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Computational Optimization and Applications manuscript No.
(will be inserted by the editor)
A Class of Benders Decomposition Methods for
Variational Inequalities
Juan Pablo Luna ·Claudia Sagastiz´abal ·
Mikhail Solodov
Received: January 31, 2019 (revised September 16, 2019) / Accepted:
Abstract We develop new variants of Benders decomposition methods for
variational inequality problems. The construction is done by applying the gen-
eral class of Dantzig–Wolfe decomposition of [14] to an appropriately defined
dual of the given variational inequality, and then passing back to the primal
space. As compared to previous decomposition techniques of the Benders kind
for variational inequalities, the following improvements are obtained. Instead
of rather specific single-valued monotone mappings, the framework includes a
rather broad class of multi-valued maximally monotone ones, and single-valued
nonmonotone. Subproblems’ solvability is guaranteed instead of assumed, and
approximations of the subproblems’ mapping are allowed (which may lead, in
particular, to further decomposition of subproblems, which may otherwise be
not possible). In addition, with a certain suitably chosen approximation, varia-
tional inequality subproblems become simple bound-constrained optimization
problems, thus easier to solve.
Keywords Variational inequalities ·Benders decomposition ·Dantzig–Wolfe
decomposition ·stochastic Nash games
Mathematics Subject Classification (2010) 90C33 ·65K10 ·49J53
J. P. Luna
COPPE-UFRJ, Engenharia de Produ¸c˜ao, Cidade Universit´aria, Rio de Janeiro - RJ, 21941-
972, Brazil.
E-mail: jpluna@po.coppe.ufrj.br
C. Sagastiz´abal
IMECC - UNICAMP, Rua Sergio Buarque de Holanda, 651, 13083-859, Campinas, SP,
Brazil.
E-mail: sagastiz@unicamp.br
M. Solodov
IMPA – Instituto de Matem´atica Pura e Aplicada,
Estrada Dona Castorina 110, Jardim Botˆanico, Rio de Janeiro, RJ 22460-320, Brazil.
E-mail: solodov@impa.br
2 Luna, Sagastiz´abal, and Solodov
1 Introduction
In applications, feasible sets of optimization or variational problems frequently
have some special structure, amenable to decomposition, which opens the po-
tential to handle larger problems’ instances and to better computational per-
formance. In this direction, two major classes of techniques are the Dantzig-
Wolfe decomposition [8] and Benders decomposition [3]; see also [4, Chap-
ter 11.1]. To outline the main ideas, it is sufficient to discuss the cases of two
linear programs (LPs), structured as specified below. More precisely, consider
minxf(x)
s.t. H(x)≤0
g(x)≤0,
(1)
and minxf(x) + r(y)
s.t. g(x) + h(y)≤0,(2)
where all functions are affine with appropriate dimensions. In the case of (1),
suppose the function ghas a special structure such that (1) is (much) easier
to solve if the constraint H(x)≤0 were to be removed. In the case of (2),
suppose the problem becomes (much) easier to solve if we fix the value of the
variable y. For example, those two cases occur when the Jacobian matrix g0
defining the linear function ghas the following block-separable structure:
g0(x) =
A1
A2
...
Aj
(3)
where Ai,i= 1, . . . , j, are matrices of appropriate dimensions.
Problem (1) can be solved using Dantzig-Wolfe decomposition introduced
in [8]. Without going into details, the Dantzig-Wolfe algorithm consists in iter-
atively solving two types of problems, called master program and subproblem.
As a result of the master solution, an estimation of the Lagrange multiplier
µkfor the constraint H(x)≤0, computed over the convex hull of past iter-
ates, is obtained. The subproblem uses that estimate to solve the Lagrangian
relaxation (with respect to the constraint H(x)≤0) of the original problem
(1), i.e., minxf(x)+[µk]>H(x)
s.t. g(x)≤0.(4)
As all the functions in the above are linear and ghas the separable structure
according to (3), subproblem (4) decomposes into jsmaller ones. Having solved
the subproblem, the iterative process continues with a new master program,
where the multiplier approximation is improved by incorporating in the convex
hull of previous iterates the solution provided by the last subproblem (e.g., see
(8) below in the setting of the Benders approach).
Benders Decomposition for VIs 3
Concerning the second problem (2), its solution using Benders decompo-
sition comes from distinguishing two kinds of variables, in a way that if we
fix one of them (say, y), then the problem becomes much easier to solve. To
motivate our approach for variational inequalities presented below, we recall
that when solving the LP (2), Benders decomposition can be derived by ap-
plying the Dantzig-Wolfe decomposition algorithm to the dual of (2). Assume
for simplicity (and without loss of generality), that f(0) = 0. Then the dual
LP of (2) is given by
maxµr(0) + hµ, g(0) + h(0)i
s.t. f0(0) + [g0(0)]>µ= 0,
r0(0) + [h0(0)]>µ= 0,
µ≥0.
(5)
Note that if g0(0) ≡g0(·) is block-separable like in (3), then the transposed
Jacobian [g0(0)]>inherits the decomposable structure. Hence, without the con-
straint r0(0) + [h0(0)]>µ= 0, solving (5) would be easier. The situation is thus
similar to problem (1), i.e., it is suitable for the Dantzig-Wolfe decomposition.
In the considered dual setting, given at iteration ka Lagrange multiplier es-
timate yk
Mand using Lagrangian relaxation of the corresponding “difficult”
constraint r0(0) + [h0(0)]>µ= 0, the resulting Dantzig-Wolfe subproblem is
maxµr(yk
M) + hµ, g(0) + h(yk
M)i
s.t. f0(0) + [g0(0)]>µ= 0, µ ≥0.(6)
With the structure at hand of g0, the latter is easy to solve, to obtain a
subproblem solution µk+1
Sto be included in the master program. However,
instead of solving directly the LP (6), we solve its dual:
minxf(x) + r(yk
M)
s.t. g(x) + h(yk
M)≤0,(7)
obtaining an optimal Lagrange multiplier µk+1 associated to the constraint.
The latter is precisely the subproblem of the Benders decomposition scheme.
Recall that in the setting of (2), for the fixed y=yk
M, the subproblem (7)
is easy to solve. Let a solution of (7) be denoted by xk+1
S. At this stage, we
have already computed Lagrange multipliers µ0
S, µ1
S, . . . , µk
Sassociated to the
previous subproblems. We can then perform the Dantzig-Wolfe master step,
which consists in solving
maxµr(0) + hµ, g(0) + h(0)i
s.t. r0(0) + [h0(0)]>µ= 0,
µ∈conv{µ0
S, µ1
S, . . . , µk
S},
(8)
where convDstands for the convex hull of the set D. Solving (8) gives a La-
grange multiplier yk
Massociated to the constraint r0(0)+[h0(0)]>µ= 0, so that
4 Luna, Sagastiz´abal, and Solodov
we can continue with the next subproblem in the Dantzig-Wolfe framework.
However, again, instead of solving directly the LP (8), we solve its dual
min(y,t)t+r(y)
s.t. hµi
S, g(0) + h(y)i ≤ t, i = 0,1, . . . , k , (9)
which can be rewritten, by using optimality conditions of (7), as
min(y,t)t+r(y)
s.t. f(xi
S) + h[h0(yi−1
M)]>µi
S, y −yi−1
Mi ≤ t, i = 1,2, . . . , k. (10)
In summary, applying to the dual (5) of the original LP (2) the Dantzig-
Wolfe algorithm given by (6) and (8), is equivalent to solving iteratively (7)
and (9). The latter is precisely the Benders decomposition method for LPs [3].
We emphasize that viewing Benders algorithm as an application of the
Dantzig-Wolfe approach to the dual problem (as above), is not the only pos-
sible interpretation; see, e.g., [4, Chapter 11.1]. But this point of view has the
advantage of possibly using those ideas for other problem classes. And this
indeed would be our approach for deriving Benders decomposition for varia-
tional inequalities (VIs) in the sequel, by applying the Dantzig-Wolfe method
for VIs of [14] to the appropriately defined dual of the original VI.
To define the class of variational problems in question, let F:Rn⇒Rnbe
a set-valued mapping from Rnto the subsets of Rn, and let H:Rn→Rland
g:Rn→Rqbe continuous functions with convex components. Define the two
convex sets SH={x|H(x)≤0}and Sg={x|g(x)≤0}. Consider the
variational inequality problem VI(F , SH∩Sg) [10], which means to find
¯x∈SH∩Sgsuch that h¯w, x −¯xi ≥ 0
for some ¯w∈F(¯x) and all x∈SH∩Sg.(11)
As in the discussion of the Dantzig–Wolfe decomposition for the LP (1),
we assume that the VI (11) would be much easier if the constraints H(x)≤0
(the set SH) are not present. A Dantzig–Wolfe method for this family of prob-
lems had been introduced for single-valued monotone VIs in [7, 11]. A more
general framework, convergent under much weaker assumptions, is [14]. In par-
ticular, [14] is applicable to (single-valued) nonmonotone Fand multi-valued
(maximally monotone) F, allowing in addition a rich class of approximations
of Fand of the derivative of H, as well as inexact solution of subproblems.
The simplest form of Dantzig–Wolfe method for (11) comes from the follow-
ing consideration. Under an appropriate constraint qualification [20], solving
problem (11) is equivalent to finding (¯x, ¯µ) such that
(¯x, ¯µ)∈SH×Rq
+,¯µ⊥H(¯x),
¯xsolves VI(F(·)+[H0(·)]>¯µ, Sg),(12)
where the notation u⊥vmeans that hu, vi= 0. In view of (12), having a
current multiplier estimate µk
Mfor the H(x)≤0 constraint, we may consider
solving the subproblem
VI(F(·)+[H0(·)]>µk
M, Sg).(13)
Benders Decomposition for VIs 5
According to the discussion above, in the given context VI (13) is simpler than
the original (11). And (13) is indeed one of the possible algorithmic options.
However, it is not the only one. Algorithm 1 below solves subproblems with
the structure in (13), but allows useful approximations of Fand H0, as well as
proximal regularization of the subproblem operator (to induce solvability). We
refer the reader to [14] for details on all the possible options. Here, we shall
not discuss them in connection with Dantzig-Wolfe Algorithm 1; but some
possibilities will be given for the Benders scheme developed in the sequel.
Algorithm 1 (Dantzig-Wolfe Decomposition for VIs)
1. Choose x0
S∈Sg∩SH, such that H(x0
S)<0 if His not affine. Set x0
M=x0
S.
Choose µ0
M∈Rq
+and w0
M∈F(x0
M). Set k:= 0.
2. Subproblem solution: Choose an approximation Fk:Rn⇒Rnof F(·),
an approximation Hk:Rn→Rq×nof H0(·), and a positive (semi)definite
matrix Qk∈Rn×n. Find xk+1
S, an approximate solution of the problem
VI( ˆ
Fk, Sg),(14)
ˆ
Fk(x) = Fk(x)+[Hk(x)]>µk
M+Qk(x−xk
M).(15)
3. Master program: Choose finite set Xk+1 ⊂Sgcontaining {x0
S, . . . , xk+1
S}.
Find a solution xk+1
Mof the problem
VI(F, SH∩conv Xk+1),(16)
with the associated wk+1
M∈F(xk+1
M) and a Lagrange multiplier µk+1
Masso-
ciated to the constraint H(x)≤0.
4. Set k:= k+ 1 and go to Step 2.
Note that although the master problem (16) formally involves the difficult
constraint set SH, by the change of variables, it is parametrized by the unit
simplex associated to the convex hull of the set Xk+1 (which again often gives
a relatively easy problem). To alleviate potential computational burden, em-
pirical approximations in the Dantzig-Wolfe master program were considered
in [6], showing an improved performance for two energy market models.
In the subproblem step of the algorithm, the regularization matrix Qk
should be taken as zero if F(and then also Fk, for natural choices) is known
to be strongly monotone. If strong monotonicity does not hold then Qkshould
be positive definite, to guarantee that subproblems are solvable.
The rest of this paper is organized as follows. In Section 2, we discuss the
type of VIs to which our Benders decomposition approach is applicable, cite
some previous developments, and state our contributions. Section 3 is devoted
to subproblems of Benders decomposition, in particular showing how appro-
priate approximations of the data can reduce them to simple optimization
problems, and how further decomposition of them can be achieved in the case
of stochastic Nash games. The master problem of Benders decomposition is
stated in Section 4. The overall algorithm and its convergence analysis are
6 Luna, Sagastiz´abal, and Solodov
presented in Section 5. The paper finishes with some concluding remarks in
Section 6.
A few words on our terminology are in order. We say that the set-valued
mapping T:Rl⇒Rlis monotone if hu−v, x −yi ≥ 0 for all u∈T(x)
and v∈T(y) and all x, y ∈Rl. Such a mapping is further called maximally
monotone if its graph is not properly contained in the graph of any other
monotone mapping. Tis c−strongly monotone if there exists c > 0 such that
hu−v, x −yi ≥ ckx−yk2for all u∈T(x) and v∈T(y) and all x, y ∈Rl.
The mapping T:Rl⇒Rlis outer semicontinuous if for any sequences {xk},
{yk}such that {xk} → ¯xand {yk} → ¯ywith yk∈T(xk), it holds that
¯y∈T(¯x). Recall that if Tis either continuous or maximally monotone, then
it is outer semicontinuous. We say that a family of set-valued mappings {Tk}
is equicontinuous on compact sets if for every compact set Dand every > 0
there exists δ > 0 such that for any x, y ∈Dwith kx−yk< δ, for every kit
holds that dH(Tk(x), T k(y)) < , where dHis the Hausdorff distance between
the sets, defined by
dH(C, D) = inf{t > 0 : C⊂D+B(0, t) and D⊂C+B(0, t)},
B(0, t) being the closed ball centered at the origin with radius t.
2 Primal-dual variational inequality framework for Benders
decomposition
We start with specifying the VI structure (and its dual) suitable for Benders
decomposition. Consider the given (primal) problem
VI(FP, SP) (17)
with the following structure: FP:Rn×Rm⇒Rn×Rmhas the form
FP(x, z) = F(x)×G(z),
with F:Rn⇒Rn,G:Rm⇒Rmand
SP={(x, z)∈Rn×Rm:Ax +Bz ≤d},
where Aand Bare matrices of appropriate dimensions. We assume that after
fixing the value of the variable z, the problem
VI (F, SP(z)), SP(z) = {x∈Rn:Ax +Bz ≤d},
is much easier to solve than (17). As in the LP case discussed in Section 1,
this happens when Ahas block separable structure, something very common
in various applications. For the eventual convergence analysis, we shall as-
sume that Fand Gare outer semicontinuous mappings. The latter holds, in
particular, if they are maximally monotone (possibly multi-valued) [5, Propo-
sition 4.2.1], which is in particular the case (but not only) of subdifferentials
Benders Decomposition for VIs 7
of convex functions. For solvability of iterative problems involved in the con-
struction, one of Fand Gshould also either be surjective or its inverse has
to be surjective. The latter is a technical assumption needed to ensure the
maximal monotonicity of the mapping of the dual VI problem (defined in
(19) further below). When For Ghas domain or image bounded, then this
technical assumption is automatic [5, Corollary 4.5.1].
The only other Benders type method for VIs that we are aware of is the
one proposed in [12]; see [13] for an application. The differences between our
development and [12], and our contributions, are summarized as follows. First,
we consider VIs with a possibly multi-valued (maximally monotone) mapping,
whereas in [12] the mapping is single-valued monotone and, even more im-
portantly, it has a rather specific form, like (F(x), u, v) with Fcontinuous
and invertible and u, v constant vectors. Clearly, our setting is much more
general and covers far more applications. Moreover, as it will be seen below,
this generalization does not complicate too much the iterations of the algo-
rithm, i.e., the subproblems will still be computationally tractable. In fact,
the iterative subproblems in our method related to the multi-valued part, that
replaces the constant part in the previous work, are independent of the vari-
able xand can be solved relatively easily. Second, the existence of solutions
of primal subproblems in the previous work was an assumption. Here, we use
regularization to ensure solvability, and thus dispense with such assumptions.
Moreover, we allow approximations for the dual subproblem VI mapping, as
in the Dantzig-Wolfe algorithm in [14], while previous work required the use
of exact information. Apart from general importance of using approximations
in many real-world applications, this is of special significance here, because it
allows us to express the corresponding subproblem as a simpler minimization
problem (instead of a VI), if an appropriate type of approximation is chosen.
As in the LP case discussed in Section 1, the VI Benders method is defined
by applying the Dantzig-Wolfe technique to an appropriately defined dual
formulation. Because (17) has a special structure, it is possible to define the
dual problem by rearranging the corresponding rush-Kuhn-Tucker (KKT)-
type conditions. It turns out that the resulting dual problem fits well the
Dantzig-Wolfe decomposition scheme. Here, we have to mention that there
exist a good number of other ways of attaching a dual VI problem to a given
(primal) VI. This involves various degrees of abstraction, e.g., [1,2,9,16], among
others. All these developments have their purpose and usefulness. However,
they do not seem helpful for our task here, as they do not appear to connect
to decomposition ideas.
We thus start with defining our own dual problem for VI (17) as follows:
VI(FD, SD),(18)
where FD:Rn×Rm×Rp⇒Rn×Rm×Rpis given by
FD(w, ζ , µ) = F−1(w)×G−1(ζ)× {d},(19)
8 Luna, Sagastiz´abal, and Solodov
and
SD=
(w, ζ , µ)∈Rn×Rm×Rp:
w+A>µ= 0
ζ+B>µ= 0
µ≥0
.
In the above, for a set-valued mapping T:Rl⇒Rl, we denote its inverse at
u∈Rlby T−1(u) = {v∈Rl:T(v) = u}.
Again, if Ahas block-decomposable structure, the constraint w+A>µ= 0
also has this property, which means that the problem would be easier to deal
with if the constraint ζ+B>µ= 0 were to be removed. We then immediately
recognize that (18) is amenable to the Dantzig-Wolfe decomposition for VIs
in Algorithm 1.
But first, let us show that the original VI (17) and its dual given by (18)
are equivalent in a certain sense.
Proposition 1 For the data defined above, the following holds.
1. If the elements (¯x, ¯z)together with ¯w∈F(¯x)and ¯
ζ∈G(¯z)solve the primal
problem (17) with a Lagrange multiplier ¯µ, then ( ¯w, ¯
ζ, ¯µ)together with
¯x∈F−1( ¯w)and ¯z∈G−1(¯
ζ)solve the dual problem (18) with multipliers
(−¯x, −¯z, −A¯x−B¯z+d).
2. If ( ¯w, ¯
ζ, ¯µ)together with ¯x∈F−1( ¯w)and ¯z∈G−1(¯
ζ)solve the dual problem
(18) with multipliers (−¯α, −¯
β, −¯γ), then
(a) ¯α= ¯x,¯
β= ¯zand ¯γ=A¯x+B¯z−d,
(b) (¯x, ¯z)together with ¯w∈F(¯x)and ¯
ζ∈G(¯z)solve the primal problem
(17) with multiplier ¯µ.
Proof Since the constraints defining SPare linear, we have that the KKT type
conditions hold for any solution of the primal problem (17):
¯w+A>¯µ= 0,(20a)
¯
ζ+B>¯µ= 0,(20b)
0≤¯µ⊥A¯x+B¯z−d≤0.(20c)
This shows that ( ¯w, ¯
ζ, ¯µ)∈SD.
On the other hand, the following inclusion is immediate:
0∈
F−1( ¯w)
G−1(¯
ζ)
d
+
I
0
A
(−¯x) +
0
I
B
(−¯z) +
0
0
−I
(d−A¯x−B¯z).(21)
The latter, together with (20c), proves the first item.
The proof of the second item is analogous, reversing the roles of the primal
and dual variables. ut
Benders Decomposition for VIs 9
3 Benders subproblem
The full Benders decomposition algorithm for VIs is stated in Section 5 below;
we give here an initial dual view. As in the Dantzig-Wolfe approach, there
are two main blocks: the subproblem and the master program. In this section
we describe the subproblems that the Dantzig-Wolfe VI scheme would solve
if applied to the dual VI (defined above). The primal counterparts of these
Dantzig-Wolfe subproblems define the Benders subproblems in Section 5.
3.1 Dantzig-Wolfe in the dual
At iteration k, we have (wk
M, ζk
M, µk
M)∈SDwith xk
M∈F−1(wk
M) and zk
M∈
G−1(ζk
M) and a Lagrange multiplier estimate (−zk
M−θk
M) associated to the
constraint ζ+B>µ= 0. The multiplier estimate is expressed here as sum (or
difference) of two terms for technical reasons concerning its relationship with
the decision variables in the applications in Sections 3.2 and 3.3.
At the kth iteration, the Dantzig-Wolfe subproblem of Algorithm 1 applied
to the dual VI (18) consists in solving
VI( ˆ
Fk
D, SDS),(22)
where the feasible set SDSis defined by
SDS=((w, ζ , µ)∈Rn×Rm×Rp:w+A>µ= 0
µ≥0),(23)
and the VI mapping ˆ
Fk
Dis defined using some approximations F−1
kand G−1
kof
F−1and G−1, respectively, and some positive (semi)definite matrices Pk, Qk
and Rk. Specifically,
ˆ
Fk
D(w, ζ , µ) =
F−1
k(w)
G−1
k(ζ)
d
+
0
I
B
(−zk
M−θk
M) +
Qk(w−wk
M)
Rk(ζ−ζk
M)
Pk(µ−µk
M)
.(24)
The approximating functions Fkand Gkare chosen according to the re-
quirements of the Dantzig-Wolfe scheme in [14, Subsection 2.1]. In particular,
xk
M∈F−1
k(wk
M)⊂F−1(wk
M),(25)
and
zk
M∈G−1
k(ζk
M)⊂G−1(ζk
M).(26)
Next, note the following two important structural features. In the feasible
set (23) the variable ζis unconstrained, and in the VI mapping (24) the entry
corresponding to this variable is independent of the other variables. It then
follows that the subproblem (22) equivalently splits into the following two
independent steps:
10 Luna, Sagastiz´abal, and Solodov
1. Find ζ∈Rmsuch that
0∈G−1
k(ζ)−zk
M−θk
M+Rk(ζ−ζk
M).(27)
2. Solve
VI( ˆ
Fk
D2, SDS2),(28)
where
ˆ
Fk
D2(w, µ) = F−1
k(w) + Qk(w−wk
M)
d−B(zk
M+θk
M) + Pk(µ−µk
M),(29)
and
SDS2=((w, µ)∈Rn×Rp:w+A>µ= 0
µ≥0).(30)
At least in full generality, the two problems above have the disadvantage in
that they are defined in terms of the inverse operators that in practice could
be difficult to deal with. In fact, in most applications those inverses would
not be known explicitly. Fortunately, in the dual framework that we defined,
it is possible to solve (27) and (28) via their duals, not involving the inverse
operators. The following propositions show those dual relations.
Proposition 2 If zk+1
Stogether with ζk+1
S∈Gk(zk+1
S)solve the problem
0∈z−zk
M−θk
M+Rk(Gk(z)−ζk
M),(31)
then ζk+1
Ssolves (27) with zk+1
S∈G−1
k(ζk+1
S).
Also, the existence of solutions of (27) implies the existence of solutions
of (31). In particular, solutions exist if Gkis maximal monotone and Rkis
positive definite.
Proof The first assertion is obtained by direct inspection. The existence of
solutions is by [18, Theorem 5]. ut
Proposition 3 Let (xk+1
S, uk+1
S)be any solution of
V I(ˆ
Fk
P, SP(zk
M, θk
M)),(32)
where
ˆ
Fk
P(x, µ) = Fk(x)
(AQkA>+Pk)µ,(33)
and
SP(zk
M, θk
M) = {(x, µ) : Ax+B(zk
M+θk
M)≤d+(AQkA>+Pk)(µ−µk
M)}.(34)
Let wk+1
S∈Fk(xk+1
S)and the Lagrange multiplier µk+1
Sverify the KKT con-
ditions for (32) at its solution (xk+1
S, uk+1
S).
Then the following holds.
1. We have that (AQkA>+Pk)(µk+1
S−uk+1
S)=0; and, when Pkis symmetric
positive define, then µk+1
S=uk+1
S.
Benders Decomposition for VIs 11
2. The element (xk+1
S, µk+1
S)solves VI( ˆ
Fk
P, SP(zk
M, θk
M)) with wk+1
S∈Fk(xk+1
S)
and multiplier µk+1
S.
3. The element (wk+1
S, µk+1
S)solves (28) with xk+1
S∈F−1
k(wk+1
S)and multi-
pliers (−xk+1
S−Qk(wk+1
S−wk
M), d + (AQkA>+Pk)(µk+1
S−µk
M)−Axk+1
S−
B(zk
M+θk
M)).
Also, the existence of solutions of (28) implies the existence of solutions
of (32). In particular, solutions exist if Fkis maximal monotone, with the
following holding: {−A>µ:µ≥0} ∩ int (Fk(Rn)) 6=∅and Pk, Qkare positive
definite.
Proof Writing the KKT conditions corresponding to (32), which hold by the
linearity of constraints in this problem, we have that
0 = wk+1
S+A>µk+1
S,(35a)
0=(AQkA>+Pk)uk+1
S−(AQkA>+Pk)>µk+1
S,(35b)
0≤µk+1
S⊥Axk+1
S+B(zk
M+θk
M)−d−(AQkA>+Pk)(uk+1
S−µk
M)≤0.
(35c)
Then (35b) implies the first item of the proposition, which together with the
system above shows also the second item. The third item is obtained in a way
similar to Proposition 1; we omit the details.
The existence assertion follows by applying [18, Theorem 5] to (28). ut
In Dantzig-Wolfe decomposition for VIs, the role of using approximations
is (at least) two-fold. First, when Fis not monotone, using its appropriate
approximation (for example, the value of Fat the current point) would induce
monotonicity in the subproblem, which is a desirable feature both theoretically
and computationally. Second, when using Fitself yields a subproblem which
does not decompose further, an appropriate approximation may induce this
further decomposition. Again, we refer to [14] for details and discussions.
In the Benders approach above, we face some difficulties. Note that the
function F−1
kis defined as an approximation of F−1around wk
M. In practice,
it may be difficult to obtain enough information about F−1, and so choosing
Fkmay not be so easy. In the case when Fis monotone, we can choose Fkas F
itself. However, Fkwould not have further decomposable structure, if Fdoes
not have it. Also, there exist important classes of problems (e.g., Nash games),
where the corresponding operator Fis typically nonmonotone. In Sections 3.2
and 3.3, we develop suitable constructions for stochastic Nash games with risk
aversion and with smoothed AVaR risk measures, respectively.
The option that is always available is to take
F−1
k(w) = xk
M,for all w.
Observe that in this case, the subproblem (28) is equivalent to
VI(Φk(µ),Rp
+),(36)
12 Luna, Sagastiz´abal, and Solodov
where
Φk(µ) = d−Axk
M−B(zk
M+θk
M)+(AQkA>+Pk)(µ−µk
M).
The important issue is that, in this case, (36) is the optimality condition for
the following strongly convex (if Pkand Qkare symmetric positive definite)
quadratic programming problem with simple bounds:
min 1
2h(AQkA>+Pk)(µ−µk
M), µ −µk
Mi+hd−Axk
M−B(zk
M+θk
M), µi
s.t. µ≥0.
This approach is thus quite appealing, as there is a wealth of powerful software
to solve such simple problems. Also in this case, wk+1
S=−A>µk+1
S.
Observe that we can always choose the matrices Qkand Pkin a convenient
way, preserving any structure of the problem data. For example, as diagonal
matrices, or block diagonal. This is particularly important when the matrix
Ahas structure which we would like to exploit. Suitable choices maintain
the decomposability in the matrix AQkA>+Pk, thus allowing (36) to split
into smaller problems. Also, the set (34) splits according to the given pattern,
which makes solving (32) much easier. Of course, in (33) the mapping Fk(x)
may not be decomposable for some choices (the entry corresponding to µ
is clearly decomposable). However, even in that case we can still use special
methods in order to take advantage of the structure of the feasible set (e.g., the
parallel variable distribution coupled with sequential quadratic programming
[19], if we are in the optimization setting). Of course, it is desirable to have
Fkdecomposable. In [14] various decomposable monotone operators FP k that
approximate Faround xk
Mare stated, even when Fis nondecomposable or
nonmonotone. Taking such approximations, we can then define Fkas
Fk(x) = FP k(x) + Uk(x−xk
M),(37)
where Ukis a positive definite matrix chosen in such a way that Fkis still
decomposable and strongly monotone. Now, since Fkis injective we can ensure
that F−1
kapproximates F−1in the sense of (25). This way of defining Fkis
especially useful when we are dealing with nontrivial functions, as is the case
of risk-measures considered in the sequel.
3.2 Decomposition Scheme for Stochastic Nash Games
In this subsection, we describe in detail a decomposition scheme for Stochas-
tic Nash Games (see, e.g., [15]), resulting from the Benders approach above.
Assume we have Nplayers and Wscenarios. Each player i∈ {1, . . . , N }solves
min(qi,zi)Ii(zi, z−i) + Rifi
ω(qi
ω, q−i
ω)W
w=1
s.t. Aωqω+Bωz≤dω,for ω= 1, . . . , W, (38)
where (qi,zi) denote the decision variables of player i, while (q−i,z−i) are the
decision variables of the other players, and Iiand fi
ωare real-valued functions.
Benders Decomposition for VIs 13
In this kind of settings, zmodel “here and now” variables that must be decided
before any future event takes place, while qi
ωare “wait and see” variables which
depend on future uncertain events (scenario ω). That is why each player uses
a risk measure Rito hedge against uncertain events. Note that the scenario-
dependent variables are coupled by the risk measure and the constraints.
To compute a variational equilibrium of this game we need to solve
V I(F(q), G(z)),{(q, z) : Aq +B z ≤d)}
where q= (q1, . . . , qW),
A=
A1
A2
...
AW
, B =
B1
B2
.
.
.
BW
, d =
d1
d2
.
.
.
dW
and for ω= 1, . . . , W we have qω= (q1
ω, . . . , qN
ω),
G(z) = ∂ziIi(z)N
i=1
and
F(q) = ∂qi
ωhRifi
l(ql)W
l=1iN
i=1W
ω=1
.
The notation ∂φ stands for the Convex Analysis subdifferential.
The matrix Anaturally decomposes along scenarios, the risk measure Ri,
on the other hand, couples inter-scenarios variables. For this reason, it is im-
portant to construct approximations for Faround a point qMthat are decom-
posable along scenarios (and monotone). In order to do so, first note that
∂qi
ωhRifi
l(ql)W
l=1i=∂ωRifi
l(ql)W
l=1∂qi
ωfi
ω(qω).
For each scenario ω, the corresponding VI operator Fω(qω) giving a variational
equilibrium is
Fω(qω) = ∂qi
ωfi
ω(qω)N
i=1.
The decomposable monotone approximation of Fω(qω) around qMω
ˆ
Fω(qω) = ˆ
Fi
ω(qi
ω)N
i=1 (39)
can be defined in various ways, the simplest is the constant approximation
ˆ
Fi
ω(qi
ω) = Fi
ω(qi
Mω).
More sophisticated options can be obtained proceeding in two steps, as follows.
14 Luna, Sagastiz´abal, and Solodov
1. First, a monotone approximation Fωis built. This is a tricky part, since
building a good monotone approximation depends on the nature of the
problem at hand, to which it must be tailored accordingly. But general
approaches are also possible. For example:
–Fω(qω) = Fω(qMω)
–Fω(qω) = Fω(qMω)+Mω(qω−qMω), where Mωis some positive semidef-
inite matrix.
2. Next, the decomposable monotone operator ˆ
Fωis defined by
ˆ
Fi
ω(qi
ω) = Fω(qi
ω, q−i
Mω),for i= 1,2, . . . , N.
Using the constructions above, we build the monotone approximation
e
F(q) = e
Fi
ω(qi
ω)N
i=1W
w=1
of Fby setting
e
Fi
ω(qi
ω) =
∂ωRifi
l(qMl)W
l=1ˆ
Fi
ω(qi
ω) if ∂ωRifi
l(qMl)W
l=1≥0,
∂ωRifi
l(qMl)W
l=1ˆ
Fi
ω(qMi
ω) otherwise.
Taking block-diagonal matrices Qkand Pk,
Qk=
Qk1
Qk2...
QkW
and Pk=
Pk1
Pk2
...
PkW
,
the structure is preserved in the product below
AQkA>+Pk=
A1Qk1A1
>+Pk1
A2Qk2A2
>+Pk2
...
AWQkWAW
>+PkW
.
As a result, solving the Benders subproblem (32) is equivalent to solving W
smaller problems
V I(ˆ
Fk
Pω, SPω(zk
M, θk
M)),(40)
ω= 1, . . . , W , where
ˆ
Fk
Pω(qω, µω) = e
Fi
ω(qi
ω)N
i=1
(AωQkA>
ω+Pkω)µω!,(41)
and
SPω(zk
M, θk
M) = (qω, µω) : Aωqω+Bω(zk
M+θk
M)≤dω+
(AωQkωA>
ω+Pkω)(µω−µMk
ω).(42)
Benders Decomposition for VIs 15
3.3 Nash Game with Explicit Smoothed AVaR.
One special case associated to the stochastic Nash equilibrium is when the
risk measure Riin (38) is a smoothing of the average value-at-risk (AVaR),
see [15]. Specifically, let
Ri(u) = min
vivi+1
1−εi
Eσi
τi(uω−vi),
where σi
τ(·) is an appropriate smoothing of the plus-function max{0,·}. In this
case we can write (38) as
min(qi,zi,vi)Ii(zi, z −i) + vi+1
1−εi
Ehσi
τifi
ω(qi
ω, q−i
ω)W
ω=1 −vii
s.t. Aωqω+Bωz≤dω,for ω= 1, . . . , W, (43)
and the associated VI is
V I(F(q, v), G(z)),{(q , v, z):[A, 0] q
v+Bz ≤d)},
where v= (v1, v2, . . . , vN)∈RNand
F(q, v) =
∂qi
ω1
1−εi
Ehσi
τifi
l(ql)−viiN
i=1!W
ω=1
,
∂vivi+1
1−εi
Ehσi
τifi
ω(qω)−viiN
i=1!
=
Pω
1−εi
(σi
τi)0fi
ω(qω)−vi∂qi
ωfi
ω(qω)N
i=1!W
ω=1
,
1−1
1−εi
Eh(σi
τi)0fi
ω(qω)−viiN
i=1!.
(44)
Observe that variables viare coupling the scenario-dependent variables qω.
To overcome this difficulty, we can take the monotone approximation (39) of
Fω(qω) = ∂qi
ωfi
ω(qω)N
i=1. Using the approximation ˆ
Fω(qω) = ˆ
Fi
ω(qi
ω)N
i=1
defined in the previous section, we construct the monotone approximation
e
F(q, v) = ((( e
Fi
ω(qi
ω))W
ω=1)N
i=1,(e
Fi
v(vi))N
i=1) of F(q, v) by
e
Fi
ω(qi
ω) =
Pω
1−εi(σi
τi)0fi
ω(qMω)−vi
Mˆ
Fi
ω(qi
ω) if (σi
τi)0fi
ω(qMω)−vi
M≥0
Pω
1−εi(σi
τi)0fi
ω(qMω)−vi
Mˆ
Fi
ω(qMi
ω) otherwise,
(45)
16 Luna, Sagastiz´abal, and Solodov
and
e
Fi
v(vi)=1−1
1−εi
Eh(σi
τi)0fi
ω(qMω)−vii.
We are interested in the structure of the Benders subproblem (32) in this par-
ticular case. The corresponding feasible set (34) is given by elements (q, v, µ)
that satisfy
[A, 0] q
v+B(zk
M+θk
M)≤d+[A, 0] QkQ12
k
Q21
kQ22
kA>
0+Pk(µ−µMk)},
where the matrices Qij
kare chosen so that QkQ12
k
Q21
kQ22
kis positive definite. Next,
performing the matrix products that define this feasible set, we obtain that it
actually has the following form:
{(q, v, µ) : Aq +B(zk
M+θk
M)≤d+ (AQkA>+Pk)(µ−µMk)}.
In particular, we see that the variable vis free, and this set has decomposable
structure along scenarios. Even more, since the approximating operator e
F(q, v)
is decomposable by scenarios, and since the variable vis free, it turns out that
solving the subproblem (32) is equivalent to solving the smaller problems (40)
– (42), where VI operators are defined by (37) and (45). The output of these
subproblems gives the values of qS, µSthat will be required by the master
problem. Note also that formally, for solving “in full” the Benders subproblem
(32), since the variables viare free, we need also to solve the equation
e
Fi
v(vi) + U(vi−vi
M)=0.
Even though of easy solution (U > 0), in practice the equation is not solved,
because the master problem does not require that information.
4 Benders master program
As already mentioned, the full Benders decomposition for VIs is stated in
Section 5 further below. In Section 3, we discussed the Benders subproblems
block. In this section, we deal with the second block; in particular, we re-
late Benders master problem for the primal VI to the Dantzig-Wolfe master
program applied to the dual VI.
At iteration k≥0, given Xk+1 ={(wi
S, ζi
S, µi
S)}k+1
i=0 , the kth master prob-
lem consists in solving
VI(FD, SDM), SDM={(w, ζ , µ)∈conv Xk+1 :0=ζ+B>µ}.(46)
Under the technical assumptions stated in the beginning of Section 2, it holds
that FDis maximally monotone. Then, since SDMis compact, the master
problem (46) is solvable [18, Theorem 5].
Let us remind that the points (wi
S, ζi
S, µi
S), for i≥1, are computed by
solving the subproblems described in Section 3 above, whereas (w0
S, ζ0
S, µ0
S) is
Benders Decomposition for VIs 17
a feasible point of (18) chosen at the beginning of the algorithm (as prescribed
by the Dantzig-Wolfe framework in [14]).
Using the matrices
Wk+1 = [w0
S|w1
S|···|wk+1
S],
Zk+1 = [ζ0
S|ζ1
S|···|ζk+1
S],
Mk+1 = [µ0
S|µ1
S|···|µk+1
S],
problem (46) can be reformulated as
VI(FD4, SD4),(47)
where
FD4(w, ζ , α) = F−1(w)×G−1(ζ)× {M>
k+1d},
and SD4is the set of points (w, ζ , α)∈Rn×Rm×Rk+2 that satisfy the
following constraints:
w−Wk+1α= 0,(48a)
ζ+B>Mk+1α= 0,(48b)
ζ−Zk+1α= 0,(48c)
1−1>α= 0,(48d)
α≥0,(48e)
where 1is the vector of ones of the appropriate dimension.
It is worth to note that since all points (wi
S, ζi
S, µi
S) are feasible for (22),
we have that Wk+1 =−A>Mk+1 .
For the same reasons as above (i.e., the involvement of the likely difficult
inverse functions), instead of solving directly (47), we shall solve instead its
dual, described next.
Proposition 4 Defining
FP(x, z, β , θ) = F(x)×G(z)× {−1}×{0},(49)
and
SPk+1 ={(x, z, β , θ) : M>
k+1[Ax +Bz −d]≤ −1β−[Z>
k+1 +M>
k+1B]θ},(50)
we have that if (xk+1
M, zk+1
M, βk+1
M, θk+1
M)solves
VI(FP, SPk+1 ),(51)
with some wk+1
M∈F(xk+1
M),ζk+1
M∈G(zk+1
M)and some Lagrange multiplier
αk+1
M, then (wk+1
M, ζk+1
M, αk+1
M)solves VI(FD4, SD4) with xk+1
M∈F−1(wk+1
M),
zk+1
M∈G−1(ζk+1
M)and Lagrange multipliers −xk+1
M,−zk+1
M−θk+1
M,θk+1
M,βk+1
M
and M>
k+1[d−Axk+1
M−Bzk+1
M]−[Z>
k+1 +M>
k+1B]θk+1
M−1βk+1
M.
Also, VI(FP, SPk+1) has solutions if, and only if, VI(FD4, SD4) has solu-
tions.
Proof The proof is analogous to that of Proposition 1. ut
18 Luna, Sagastiz´abal, and Solodov
5 Benders decomposition algorithm for VIs and its convergence
analysis
We now state formally the algorithm and then analyze its convergence prop-
erties. The Benders decomposition for VI (17) follows the following pattern.
Algorithm 2 (Benders Decomposition for VIs)
1. Choose a feasible dual point (w0
M, ζ0
M, µ0
M)∈SD. Choose x0
M∈F(w0
M),
z0
M∈G(ζ0
M) and θ0
M∈Rm. Set w0
S=µ0
M,µ0
M=µ0
S,ζ0
M=ζ0
Sand k:= 0.
2. Subproblem solution: Choose mappings approximations Fk:Rn⇒Rn
and Gk:Rm⇒Rmof F(·) and G(·), and symmetric positive (semi)definite
matrices Qk∈Rn×n,Rk∈Rm×mand Pk∈Rp×p. Find the primal-dual
points (xk+1
S, zk+1
S) and (wk+1
S, ζk+1
S, µk+1
S) by solving the problems (27) or
(31), and (32) or (36), according to the approximation functions chosen.
3. Master program: Find (xk+1
M, zk+1
M, θk+1
M) and (wk+1
M, ζk+1
M, µk+1
M) by solv-
ing (51), with a Lagrange multiplier αk+1
Massociated to the constraint in
(50). Compute µk+1
M=Mk+1αk+1
M.
4. Set k:= k+ 1 and go to Step 2.
We proceed to analyze convergence properties of Algorithm 2. We start
with the associated convergence gap quantity. For the Dantzig-Wolfe method,
the gap of convergence ∆kis defined by [14, Eq (19)]. This quantity drives
convergence of the algorithm, closing the gap as it tends to zero. In the Benders
setting, this would be clear, for example, from Proposition 5 below (which
shows that as ∆ktends to zero, so is the distance between the master problems
and the subproblems solutions); see also the subsequent Theorem 1 for the
eventual consequences of this fact.
In the case under consideration, ∆kdefined by [14, Eq (19)] has the form
∆k=D(xk
M,−θk
M, d −Bzk
M−Bθk
M),(wk+1
S, ζk+1
S, µk+1
S)−(wk
M, ζk
M, µk
M)E
=hxk
M, wk+1
S−wk
Mi+h−θk
M, ζk+1
S−ζk
Mi
+hd−Bzk
M−Bθk
M, µk+1
S−µk
Mi,
(52)
where we have used the fact that
(xk
M, zk
M, d) + (0,−zk
M−θk
M,−Bzk
M−Bθk
M)∈ˆ
Fk
D(wk
M, ζk
M, µk
M),
which follows from (24).
Since (wk+1
S, ζk+1
S, µk+1
S)∈SDSand (wk
M, ζk
M, µk
M)∈SD, we further obtain
that
∆k=hAxk
M, µk
M−µk+1
Si+h−θk
M, ζk+1
S−ζk
Mi+hd−Bzk
M−Bθk
M, µk+1
S−µk
Mi
=hAxk
M+Bzk
M−d, µk
M−µk+1
Si+h−θk
M, ζk+1
S−ζk
Mi−hBθk
M, µk+1
S−µk
Mi
=hd−Axk
M−Bzk
M, µk+1
S−µk
Mi−hθk
M, ζk+1
S+B>µk+1
Si.
Benders Decomposition for VIs 19
Proposition 5 If for each kthe function ˆ
Fk
Dis chosen ck-strongly monotone,
then
ckk(wk+1
S, ζk+1
S, µk+1
S)−(wk
M, ζk
M, µk
M)k2+∆k≤0.(53)
In particular, ∆k≤0for every k.
Proof Using the ck-strong monotonicity of ˆ
Fk
D, since
(xk
M,−θk
M, d −Bzk
M−Bθk
M)∈ˆ
Fk
D(wk
M, ζk
M, µk
M)
and since (wk+1
S, ζk+1
S, µk+1
S) solves VI( ˆ
Fk
D, SDS) with some
yk+1
S∈ˆ
Fk
D(wk+1
S, ζk+1
S, µk+1
S),
we have that
ckk(wk+1
S, ζk+1
S, µk+1
S)−(wk
M, ζk
M, µk
M)k2
≤ hyk+1
S−(xk
M,−θk
M, d −Bzk
M−Bθk
M),(wk+1
S, ζk+1
S, µk+1
S)−(wk
M, ζk
M, µk
M)i.
We then further obtain
ckk(wk+1
S, ζk+1
S, µk+1
S)−(wk
M, ζk
M, µk
M)k2
≤ hyk+1
S,(wk+1
S, ζk+1
S, µk+1
S)−(wk
M, ζk
M, µk
M)i − ∆k.
Using the latter relation and (wk
M, ζk
M, µk
M)∈SDS, (53) follows. ut
We are now in position to state convergence properties of Benders decom-
position of VIs with the given structure. Theorem 1 below assumes that F
and Gare outer semicontinuous (the definition is given in the end of Sec-
tion 1), which is the only property used in the proof, if the existence of solu-
tions/iterations is a given. Outer semicontinuity is in fact automatic from the
initial assumptions stated in the beginning of Section 2, which also guarantee
the existence of solutions of all the (sub)problems along the iterations. We
also assume the equicontinuity (the definition is given in the end of Section 1)
of the approximating families {F−1
k}and {G−1
k}. Again, there is a number of
ways to ensure the latter. Constant approximations is one option. If first-order
(Newtonian) approximations are used for single-valued smooth data, choosing
bounded {H−1
Fk}does the job. Finally, if the exact information Fand Gis em-
ployed (no approximations) in the single-valued case, the continuity of those
functions is sufficient.
Theorem 1 Suppose that Fand Gare outer semicontinuous (which holds,
in particular, if they are maximally monotone). For the iterative sequences
generated by Algorithm 2, the following holds.
1. If the sequences {µk+1
S},{zk+1
S},{zk
M}and {θk
M}are bounded and if the
family of matrices {Rk}is uniformly positive definite, then the sequences
{(wk+1
S, ζk+1
S, µk+1
S)}and {(wk
M, ζk
M, µk
M)}are bounded.
20 Luna, Sagastiz´abal, and Solodov
2. If the sequences {xk
M},{µk+1
S},{zk+1
S},{zk
M}and {θk
M}are bounded, the
family of matrices {Rk}is uniformly positive definite, and the approxima-
tions {ˆ
Fk
D}are chosen monotone, then
lim
k→∞
∆k= 0.
In particular, if the elements of {ˆ
Fk
D}are chosen uniformly strongly mono-
tone, then
lim
k→∞
k(wk+1
S, ζk+1
S, µk+1
S)−(wk
M, ζk
M, µk
M)k= 0.(54)
3. Suppose that condition (54) holds; and the sequences {Pk},{Qk},{Rk},
{(wk+1
S, ζk+1
S, µk+1
S)}and {(xk+1
S, zk+1
S)}are bounded. Then, if the approxi-
mations {F−1
k}and {G−1
k}are equicontinuous on compact sets, every clus-
ter point of {(xk+1
S, zk+1
S)}is a solution of VI (17).
Proof 1. Let {µk+1
S}be bounded. Since µk
M∈conv({µj
S}), it follows that the
sequence {µk
M}is bounded. Furthermore, as
wk+1
S=−A>µk+1
S, wk
M=−A>µk
M, ζk
M=−B>µk
M,
it follows that the sequences {wk+1
S}and {(wk
M, ζk
M, µk
M)}are bounded.
Now boundedness of {ζk+1
S}follows from
zk+1
S−zk
M−θk
M+Rk(ζk+1
S−ζk
M)=0
and the uniform positive definite property of the family {Rk}.
2. By the first item, we have that the sequences {(wk+1
S, ζk+1
S, µk+1
S)}and
{(wk
M, ζk
M, µk
M)}are bounded.
Using (53) with ck= 0 (i.e., monotonicity instead of strong monotonicity),
we have that ∆k≤0. Hence,
¯
∆= lim inf
k→∞
∆k≤lim sup
k→∞
∆k≤0.
We take a subsequence {∆kj}such that limj→∞ ∆kj=¯
∆. Without loss of
generality, we can assume convergence of the corresponding subsequences,
in particular: {(wkj
M, ζkj
M, µkj
M)} → ( ¯w, ¯
ζ, ¯µ), {(xkj
M, zkj
M, θkj
M)} → (¯x, ¯z, ¯
θ),
{(wkj+1
S, ζkj+1
S, µkj+1
S)} → ( ˆw, ˆ
ζ, ˆµ). Then from (52), we have that
lim
j→∞
∆kj=¯
∆=D(¯x, −¯
θ, d −B( ¯z+¯
θ)),( ˆw, ˆ
ζ, ˆµ)−( ¯w, ¯
ζ, ¯µ)E.
Consider the problem that results from VI(FD, SDM) given by (46), after
relaxing the constraint ζ+B>µ= 0 using the multiplier (−zki
M−θki
M). Fix
any index j. Since (wki
M, ζki
M, µki
M) is a solution in (46), and recalling the
definition (19) of FD, we have that
*
xki
M
zki
M
d
+
0
I
B
(−zki
M−θki
M),
w
ζ
µ
−
wki
M
ζki
M
µki
M
+≥0,
Benders Decomposition for VIs 21
for any (w, ζ, µ)∈SDM. Note further that for every i>j, we have that
(wkj+1
S, ζkj+1
S, µkj+1
S)∈SDM. Therefore,
D(xki
M,−θki
M, d −B(zki
M+θki
M)),(wkj+1
S, ζkj+1
S, µkj+1
S)−(wki
M, ζki
M, µki
M)E≥0,
and passing onto the limit as i→ ∞, we obtain that
D(¯x, −¯
θ, d −B( ¯z+¯
θ)),(wkj+1
S, ζkj+1
S, µkj+1
S)−( ¯w, ¯
ζ, ¯µ)E≥0.
Passing onto the limit again, now as j→ ∞, we obtain that
¯
∆=D(¯x, −¯
θ, d −B( ¯z+¯
θ)),( ˆw, ˆ
ζ, ˆµ)−( ¯w, ¯
ζ, ¯µ)E≥0,
which shows that limk→∞ ∆k= 0. Finally, (53) implies the last assertion
of this item.
3. Suppose that (¯x, ¯z) is an accumulation point of {(xk+1
S, zk+1
S)}and that
the subsequence {(xkj+1
S, zkj+1
S)}converges to it as j→ ∞. Since
zkj+1
S−zkj
M−θkj
M+Rkj(ζkj+1
S−ζkj
M) = 0,
using the stated hypotheses we conclude that
lim
j→∞(zkj
M+θkj
M) = ¯z .
On the other hand, taking into account (54) and passing onto further sub-
sequences if necessary, we can assume that
lim
j→∞
wkj+1
S= lim
j→∞
wkj
M= ¯w,
lim
j→∞
ζkj+1
S= lim
j→∞
ζkj
M=¯
ζ,
lim
j→∞
µkj+1
S= lim
j→∞
µkj
M= ¯µ.
Since the families of approximations {F−1
k}and {G−1
k}are equicontinuous
on compact sets, there exists, for each t, some kjtsuch that for every kit
holds that
dHF−1
k(wkjt+1
S), F −1
k(wkjt
M)<1
t.
In particular, there exists ˆxkjt
M∈F−1
k(wkjt
M)⊂F−1(wkjt
M) such that
kxkjt+1
S−ˆxkjt
Mk<1
t.
Hence,
lim
t→∞ ˆxkjt
M= ¯x.
Then, since Fis outer semicontinuous, we conclude that ¯x∈F−1( ¯w). In
a similar way we can conclude that ¯z∈G−1(¯
ζ). Also, note that since
22 Luna, Sagastiz´abal, and Solodov
(wkj+1
S, ζkj+1
S, µkj+1
S)∈SDS, we have that ( ¯w, ¯
ζ, ¯µ)∈SDS, and since
(wkj
M, ζkj
M, µkj
M)∈SDM, also ( ¯w, ¯
ζ, ¯µ)∈SDM. Thus ( ¯w, ¯
ζ, ¯µ)∈SD.
Finally, since (wkj+1
S, µkj+1
S) solves (28) with xkj+1
S∈F−1
kj(wkj+1
S), we have
that for every (w, µ)∈SDS2the following inequality holds:
hxkj+1
S+Qkj(wkj+1
S−wkj
M), w −wkj+1
Si
+hd−B(zkj
M+θkj
M) + Pkj(µkj+1
S−µkj
M), µ −µkj+1
Si ≥ 0.(55)
Thus, passing onto the limit as j→ ∞, we obtain that
h¯x, w −¯wi+hd−B¯z, µ −¯µi ≥ 0,
that is,
h¯x, w −¯wi+hd, µ −¯µi+h¯z, −¯
ζ−B>µi ≥ 0.
So, for each (w, ζ, µ)∈SDwe have that
h¯x, w −¯wi+hd, µ −¯µi+h¯z, ζ −¯
ζi ≥ 0.
Since also ¯x∈F−1( ¯w) and ¯z∈G−1(¯
ζ), we conclude that ( ¯w, ¯
ζ, ¯µ) is a
solution of VI(FD, SD). The latter implies, by Proposition 1, that (¯x, ¯z) is
a solution of (17).
ut
6 Concluding remarks
Whereas Dantzig-Wolfe decomposition deals with coupling constraints (a set-
ting that fits naturally VIs arising from generalized Nash equilibrium problems,
for example), the Benders decomposition algorithm is intended for VIs where
we recognize the existence of “coupling variables” (denoted by zin this work).
These coupling variables have the property that whenever they are fixed to
some values, the resulting problem on the remaining variables (denoted by qin
this work), becomes much easier to solve. This frequently occurs when the vari-
ables qhave a decomposable structure that allows the problem to be split in
several smaller ones. This is the case in a wide range of diverse applications;
for example: power management, investment planning, capacity expansion,
etc. (see [17]). To deal with such decomposable structures, we derived a fairly
general Benders decomposition scheme for VIs by applying Dantzig-Wolfe de-
composition to a suitably defined dual VI, and then passing back to the primal
space. Among the specific contributions of our development is the ability to
handle multi-valued mappings (previously only rather specific single-valued
case was considered), the possibility to further decompose the subproblems
using appropriate approximations of the VI data, as well as the possibility to
reduce the subproblems to simple optimization problems. Finally, such devel-
opments also open the possibility of combining Benders decomposition (say,
along scenarios) with Dantzig-Wolfe decomposition (say, along the players in
Benders Decomposition for VIs 23
the game). As an example, we considered the case of the Stochastic Nash
Equilibrium Problem. There, when using the Benders decomposition scheme,
at each iteration we fix the variable zto zk
Mand solve the subproblem (32).
This problem can be split in smaller problems, along scenarios. Each of these
smaller problems constitutes a generalized Nash equilibrium problem where
the uncertainty has realized. If those smaller problems are still too hard to
solve because of their size, they can be decomposed further, this time along
the agents, via the Dantzig-Wolfe algorithm.
Acknowledgments
The authors thank the two referees for their useful comments.
All the three authors are supported by the FAPERJ Grant 203.052/2016
and by FAPERJ Grant E-26/210.908/2016 (PRONEX–Optimization). The
second author’s research is also supported by CNPq Grant 303905/2015-8,
by Gaspard Monge Visiting Professor Program, and by EPSRC grant num-
ber EP/ R014604/1 (Isaac Newton Institute for Mathematical Sciences, pro-
gramme Mathematics for Energy Systems). The third author is also supported
by CNPq Grant 303724/2015-3, and by Russian Foundation for Basic Research
Grant 19-51-12003 NNIOa.
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