Convergence of Adaptive Forward-Reflected-Backward Algorithm for Solving Variational Inequalities

Abstract

The article deals with the problem of the approximate solution of variational inequalities. A novel iterative algorithm for solving variational inequalities in a real Banach space is proposed and studied. The proposed algorithm is an adaptive variant of the forward-reflectedbackward algorithm (Malitsky, Tam, 2020), where the used rule for updating the step size does not require knowledge of the Lipschitz continuous constant of the operator. In addition, the Alber generalized projection is used instead of the metric projection onto the feasible set. For variational inequalities with monotone and Lipschitz continuous operators, acting in a 2uniformly convex and uniformly smooth Banach space, a theorem on the weak convergence of the method is proved.

Keywords

• 1 Variational inequality
• monotone operator
• Lipschitz continuous operator
• forward-reflectedbackward algorithm
• 2-uniformly convex Banach space
• uniformly smooth Banach space
• convergence

1. Introduction

a variant of the extra-gradient method with projection understood in the sense of Bregman divergence [ 24 ]. Also, an interesting method of dual extrapolation for solving variational inequalities was proposed by Yu. Nesterov [ 25 ]. Adaptive variants of the Nemirovski mirror-prox method were studied in [ 19–23 ].

In the early 1980s, L. D. Popov proposed an interesting modification of the classical ArrowHurwitz algorithm for finding saddle points of convex-concave functions [ 26 ]. Let X and Y are closed convex subset of spaces Rd and R p , respectively, and L : X Y  R be a differentiable convex-concave function. Then, the algorithm [ 26 ] approximation of saddle points of L on X Y can be written as x0 , x1  X , y0 , y1 Y ,   0, xn  PX  xn  1L  xn1, yn1  ,    yn  PY  yn  2L  xn1, yn1  ,  xn1  PX  xn  1L  xn , yn  ,   yn1  PY  yn  2L  xn , yn  , where PX and PY are metric projection onto X and Y , respectively, 1L and 2L are partial derivatives. Under some suitable assumptions, L. D. Popov proved the convergence of this algorithm.

A modification of Popov's method for solving variational inequalities with monotone operators was studied in [ 27 ]. And in the article [ 27 ], a two-stage proximal algorithm for solving the equilibrium programming problem is proposed, which is an adaptation of the method [ 26 ] to the general Ky Fan inequalities. The mentioned equilibrium problem (Ky Fan inequality) has the form find x С : F  x, y  0  y  С , where С is nonempty subset of vector space H (usually Hilbert space), F : C C  R is function such that F  x, x  0  x  С (called bifunction). And the two-stage proximal algorithm is written like this  yn  proxnF yn1, xn ,  xn1  proxnF yn , xn , where n 0,  , prox is proximal operator for function  : C  R is defined by prox x  arg min yC   y  12 y  x 2  .

In [ 28, 29 ], the two-stage proximal mirror method was studied, which is a modification of the twostage proximal algorithm [ 27 ] using Bregman divergence instead of the Euclidean distance. Note that recently Popov's algorithm for variational inequalities has become well known among machine learning specialists under the name “Extrapolation from the Past” [ 9 ]. Further development of this circle of ideas led to the emergence of the so-called forward-reflected-backward algorithm [ 30 ] and related methods [ 31, 32 ]. The forward-reflected-backward algorithm generates a sequence  xn  iteratively defined by

xn1  PC  xn n Axn n1  Axn  Axn1  , with infn n ,supn n  0, 21L  , where L is the Lipschitz constant of A .

In this paper, we propose a novel algorithm for solving variational inequalities in a Banach space. Variational inequalities in Banach spaces arise and are intensively studied in mathematical physics and the theory of inverse problems [ 1, 2, 4 ]. Recently, there has been progress in the study of algorithms for problems in Banach spaces [ 4, 15–18 ]. This is due to the wide involvement of the results and constructions of the geometry of Banach spaces [ 33–35 ]. The proposed algorithm is an adaptive variant of the forward-reflected-backward algorithm [ 30 ], where the rule for updating the step size does not require knowledge of the Lipschitz constant of operator. Moreover, instead of the metric projection onto the feasible set, the Alber generalized projection is used [ 35 ]. An attractive feature of the algorithm is only one computation at the iterative step of the projection onto the feasible set. For variational inequalities with monotone Lipschitz operators acting in a 2-uniformly convex and uniformly smooth Banach space, a theorem on the weak convergence of the method is proved.

2. Preliminaries

We recall several concepts and facts of the geometry of Banach spaces that are necessary for the formulation and proof of the results [ 33–37 ].

Everywhere E denotes a real Banach space with the norm  , E dual to E space, x, x is value of functional x  E on element x  E . We denote norm in E as   .

Let SE  x  E : x  1 . Banach space E is strictly convex if for all x, y  SE and x  y we have for all  0, 2 . Obviously, a 2-uniformly convex space is uniformly convex. It is known that a uniformly convex Banach space is reflexive.

A Banach space E is called smooth if the limit The modulus of convexity of the space E is defined as follows

  E    inf 1  x  y 2

 : x, y  BE , x  y      0, 2 .

Banach space E is uniformly convex if  E    0 for all  0, 2 . Banach space E is called 2-uniformly convex if exists c  0 that x  y 2  E    c 2 lim t0 x  ty  x t (1) exists for all x, y  SE . A Banach space E is called uniformly smooth if the limit (1) exists uniformly in x, y  SE . There is a duality between the convexity and smoothness of the Banach space E and its dual E [ 33, 34 ]:   

E is strictly convex space  E is smooth space; E is smooth space  E is strictly convex space;

E is uniformly convex space 

E is uniformly smooth space;

 E is uniformly smooth space  E is uniformly convex space.

Note that if the space E is reflexive, the first two implications can be reversed. It is known that Hilbert spaces and spaces Lp (1  p  2 ) are 2-uniformly convex and uniformly smooth (spaces Lp are uniformly smooth for p 1,  ) [ 33, 34 ].

 Multivalued operator J : E  2E , acting as follows

Jx  x  E : x, x  x 2  x 2 ,  is called the normalized duality mapping. It is known that [ 36 ]:  if the space E is smooth, then the mapping J is single valued;  if the space E is strictly convex, then the mapping J is injective and strictly monotone;  if the space E is reflexive, then the mapping J is surjective;  if the space E is uniformly smooth, then the mapping J is uniformly continuous on bounded subsets of E .

Let E be a smooth Banach space. Consider the functional introduced by Yakov Alber [ 35 ]   x, y   x 2  2 Jy, x  y 2 x, y  E .

A useful identity follows from the definition of  :   x, y   x, z    z, y  2 Jz  Jy, x  z x, y, z  E .

If the space E is strictly convex, then for x, y  E we have   x, y  0  x  y .

Lemma 1 ([ 35 ]). Let E be a uniformly convex and uniformly smooth Banach space,  xn  and  yn  are bounded sequences of E elements. Then xn  yn  0 

Jxn  Jyn   0    xn , yn   0 .

Lemma 2 ([ 37 ]). Let E be a 2-uniformly convex and smooth Banach space. Then, for some number   1, the inequality holds   x, y   1 x  y 2 x, y  E .



Let K be a non-empty closed and convex subset of a reflexive, strictly convex and smooth space E . It is known [ 35 ] that for each x  E there is a unique point z  K such that   z, x  inf   y, x .

yK

This point z is denoted by K x , and the corresponding operator K : E  K is called the generalized projection of E onto K (Alber generalized projection) [ 35 ]. Note that if E is a Hilbert space, then K coincides with the metric projection onto the set K .

Lemma 3 ([ 35 ]). Let K be a closed and convex subset of a reflexive, strictly convex and smooth space E , x  E , z  K . Then z  K x 

Jz  Jx, y  z  0 y  K .

(2) Remark 1. The inequality (2) is equivalent to the following [ 35 ]:   y, K x  K x, x   y, x y  K .

Basic information about monotone operators and variational inequalities in Banach spaces can be found in [ 1, 2, 4, 35, 36 ]. We mention only two interesting examples of monotone operators acting in a Banach space [ 4 ].

For p  2 , define the operator A by

Au  u  x p2 u  x   dy .

R3 x  y 2 The operator A is potential and monotone, and acts from Lp  R3  to Lq  R3  , where p1  q1  1. Note that A is the gradient of the functional u  y  p dxdy .

Let G  Rn be a bounded domain. Differential expression

Au   xi  ai  x, xui p1  xui p2 xui  a0  x, u p1  u p2 u , p  1, n i1 where the function ai  x, s , i  0,1,..., n , is measurable as a function on x for every s 0,  and continuous for almost all x G as a function on s , ai  x, s   M for all s 0,  and for almost all x G , specifies a monotone operator acting from Sobolev space W01, p G to W01, p G  .

3. Algorithm

(3) (4) (5)

Let E be 2-uniformly convex and uniformly smooth Banach space, C be non-empty subset of space E , A be an operator from E to E . Consider variational inequality: find x C :

Ax, y  x  0 y  C .

We denote set of solutions of (3) by S .

Assume that the following conditions are satisfied:  set C  E is convex and closed;  operator A : E  E* is monotone and Lipschitz -type with L  0 on C ;  set S is non-empty.

Remark 2. We can formulate (3) as fixed-point problem [ 35 ]: x   J 1  Jx  Ax ,

C where   0 . Formulation (4) is useful because it contains an obvious algorithmic idea.

Consider dual variational inequality:

find x C :

Ay, x  y  0 y  C .

We denote set of solutions of (5) by S d . Note that set S d is closed and convex [ 2 ]. Inequality (5) is sometimes called weak or dual formulation of (3) (or Minty inequality) and solutions (5) are weak solutions (3). For monotone operators A we always have S  S d . In our conditions S d  S [ 2 ].

We assume that the following is satisfied:  normalized duality mapping J : E  E sequentially weakly continuous, i.e., from xn  x weak in E then Jxn  Jx weak* in E .

Remark 3. In our situation, when the space E (and of course E ) is reflexive, the weak* and weak convergence coincide in E .

Consider now a novel algorithm for solving the variational inequality (3). We will use a simple rule for updating the parameters n without information about the Lipschitz constant of the operator A . The proposed algorithm is a modification of the forward-reflected-backward algorithm recently proposed in [ 30 ] for solving operator inclusions with the sum of the maximal monotone and Lipschitz continuous monotone operators acting in a Hilbert space.

Let us know the constant   1 from the Lemma 2. min1, L1 . Then exists lnimn  0 .

The sequence  xn  generated by Algorithm 1 satisfies the inequality 2 n Axn  n1  Axn  Axn1 , y  xn1    y, xn    xn1, xn    y, xn1  y  С . (6)

Inequality (6) shows a rule of finishing the algorithm. Indeed if then from (6) it follows then for all y С , i.e., xn  S .

Now we go to the proof of convergence of Algorithm 1.

4. Main inequality

xn1  xn  xn1

Axn, y  xn  0

In this section, we state and prove the inequality on which the proof of Algorithm 1 weak convergence is based.

Lemma 4. For the sequence  xn  generated by Algorithm 1, the following inequality holds    z, xn1   2n Axn  Axn1, xn1  z  n   xn1, xn  

n1   z, xn   2n1 Axn1  Axn , xn  z  n1   xn , xn1   n where z S .

Proof. Let z S . We have 1 n1  n   xn1, xn  ,   n n1    z, xn1    z, xn    xn1, xn   2 n Axn  n1  Axn  Axn1 , z  xn1 . (7) From monotonicity of operator A we have n Axn  n1  Axn  Axn1  , z  xn1  n

Axn  Axn1, z  xn1  n1

Axn  Axn1, z  xn1  n

Axn1, z  xn1   n

Axn  Axn1, z  xn1  n1

Axn  Axn1, z  xn  0 n1

Axn  Axn1, xn  xn1 . (8)

Applying (8) to (7) we obtain

  z, xn1    z, xn    xn1, xn   2n

Axn  Axn1, z  xn1  2n1

Axn  Axn1, z  xn  2n1

Axn  Axn1, xn  xn1 .

(9) From rule of calculation n we have upper estimation for 2n1 Axn  Axn1, xn  xn1 in (9). We have 2n1

Axn  Axn1, xn  xn1   2n1 Axn  Axn1 * xn  xn1  2 nn1 xn  xn1 xn1  xn   nn1 xn  xn1 2  nn1 xn  xn1 2 

We obtain

  z, xn1   2n

Axn  Axn1, xn1  z   nn1   xn , xn1   nn1   xn1, xn  . 

n   xn1, xn   n1    z, xn   2n1

Axn1  Axn , xn  z  nn1   xn , xn1    1 n1   n

n   xn1, xn  .  n1 

The proof is complete. ■ Remark 4. We can change rule of updating for step 3 of Algorithm 1 to the following:

  min n , n1     n ,    xn1, xn  

 , if Axn1  Axn , Axn1  Axn *  otherwise.

Lemma 4 holds also for variant of Algorithm 1 with the rule (10).

5. Convergence

Let us formulate the main result. (10)

Theorem 1. Let C be a non-empty convex and closed subset of 2-uniformly convex and uniformly smooth Banach space E , A : E  E is monotone Lipschitz continuous operator, S   . Assume that normalized duality mapping J is sequentially weakly continuous. Then sequence generated by Algorithm 1  xn  converge weakly to z  S .

Proof. Let z S . Assume

an    z, xn   2n1 Axn1  Axn , xn  z  n1   xn , xn1  , n bn  1 n1  n   xn1, xn  .

  n n1 

Inequality from Lemma 4 takes form

Since there exists lnim n  0 , then 1 n1  n

n n1 Show that an  0 for all large n  N . We have an1  an  bn .

 1 2  0,1 , n  . an    z, xn   2n1 Axn1  Axn , xn  z  n1   xn , xn1   n 1  

xn  z 2  2n1 Axn1  Axn * xn  z  nn1 xn1  xn 2   1 xn  z 2  2 nn1 xn  xn1 xn  z  nn1 xn1  xn 2   1  nn1  xn  z 2 . Since there exists such n0  N that then an  0 starting from n0 .

So, we came into conclusion that there exists a limit 1  n1  0 for all n  n0 ,  n and

Since

Hence, we obtain that the sequence  xn  is bounded and   lim   z, xn   2n1 Axn1  Axn , xn  z  n1   xn , xn1   n  n  1 n1   n1  n

n   xn1, xn    .

n1  lim  xn1, xn   lim xn1  xn  0 .

n n   lim  2n1 Axn1  Axn , xn  z  n1   xn , xn1    0 , n  n  then sequences   z, xn  converge for all z S .

Hence,

From other side

Hence using monotonicity of operator A we have an inequality

Ay, y  xn  Axn , xn  xn1  Axn , y  xn1   1 n

Jxn  Jxn1, y  xn1  n1 Axn  Axn1, y  xn1 n y  С .

From lim x  xn1  0 and Lipschitz property of operator A it follows n n From uniform continuity of normalized duality mapping J on bounded sets we get lim Axn  Axn1 *  0 . n lim Jxn  Jxn1 *  0 .

n lim Ay, y  xn  0 n

y  С .

Ay, y  z  lkim Ay, y  xnk  lnim Ay, y  xn  0 y  С .

Then it follows that z  S .

Show that sequence  xn  converges weakly to z . Arguing by contradiction. Let exists the subsequence  xmk  such that xmk  z weakly and z  z . Easy to see that zS . We have 2 Jxn , z  z    z, xn    z, xn   z 2  z 2 .

From that we see the existence of limit lim Jxn , z  z . From sequentially weak continuity of n normalized duality mapping J we get

Jz, z  z  lim Jxnk , z  z  lim Jxmk , z  z  Jz, z  z ,

k k i.e., Jz  Jz, z  z  0 . Then it follows that z  z . The proof is complete. ■

The weak convergence of the variant of the algorithm with a constant parameter   0 is similarly proved. __________________________________________________________________________________ Algorithm 2.

 1  Initialization. Choose x0  E , x1  E ,   0,  . Let n  1.

 2 L 

Calculate

xn1   J 1  Jxn  2 Axn  Axn1  .

C 2. If xn1  xn  xn1, then STOP and xn  S , else let n : n 1 and go to 1. __________________________________________________________________________________

Remark 5. A special case of Algorithm 2 is the optimistic gradient descent ascent (OGDA) algorithm, popular among machine learning specialists [ 8, 9 ].

Lemma 5. For the sequence  xn  generated by Algorithm 2, the following inequality holds   z, xn1   2 Axn  Axn1, xn1  z  L  xn1, xn  

  z, xn   2 Axn1  Axn , xn  z  L  xn , xn1   1 2 L  xn1, xn  , where z  S .

Theorem 2. Let C be a nonempty convex and closed subset of 2-uniformly convex and uniformly smooth Banach space E , operator A : E  E is monotone and Lipschitz continuous with constant L  0 . Let S   and normalized duality mapping J is sequentially weakly continuous. Then sequence generated by Algorithm 2  xn  converge weakly to z  S .

6. Conclusions

In this paper, we have proposed and studied a new algorithm for solving variational inequalities in a Banach space. The proposed algorithm is an adaptive variant of the forward-reflected-backward algorithm [ 30 ], where the rule for updating the step size does not require knowledge of the Lipschitz continuous operator constant. Moreover, instead of the metric projection onto the admissible set, the Alber generalized projection is used [ 35 ]. An attractive feature of the algorithm is only one computation at the iterative step of the generalized projection onto the feasible set. For variational inequalities with monotone Lipschitz continuous operators acting in a 2-uniformly convex and uniformly smooth Banach space, a theorem on the weak convergence of the method is proved.

Based on the technique [ 38 ], similar results can most likely be obtained for problems with pseudomonotone, Lipschitz continuous, and sequentially weakly continuous operators acting in a uniformly convex and uniformly smooth Banach space. Also, in a future article we will present a proof of the convergence of a modification of the algorithm using the Bregman projection.

Note that a problem of significant interest in nonlinear analysis applications is to find x   A1  A2 1 0 , where A1 : E  2E is a maximal monotone operator and A2 : E  E is a monotone and Lipschitz operator. Based on the results of this work and [ 30 ] for solving this problem, we can construct the following adaptive splitting method xn1   J  n A1 1  Jxn  n A2 xn  n1  A2 xn  A2 xn1  ,   min n , n1     n , xn1  xn  , if A2 xn1  A x ,

2 n A2 xn1  A x 2 n *  otherwise, where   0, 21  . The proof of its convergence will be presented in another work soon.

Acknowledgements

This work was supported by Ministry of Education and Science of Ukraine (project “Mathematical modeling and optimization of dynamical systems for defense, medicine and ecology”, 0119U100337).

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