Vietnam J. Math (2015)43.229-238 DOI 10,10O7A!0013-0i4-01l5-x
On Existence and Solution Methods for Strongly Pseudomonotone Equilibrium Problems
Le Dung Muu • Nguyen Van Quy
Received. 19 December 2013 / Accepted: 27 June 2014 / Published online: 4 January 2015
© Vietnam Academy of Science and Technology (VAST) and Springer Science-i-Business Media Singapore 2015
Abstract We study the equilibriuin problems with strongly pseudomonotone bifunctions in real Hilbert spaces We show the existence of a unique solution We then propose a strongly convergent generahzed projection method for equilibrium problems with strongly pseu- domonotone bifunctions. The proposed method uses only one projection without requiring Lipschitz continuity. Application to variational inequalities is discussed.
Keywords Strongly pseudomonotone equihbria • Solution existence - Generahzed projection method
Mathematics Subject Classification (2010) 47J20 • 47H09 • 90C25
I Introduction
Throughout Ihe paper, we suppose that ,1^ is a real Hilbert space endowed with weak topol- ogy defined by the inner product (-, •) and its reduced norm || ||.LetC c J ^ be a nonempty closed convex subset and / : C x C - * ] R b e a bifunction satisfying fix.x) = 0 for every ;t e C. As usuaL we call such a bifunction an equilibrium bifunction We consider the following equilibrium problem:
Find X* eC : fix*, x)>0 Vx e C (EP)
This paper is dedicated lo Professor Nguyen Khoa Son on the occasion of his 65ih birthday.
This work is supported by the National Foundation for Science and Technology Development of Vietnam (NAFOSTED).
L D Muu (EKI)
Institule of Mathematics. VAST, 18 Hoang Quoc Viet, Cau Giay. Hanoi, Vielnam e-mail ldmuu@math ac vn
N. V. Quy
Academy of Finance. 8 Phan Huy Chu, Hoan Kiem, Hanoi, Vietnam e-mad: quynv2002@yahoo com
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230 L. D Muu and N. V Quy This problem is also often called Ky Fan's inequality due to his contribution to the subject.
Equilibnum problem (EP) gives a unified formulation for some problems such as opti- mizanon problems, saddle point, variational inequalities, fixed point, and Nash equilibria, in the sense that it includes these problems as particular cases (see for instance [4, 6, 22]).
An imponant approach for solvmg equilibrium problem (EP) is the subgradient projec- tion method which can be regarded as an extension of the steepest descent projection method in smooth optimization. It is well known that when the bifunction / is convex subdifferen- tiable with respect to the second argument and Lipschitz, strongly monotone on C, one can choose rcgularizalion parameters so that this method Hnearly convergent (see, e.g., [23]).
However, when / is monotone, the method may not be convergent (see Example 12.1.3 in [9] for monotone variational inequality), in recent years, the extragiadient (or double projecuon) method developed by Korpelevich in [17] has been extended to obtain conver- gent algorithms for pseudomonotone equilibrium problems [26]. The extragradient method requires two projections onto the strategy set C, which in some cases is computational cost.
Recently, in [8,28], inexact subgradient algorithms using only one projection have been pro- posed for solving equilibrium problems with paramonotone equilibrium bifunctions. Other methods such as auxiliary problem principle [18, 24], penalization techmque [5, 22], gap and merit functions [19, 25], and the Tikhonov and proximal point regularization meth- ods [12-14, 16, 20] are commonly used for equilibrium problems. SoluUon existence for equilibnum problems can be found in some papers (see, e.g., [2-4, 6]).
In this paper, we itudy equilibrium problem (EP) with strongly pseudomonotone bifunc- Uons. We show the existence of a unique solution of the problem We then propose a generalized projection method for strongly pseudomonotone equilibrium problems. Three main features of the proposed method are:
it uses only one projection without requiring Lischitz continuity allowing strong convergence;
It allows that moving directions can be chosen in such a general way taking both the cost bifunction and the feasible set into account;
It does not require that the bifunction is subdifferentiable with respect (o the second argument everywhere.
2 Solution Existence
As usual, by Pc, we denote the projection operator onto the closed convex set C with tl norm || • ]|, that is,
Pcix) € C : \\x- Pcix)\\ < \\x - y\\ Vv e C
The following well-known results on the projection operator will be used in the sequel Lemma 1 ([1]) Suppose that C is a nonempty closed convex set in J T Then.
(i) Pcix) is singleton and well defined for every x;
(ii) n = Pcix) if and only if{x ~ TT, y - n) <OVy e C-
(iii) \\Pcix)~Pciy)f<\\x-y\\^-\\Pcix)-x + y~Pc(y)fVx,yeC.
We recall some weU-known definitions on monotonicity (see, e.g., [2, 3. 6]).
Definition I A bifunction ^ • C x C -> K is said to be
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On Existence and Soluiion Methods for Strongly Pseudomonotone
(a) strongly monotone on C with modulus ^ > 0 (shortly ^-strongly monotone) if 0(A-. y) + <Piy. X) < -p\\x - y\\- Vx. y e C:
(b) monotone on C. if
4>ix,y)-\-4>iy.x)<0 Vx.yeC:
(c) strongly pseudomonotone on C with modulus ^ > 0 (shortly ^-strongly pseudomono- tone), if
0 ( j : , y ) > O = > 0(y..() < - ^ | | j i : - y | | - Vx.yEC.
(d) pseudomonotone on C, if
cpix,y)>0 = > 0 ( y , . v ) < O Vx.yeC.
From the definitions, il follows that (a) ^ (b) =>- (d) and (a) ^ (c) ^ (d), but there is no relationship between (b) and (c). Furthermore, if / is strongly monotone (respec- tively pseudomonotone) with modulus ^ > 0, then it is strongly monotone (respectively pseudomonotone) with modulus fi' for every 0 < ^' < ^.
The following example, for strongly pseudomonotone, bifunction is a generalization of that for strongly pseudomonotone operator in [15]. Let
fix,y):=iR-\\x\\)gix,y). e , := [x-.e-J^ : \]x\\ <r}, where ^ is strongly monotone on Br with modulus^ > 0, for instance g(j:, y) = (a:, >' — ;»:), and R > r > 0. We see that / is strongly pseudomonotone on Br. Indeed, suppose that fix. y) > 0. Since X e Br- we have gix, y) > 0 Then, by ^-strong monotonicity of g on Br, giy, x) < —fi\\x - y\\- for every x.y € B,. From the definition of / and y e Br, it follows that
fiy, x) = iR- \\y\\)giy, x) < -fiiR - ||y||)||jr - yf < -^(R - r)\\x - yf Thus, / is strongly pseudomonotone on Sr with modulus ^iR — r)
The following lemma, that will be used to prove Proposition 1 below, is a direct consequence of Theorem 3.2 in [3].
Lemma 2 Lettf) : C xC ^>- Wbean equilibrium bifiinclion such that <j>i-. y) is hemicontin- uousfor each y € C and(i>ix, •) is lower semicontinuous convex for each x & C. Suppose that the following coercivity condition holds
3 compact set W : iVxeC\W, 3r € C : tPix. y) < 0).
Then, the equilibrium problem (EP) has a solution.
In what follows, we need the following blanket assumptions on the bifunction / : (Al) For each x e C, the function fix. •) is lower semicontinuous, convex (not neces-
sarily subdifferentiable everywhere), and for each y e C, the function / ( • , y) is hemiconiinuous on C,
(A2) / is ^-strongly pseudomonotone on C.
It is well known that if / is strongly monotone on C, then under assumption (Al), equilibnum problem (EP) has a unique solution. The following lemma extends this resull to (EP) wilh strongly pseudomonotone bifunctions.
^ S p r h
L, D, Muu and N, V. Quy Proposition 1 Suppose that f is ^-strongly pseudomonotone on C. then under assumption (Al), Problem (EP) has a unique solution.
Proof First, suppose that C is unbounded. Then, by Lemma 2, it is sufficient to prove Ihe following coercivity condition:
3 closed ball B : (V.v € C \ fi. 3y e C O B : fix, y) < 0). (CO) Indeed, otherwise, for every closed ball B, around 0 with radius r, there is .x'' e C\Br such that/(jc.y) > O V y e C H B,
Fix ro > 0, then for every r > ro. there exists .v"" e C \ Br such that fix'', y") > 0 with v" e C n Brg. Thus, since / is ^-strongly pseudomonotone, we have
fiy'',x')-^fi\\x' -y^f <0 V/. (1) On the other hand, since C is convex and fiy", •) is convex on C, it is well known from
convex analysis that there exists x'-' e C such that difiy^.x'^) / 0, where d2fiy^.x°) stands for the subdifferential of the convex function fiy°, •) at x". Take w* e difiy^, x^), by the definition of subgradient one has
{w\x~x"}-\-fiy".x'')<fiy°,x) Vx.
Widi A- — x'' it yields
/(y", x') -F ^IJA'- - yY > / ( / . x'>) + (u;*, x^ - x") -^ fiWx'' - yY - f(y''.x'>)-\\w*\\\\x'-x°\\-\-fi\\x'-y<'f.
Letting r -^ oo, since ||.v'"|| -^ oo, we obtain /(y^.x"") -f fiWx'' - y'^f -> oo which contradicts (1). Thus, the coercivity condition (CO) must hold true, then by virtue of Lemma 2. equilibrium problem (EP) admits a solution.
In the case, C is bounded, the proposition is a consequence of Ky Fan's theorem [10].
The uniqueness of the solution is immediate from the strong pseudomonotonicity of
•^- D
We recall [11] that an operator F • C -* .J^ is smd lo be strongly pseudomonotone or\
C with modulus ^ > 0, shortly /5-strongly pseudomonotone, if
{Fix),y-x}>0 = > {Fiy).y~x)^fi\\y-xf Vx.yeC.
In order to apply the above proposition to the variational inequality problem
Find x* e C ' ( F ( J : * ) , y - x * ) > 0 Vy e C, (VI) where f is a strongly pseudomonotone operator on C, we define the bifunction / by taking
fix.y):={Fix),y-x). (2) It is obvious that X* is a solution of (VI) if and only if it is a solution of equilibnum problem
(EP) with / defined by (2). Moreover, it is easy to see that F is ^-strongly pseudomonotone and hemicontmuous on C if and only if so is / . The following solution existence result, which IS an immediate consequence of Proposition I. seems to have not appeared in the literature.
Corollary 1 Suppose that F is hemicontimious and strongly pseudomonotone on C Then the variational inequality problem (VI) has a unique solution.
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On Existence and Solution Methods for Strongly Pseudomonotone 233 3 A Generalized Projection Method for Strongly Pseudomonotone EPs
For e > 0, we call a point -r*^ € C an e-solution to Problem (EP), if fix^.y) > -s for every y € C. The following well-known lemma will be used in die proof of the convergence theorem below.
Lemma 3 [21] Suppose that [ a t ) ^ is an infinite sequence of positive numbers sansfying
with X^t^o ^k < C'O. Then, the sequence {iXk\ is convergent Algorithm 1
Step 1 iChoosmg a starting point and step size) 5&. x^ e C, choose a tolerance £ > Oand a sequence {pk ] of positive numbers such that
'Y^Pk = oD, ^pf-cco. (3) Letfc := 1.
Step 2 iFinding a moving direction) Find g'' e Jff such diat
/{jr*.y) + { / , J C * - y ) > - p * VyeC. (4) (a) If ^* = 0 and pk < e, terminate: x* is an e-soluuon.
(b) Otherwise, execute Step 3.
Slep3 iProjeclion) Compulex''+^ := Pcix'' - Pkg'')- (a) If x*"*"' = X* and pk < s. terminate: x* is an e-solution.
(b) Otherwise, go back to Step 2 with k replaced by /: -i- 1.
Theorem 1 Suppose that assumptions iAl) and iA2) are satisfied. Then.
(i) if the algorithm terminates at iteration k. then x* is an E-soiiilion;
(ii) I'r holds thai
||x*+' -x'f < il -2ppk)\\x'' - x'f +2PI -\- plWg'f Vk. (5) where x* is the unique solution of (EP). Furthermore, if the algorithm does not ter- minate, then the sequence {x*) strongly converges lo the solution x" whenever the sequence (g*) is bounded.
Proof (i) If the algorithm terminates at Step 2, then g* — 0 and pk < E. Then, by (4), fix'', V) > —pk > —£ for every y e C. Hence, x* is an £-solution. If the algorithm terminates at Step 3, then x* is an e-solution because of Lemma 1 (ii) and (4).
(ii) Since x*+' = Pcix'' - Pkg''), one has
= \\x''-x*f-2pk(g'',x''-x-} + pl\\g'^f. (6)
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L D Muu and N V Quy Applying (4) with y — x*. we obtain
/ ( x * , x * ) + ( g * , x * - x * ) > - / J i . which imphes
- ( g * . . v ^ - x ' ) < / ( - v * . x ' ) - l - / > t . O) Then, il follows fiom (4) diat
y^t+l _ ^*||2 ^ ||_^A _ ^*||2 _^ 2pft ( / ( x ^ x ' ) + p;,) + p | | l / l l ^ (8) Since x ' is a solution, / ( x * , x*-) > 0, il follows from ^-strong pseudomonotonicity of / that
fix'',x')<-^\\x''^.x'\\\
Combining the lasl inequality with (8), we obtain
ll^*-+i _ ^ . | | 3 ^ | | . v ' - - - x ' | | ^ - 2 ^ / ) s | | x * - x * | | - + 2 p | + Pi^l|g*f
= (1 - 2 ^ p , ) l | x * -x*f+2pl-^pl\\g'f. (9) Now suppose that the algorithm does not terminate, and that Wg'- jj < C for every k. Then,
it follows from (9) that
ll_^i+i_^*||2 ^ (]-2ppk)\\x''-x*f-\-i2 + C^)pl
= | | x * - x * | | - - A i | | x ^ - x * | | ^ - t - ( 2 - | - C
where Ai- :— Ifipk. Smce ^ ^ i pj < oo, in virtue of Lemma 3, we can conclude that the sequence {||x* -x*||-^l is convergent In order to prove that the limhof this sequence is 0, we sum up inequality (10) from 1 to fc -f I to obtain
| | x * + ' - x * | | ^ < | | x ' - x * | | ^ - ^ X ; | | x ^ - x * f + ( 2 - H C 2 ) ^ P j ^ which implies
^l^''^^ ^ ^*f + J2xj\\xJ - x*f < \\x' - x^f + i2-\- C^)J2pr (!')
; = l j=\
Since Xj :— 2fipj, we have
Y^Xj=2fiY,pj=<x,. (12)
Note that { x ' | is bounded and that YIJLQP^ < oOi we can deduce from (11) and (12) that
||.v'_A-'^||^^Oasyt ^ oc. D The algorithm described above can be regarded as an extension of the one in [28] in a
Hilbert space setting The main difference lies in the determination of g^ given by formula (4). This formula is motivated from the projection-descent method in optimization, where a moving direction must be both descent and feasible. Such a direction thus involves both the objective function and the feasible domain. In fact moving directions defined by (4) rely not only upon the gradient or a subgradient as in [28] and other projection algonthms for equilibrium problems, but also upon the feasible set. Pomts (i), (ii), and (ni) in Remark 1 below discuss more detail on formula (4).
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On Existence and Solution Methods for Strongly Pseudomonotone
Remark I (i) A subproblem in this algorithm is to find a moving du-ection g* satisfying (4). It is obvious that if g* is a p*-subgradient of the convex function / ( x * , -) at x*, then g* satisfies (4).
When mk : = infj.gc fix'^. y) > —oo, it is easy lo see that if g* is any vector satisfying
( g ^ y - X*) < mk -\- Pk : - fA Vy e C.
I.e., g* IS a vector in rt-normal set N'^ ix'') of C at x*, then (4) holds true, (n) For variational inequality (VI) with fix, y) defined by (2), formula (4) takes die form
(F(x*), V - X * ) + (g*,x* - y) > -Pk Vy e C. (13) which means that g* - F(x*) e A'^^(x*), where A'^'(x*) denotes the pk-nounal set of C atx*, Ihat is,
N^' (X*) : = (u;* : (w'', y - x*) < pk Vy e C).
In an usual case, when C is given by C : - (x e J f : g(x) < 0} with g being a subd- ifferentiable continuous convex function, one can take g'' = Fix'') when g(x*) < 0, and g* may be any vector such that g* - Fix'^) e 9g(x*) when g(x*) = 0. Since
Ncix'') = [0] if gix'') < 0, and Ndx'') ^ 9g(x*) if gix'') - 0, in both cases g* ~ F(x*) e Ndx'') a N^'-ix'') for any pk > 0.
(iii) The direction g* satisfying g* - F{x*) e A'^*(x'^) takes not only the cost operator F. but also the constrained set C into account This is helpful in certain cases, for example, it allows avoiding the projection onto C Indeed, it may happen that —Fix'') is not a feasible direction of C at x*, but —g* is it
(iv) Note that since fix''. •) is convex, for every pt > 0, a pi-subgradient of the function / ( x * , •) does always exist at every point in d o m / ( x * , -), but it may fail to exist if Pk = 0. Thus, the proposed algonthm does not require that the convex function / ( x * , •) is subdifferentiable ai every point in its domain,
(v) For implementing the algonthm, one suggests that we take pk :— ep^, where p'g. is a decreasingly monotone positive sequence satisfying (3).
Remark 2 If /" is jointly continuous on an open set A x A containing CxC, then {g*}
IS bounded whenever Pk -> 0 (see. e.g.. Proposition 3.4 in [29]). In the case of variational inequality (VI) with fix. y) defined by (2), if g* - f (x*^) and F is contmuous, then (g*}
is bounded because of boundedness of |x*|.
4 A Numerical Example
We consider an oligopolistic equilibnum model of the electricity markets (see, e.g., [7,27]).
In this model, there are M' companies, each company / may possess /, generating umts. Let X denote the vector whose entry x, stands for the power generating by unit /. Following [7], we suppose that the price p is a decreasing affine function of a with a — X]"=j x, where n* is the number of all generating units, that is,
p(x) = a o - 2 ^ x ,
L, D Muu and N V. Quy Table 1 The lower and upper
bounds for die power gen and companies
where OQ > 0 is a constant (in general is large). Then, the profit made by company f is given by
fix)^piG)J^Xj-J2'^A^-J^-
where Cjixj) is the cost for generating Xj. Unlike [7], we do not suppose that the cost Cj (xy) is differentiable, but
CjiXj): .Mc^jiXj),c)iXj)}
c«(.;) :- "-^x] + ^^x, + y], c)ixj) := a]x, + ^ y / ' ' ' ' ( . ; ) ' ^ > " ^ ^ J , where " J . j6j. y^ (fe — 0. 1) are given parameters.
Let x""" and x""^* be the lower and upper bounds for the power generating by the unit j . Then, the strategy set of the model takes the form
C : - [ x - ( x , . . Define the matnces A. 6 by taking
,x„.)' '^Vj).
^ ^2j](l-9')(f/)^ B:=2j2^'iq')^.
where?' . ^ ( ? j , . . . .g,',,)^ with
1 i f j e / , , I 0 otherwise.
Table 2 The parameters of the generating unit cost functions Gen.
1 2 3 4 5 6
a-' 0 04 0 04 0 12 0 02 0.05 0.05
P'j 2.00 1.75 1 00 3 25 300 3.00
Yj
0 00 0 00 0.00 0.00 0.00 0-00
«;
2-00 1.75 LOO 3.25 3-00 3,00
f>]
1 00 1 00 1 00 1.00 1.00 1-00
yJ 25,00 28 57
8.00 86,2! 20.00 20.00
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On Existence and Solution Methods for Strongly Pseudomonotone
Table 3 The power made by three companie.s X | X2 Xi
46.543 31.947 15.019 -V4
20 989
.>^
12.581
•Ve 12-612
Cpu(sl 19-991
a.= -aoY2q', c(x) : ^ ^ C j ( X y ) (16) Then, by Lemma 7 m [27], the equilibrium problem being solved can be formulated as
X € C : / ( x , > - ) : - ( M - ( - - B J x - | - ^ B y - | - a j (,• _ ; f ) + c ( y ) - c(x) > 0 VyeC (EP) Note t h a t / ( x . y ) - | - / ( y , x ) = - ( v - x ) ^ ( A - l - f i ) ( i - v ) ' " . Thus, since 4 - f B is not positive semidefinite, / may not be monotone on C However, if we replace / by f\ defined as
1 then / i is strongly pseudomonotone on C In fact, one has
fiix. y) -i- fiiy.x) ^ -iy -x)^iA -\-2B)iy - X).
Thus, i f / i (x.y) > 0 . t h e n
/ i ( v . x ) < - ( y - x ) ^ ( A - | - 2 S ) ( y - x ) < - X | | y - x | | -
for some A > 0. The following lemma is an immediate consequence of the auxiliary principle (see, e.g., [23, 24]).
Lemma 4 The problem
Find X* eC: fl (x*, y) > 0 V\ eC is equivalent lo the one
Find x* e C : / i ( x * , y ) - | - - ( y - x * ) ^ B ( y - x ' ) > 0 VyeC (EPl) in the sense thai their solution sets coincide.
In virtue of this lemma, we can apply the proposed algonthm to the model by solving the equilibnum problem (EPl), which in turns, is just equilibrium problem (EP).
We test the proposed algorithm for this problem which corresponds to the first model in [7] where three companies (n'" = 3) are considered with ao := 387 and the parameters are given in Tables 1 and 2.
We implement Algorithm 1 in Matlab R2008a running on a Laptop with Intel(R) Core(TM) i3CPU M330 2.13 GHz with 2 GB Ram. We choose e - 10' and pk := f for every k. The computational results are reported m Table 3 with ihe starting pomt x ' = 0.
Acknowledgments We would like to thank Ihe referees for their useful remarks and comments thai helped us very much in revising the paper.
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L, D. Muu and N. V. Quy
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