On proximal-like methods for equilibrium programming

Transcription

1 On proximal-lie methods for equilibrium programming Nils Langenberg Department of Mathematics, University of Trier Trier, Germany, Abstract In [?] Flam and Antipin discussed a proximal-lie method for equilibrium programming problems. In fact, they mae use of a Bregman function with Lipschitz continuous gradient to generate well-posed subproblems. However, they require the feasible set i.e. the set of all possible strategies to be compact. Further, as a sort of inexactness, they only allow the use of ε-subgradients, but no inexact solution of the subproblems in its proper sense. The present paper extends their method to the case of unbounded strategy sets and thus permits to treat a much broader class. Further, besides the use of ε-subgradients, we admit the inexact solution of the auxiliary problems by means of a summable-error-criterion in order to increase the numerical attraction of our method. Further, we develop another proof technique which permits to omit the frequently assumed Lipschitz continuity of the gradient of the Bregman function in almost all of our results. Thus, under rather restrictive but not unusual additional assumptions also zone-coercive Bregman-lie functions can be used. These functions do not have Lipschitz continuous gradients; their advantage consists in the obtained interior-pointeffect, i.e. the generated subproblems may be treated as unconstrained. Key words: Saddle point problems, Nash equilibria, Bregman distances, unbounded feasible sets, inexact solution AMS Subject Classifications: 91A06, 91A10, 90C25, 90C30, 90D10 1 Introduction In the 1950 s Nash [?,?] introduced and studied a concept of equilibria in non-cooperative games. It mainly consists in the idea that a given strategy tuple should be considered as an equilibrium if no player can improve his situation i.e. reduce his costs, increase his profits etc. by a unilateral deviation of this strategy tuple.

2 N. Langenberg: Proximal-lie methods for equilibrium programming 2 For the moment, denote K i R n i the set of all strategies of each player i {1,..., N} and N K := K i R n, 1 i=1 where n = N i=1 n i denotes the dimension of the game. For the convergence analysis below, K does not have to be of product structure as in 1. Further, each player i has an objective function f i : K R; without loss of generality we agree that we deal with the minimization e.g. of costs. Finally, denote y i, x i := x i, y i := y 1,..., y i 1, x i, y i+1,..., y N, meaning the situation that player i chooses strategy x i and all the other players j choose y j, where x, y K are arbitrary strategy tuples. Now we are able to reformulate the Nash equilibrium problem in mathematical terms. Given a strategy tuple x K, some player i would not deviate from this tuple unilaterally, i.e. from his strategy x i, as far as his costs would increase otherwise, i.e. as far as for each strategy y i K i we have f i x f i x i, y i. 2 Thus, some strategy tuple x is a Nash equilibrium if and only if 2 holds for each player; in other words, if and only if f i x f i x i, y i y i K i, i = 1,... N. 3 Definition 1. Niaido-Isoda-function / Ky-Fan-function The Niaido-Isoda-function sometimes also called Ky-Fan-function related to the objective functionals f i is defined by Fa, b := N fi a i, b i f i a a, b K. 4 i=1 One easily observes the following facts: Remar Fx, x = 0 for each x K. Flam and Antipin [?] used a function without this property, namely F a, b = N i=1 f ia i, b i. 2. In standard Nash games, i.e. games with property 1, some x K is a Nash equilibrium iff one has Fx, x 0 for each x K, respectively F x, x 0 for each x K. In generalized Nash games, i.e. games without property 1, x K fulfilling Fx, x 0 for each x K is a so-called normalized or variational Nash equilibrium [?].

3 N. Langenberg: Proximal-lie methods for equilibrium programming 3 3. Combining both the latter properties we have that x K is a Nash equilibrium iff x Arg min{fx, x : x K}. 1 The last remar leads to the following general problem: Definition 2. Statement of Problem Given a set K R n and a bivariate function F : K K R, find some x K such that x Arg min{fx, x : x K}. 5 As the following examples show, besides Nash equilibrium problems, several other problem classes are covered by the discussed problem 5. Example 1. see [?] for further examples and some detail 1. Nonlinear optimization problems: Given some f : K R, let Fx, y := fy fx. Then x minimizes f on K iff it solves Variational inequality problems: Given some T : K R n, let Fx, y := T x, y x. Then some x solves 5 iff it solves the related variational inequality problem in the common sense, i.e. T x, y x 0 y K, see e.g. [?]. This case is intensively investigated in [?] using assumptions similar to those made below. 3. Convex-concave saddle point problems: Let K i R n i for i = 1, 2 and K = K 1 K 2 be the product of two nonempty convex compact subsets. Suppose that the Lagrangian Lx, y of some convex saddlepoint-problem is convex-concave. Let Fx, y := Ly 1, x 2 Lx 1, y 2, where x = x 1, x 2, y = y 1, y 2 K. Then x is a min-max saddlepoint of L iff it solves 5. Remar 2. It is also possible to formulate Nash equilibrium problems in terms of variational inequalities. Define T x := xi f i x N i=1, then x is a Nash equilibrium iff it solves the V IT, K see [?], Proposition However, this approach has two central disadvantages: 1 Argmin is written with capital A to indicate that this set is possibly not a singleton in fact, in many problems it is even unbounded. When uniqueness is ensured, we will use the notation argmin with a small a.

4 N. Langenberg: Proximal-lie methods for equilibrium programming 4 Only a subset of the set of Nash equilibria, namely the so-called variational equilibria, can be computed see for example [?], Example 1. Maybe even worse, efficient solution methods for variational inequality problems all require certain monotonicity conditions in the sense of VI theory, which is a rather restrictive assumption here, as we will setch briefly. Just consider the case that each f i is twice differentiable, then T is differentiable and thus monotone iff its Jacobian J T is positive semidefinite. But this, loosely speaing, corresponds to the situation that each player has more influence on his costs than all the other players together, and thus it seems to be rather stringent. Sufficient criteria for some maybe weaened monotonicity properties of the operator T seem to be unnown in literature. For these reasons, we discard the above variational inequality approach and turn bac to the introduced fixed-point approach, which, admittedly, also only covers normalized Nash equilibria in games without property 1. This paper is organized as follows: In the following section we will shortly discuss the notion of Bregman-lie functions which will be used in the algorithms below. In Sections 3-5 the method of Flam and Antipin [?] is investigated in various extensions. Besides some improvements concerning the admittance of inexact solutions of the generated subproblems, the central purposes of these sections are to allow also unbounded strategy sets K and to obtain new proofs which also permit to omit the hypothesis of Lipschitz continuity of the gradient of the regularizing functional nearly everywhere. Section 6 is dedicated to possible conditions which allow to omit the above mentioned Lipschitz hypothesis really everywhere. Some concluding remars can be found in Section 7. 2 Bregman-lie functions Both as a regularization and a penaltization term, we will mae use of a Bregman-lie function. This concept is frequently used in proximal-lie methods see e.g. [?,?,?,?,?]. For a better understanding, we give a short definition and some remars on Bregman-lie functions. Definition 3. Bregman-lie functions Let = S R n be an open and convex set. A function h : cls R is said to be a Bregman-lie function with zone S, when the following holds:

5 N. Langenberg: Proximal-lie methods for equilibrium programming 5 B.1 h is continuous, strictly convex on cls and differentiable on S. B.2 The set Mx, α := {y S : D h x, y α} is bounded for all fixed α R and x cls, where the Bregman distance is defined by when x cls, y S. D h x, y := hx hy hy, x y, B.3 If {z } N is a sequence in S, converging to z cls, at least one of the following statements holds: a D h z, z 0 for. b If z z is another point in cls, then D h z, z. B.4 Let {z } cls and {y } S be two sequences and assume that one of these sequences is convergent. If further D h z, y 0 holds, then the other sequence converges to the same limit as well. One recognizes that the difference between such Bregman-lie functions and standard Bregman functions can be found in condition B.3, where property a corresponds to the standard Bregman case which is used in [?]. Property b was introduced in [?], see also [?]. Solodov and Svaiter [?] showed that B.4 is a consequence of B.1. It is nown that D h is non-negative and D h x, y = 0 iff x = y, but D h is not a true distance. The nown three-point-formula [?] reads as follows: D h x, x D h x +1, x D h x, x +1 = hx +1 hx, x x Definition 4. Additional assumptions on Bregman-lie functions B.5 Zone-coerciveness: hs = R n. B.6 Cone property: For each z K there are αz > 0, cz R such that D h z, w + cz αz z w. Assume that αz > α > 0 and + > c cz for all z cls. Property B.5 can be used to guarantee an interior-point-effect. This can be explained by dom h = S, and h cannot be extended to the boundary of S. Clearly, the next iterate of corresponding algorithms has to belong to dom h = S and thus, the generated subproblems might be treated as unconstrained ones with certain precaution in numerical experiments.

6 N. Langenberg: Proximal-lie methods for equilibrium programming 6 However, B.5 contradicts the Lipschitz continuity of h when S R n. Property B.6 does not appear frequently in the literature. It was introduced in [?] and strengthens B.2. In the referred article it is shown that B.6 is fulfilled, for example, for strongly convex and other functions. Using this property, we can also admit unbounded feasible sets K as well as inexact solutions of the generated subproblems. In the sequel we assume that h is a Bregman-lie function with zone intk, where K denotes the feasible set of the problem under consideration. When it is assumed to be strongly convex which is useful to chec some smoothness hypothesis below κ denotes its modulus of strong convexity. Zonecoercive and strongly convex Bregman-lie functions are nown for a broad class 2 of convex sets K [?]. However, apart from B.5 when K R n, the function hx := 1 2 x 2 obviously has all the above properties, and it is the only function explicitly mentioned in [?]. It corresponds to the classical proximal-point-regularization and has a Lipschitz continuous gradient. 3 A Proximal-lie Method Algorithm 1 illustrates the method to be discussed. Algorithm 1: A Proximal-lie method 1. Let a start-iterate x 0 K be given. Choose some χ 0 > 0, ε 0 0, δ 0. Set := If x is an equilibrium point STOP. 3. Calculate the next iterate x +1 K: x +1 = arg min{fx +, x + χ D h x, x : x K}, 7 where the argmin operation can be executed inexactly: f +,+1 + χ hx +1 hx, x x +1 δ x x +1 8 for some f +,+1 ε 2 Fx+, x +1 and all x K. 4. Choose χ +1 > 0, ε +1 0, δ Set := + 1 and go to 2. 2 By the way, [?] contains an example of a set K which is described by convex inequalities but which does not have some required additional structure. In such case, the customary prox-regularization hx = 1 2 x 2 still might be chosen as in [?].

7 N. Langenberg: Proximal-lie methods for equilibrium programming 7 This method mainly coincides with the one discussed in [?]. x + denotes the so-called foresight point, see Definition 5 below, and ε 2 F, stands for the partial ε -subdifferential of F with respect to the second argument. 8 is fulfilled if there is some normal vector ξ +1 N K x +1 such that f +,+1 + χ hx +1 hx + ξ +1 = e +1 9 for some error vector e +1 fulfilling e +1 δ, i.e. equation 9 can be solved inexactly, whereas Flam and Antipin [?] required e +1 0 for all N no errrors are admitted. Definition 5. The point x + intk in 7 is called foresight point. As done by Flam and Antipin [?] we will consider the following cases: 1. perfect foresight: x + := x +1 for all N. 2. imperfect foresight: The foresight point is calculated by x + := arg min{fx, x + χ D h x, x : x K}. 10 The concept of foresight was introduced by Antipin, see e.g. [?,?]. Besides a non-mathematical interpretation this will also provide convenient mathematical properties. As will be seen below, the imperfect foresight which should, from a point of view beyond mathematics, be the standard case leads to subproblems of different type in comparison to perfect foresight. Imperfect foresight might be considered as a generalization of extragradient methods; especially for the case of nonlinear optimization problems see Example 1 this reminds of the extragradient method of Korpelevich [?]. Before maing some assumptions for the convergence analysis, let us first define two essential properties of the considered bivariate function F. Definition 6. Monotonicity properties of F 1. F is said to be monotone on K, if holds true for all a, b K. Fa, b Fa, a Fb, b + Fb, a F is said to be pseudomonotone with respect to the solution set on K, if Fa, x Fa, a 12 holds true for all a K, where x is an arbitrary solution of 5.

8 N. Langenberg: Proximal-lie methods for equilibrium programming 8 These notions of monotoncity stem from the theory of variational inequalities. Indeed, given some T : K R n, let Fx, y := T x, y x. Then it is easily seen that monotonicity properties of F as defined above correspond to analogue properties in the theory of variational inequalities see e.g. [?]. This gives a motivation to define the notions as above. Flam and Antipin used the notion of monotonicity for property 12, whereas in several other wors of Antipin property 11 is called sew-symmetry see [?] as well as Definition 1 and Properties 1,2 in [?]. 12 is weaer than 11, at least in the case of Fz, z = 0 for all z K, which can be assumed without loss of generality, see Section 1. We will mae the following general assumptions for the discussion below. Assumption A A.1 Existence of solutions: x K is an arbitrary but fixed solution. A.2 The function F is pseudomonotone w. r. t. the solution set on K. A.3 For fixed first argument, F convex in its second argument. A.4 =0 max{δ, ε } < with δ 0, ε 0 for each N. A.5 There are χ, χ > 0 such that χ χ χ < for all N. Let us briefly discuss the above, rather mild assumptions. A.1 surely is a rather natural assumption and guaranteed when K is nonempty, convex and compact and F is lower semicontinuous and continuous in its first argument [?]. Other sufficient conditions for solvability are e.g. discussed in [?]. In the present paper, we do not distinguish between solution and equilibrium. Obviously, in the covered case of variational inequalities A.2 is weaer than the hypothesis of a monotone operator which is often used in the discussion of interior-point proximal-lie methods see [?,?,?]. In [?] the convergence of the Bregman-PPA is analyzed for pseudomonotone problems. Quite mild sufficient conditions for A.2 are proven in [?], this holds for example true if F is continuous, concave in its first argument and convex in its second argument. Also results concerning quadratic objective functions f i in the case of Nash games can be found in [?]. We observe that, discussing saddle-point-problems as mentioned above, F always fulfills A.2.

9 N. Langenberg: Proximal-lie methods for equilibrium programming 9 In Nash games A.3 especially holds true for player convex games, i.e. when f i is convex in the variables x i of player i see also Lemma 6 in Section 6. Concerning the summability of ε in A.4 we note that this is a weaer hypothesis than summability of ε as required in [?]. Clearly, in contrast to [?], our condition A.4 also permits to use ε = 2. As a resumen of the above discussion, we state that our assumptions here are rather mild and convenient. The next results will be required below. Lemma 1. see [?] Assume that {a }, {b }, {c }, {d } are sequences of non-negative numbers. If the relation a b a + c d 13 holds true and =0 max{b, c } <, then the sequence {a } is convergent and further =0 d < also holds true. We also need the following inequality: If 0 r 1 2, then 4 Case of perfect foresight 1 1 r r Let us turn to the case of perfect foresight first. Then, due to x + = x +1, Algorithm 1 turns to the following method: Algorithm 2: The method in case of perfect foresight 1. Let a start-iterate x 0 K be given. Choose some χ 0 > 0, ε 0 0, δ 0. Set := If x is an equilibrium point STOP. 3. Calculate the next iterate x +1 K by x +1 = arg min{fx +1, x + χ D h x, x : x K}, 15 again inexactly in the sense that f +1,+1 +χ hx +1 hx, x x +1 δ x x for all x K and some appropriate f +1,+1 ε 2 Fx+1, x Choose χ +1 > 0, ε +1 0, δ Set := + 1 and go to 2.

10 N. Langenberg: Proximal-lie methods for equilibrium programming 10 In view of 15 it is easy to see that the generated subproblem is an implicit one, since the searched vector x +1 appears on both sides of the equation. As mentioned above, deviating from the strategy of Flam and Antipin [?], here we also admit to solve 15 inexactly by means of a summableerror-criterion, i.e. according to A.4, =0 δ < +, whereas Flam and Antipin required δ = 0 for every N. Now let us turn to the convergence analysis of Algorithm 2. Since the entire convergence analysis in [?] is based on Lemma 4.1 therein and the latter result essentially requires Lipschitz continuity of h, we mae a different approach here. Insert the three-point-formula 6 in 16 and divide by χ to obtain δ χ x x +1 1 χ f +1,+1, x x D h x, x D h x, x +1 D h x +1, x Since Fx +1, is convex, we have due to the ε-subgradient inequality: Fx +1, x Fx +1, x +1 f +1,+1, x x +1 ε, 18 or after a rearrangement f +1,+1, x x +1 Fx +1, x Fx +1, x +1 + ε. 19 Resuming the above discussion for x = x being any solution of the considered problem, via A.2 we obtain the following: D h x, x +1 D h x, x D h x +1, x + δ x x χ + 1 Fx +1, x Fx +1, x +1 + ε χ χ To apply Lemma 1, it remains to investigate the last term in line 20. Theoretically this can be done similarly to Flam and Antipin [?]: Suppose boundedness of K and use A.4, A.5 to obtain =1 δ χ x x +1 diamk which would be sufficient to apply Lemma 1. =1 δ χ <, 21 Here we mae another approach which also covers unbounded sets K. In view of the Bregman axiom B.6 the existence of cx, αx with x x +1 1 Dh αx x, x +1 + cx 22

11 N. Langenberg: Proximal-lie methods for equilibrium programming 11 is guaranteed. Therefore 20 turns to D h x, x +1 D h x, x D h x +1, x + ε χ 23 which in turn implies + δ αx χ D hx, x +1 + δ cx αx χ + 1 Fx +1, x Fx +1, x +1, χ δ 1 Dh αx x, x +1 D h x, x D h x +1, x + ε + δ cx χ χ αx χ + 1 Fx +1, x Fx +1, x χ Due to A.4 we now that there is some 0 N such that 0 r := δ αx χ < Multiply 24 with 1 r 1 and obtain D h x, x +1 1 r 1 D h x, x 1 r 1 D h x +1, x r 1 ε + 1 r 1 δ cx χ αx χ 1 r 1 + Fx +1, x Fx +1, x +1. χ Remembering 14, 26 can be rewritten as follows: D h x, x δ Dh αx x, x 1 r 1 D h x +1, x χ δ ε αx χ χ 2δ δ cx αx χ αx χ 1 r 1 + Fx +1, x Fx +1, x +1 χ Dh x, x D h x +1, x δ αx χ δ ε αx δ δ cx χ χ αx χ αx χ + 1 Fx +1, x Fx +1, x +1 χ At this place, an application of Lemma 1 with

12 N. Langenberg: Proximal-lie methods for equilibrium programming 12 a := D h x, x, b := 2δ αx χ, c := 1 + 2δ ε αx χ χ δ δ cx αx χ αx χ, d := D h x +1, x 1 χ Fx +1, x Fx +1, x +1, remember A.2 directly proves the following Theorem 1. Theorem 1. Under the assumptions A.1-A.5 the following is valid: 1. The sequence {D h x, x } converges for any solution x. 2. The series =0 D hx +1, x is convergent. 3. The series =0 1 χ Fx +1, x Fx +1, x +1 is convergent. Let us state some important consequences. Corollary 1. Directly from Theorem 1 we obtain: 1. In view of the Bregman axiom B.2 the sequence {x } generated by Algorithm 2 is bounded. 2. In view of the three-point-formula 6 it is true that hx +1 hx, x x D h x +1, x 0 and Fx +1, x Fx +1, x +1 0 for. Since {x } is bounded, it has at least one cluster point x which belongs to K since the latter set is closed. From now on, x j x denote a convergent subsequence and a cluster point, respectively. In view of the Bregman axiom B.4 and Theorem 1 it is clear that also x j+1 x has to hold. Then, we might without loss of generality assume that the corresponding subsequence {χ j } is convergent to, say, χ > 0. Next, let us conclude that every cluster point of {x } solves the considered problem under the additional hypothesis of Lipschitz continuity of h. In this case the following result is useful.

13 N. Langenberg: Proximal-lie methods for equilibrium programming 13 Lemma 2. see [?], Lemma 4.2 Suppose some function η : R n R {+ } is lower semicontinuous, proper convex. If M R n is nonempty closed convex, satisfying the domain qualification ri dom η rim. If the used Bregman-lie function h is real-valued, convex on an open set containing M and has a Lipschitz continuous gradient, the following statements are equivalent: 1. x arg min{ηx + ρd h x, x : x M} for some ρ > x arg min{ηx : x M} Remar 3. Since we do not mae use of the ε-subdifferential with respect to h, no application of the Brøndsted-Rocafellar-property is necessary with respect to h. Hence the condition that h is real-valued convex on an open set containing K used in [?] can be left out. It is sufficient to require these properties on K. Lemma 3. Let A.1-A.5 as well as the assumptions of Lemma 2 hold true. Then every cluster point of the generated sequence {x } is a solution. Proof. Consider some subsequence {x j } and the corresponding subsequence {χ j } which are both assumed to converge to x and χ, respectively. Passing to the limit in the iteration scheme 15 we obtain that x = arg min{fx, x + χ D h x, x : x K}. 29 Now Lemma 2 implies x Arg min{fx, x : x K}. Whereas Lipschitz continuity of h is required throughout all results in [?], we only require it here to prove Lemma 2 and its consequence, Lemma 3. In Section 6 some other properties are discussed which allow to omit the Lipschitz assumption. The next result is well-nown in the theory of Bregman-based proximal methods and it contains the conclusion of convergence of the sequence {x } generated by Algorithm 2. Theorem 2. see e.g. [?,?] Under the assumptions of Lemma 3 the sequence {x } generated by Algorithm 2 converges to an equilibrium point.

14 N. Langenberg: Proximal-lie methods for equilibrium programming 14 5 Case of imperfect foresight Here, Algorithm 1 has the following form: Algorithm 3: The method in case of imperfect foresight 1. Let a start-iterate x 0 K be given. Choose some χ 0 > 0, ε 0 0, δ 0. Set := If x is an equilibrium point STOP. 3. First, calculate the foresight point x + by x + = arg min{fx, x + χ i D hx, x : x K}, 30 inexactly in the sense that f,+ + χ i hx + hx, x x + δ i x x+ 31 for all x K and some appropriate f,+ εi 2 Fx, x + which is again the partial ε -subdifferential of F w.r.t. the second argument. 4. Use the calculated foresight point x + to determine x +1 K by x +1 = arg min{fx +, x + χ o D hx, x : x K}, 32 again inexactly in the sense that f +,+1 +χ o hx +1 hx, x x +1 δ o x x+1 33 for some f +,+1 εo 2 Fx+, x +1 and all x K. 5. Choose χ +1 > 0, ε +1 0, δ Set := + 1 and go to 2. Concerning the parameters χ, δ and ε a superscript i indicates that the concerning parameters are those for the inner problem i.e. the determination of the foresight point x +, whereas a superscript o indicates the relation to the outer problem, i.e. calculating the next iterate x +1. The parameters to be chosen in Steps 1 and 5, respectively, are understood as vectors in the sense that χ +1 = χ i +1, χo +1, for example, and nonnegativity clearly is meant componentwise. It can be seen easily that both essential steps in Algorithm 3 consist in the solution of an optimization problem with strictly convex objective function

15 N. Langenberg: Proximal-lie methods for equilibrium programming 15 which is even strongly convex if h is strongly convex. Further, and this is a significant advantage to the case of exact foresight, the subproblems are explicit ones. However, to calculate x +1, here two subproblems have to be solved instead of one implicit subproblem in Algorithm 2. Now let us turn to the analysis of Algorithm 3 which needs two additional assumptions. Both are taen from Flam and Antipin [?]. Assumption A continuation A.6 Hypothesis on Smoothness: There is some Λ > 0 such that 1 2 Fa, b Fc, b Fa, d+fc, d 2Λ D h c, a D h b, d 34 holds true for all a, b, c, d K. A.7 The regularization parameters are chosen such that χ i, χo χ Λ holds true, where Λ fulfills the smoothness property A.6. While special conditions on the regularization parameters lie A.7 are not unusual in the literature concerning proximal-lie methods, condition A.6 should be discussed in some detail. This rather strange appearing condition simplifies in the case hx = 1 2 x 2 to Fa, b Fc, b Fa, d + Fc, d Λ 0 a c b d, 35 which holds true whenever F is twice continuously differentiable [?]. The reason for choosing Λ 0 instead of Λ will emerge from 37 below. Remar For any strongly convex regularizing functional h we obviously have D h v, w κ 2 v w 2 for every v, w such that D h v, w is well-defined. In other words, v w 2 κ D h v, w Assume that 35 holds with modulus Λ 0. Combining this with 36 we obtain for strongly convex Bregman-lie functions: Fa, b Fc, b Fa, d + Fc, d Λ 0 a c b d Λ 0 2 κ D h a, c 12 = 2Λ 0 κ 1 D h a, cd h b, d κ D h b, d

16 N. Langenberg: Proximal-lie methods for equilibrium programming 16 Consequently, when 35 holds true with Λ 0 and h is strongly convex with modulus κ, then A.6 holds true with Λ := Λ 0 κ 1. The previous result extends the discussion of A.6 made by Flam and Antipin [?] significantly since it gives a sufficient and convenient criterion to chec validity of A.6 which is resumed in the following. Lemma 4. Assume that F is twice continuously differentiable and that h is strongly convex. Then A.6 is always fulfilled. Of course we should note that when h is zone-coercive we only require A.6 on the interior of K. This is necessary since D h v, w is not defined if w / intk, but also sufficient, as will be seen in the sequel, since the second arguments of Bregman distances will be some iterates which, due to the interior-point-effect, belong to intk and thus the corresponding Bregman distances will be well-defined. We return to the convergence analysis of Algorithm 3 and start with the discussion of the inner subproblem, i.e. the determination of the foresight point. 31 implies, jointly with the three-point-formula 6 and A.2 3 : δi χ i x x + 1 χ i f,+, x x D h x, x D h x, x + D h x +, x 1 χ i Fx, x Fx, x D h x, x D h x, x + D h x +, x, which implies for x = x +1 K the following relation: δi χ i x +1 x + 1 Fx χ i, x +1 Fx, x D h x +1, x D h x +1, x + D h x +, x. An analogue result holds true for the outer problem, i.e. the determination of the next iterate remember that both the problems have the same structure: 3 It is also possible to omit the following division by the regularization parameters χ i, χ o. In this case, the convergence can be analyzed analogeously, and when assuming χ i < χ o one might prove D h x +1, x 0 as well. However, since the latter property is not necessary for the further analysis we refrain from a restriction on the parameters.

17 N. Langenberg: Proximal-lie methods for equilibrium programming 17 δo χ o x x +1 1 χ o f +,+1, x x D h x, x D h x, x +1 D h x +1, x 1 χ o Fx +, x Fx +, x D h x, x D h x, x +1 D h x +1, x, which in turn implies for x = x K the following relation: δo χ o x x +1 1 Fx + χ o, x Fx +, x D h x, x D h x, x +1 D h x +1, x. Now sum up inequalities 40 and 43 and obtain δo χ o x x +1 δi χ i x +1 x + 1 Fx + χ o, x Fx +, x Fx χ i, x +1 Fx, x + +D h x, x D h x, x +1 D h x +1, x + D h x +1, x D h x +1, x + D h x +, x. Due to pseudomonotonicity with respect to the equilibrium set and the smoothness hypothesis A.6 we may deduce: 1 χ Fx + o, x Fx +, x χ Fx, x +1 Fx, x + i 1 min{χ Fx + i,χo }, x Fx +, x +1 + Fx, x +1 Fx, x + 1 min{χ i,χo } Fx +, x + Fx +, x +1 + Fx, x +1 Fx, x + Now 44 turns to 44 2Λ min{χ i,χo } Dh x +, x D h x +1, x

18 N. Langenberg: Proximal-lie methods for equilibrium programming 18 D h x, x +1 D h x, x D h x +1, x + D h x +, x 46 2Λ + min{χ i, χo } Dh x +, x D h x +1, x Rearranged, this is equivalent to D h x +, x 2D h x +, x δo χ o x x +1 + δi χ i x +1 x + 47 D h x, x D h x +1, x + D h x +, x 48 +2D h x +, x 1 Λ 2 min{χ i, χo }D hx +1, x δo χ o x x +1 + δi χ i x +1 x Λ min{χ i, χo }D hx +1, x Dh x +1, x + D h x, x D h x, x +1 + δo χ o x x +1 + δi χ i x +1 x +. Now add Λ 2 min{χ i,χo }2 1 D h x +1, x + to both sides of the latter inequality. Then D h x +, x 2D h x +, x Λ 2 min{χ i,χo }2 Dh x +1, x + Λ min{χ i,χo }D hx +1, x D h x, x D h x, x +1 + δo χ o x x +1 + δi x +1 x +. χ i + Λ 2 1 D min{χ i h x +1, x +.,χo }2 At this place, we mae use of the elementary binomial theorem a 2 2ab + b 2 = a b Letting a := D h x +, x 1 Λ 2 and b := min{χ i,χo}d hx +1, x + 1 2, we obtain Dh x +, x 1 Λ 2 min{χ i,χo}d hx +1, x = 51 D h x +, x 2D h x +, x Λ 2 min{χ i,χo }2 Dh x +1, x + Λ min{χ i,χo }D hx +1, x D h x, x D h x, x +1 + δo χ o x x +1 + δi x +1 x + χ i + Λ 2 1 D min{χ i h x +1, x +.,χo }2

19 N. Langenberg: Proximal-lie methods for equilibrium programming 19 Rearranging the latter inequality the following holds: D h x, x +1 D h x, x D h x +, x 1 Λ 2 min{χ i, χo }D hx +1, x δo χ o x x +1 + δi χ i x +1 x + + Λ 2 min{χ i, 1 D h x +1, x +. χo }2 Remembering the Bregman axiom B.6 we further now the existence of reals α and c such that for every N x x +1 D hx, x +1 + c α For this reason, and x +1 x + D hx +1, x + + c. α 53 D h x, x +1 D h x, x D h x +, x 1 Λ 2 min{χ i, χo }D hx +1, x Hence, + δo χ o α Dh x, x +1 + c + δi χ i α Dh x +1, x + + c + Λ 2 min{χ i, χo }2 1 D h x +1, x + = D h x, x D h x +, x 1 2 Λ min{χ i, χo }D hx +1, x δo χ o αd hx, x +1 + δi χ i α Dh x +1, x δ o χ o + δi c χ i α + Λ 2 min{χ i, 1 D h x +1, x +. χo }2 2 δ o 1 χ o α Dh x, x +1 D h x, x + δ o χ o + δi c χ i 55 α D h x +, x 1 Λ 2 min{χ i, χo }D hx +1, x Λ 2 min{χ i, + δi χo }2 χ i α 1 D h x +1, x +. }{{} =:θ

20 N. Langenberg: Proximal-lie methods for equilibrium programming 20 We next shall investigate the term θ. Due to Assumptions A.4, A.5 and A.7 there is some 0 N such that it holds θ χ δi χ i }{{}}{{} α <0 0 Λ2 < 0 56 for all 0. Further, if 0 is large enough, from A.4 we also have that r := δo χ o α < As in the case of perfect foresight, this allows to apply inequality 14: 1 1 δo χ o α δ o χ o α 58 Thus, we multiply 55 with 1 r 1 > 0 and obtain: D h x, x +1 1 δo 1Dh χ o α x, x + 1 δo 1 δ o χ o α χ o + δi c χ i α 1 δo χ o α 1 Dh x +, x 1 2 Λ min{χ i, χo }D hx +1, x δo χ o α 1 Λ 2 min{χ i, χo }2 + δi χ i α 1 D h x +1, x δ o χ o α Dh x, x D h x +, x 1 2 δ o δ o χ o α χ o + δi Λ min{χ i, χo }D hx +1, x Λ 2 min{χ i, χo }2 + δi χ i α 1 D h x +1, x +. χ i 2 c α 2 59 Now we are ready for an application of Lemma 1. Therefore, let a := D h x, x, b := 2 δo χ o α, c := δ o δ o χ o α χ o + δi c χ i α and d := D h x +, x 1 Λ 2 min{χ i,χo}d hx +1, x Λ 2 min{χ i, χo }2 + δi χ i α 1 D h x +1, x +.

21 N. Langenberg: Proximal-lie methods for equilibrium programming 21 Theorem 3. Under the Assumptions A.1-A.7 the following holds true: 1. The sequence {D h x, x } is convergent for any equilibrium x. 2. The series =0 D hx +1, x + is convergent. D h x +, x is con- 3. The series =0 vergent. We gather some consequences of the latter theorem. Λ min{χ i,χo }D hx +1, x Corollary 2. Under the assumptions of Theorem 3 the following is valid: 1. Due to the convergence of {D h x, x } the generated sequence {x } is bounded in view of the Bregman axiom B D h x +1, x + 0 for. 3. D h x +, x 1 Λ 2 min{χ i,χo}d hx +1, x due to 2., D h x +, x 0. 0 for, especially, 4. Several other results hold analogeously to the previous section, lie for example hx + hx, x x + 0 and also for the outer problem hx +1 hx +, x x If {x j } x is a convergent subsequence, we directly obtain from the preceding corollary that also {x j+ } x and {x j+1 } x has to hold true. Now if h is Lipschitz continuous, the conclusion of the convergence of {x } to a solution is the same as in Section 4 using Lemma 3 and Theorem 2. 6 Case of zone-coercive regularizing functionals In the previous sections we proved various important auxiliary results lie boundedness of {x } and convergence of {D h x, x } without using Lipschitzcontinuity of h. The latter assumption was only required to show that each cluster point of {x } and {x + }, respectively, is an equilibrium of the considered problem. The present section is devoted to some alternative conditions on F which allow the application of zone-coercive Bregman-lie functions, i.e. those functions without Lipschitz-continuous gradient. The above argumentation cannot be repeated for a zone-coercive Bregman- lie function h since dom h = intk, i.e. hx and thus also D h x, x is not defined whenever x belongs to the boundary of the feasible set K.

22 N. Langenberg: Proximal-lie methods for equilibrium programming 22 In the considerations below we focus on the situation of Nash games, where Fx, x = 0 can be assumed without loss of generality, see Section 1. Concerning the covered problem class of nonlinear optimization, no analogue property is necessary, and for the related discussion of variational inequalities, we refer the reader to [?]. Now let us impose two further conditions which permit to conclude that every cluster point of {x } is a solution. Then, Theorem 2 can be applied to obtain the convergence of the entire sequence. In the following discussion we only consider Algorithm 2. However, it will be easy to derive analogue assumptions and results concerning Algorithm 3. Assumption A continuation A.8 F is continuous and pseudomonotone 4 not just with respect to the solution set, meaning that the implication Fa, b 0 Fb, a 0 60 holds true for arbitrary a, b K. Further Fz, is strictly convex. A.9 F is continuous and F, z is strictly concave for any z K. Let us postpone a discussion of these additional assumptions and first conclude the desired result. Lemma 5. Assume that in addition to the assumptions of Lemma 3 also A.8 holds true. Then each cluster point of {x } is an equilibrium. Proof. Continuity of F and Corollary 1 imply Fx, x = 0, from which Fx, x 0 and thus also Fx, x = 0 follow, since x is assumed to be an equilibrium. Now since Fx, is strictly convex we obtain for x x 0 Fx, 1 2 x + x < 1 2 Fx, x + Fx, x = 0, 61 which is an obvious contradiction. Before turning to the discussion of A.9 we give a rather mild sufficient condition for Fz, to be strictly convex in the situation of Nash problems. 4 This holds true in the case of saddle point problems of the Lagrangian of a convex program.

23 N. Langenberg: Proximal-lie methods for equilibrium programming 23 Lemma 6. Fx, is strictly convex already if there is at least one objective function f l being strictly convex in x l and all the other objective functions f i are assumed just to be convex in x i. Proof. Tae some v w K and λ [0, 1] arbitrary. Using the notation M l := {1,..., N} \ {l} we obtain: Fx, λv + 1 λw = N i=1 f i x i, λv i + 1 λw i f i x = f i x i, λv i + 1 λw i f i x i M l +f l x l, λv l + 1 λw l f l x λ f i x i, v i f i x + 1 λ f i x i, w i f i x i M l i M l +f l x l, λv l + 1 λw l f l x < λ f i x i, v i f i x + 1 λ i M l i M l f i x i, w i f i x +λ f l x l, v l f l x + 1 λ f l x l, w l f l x = λfx, v + 1 λfx, w Hence, the assertion follows. Clearly, the proof is rather simple; nevertheless, the result can easily be used to chec strict convexity of F when all the functions f i are nown. Lemma 7. Assume that in addition to the above assumptions also A.7 holds true. Then each cluster point of {x } is an equilibrium. Proof. Pseudomonotonicity with respect to the solution set and strict concavity of F, x yield for x x the contradiction 0 F 1 2 x + x, x > 1 Fx, x + Fx, x = 0, 62 2 where the last equation follows from Corollary 1 and continuity of F. Now let us shortly discuss assumptions A.8 and A.9. First, note that pseudomonotonicity in the sense of A.8 lies between pseudomonotonicity w.r.t. the solution set and monotonicity sew-symmetry in the wors of

24 N. Langenberg: Proximal-lie methods for equilibrium programming 24 Antipin [?,?], at least when Fz, z = 0 for any z K. Clearly, strict convexity concavity is rather restrictive since it implies uniqueness of solutions. Nevertheless, it is not unusual in the literature with respect to Nash equilibrium problems, see e.g. [?]. We should mention that the latter authors require this type of strict convexity assumption for a constrained reformulation of the Nash game whereas in the present paper we only require it to mae use of zone-coercive Bregmanlie functions to obtain unconstrained subproblems. Further, they mae use of a regularization term with Lipschitz continuous gradient; to do this, we did not require such a hypothesis. Another assumption is called strict complementarity, see e.g. [?,?]. However, this assumption seems to be harder to chec a priori. An optimal problem-tailored condition would loo lie A.* If F is pseudomonotone with respect to the solution set, x K is an equilibrium and x K is such that Fx, x = 0, then x is an equilibrium as well. This condition is closely related to the so-called cut property see [?]. In [?] the convergence of the BPPA for pseudomonotone variational problems is studied under an additional condition which seems to be rather mild [?,?]. However, sufficient conditions on the objective functions f i to fulfill this property seem to be unnown in the literature. Discussing nonlinear optimization problems, letting Fx, y = fy fx we already obtain fx fx fx. For the covered situation of variational inequality problems the concept of pseudomonotonicity turns out to be appropriate [?,?]. With respect to Algorithm 3 the discussion is quite the same. By continuity of F and passing to the limit in 44, again Fx, x = 0 is obtained such that the argumentation concerning Algorithm 2 can be applied again. Finally, we proof another result. The underlying hypothesis is the following: A.10 There is an equilibrium x intk and 2 F is locally bounded on an open set containing K. Of course, A.10 is quite hard to chec a priori nevertheless, there is a sufficient condition in [?], where A.10 is investigated for the case of a rather general scheme of variational inequalities. However, we can prove:

25 N. Langenberg: Proximal-lie methods for equilibrium programming 25 Lemma 8. Assume that in addition to the above assumptions also A.10 holds true. Then each cluster point of {x }, generated by Algorithm 2, is an equilibrium. Proof. As a consequence of the iteration scheme and Corollary 1 we have lim f j+1, j +1, x x j j Owing to the Brøndsted-Rocafellar-property cf. [?] of the ε subdifferential and convexity of Fx j+1, we now that for each j there is x j+1 and some f j+1, j +1 2 Fx j+1, x j+1 such that x j+1 x j+1 ε j +1 and f j+1, j +1 f j+1, j +1 ε j Since 2 F is locally bounded on an open set containing K, we can without loss of generality assume the existence of some f 2 Fx, x such that lim j f j+1, j +1 = lim j f j+1, j +1 = f. 65 Recall that as a consequence of zone-coerciveness also the so-called boundary coerciveness holds true, i.e. hx j, x x j 66 is true when {x j } x and x belongs to the boundary of K. In this situation 66 obviously contradicts the nown convergence of D h x, x j. Therefore, any cluster point of {x } has to belong to intk. This in turn implies the existence of hx also for the case of zone-coercive regularizing functionals. In consequence, hx j+1 hx j 0, 67 for j. Now due to the boundedness of the generated sequence {x }, we obtain by passing to the limit j in the iteration scheme: f, x x = lim f j+1, j +1, x x j+1 0 x K. 68 j Therefore x Arg min Fx, follows, i.e. each cluster point of {x } is a solution. Again, a similar result for Algorithm 3 can be derived quite analogeously.

26 N. Langenberg: Proximal-lie methods for equilibrium programming 26 7 Concluding Remars We investigated a proximal-lie scheme to solve a class of fixed-point-problems, which in turn covers several other important classes as Nash equilibrium problems, variational inequalities or also nonlinear optimization problems. In comparison to the original wor [?] the proof techniques demonstrated in the present paper permit to achieve several essential advances: Theoretical advances as for example the applicability of the method for unbounded sets K, numerical advances as for example the allowance to solve the generated subproblems inexactly as well as a relaxation concerning the sequence of subgradient parameters ε. From now on, for example ε = 2 might be used which probably contributes to a better numerical performance of the discussed algorithms. Also, in the case of imperfect foresight the regularization parameters χ i, χo do not have to be the same in our method. Further, we proved some auxiliary results lie the convergence of {D h x, x } without the hypothesis of a Bregman function with Lipschitz continuous gradient, which enables to prove convergence also for zone-coercive Bregmanlie functions although, admittedly, some rather restrictive additional assumption on the problem data is required. Possibly one might construct singular examples which can do without such a property of strict convexity type, but in general such an additional assumption which does not imply uniqueness of the solution seems to be unnown in the literature.