In this section, we assume thatX is finite-dimensional and we consider the minimization problem on Xad(2.3), that we recall hereafter.
min
x∈XadJ (x).
In the rest of this section, we assume that J is differentiable and pseudoconvex on an open convex subset C that containsXad.
We consider the algorithm built on the auxiliary problem principle where the auxiliary function and the step size may vary with the iteration index k, as presented in [13]. Let a sequence of differen-tiable and strongly convex functions {Mk, k ∈ N} and positive numbers {εk, k ∈ N} be chosen.
Algorithm 4.4.1. (i) Start from some initialx0inX.
(ii) At stagek, knowing xk, computexk+1by solving the auxiliary problem min
x∈Xad
Mk(x) +εkJ0(xk) − (Mk)0(xk), x
. (4.7)
(iii) Stop ifkxk+1− xkk is below some threshold. Otherwise, go back to (ii) with k ← k + 1.
Assumptions 4.4.1.
(i) J is coercive, lower semicontinuous onXadand there exists u ∈Xadwhere J is finite.
(ii) J0is Lipschitz continuous with constant L onXad. (iii) J is pseudoconvex on C.
(iv) (Mk)0 is strongly monotone with constant bk and Lipschitz continuous with constant Bk on Xad. Moreover, ∃b > 0 and B > 0 such that, ∀k ∈ N, bk ≥ b and Bk≤ B.
Theorem 4.4.1. Assume that J is coercive, lower semicontinuous onXad, and that there existsu ∈ XadwhereJ is finite; then there exists a solution x∗to (2.3). In addition, if(Mk)0is strongly monotone with constantbk, with bk ≥ b > 0, ∀k ∈ N, on Xad, then there exists a unique solutionxk+1to the auxiliary problem (4.7).
Moreover, ifJ0is Lipschitz continuous with constantL onXad, and if theεkare such that
∀k ∈ N, α < εk< 2bk/(L + β), where α > 0, β > 0, (4.8) then the sequence{J (xk)} is strictly decreasing, unless xk is a solutionx∗ to (2.3) for somek; the sequence{xk} is bounded and kxk+1− xkk converges to zero.
In addition, if we assume that(Mk)0is Lipschitz continuous with constantBkonXad, that∃B > 0 such that,∀k ∈ N, Bk≤ B, and that J is pseudoconvex on C, then every cluster point of the sequence {xk} is a solution of problem (2.3).
The proof of this theorem can be found in [65].
Remark 4.4.1. IfX is not supposed to be finite-dimensional in Theorem 4.4.1, then in order to have the optimality condition (2.4), it seems to us necessary to assume that the gradientJ0(x) is weakly continuous, which is a strong condition. This is the reason why the convergence results in Theorem 4.4.1 are limited to the finite-dimensional case.
Remark 4.4.2. If, ∀k ∈ N, εk = ε in Algorithm 4.1.1 (respectively, Algorithm 4.4.1), then the con-dition onε in Theorem 4.2.1 and Corollary 4.3.1 (respectively, Theorem 4.4.1) may be simplified by settingα = β = 0.
4.5 Conclusions
In this paper, we proved the convergence of the auxiliary problem method in Hilbert spaces, when the operator involved in the variational inequality problem has the pseudo-Dunn property or is strongly pseudomonotone. We also proved the convergence in finite-dimension when the operator is the gradi-ent of a pseudoconvex function. The counterexample of the operator of rotation by π/2 shows that this cannot be extended to general pseudomonotone and even monotone variational inequalities. In [4], we show how the Moreau-Yosida regularization and the progressive regularization method (developed in [1]) can still be extended to pseudomonotone variational inequalities involving general operators.
Chapter 5
Convergent Algorithm Based on
Progressive Regularization for Solving Pseudomonotone Variational Inequalities
This chapter is devoted to the paper: Na¨ıma El Farouq, Convergent Algorithm Based on Progressive Regularization for Solving Pseudomonotone Variational Inequalities, JOTA, Vol. 120, No. 3, pp.
455–485, 2004.
In this paper, we extend the Moreau-Yosida regularization of monotone variational inequalities to the case of weakly monotone and pseudomonotone operators. With these properties, the regularized operator satisfies the pseudo-Dunn property with respect to any solution of the variational inequality problem. As a consequence, the regularized version of the auxiliary problem algorithm converges.
In this case, when the operator involved in the variational inequality problem is Lipschitz continuous (a property stronger than weak monotonicity) and pseudomonotone, we prove the convergence of the progressive regularization algorithm introduced in [1, 36].
5.1 Moreau-Yosida Regularization and Progressive Regularization of Monotone Variational Inequalities
In this section, we recall the Moreau-Yosida regularization, and the progressive regularization of monotone variational inequalities as introduced in [1, 36]. Let Ψ be a monotone operator and in addition assume that it satisfies one of the following conditions:
(a) Ψ is single-valued and hemicontinuous.
(b) Ψ is maximal monotone and int(D(Ψ)) ∩Xad 6= ∅.
(c) Ψ is maximal monotone and D(Ψ) ∩ int(Xad) 6= ∅.
Here, D(.) denotes the domain of the operator and int denotes the interior of the set.
The Moreau-Yosida regularized operator Ψγof the operator Ψ overXadis defined as follows:
Ψγ(x) = γ(x −y(x)) ,e (5.1)
where γ is a positive number andy(x) is the unique solution of the following variational inequality:e y(x) ∈e Xad, hΨ(y(x)) + γ(e ey(x) − x), y −y(x)i ≥ 0 ,e ∀y ∈Xad. (5.2)
The existence and uniqueness ofy(x) are ensured by the fact that the operator Ψ+γI satisfies also onee of the three items above and is strongly monotone; see for example [63]. Then, solving the variational inequality (1.1) is equivalent to finding a zero of Ψγ, that is, to solving
find x∗ ∈X : Ψγ(x∗) = 0 . (5.3)
The regularization algorithm is the following:
Algorithm 5.1.1. At stage k, knowing xk, computexk+1as the solution of the auxiliary problem min
We recall hereafter the progressive regularization algorithms that we investigate:
Algorithm 5.1.2. Algorithm Parallel in (x, y). At stage k, knowing xk andyk, computexk+1 and yk+1as the respective solutions of the auxiliary problems
minx∈X
Algorithm 5.1.3. Sequential Algorithm, y before x. In the previous algorithm, (5.7) is solved first andykis replaced in (5.6) byyk+1,
Up to the fact that two different step sizes are used, Algorithm 5.1.2 may be viewed as the basic algorithm applied to a composite operator (this algorithm is presented below) defined overX × X, renamedU × V hereafter. Let Γ denote this operator. Then, clearly
Γ(u, v) =
Basic Parallel Algorithm. By choosing an additive auxiliary function M , that is, M (u, v) = K(u) + L(v),
and by using two different step sizes, the basic algorithm can be reformulated as follows.
Algorithm 5.1.4. Parallel Algorithm. At stagek, knowing ukandvk, solve min