theorem establishes the bounded perturbation resilience of the extragradient method. The proof idea is to build a relationship between BPR and the convergence of the iterative step (.).
Theorem . Assume that Conditions .-. hold. Assume the sequence {v k } ∞ k= is bounded, and the scalars {λ k } ∞ k= are such that λ k ≥  for all k ≥ , and ∞ k= λ k < +∞. Then the sequence {x k } ∞ k= generated by (.) and (.) converges weakly to a solution of the variational inequality (.).
Proof Take x * ∈ SOL(C, F). From ∞ k= λ k < +∞ and that {v k } ∞ k= is bounded, we have . Identifying e k  with λ k v k in (.) and (.) and using (.), we get (.) From x k+ ∈ C, the definition of y k and Lemma ., we have We have where the last inequality comes from λ k v k < (μ)/ and μ < . Substituting (.) into (.), we get Following the proof line of Theorem ., we get {x k } ∞ k= weakly converges to a solution of the variational equality (.).
By using Theorems . and ., we obtain the convergence rate of the extragradient method with BP.

Construction of the inertial extragradient methods by BPR
In this subsection, we construct two classes of inertial extragradient methods by using BPR, i.e., identifying e k i , k = , , and λ k , v k with special values. Polyak [, ] first introduced the inertial-type algorithms by using the heavy ball method of the second-order dynamical systems in time. Since the inertial-type algorithms speed up the original algorithms without the inertial effects, recently there has been increasing interest in studying inertial-type algorithms (see, e.g., [-]). The authors [] introduced an inertial extragradient method as follows: where L is the Lipschitz constant of F. Based on the iterative step (.), we construct the following inertial extragradient method: Then the sequence {x k } ∞ k= generated by the inertial extragradient method (.) converges weakly to a solution of the variational inequality (.).
It is obvious that v k ≤ . So, it follows that {e k i }, i = , , satisfy (.) from the condition on {β (i) k }. Using Theorem ., we complete the proof.
Remark . From (.), we have x kx k- ≤  for big enough k, that is, Using the extragradient method with bounded perturbations (.), we construct the following inertial extragradient method: We extend Theorem . to the convergence of the inertial extragradient method (.).

Theorem . Assume that Conditions
Then the sequence {x k } ∞ k= generated by the inertial extragradient method (.) converges weakly to a solution of the variational inequality (.).

The extension to the subgradient extragradient method
In this section, we generalize the results of extragradient method proposed in the previous sections to the subgradient extragradient method.
Censor et al.
[] presented the subgradient extragradient method (.). In their method the step size is fixed γ ∈ (, /L), where L is a Lipschitz constant of F. So, in order to determine the stepsize γ k , one needs first calculate (or estimate) L, which might be difficult or even impossible in general. So, in order to overcome this, the Armijo-like search rule can be used: To discuss the convergence of the subgradient extragradient method, we make the following assumptions.

The subgradient extragradient method with outer perturbations
In this subsection, we present the subgradient extragradient method with outer perturbations.
Algorithm . The subgradient extragradient method with outer perturbations: Step : Select a starting point x  ∈ H and set k = .
Step : Given the current iterate x k , compute where γ k = σρ m k , σ > , ρ ∈ (, ) and m k is the smallest nonnegative integer such that (see []) Construct the set and calculate Step : If x k = y k , then stop. Otherwise, set k ← (k + ) and return to Step . and

The BPR of the subgradient extragradient method
In this subsection, we investigate the bounded perturbation resilience of the subgradient extragradient method (.).
To this end, we treat the right-hand side of (.) as the algorithmic operator A of Definition ., namely, we define, for all k ≥ , where γ k satisfies (.) and Identify the solution set with the solution set of the variational inequality (.) and identify the additional set with C. According to Definition ., we need to show the convergence of the sequence {x k } ∞ k= that, starting from any x  ∈ H, is generated by which can be rewritten as where γ k = σρ m k , σ > , ρ ∈ (, ) and m k is the smallest nonnegative integer such that The sequences {v k } ∞ k= and {λ k } ∞ k= obey conditions (i) and (ii) in Definition ., respectively, and also (iii) in Definition . is satisfied.
The next theorem establishes the bounded perturbation resilience of the subgradient extragradient method. Since its proof is similar to that of Theorem ., we omit it. We also get the convergence rate of the subgradient extragradient methods with BP (.) and (.).
Theorem . Assume that Conditions ., . and . hold. Assume the sequence {v k } ∞ k= is bounded, and the scalars {λ k } ∞ k= are such that λ k ≥  for all k ≥ , and ∞ k= λ k < +∞. Let the sequences {x k } ∞ k= and {y k } ∞ k= be generated by (.) and (.). For any integer t > , we have y t ∈ C which satisfies and

Construction of the inertial subgradient extragradient methods by BPR
In this subsection, we construct two classes of inertial subgradient extragradient methods by using BPR, i.e., identifying e k i , k = , , and λ k , v k with special values. Based on Algorithm ., we construct the following inertial subgradient extragradient method: where γ k satisfies (.) and Similar to the proof of Theorem ., we get the convergence of the inertial subgradient extragradient method (.).
Theorem . Assume that Conditions ., . and . hold. Assume that the sequences Then the sequence {x k } ∞ k= generated by the inertial subgradient extragradient method (.) converges weakly to a solution of the variational inequality (.).
Using the subgradient extragradient method with bounded perturbations (.), we construct the following inertial subgradient extragradient method: where γ k = σρ m k , σ > , ρ ∈ (, ) and m k is the smallest nonnegative integer such that We extend Theorem . to the convergence of the inertial subgradient extragradient method (.).
Theorem . Assume that Conditions ., . and . hold. Assume that the sequence {β k } ∞ k= satisfies ∞ k= β k < ∞. Then the sequence {x k } ∞ k= generated by the inertial subgradient extragradient method (.) converges weakly to a solution of the variational inequality (.).
Example . Let x  ∈ R n be a K -sparse signal, K n. The sampling matrix A ∈ R m×n (m < n) is stimulated by the standard Gaussian distribution and a vector b = Ax  + e, where e is additive noise. When e = , it means that there is no noise to the observed data. Our task is to recover the signal x  from the data b.
It is well known that the sparse signal x  can be recovered by solving the following LASSO problem []: where t > . It is easy to see that the optimization problem (.) is a special case of the variational inequality problem (.), where F(x) = A T (Axb) and C = {x | x  ≤ t}. We can use the proposed iterative algorithms to solve the optimization problem (.). Although the orthogonal projection onto the closed convex set C does not have a closed-form solution, the projection operator P C can be precisely computed in a polynomial time. We include the details of computing P C in Appendix. We conduct plenty of simulations to compare the performances of the proposed iterative algorithms. The following inequality was defined as the stopping criterion: where >  is a given small constant. 'Iter' denotes the iteration numbers. 'Obj' represents the objective function value and 'Err' is the -norm error between the recovered signal and the true K -sparse signal. We divide the experiments into two parts. One task is to recover the sparse signal x  from noise observation vector b, and the other is to recover the sparse signal from noiseless data b. For the noiseless case, the obtained numerical results are reported in Table . To visually view the results, Figure  shows the recovered signal compared with the true signal x  when K = . We can see from Figure  that the   recovered signal is the same as the true signal. Further, Figure  presents the objective function value versus the iteration numbers.
For the noise observation b, we assume that the vector e is corrupted by Gaussian noise with zero mean and β variances. The system matrix A is the same as in the noiseless case, and the sparsity level K = . We list the numerical results for different noise level β in Table . When the noise β = ., Figure  shows the objective function value versus the iteration numbers. Figure  shows the recovered signal vs the true signal in the noise case.  Example . Let F : R  → R  be defined by F(x, y) = x + y + sin(x), -x + y + sin(y) , ∀x, y ∈ R.
(  .  ) The authors [] proved that F is Lipschitz continuous with L = √  and -strongly monotone. Therefore the variational inequality (.) has a unique solution, and (, ) is its solution. Let where z ij ∈ (, ) and b i ∈ (, ) are generated randomly.
It is easy to verify that F is L-Lipschitz continuous and η-strongly monotone with L = max(eig(A)) and η = min(eig(A)). (ii) the performances of the inertial extragradient methods (.) and (.) are almost the same; (iii) the inertial subgradient extragradient method (.) performs better than the inertial subgradient extragradient method (.) for Example ., while they are almost the same for Example .; (iv) the (inertial) extragradient methods behave better than the (inertial) subgradient extragradient methods since the sets C in Examples . and . are simple, and hence the computation load of the projection onto it is small; (v) the inertial extragradient method (.) has an advantage over the inertial extragradient methods (.) and (.). The reason may be that it takes a bigger inertial parameter α k .

Conclusions
In this research article we study an important property of iterative algorithms for solving variational inequality (VI) problems which is called bounded perturbation resilience. In particular, we focus on extragradient-type methods. This enables us to develop inexact versions of the methods as well as apply the superiorization methodology in order to obtain a 'superior' solution to the original problem. In addition, some inertial extragradient methods are also derived. All the presented methods converge at the rate of O(/t), and three numerical examples illustrate, demonstrate and compare the performances of all the algorithms.

Appendix
In this part, we present the details of computing a vector y ∈ R n onto the  -norm ball constraint. For convenience, we consider projection onto the unit  -norm ball first. Then we extend it to the general  -norm ball constraint.
The projection onto the unit  -norm ball is to solve the optimization problem The above optimization problem is a typical constrained optimization problem, we consider solving it based on the Lagrangian method. Define the Lagrangian function L(x, λ) as Let (x * , λ * ) be the optimal primal and dual pair. It satisfies the KKT conditions of It is easy to check that if y  ≤ , then x * = y and λ * = . In the following, we assume y  > . Based on the KKT conditions, we obtain λ * >  and x *  = . From the first order optimality, we have x * = max{|y|λ * , } ⊗ Sign(y), where ⊗ represents elementwise multiplication and Sign(·) denotes the symbol function, i.e., Sign(y i ) =  if y i ≥ ; otherwise Sign(y i ) = -.
To find λ * such that f (λ * ) = , we follow the following steps.
Step . Define a vector y with the same element as |y|, which was sorted in descending order. That is, y  ≥ y  ≥ · · · ≥ y n ≥ .
Step . For every k = , , . . . , n, solve the equation k i= y ikλ = . Stop search until the solution λ * belongs to the interval [y k+ , y k ].
In conclusion, the optimal x * can be computed by x * = max{|y|λ * , } ⊗ Sign(y). The next lemma extends the projection onto the unit  -norm ball to the general  -norm ball constraint. Lemma A. Let C  = {x | x  ≤ }. For any t > , define a general  -norm ball constraint set C = {x | x  ≤ t}. Then, for any vector y ∈ R n , we have P C (y) = tP C  y t .
Proof To compute the projection P C (y) is to solve the optimization problem P C (y) = arg min For any x ∈ C, let x = x t , it follows that x ∈ C  . The optimal solution x * of the above optimization problem satisfies x * = P C (y) = tx * , where x * is the optimal solution of the optimization problem of It is observed that x * is an exact projection onto the closed convex set C  . That is, x * = P C  ( y t ). This completes the proof.