Convergence of Adaptive Methods for Equilibrium Problems in Hadamard Spaces Vladimir Semenov, Yana Vedel Taras Shevchenko National University of Kyiv, 64/13 Volodymyrska Street, Kyiv, 01161, Ukraine Abstract In this paper we consider equilibrium problems in metric Hadamard spaces. We propose and study new adaptive algorithms for their approximate solution. For pseudomonotone bifunctions of Lipschitz type, theorems on the weak convergence of sequences generated by the algorithms are proved. The proofs are based on the use of Fejer properties of algorithms with respect to the set of solutions to the problem. A new regularized adaptive extraproximal algorithm is also proposed and studied. To regularize the basic extraproximal scheme, the classical Halpern scheme was used. The proposed algorithms are applicable to pseudomonotone variational inequalities in Hilbert spaces. Keywords 1 Equilibrium problems, Hadamard space, pseudomonotonicity, adaptability, regularization, convergence, extraproximal algorithm 1. Introduction A popular direction of modern nonlinear analysis is the study of equilibrium problems (Ky Fan inequalities) of the form [1-9]: find x  С : F  x, y   0  y  С , (1) 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). We can formulate mathematical programming problems, variational inequalities, and many game theory problems in form (1). The study of algorithms for solving equilibrium and related problems is actively continuing [5-8, 10-30]. In this article, we will focus only on methods of the extraproximal type. The following analogue of G. Korpelevich extragradient method [15] for equilibrium problems [16] is called extraproximal  yn  prox n F  xn , xn ,    xn 1  prox n F  yn , xn ,  where n   0,   , prox is proximal operator for function  . In [19] two step proximal method for solving equilibrium problems in Hilbert space was proposed  yn  prox n F  yn1 , xn ,    xn 1  prox n F  yn , xn ,  where n   0,   , which is adaptation for L. D. Popov method [20] for general equilibrium programming problems (see also [21, 22]). Note that a version of this algorithm for variational inequalities became known among machine learning specialists under the name “Extrapolation from the Past” [31]. IT&I-2020 Information Technology and Interactions, December 02–03, 2020, KNU Taras Shevchenko, Kyiv, Ukraine EMAIL: semenov.volodya@gmail.com (V. Semenov); yana.vedel@gmail.com (Y. Vedel) ORCID: 0000-0002-3280-8245 (V. Semenov); 0000-0002-7160-9994 (Y. Vedel) ©️ 2020 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) 321 Recently, there has been an increased interest in the construction of theory and algorithms for solving mathematical programming problems in metric Hadamard spaces [32] (also known as CAT  0 spaces). A strong motivation for studying these problems is the ability to rewrite some nonconvex problems in the form of convex (more precisely, geodesically convex) in a space with a specially selected metric structure [32, 33]. Some authors began to study equilibrium problems in Hadamard spaces [33-37]. For example, in [35], concluding from the results of [16], the authors proposed and substantiated an analogue of the extraproximal method for pseudomonotone equilibrium problems in Hadamard spaces. In this paper, which continues and refines articles [36, 37], two new adaptive two-stage proximal algorithms for the approximate solution of equilibrium problems in Hadamard spaces are described and studied. The proposed rules for choosing the step size do not calculate the values of the bifunction at additional points and do not require knowledge of the Lipschitz constants of the bifunction. For pseudo-monotone bifunctions of Lipschitz type, theorems on the weak convergence of sequences generated by the algorithms are proved. The proofs are based on the use of Fejer properties of algorithms with respect to the set of solutions to the problem. A new regularized adaptive extraproximal algorithm is also proposed and studied. To regularize the basic adaptive extraproximal scheme [37], the classical Halpern scheme [38] was used, a version of which for Hadamard spaces was studied in [32]. It is shown that the proposed algorithms are applicable to pseudomonotone variational inequalities in Hilbert spaces. 2. Preliminaries Here are some concepts and facts related to metric Hadamard spaces. Details can be found in [32, 39, 40]. Let  X , d  be a metric space and x , y  X . Geodesic path connecting points x and y is    isometry  : 0, d  x, y    X such that   0   x ,  d  x, y   y . Set  0, d  x, y    X is  denoted by  x, y  and called geodesic segment with ends x and y (or simply geodesic). Metric space  X , d  is called geodesic space if it is possible to connect any two points of X by geodesic and it is unique geodesic space if for any two points from X there exists exactly one geodesic to connect them. Geodesic space  X , d  is called CAT  0 space if for any three points y0 , y1 , y2  X such that d 2  y1 , y0   d 2  y2 , y0   12 d 2  y1 , y2  the following inequality holds: 1 2 1 1 d 2  x, y0   d  x, y1   d 2  x, y2   d 2  y1 , y2  x  X . 2 2 4 It is known that CAT  0 space is unique geodesic [32]. For two points x and y from CAT  0 space  X , d  and t  0,1 we denote by tx  1  t  y unique point z of  x, y  such that d  z, x   1  t  d  x, y  and d  z, y   td  x, y  . Set C  X is called convex if for all x , y  C and t  0,1 holds tx  1  t  y  C . The following inequality is useful property for CAT  0 space  X , d  d 2  tx  1  t  y, z   td 2  x, z   1  t  d 2  y, z   t 1  t  d 2  x, y  , x, y, z  X , t 0,1 . (2) Important examples of CAT  0 spaces are Euclidean spaces, R -trees, Hadamard manifolds (complete connected Riemannian manifolds of non-positive curvature) and Hilbert sphere with hyperbolic metric [32, 39, 40]. Complete CAT  0 space is called Hadamard space. 322 As in a Hilbert space, the operator of metric projection onto a closed convex set is well defined in Hadamard spaces C [32]. For each x  X there exists unique element PC x from set C with the property d  PC x, x   min d  z, x  , moreover the characterization takes place [32]: zC y  PC x  y  C and d 2  y, z   d 2  x, z   d 2  y, x  z  C . Let  X , d  be a metric space and  xn  be a bounded sequence of elements from X . Let r  x,  xn    lim d  x, xn  . The number r   xn    inf xX r  x,  xn   is called asymptotical radius n   xn  and set A   xn    x  X : r  x,  xn    r   xn   is asymptotic center  xn  . It is known that in Hadamard space A   xn   it consists of one point [32]. Sequence  xn  of elements from Hadamard space  X , d  converges weakly to an element x  X if A   xn     x for any k   sequence xnk . It is known that any sequence of elements from closed convex bounded subset K of Hadamard space has subsequence which converges weakly to element from K [32, 39]. The well- known analogue of Opial lemma is useful in proving the weak convergence of sequences of elements of the Hadamard space. Lemma 1 ([32, p. 60]). Let sequence  xn  of elements from Hadamard space  X , d  converges weakly to an element x  X . Then for all y  X \ x we have lim d  xn , x   lim d  xn , y  . n  n  Let  X , d  be an Hadamard space. Function  : X  R  R   is called convex if for all points x , y  X and t  0,1 holds  tx  1  t  y   t  x   1  t    y  . For example, in Hadamard space functions y d  y, x  are convex [32]. If there exists   0 such that for all x , y  X and t  0,1 the following inequality is satisfied  tx  1  t  y   t  x   1  t   y   t 1  t  d 2  x, y  , then function  is called strongly convex. It is known that for convex functions lower semicontinuity and weakly lower semicontinuity are equivalent [32, p. 64] and strongly convex semicontinuous function reaches its minimum at unique point. For convex proper and lower semicontinuous function  : X  R  R   proximal operator is defined by [32] prox x  arg min yX   y   12 d 2  y, x   . Since functions   12 d 2 , x  are strongly convex the definition of proximal operator is correct, i.e. for all x  X there exists unique element prox  x  X . 3. Equilibrium problems in Hadamard space Let  X , d  be a Hadamard space. Consider an equilibrium problem for nonempty closed convex set С  X and bifunction F : C  C  R [34-37]: find x  С : F  x, y   0  y  С . (3) Assume that following conditions are satisfied: 1. F  x, x   0 for all x  С ; 323 2. functions F  x,  : C  R are convex and lower semicontinuous for all x  C ; 3. functions F , y  : C  R are upper weakly semicontinuous for all y  C ; 4. bifunction F : C  C  R is pseudomonotone, i.e. for all x , y  C from F  x, y   0 it follows that F  y, x   0 . 5. bifunction F : C  C  R is Lipschitz type, i.e. there exist a  0 , b  0 , such that F  x, y   F  x, z   F  z, y   ad 2  x, z   bd 2  z, y   x, y, z  C . (4) Remark 1. If F  x, y    Ax, y  x  , where A : C  H , С is nonempty subset of Hilbert space H , then problem (3) takes form of variational inequality find x  С :  Ax, y  x   0  y  С . (5) If set C  H is convex and closed and operator A : C  H pseudomonotone, Lipschitz continuous and sequential weakly semicontinuous, then for (5) conditions 3–5 are satisfied. Consider dual equilibrium problem: find x  С : F  y, x   0  y  С . (6) * We denote sets of solutions for problems (3) and (6) by S and S . If conditions 1–4 are satisfied we have S  S [34]. Moreover, set S is closed and convex. * * Further we assume that S   . 4. Adaptive algorithms For approximate solution of (3) we consider extraproximal algorithm with adaptive choice of step size [37]. Algorithm 1. Initialization. Choose element x1  С ,    0,1 , 1   0,   . Set n  1 . Step 1. Calculate   yn  prox n F  xn , xn  arg min yC F  xn , y   21n d 2  y, xn  . If xn  yn , then stop and xn  S . Otherwise, go to step 2. Step 2. Calculate   xn1  prox n F  yn , xn  arg min yC F  yn , y   21n d 2  y, xn  . Step 3. Calculate n , if F  xn , xn 1   F  xn , yn   F  yn , xn 1   0,  n 1     d 2  xn , yn   d 2  xn 1 , yn    min   ,  , otherwise.  2  F  xn , xn 1   F  xn , yn   F  yn , xn 1    n  Set n : n  1 and go to step 1. Remark 2. On each step of algorithm 1 we need to solve two convex problems with strongly convex functions. In proposed algorithm parameter n 1 depends on location of points xn , yn , xn 1 , values F  xn , xn1  , F  xn , yn  and F  yn , xn1  . No information about constants a and b from inequality 324 (4) is used. Obviously, the sequence  n  is non-decreasing. Also, it is lower bounded by    min 1 ,  . Indeed, we have  2 max a, b  F  xn , xn1   F  xn , yn   F  yn , xn1   max a, b  d 2  xn , yn   d 2  xn 1 , yn   . Let us prove the important inequality. Lemma 2. For x  С and x   prox  F  x , x , where   0 , the following inequality takes place F  x, x    F  x, y   1 2  d 2  y , x   d 2  x, x    d 2  x  , y   y  С . (7)  Proof. From the definition x  arg min yC F  x, y   21 d  y, x  it follows that  2  F  x, x   d  x , x   F  x, p   1 2  1 2 d  p, x  p  С . (8) 2 2 Setting in (8) p  tx   1  t  y , y  С , t   0,1 , we obtain F  x, x    d  x , x   F  x, tx   1  t  y   d  tx  1  t  y, x   1 2  1 2  2 2  tF  x, x    1  t  F  x, y   1 2   td 2  x  , x   1  t  d 2  y, x   t 1  t  d 2  x  , y  . Thereby, 1  t  F  x, x    1  t  F  x, y   1 2     1  t  d 2  x  , x   1  t  d 2  y, x   t 1  t  d 2  x  , y  . (9) Dividing in (9) by 1  t and passing to the limit as t  1 we obtain (7). ■ From Lemma 2 it follows that for sequences  xn  ,  yn  , generated by Algorithm 1 the following inequalities hold F  xn , yn   F  xn , y   1 2n  d 2  y, xn   d 2  xn , yn   d 2  yn , y   y  С . (10) F  yn , xn1   F  yn , y   1 2n  d 2  y, xn   d 2  xn , xn1   d 2  xn1 , y   y  С . (11) Inequality (10) provides a justification for the stopping rule for Algorithm 1. Indeed, for xn  yn from (10) it follows F  xn , y   0 y  С , i.e., xn  S . Let us prove an important estimate relating the distances between the points generated by Algorithm 1 to an arbitrary element of the set of solutions S . Lemma 3. For sequences  xn  ,  yn  , generated by Algorithm 1, the following inequality takes place 325       d 2  xn1 , z   d 2  xn , z   1   n  d 2  xn1 , yn   1   n  d 2  yn , xn  , (12)  n1   n1  where z  S . Proof. Let z  S . From pseudomonotonicity of bifunction F it follows that F  yn , z   0 . (13) From (13) and (11) 2n F  yn , xn1   d 2  z, xn   d 2  xn , xn1   d 2  xn1 , z  . (14) From the calculation rule for n 1 we conclude  F  xn , xn 1   F  xn , yn   F  yn , xn 1   2n 1  d 2  xn , yn   d 2  xn 1 , yn   . (15) Evaluating the left side of (14) from below using (15), we get n 2n  F  xn , xn 1   F  xn , yn     n 1  d 2  xn , yn   d 2  xn 1 , yn     d 2  z, xn   d 2  xn , xn1   d 2  xn1, z  . (16)   For a lower bound 2n F  xn , xn1   F  xn , yn  in (16) we use (10). We have n d 2  xn , yn   d 2  yn , xn 1   d 2  xn 1 , xn    n 1  d 2  xn , yn   d 2  xn1 , yn     d 2  z, xn   d 2  xn , xn1   d 2  xn1, z  . (17) By regrouping (17), we get (12). ■ To prove the convergence of Algorithm 1, we need an elementary lemma about number sequences. Lemma 4. Let  an  ,  bn  be two sequences of non-negative numbers which satisfy an1  an  bn for all n  N . Then exists a limit lim an and  bn   l1 . n  Let us formulate one of the main results of the work. Theorem 1. Let  X , d  be an Hadamard space, C  X be a non-empty convex closed set, for bifuntion F : C  C  R conditions 1–5 are satisfied and S   . Then sequences  xn  ,  yn  generated by Algorithm 1 converge weakly to the solution z  S of equilibrium problem (3), moreover, lim d  yn , xn   lim d  yn , xn1   0 . n n Proof. Let z  S . Assume       an  d  z, xn  , bn  1   n  d 2  xn1 , yn   1   n  d 2  yn , xn  .  n1   n1  Inequality (12) takes form an1  an  bn . Since there exists lim n  0 , n  n 1  1     0,1 , n   . n 1 From Lemma 4 we conclude that exists a limit lim d 2  z, xn  and n     d  x , y   d  y , x     . n 1 2 n 1 n 2 n n Whence we get boundedness of the sequence  xn  and 326 lim d  yn , xn   lim d  xn1 , yn   lim d  xn1 , xn   0 . (18) n n n Consider subsequence  x  which converges weakly to the point z  C . Then from (18) it nk   converges weakly to z . Let us show that z  S . We have follows that ynk F  y , y  F  y , x   nk 1 2  d  y, x   d  x , x   d  x nk nk 1 nk 2 nk 2 nk nk 1 2 nk 1 ,y      F xnk , xnk 1  F xnk , ynk      2n 1  d  x , y   d  x , y   21  d  y, x   d  x , x   d  x 2 nk nk 2 nk 1 nk nk 2 nk 2 nk nk 1 2 nk 1  ,y  k   1 2  d  x , x   d  x , y   d  y , x   nk 2 nk 1 nk 2 2  d  x , y   d  x , y   nk nk 2 nk nk 1 nk 1 2 nk nk 2 nk 1 nk  1 2  d  y, x   d  x , x   d  x , y  y  С . nk 2 nk 2 nk nk 1 2 nk 1 (19) Passing to the limit in (19) taking into account (18) and weakly upper semicontinuity of function F , y  : C  R , we get F  z, y   lim F ynk , y  0 y  С , i.e., z  S . k    Applying Opial lemma for Hadamard spaces (Lemma 1) we obtain the convergence of sequence  xn  to the point z  S . Indeed, we argue by contradiction. Let exists the subsequence  xm  , which k converges weakly to some point z  C and z  z . It is clear that z  S . We have lim d  xn , z   lim d  xnk , z   lim d  xnk , z   lim d  xn , z   lim d  xmk , z   n  k  k  n  k   lim d  xmk , z   lim d  xn , z  , k  n  which is impossible. Therefore  xn  converges weakly to z  S . From (18) it follows that sequence  yn  also converges to z  S . ■ Remark 3. We see from proof for Theorem 1 that for sequence  xn  starting from some number N Fejer condition is satisfied with respect to the set of solutions S . In recent paper [36] for solution of problem (3) the following algorithm was proposed  yn  prox  F y , xn  arg min yC F  yn1 , y   21 d 2  y, xn  ,  n  n1  n    (20)  x  prox n F  yn , xn  arg min yC F  yn , y   2n d 2  y, xn  ,  n1 1   where values n  0 were set according to the requirement inf n n ,supn n   0, 2 2 a1b . I.e. the   information about constants from condition (4) was used. Based on the scheme (20) and works [28, 29, 37], we construct a two-stage proximal algorithm with adaptive choice of the value n . Algorithm 2. Initialization. Choose element x1 , y0  C ,    0, 13  , 1   0,   . Set n  1 . Step 1. Calculate yn  prox n F  yn1 , xn  arg min yC F  yn 1 , y   21n d 2  y, xn  .   Step 2. Calculate xn1  prox n F  yn , xn  arg min yC F  yn , y   21n d 2  y, xn  .   If xn 1  xn  yn , then stop and xn  S . Otherwise, go to step 3. 327 Step 3. Calculate n , if F  yn 1 , xn 1   F  yn 1 , yn   F  yn , xn 1   0,  n 1     d 2  yn 1 , yn   d 2  xn 1 , yn    min  n ,  , otherwise.   2  F  y n 1 , xn 1   F  y n 1 , y n   F  y n , xn 1    Set n : n  1 and go to the step 1. Let us present the main results on the convergence of the Algorithm 2. Lemma 5. For sequences  xn  ,  yn  , generated by Algorithm 2 the following inequality takes place    d 2  xn1 , z   d 2  xn , z   1   n  d 2  xn1 , yn    n1       1  2 n  d 2  yn , xn   2 n d 2  xn , yn1  ,  n1  n1 where z  S . Theorem 2. Let  X , d  be a Hadamard space, C  X be a nonempty convex closed set, for bifunction F : C  C  R conditions 1–5 are satisfied and S   . Then sequences  xn  ,  yn  generated by Algorithm 2 converge weakly to the solution z  S of problem (3). 5. Regularized adaptive algorithms To ensure the convergence of the approximating sequences in the metric of space to the solution of the equilibrium problem (3), we consider the extraproximal Algorithm 1, regularized using the well-known Halpern scheme [32, 38], with adaptive choice of the step size. Algorithm 3. Initialization. Choose elements a  C , x1  С , numbers    0,1 , 1   0,   , and sequence  n  , such that  n   0,1 , lim  n  0 ,     . Set n  1 .  n  n 1 n  Step 1. Calculate yn  prox n F  xn , xn  arg min yC F  xn , y   21n d 2  y, xn  .   Step 2. Calculate zn  prox n F  yn , xn  arg min yC F  yn , y   21n d 2  y, xn  .  Step 3. Calculate xn1   n a  1   n  zn . Step 4. Calculate n , if F  xn , zn   F  xn , yn   F  yn , zn   0,  n 1     d 2  xn , yn   d 2  zn , yn    min   ,  , otherwise.  2  F  xn , zn   F  xn , yn   F  yn , zn    n  Set n : n  1 and go to step 1. 328 The following known facts have an important role in proving the convergence of Algorithm 3. Lemma 6 ([41]). Let sequence of numbers  an  has subsequence ank   with property a  a nk nk 1 for all k  N . Then exists non-decreasing sequence  mk  of natural numbers such that mk   and amk  amk 1 , ak  amk 1 for all k  n1 . Lemma 7. Let  an  be a sequence of non-negative numbers satisfying the inequality an1  1  n  an  n n for all n  N , where sequences  n  and  n  have properties:  n   0,1 ,    , lim   0 . Then n n  n lim an  0 . n  First, takes place Lemma 8. For sequences  xn  ,  yn  and  zn  generated by Algorithm 3 inequality holds d 2  xn1 , z   1  n  d 2  xn , z          1   n  1   n  d 2  zn , yn   1   n  1   n  d 2  yn , xn    n1   n1   n d 2  a, z   n 1  n  d 2  a, zn  , (21) where z  S . Proof. Let z  S . From xn1   n a  1   n  zn and inequality for strong convexity (2) the estimation follows d 2  xn1 , z   n d 2  a, z   1  n  d 2  zn , z   n 1  n  d 2  a, zn  . For upper estimation d 2  zn , z  we use Lemma 3 and get (21). ■ Lemma 9. Sequences  xn  ,  yn  and  zn  generated by Algorithm 3 are bounded. Proof. Let z  S . We have d  xn1 , z   d n a  1  n  zn , z   n d  a, z   1  n  d  zn , z  . Since exists lim n  0 , then n  n 1  1     0,1 , n   . n 1 Using inequality from Lemma 3, we obtain d  xn1 , z   n d  a, z   1  n  d  xn , z   max d  a, z  , d  xn , z   for all n  n0 . Hence d  xn1 , z   max d  a, z  , d xn0 , z   for all n  n0 . Thereby sequence  xn  is bounded. So from Lemma 3 we conclude that  yn  and  zn  are bounded. 329 Theorem 3. Let  X , d  be a Hadamard space, C  X be a nonempty convex closed set, for bifunction F : C  C  R conditions 1–5 are satisfied and S   . Then sequences  xn  ,  yn  and  zn  generated by Algorithm 3 converge to the element PS a . Proof. Consider element z0  PS a . From Lemma 9 it follows that exists number 𝑀 > 0 such that d 2  a, z0   1   n  d 2  a, zn   M for all n . Then from inequality of Lemma 8 we obtain the estimation    d 2  xn1 , z0   1   n  d 2  xn , z0   1   n  1   n  d 2  zn , yn    n1      1   n  1   n  d 2  yn , xn    n M . (22)  n1    Consider sequence d  xn , z0  . There are two options: a) there exists a number n  N such that d  xn1, z0   d  xn , z0  for all n  n ; b) there exists increasing sequence of numbers ( nk ) such that    d xnk 1 , z0  d xnk , z0 for all k  N .  First, consider option a). In that case there exists lim d  xn , z0   R . Since n n d 2  xn1 , z0   d 2  xn , z0   0 , 𝛼𝑛 → 0 and 1    1     0,1 , 𝑛 → ∞, n 1 we have d  xn , yn   0 , (23) d  zn , yn   0 . (24)  x  which converges weakly to the Since  xn  is bounded it follows that exists a subsequence nk point w  X . Then from (23), (24) it follows that  y  and  z  converge weakly to w . So nk nk w  C . Let us show that w  S . We have    F ynk , y  F ynk , znk   21  d  y, x   d  x , z   d  z , y   2 nk 2 nk nk 2 nk nk    F xnk , znk  F xnk , ynk      2n 1  d 2  xn , yn   d 2  zn , yn   k k 1 2n k k  d 2  y, xn   d 2  xn , zn   d 2  zn , y    k k k k  k k   2 1  nk d  z , x   d  x , y   d  y , z   2 nk nk 2 2  d  x , y   d  z , y   nk nk 2 nk nk nk 1 2 nk nk 2 nk nk  2  d  y, x   d  x , z   d  z , y  y  С . 1 nk 2 nk 2 nk nk 2 nk (25) Passing to the limit in (25) taking into account (23), (24) and weak upper semicontinuity of function F , y  : C  R , we get 330 F  z, y   lim F  ynk , y   0 y  С , k  i.e., z  S . Let us prove that lim  d 2  a, z0   1   n  d 2  a, zn    0 . (26) n  Consider subsequence z nk   such that k    lim d 2  a, z0   1   nk d 2 a, znk      lim  d  a, z   1    d  a, z   . n  2 0 n 2 n We can also assume that znk  w  S weakly. Then, using the weak lower semicontinuity of the function d 2  a,  , we obtain k    lim d 2  a, z0   1   nk d 2 a, znk      d  a, z   d  a, w  . 2 0 2 (27) Since z0  PS a  arg min wS d  a, w , then from (27) follows (26). Then from (26), inequality d 2  xn1 , z0   1   n  d 2  xn , z0    n  d 2  a, z0   1   n  d 2  a, zn   , which takes place for big n and Lemma 7 we conclude that d  xn , z0   0 . From (23), (24) we get d  yn , z0   0 and d  zn , z0   0 . Let us study option b). In that case consider sequence of numbers  mk  with properties (Lemma 6): i) mk   ; ii) d xmk 1 , z0  d xmk , z0    k  n ; iii) d  x 1 mk 1  , z0  d  xk , z0  k  n1 . From inequality of Lemma 8 and ii) it follows  m  2  m d 2  xm , z0   1   m  1    d  z m , ym   k k k k  m 1  k k  k  mk  2   1   mk 1         d ymk , xmk   mk d  a, z0    mk 1   mk d a, zmk   mk M . 2 2      mk 1  k      From where lim d xmk , ymk  lim d zmk , ymk  0 . Arguments similar to the above, show that the k  partial sequences weak limits  x  ,  y  and  z  belongs to set S . As before, we get mk mk mk   lim d 2  a, z0   1   mk d 2 a, zmk k      0 . For big numbers k we have    d 2 xmk 1 , z0  1   mk  d 2  xmk , z0    mk d 2  a, z0   1   mk  d 2  a, zmk      1   mk  d 2  xmk 1 , z0    mk d 2  a, z0   1   mk  d 2  a, zmk  .  331 Whence, taking into account iii), we obtain d 2  xk , z0   d 2  xmk 1 , z0   d 2  a, z0   1   mk  d 2  a, zmk  . Thereby  lim d 2  xk , z0   lim d 2  a, z0   1   mk  d 2  a, zmk   0 . k  k   So, lim d  xn , z0   0 and lim d  yn , z0   lim d  zn , z0   0 . ■ n n  n Using this technique and idea of work [36] we can construct regularized variant of Algorithm 2 with adaptive step. Algorithm 4. Initialization. Choose elements x1 , y0  C ,    0, 13  , 1   0,   and sequence  n  such that  n   0,1 , lim  n  0 ,     . Set n  1 .  n  n 1 n Step 1. Calculate zn  n a  1  n  xn .  Step 2. Calculate yn  prox n F  yn1 , zn  arg min yC F  yn 1 , y   21n d 2  y, zn  .   Step 3. Calculate xn 1  prox n F  yn , zn  arg min yC F  yn , y   21n d 2  y, zn  .  Step 4. Calculate n , if F  yn 1 , xn 1   F  yn 1 , yn   F  yn , xn 1   0,  n 1     d 2  yn 1 , yn   d 2  xn 1 , yn    min   n ,  , otherwise.    2  F  y n 1 , xn 1   F  y n 1 , y n   F  y n , xn 1    Set n : n  1 and go to step 1. Remark 4. Unfortunately, now we do not have a proof of the convergence of Algorithm 4 under the condition that the bifunction is pseudomonotone. 6. Modification of Algorithm 3 for variational inequalities Consider a particular case of the equilibrium problem: the variational inequality in the real Hilbert space H : find x  С :  Ax, y  x   0  y  С . (28) We assume that following conditions are satisfied  C  H is convex and closed;  operator A : C  H is pseudomonotone, Lipschitz continuous, and sequentially weakly continuous;  the set of solutions (28) is not empty. Let PC be a metric projection operator on convex closed set C , i.e. PC x is an unique element of set C with property PC x  x  min z  x . zC For variational inequalities (28) Algorithm 3 takes the following form. 332 Algorithm 5. Initialization. Choose elements a  C , x1  С , numbers    0,1 , 1   0,   and sequence  n  , such that  n   0,1 , lim  n  0 ,     . Set n  1 .  n  n 1 n Step 1. Calculate yn  PC  xn  n Axn  . Step 2. Calculate zn  PC  xn  n Ayn  . Step 3. Calculate xn1   n a  1   n  zn . Step 4. Calculate n , if  Axn  Ayn , zn  yn   0,  n 1     xn  yn 2  zn  yn 2  min n , 2 Ax  Ay , z  y  , otherwise.    n n n n   Set n : n  1 and go to step 1. From theorem 3 the following result follows. Theorem 4. Let H be a Hilbert space, C  X be an nonempty convex closed set, operator A : C  H pseudomonotone, Lipschitz continuous, sequentially weakly continuous and there are solutions (28). Then the sequences generated by Algorithm 5  xn  ,  yn  and  zn  strongly converge to projection of element a on the set of solutions (28). Remark 5. If operator A is monotone, then the result of Theorem 4 is valid without the assumption of the sequential weak continuity of the operator A . Similar results take place for modifications of algorithms 1, 2, and 4. 7. Conclusions In this paper, which continues and refines articles [36, 37], two new adaptive two-stage proximal algorithms for the approximate solution of equilibrium problems in Hadamard spaces are described and studied. The proposed rules for choosing the step size do not calculate the values of the bifunction at additional points and do not require knowledge of the Lipschitz constants of the bifunction. For pseudo-monotone bifunctions of Lipschitz type, theorems on the weak convergence of sequences generated by the algorithms are proved. A new regularized adaptive extraproximal algorithm is also proposed and studied. To regularize the basic adaptive extraproximal scheme [37], the classical Halpern scheme [38] was used, a version of which for Hadamard spaces was studied in [32]. It is shown that the proposed algorithms are applicable to pseudomonotone variational inequalities in Hilbert spaces. In the coming papers, we plan to consider more special versions of algorithms for variational inequalities and minimax problems on Hadamard manifolds (for example, on the manifold of symmetric positive definite matrices). The construction of randomized versions of algorithms is also of interest. 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”, 0219U008403). 333 References [1] G. Kassay, V. D. Radulescu, Equilibrium Problems and Applications, Academic Press, London, 2019. [2] C. Baiocchi, A. Capello, Variational and Quasi-Variational Inequalities. Applications to Free Boundary Problems, Wiley, New York, 1984. [3] D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, 1980. [4] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994) 123–145. [5] P. N. Anh, Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities, J. Optim. Theory Appl. 154 (2021) 303–320. [6] P. L. Combettes, S. A. Hirstoaga, Equilibrium Programming in Hilbert Spaces, J. Nonlinear Convex Anal. 6 (2005) 117–136. [7] A. S. Antipin, Equilibrium programming: Gradient methods, Autom. Remote Control 58 (1997) 1337–1347. [8] A. S. Antipin, Equilibrium programming: Proximal methods, Comput. Math. Math. Phys. 37 (1997) 1285–1296. doi:10.1134/S0965542507120044. [9] G. Mastroeni, On auxiliary principle for equilibrium problems, in: P. Daniele et al. (Eds.), Equilibrium Problems and Variational Models, Kluwer Academic Publishers, Dordrecht, 2003, pp. 289–298. doi:10.1007/978-1-4613-0239-1. [10] S. D. Flam, A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program. 78 (1997) 29–41. [11] A. N. Iusem, W. Sosa, On the proximal point method for equilibrium problems in Hilbert spaces, Optimization 59 (2010) 1259–1274. [12] A. Moudafi, Proximal point methods extended to equilibrium problems, J. Nat. Geom. 15 (1999) 91–100. [13] S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), 506–515. [14] N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006) 191–201. [15] G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Matecon. 12 (1976) 747–756. [16] T. D. Quoc, L. D. Muu, N. V. Hien, Extragradient algorithms extended to equilibrium problems, Optimization. 57 (2008) 749–776. doi:10.1080/02331930601122876. [17] P. T. Vuong, J. J. Strodiot, V. H. Nguyen, Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems, J. Optim. Theory Appl. 155 (2012) 605– 627. [18] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM Journal on Control and Optimization 38 (2000) 431–446. [19] S. I. Lyashko, V. V. Semenov, A New Two-Step Proximal Algorithm of Solving the Problem of Equilibrium Programming, in: B. Goldengorin (Ed.), Optimization and Its Applications in Control and Data Sciences, volume 115 of Springer Optimization and Its Applications, Springer, Cham, 2016, pp. 315–325. doi:10.1007/978-3-319-42056-1_10. [20] L. D. Popov, A modification of the Arrow-Hurwicz method for search of saddle points, Mathematical notes of the Academy of Sciences of the USSR. 28 (1980) 845–848. doi:10.1007/BF01141092. [21] L. Chabak, V. Semenov, Y. Vedel, A New Non-Euclidean Proximal Method for Equilibrium Problems, In: O. Chertov et al. (Eds.), Recent Developments in Data Science and Intelligent Analysis of Information, volume 836 of Advances in Intelligent Systems and Computing, Springer, Cham, 2019, pp. 50–58. doi:10.1007/978-3-319-97885-7_6. [22] D. A. Nomirovskii, B. V. Rublyov, V. V. Semenov, Convergence of Two-Stage Method with Bregman Divergence for Solving Variational Inequalities, Cybernetics and Systems Analysis. 55 (2019) 359–368. doi:10.1007/s10559-019-00142-7. 334 [23] A. Nemirovski, Prox-method with rate of convergence O(1/T) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems, SIAM J. Optim. 15 (2004) 229–251. doi:10.1137/S1052623403425629. [24] F. S. Stonyakin, E. A. Vorontsova, M. S. Alkousa, New Version of Mirror Prox for Variational Inequalities with Adaptation to Inexactness, in: Jacimovic M., Khachay M., Malkova V., Posypkin M. (Eds.), Optimization and Applications, OPTIMA 2019, volume 1145 of Communications in Computer and Information Science, Springer, Cham, 2020, 427–442. doi:10.1007/978-3-030-38603-0_31. [25] F. S. Stonyakin, On the adaptive proximal method for a class of variational inequalities and related problems, Trudy Inst. Mat. i Mekh. UrO RAN. 25 (2019) 185–197. doi:10.21538/0134- 4889-2019-25-2-185-197. [26] F. Bach, K. Y. Levy, A Universal Algorithm for Variational Inequalities Adaptive to Smoothness and Noise, arXiv preprint arXiv:1902.01637. (2019). [27] J. Diakonikolas, Halpern iteration for near-optimal and parameter-free monotone inclusion and strong solutions to variational inequalities, arXiv preprint arXiv:2002.08872. (2020). [28] S. V. Denisov, V. V. Semenov, P. I. Stetsyuk, Bregman Extragradient Method with Monotone Rule of Step Adjustment, Cybernetics and Systems Analysis. 55 (2019) 377–383. doi:10.1007/s10559-019-00144-5. [29] S. V. Denisov, D. A. Nomirovskii, B. V. Rublyov, V. V. Semenov, Convergence of Extragradient Algorithm with Monotone Step Size Strategy for Variational Inequalities and Operator Equations, Journal of Automation and Information Sciences. 51 (2019) 12–24. doi:10.1615/JAutomatInfScien.v51.i6.20. [30] S. I. Lyashko, D. A. Klyushin, D. A. Nomirovsky, V. V. Semenov, Identification of age- structured contamination sources in ground water, In: R. Boucekkline, N. Hritonenko, and Y. Yatsenko (Eds.), Optimal Control of Age-Structured Populations in Economy, Demography, and the Environment, Routledge, London–New York, 2013, pp. 227–292. [31] G. Gidel, H. Berard, P. Vincent, S. Lacoste-Julien, A Variational Inequality Perspective on Generative Adversarial Networks. arXiv preprint arXiv:1802.10551. (2018). [32] M. Bacak, Convex Analysis and Optimization in Hadamard Spaces, De Gruyter, Berlin, 2014. [33] V. Colao, G. Lopez, G. Marino, V. Martin-Marquez, Equilibrium problems in Hadamard manifolds, Journal of Mathematical Analysis and Applications. 388 (2012) 61–77. doi:10.1016/j.jmaa.2011.11.001. [34] H. Khatibzadeh, V. Mohebbi, Monotone and pseudo-monotone equilibrium problems in Hadamard spaces, J. of the Australian Math. Soc. (2019). doi:10.1017/S1446788719000041. [35] H. Khatibzadeh, V. Mohebbi, Approximating solutions of equilibrium problems in Hadamard spaces, Miskolc Mathematical Notes. 20 (2019) 281–297. doi:10.18514/MMN.2019.2361. [36] Y. I. Vedel, G. V. Sandrakov, V. V. Semenov, L. M. Chabak, Convergence of a Two-Stage Proximal Algorithm for the Equilibrium Problem in Hadamard Spaces, Cybernetics and Systems Analysis. 56 (2020) 784–792. doi:10.1007/s10559-020-00299-6. [37] Y. I. Vedel, E. N. Golubeva, V. V. Semenov, L. M. Chabak, Adaptive extraproximal algorithm for the equilibrium problem in Hadamard spaces, Journal of Automation and Information Sciences. 52 (2020) 46–58. doi:10.1615/JAutomatInfScien.v52.i8.40. [38] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967) 957–961. doi:10.1090/S0002-9904-1967-11864-0. [39] W. Kirk, N. Shahzad, Fixed point theory in distance spaces, Springer, Cham, 2014. doi:10.1007/978-3-319-10927-5. [40] D. Burago, Yu. Burago, S. Ivanov, A Course in Metric Geometry, volume 33 of Graduate Studies in Mathematics, AMS, Providence, 2001. [41] P.-E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Analysis. 16 (2008) 899–912. doi:10.1007/s11228- 008-0102-z. 335