2.5 Review on some optimization problems
2.5.2 Split variational inequality problem
where
θn:=
(minn
θ,∥x τn
n−xn−1∥
o
, if xn ̸=xn−1
θ, otherwise.
(2.5.7) Step 2: Compute
yn=PC(wn−λnAwn).
If wn =yn or Ayn = 0, then Stop and yn is a solution of the VIP. Otherwise, go to Step 3.
Step 3: Compute
zn =PTn(wn−λnAyn) where
Tn:={x∈ H |⟨wn−λnAwn−yn, x−yn⟩ ≤0}, λn :=δℓmn and mn is the smallest nonnegative integer m satisfying
δℓm⟨Ayn−Awn, yn−zn⟩ ≤ η 2
h∥wn−yn∥2+∥yn−zn∥2i . Step 4: Calculate
xn+1 =αnf(xn) + (1−αn)zn. Set n :=n+ 1 and go to Step 1.
Under some mild conditions, the authors obtained strong convergence result for the pro- posed algorithm.
[59]. This product space formation has some limitations which include: the difficulty en- countered when computing the projection onto some new product subspace formulations, the difficulty encountered when translating the method back to the original spaces H1 and H2, and the fact that it does not fully exploit the splitting structure of the SVIP (1.2.4)-(1.2.5) (see, for example [63, Page 12]). To circumvent these limitations, Censor et al. [63] proposed a projection-based method that does not require any product space formulation. This makes the projection-based method easier to implement. The pro- posed projection-based method is presented as follows: For x1 ∈ H1, the sequence {xn}is generated by
xn+1 =PC(I −λA)(xn+ηT∗(PQ(I−λF)−I)T xn), n≥1, (2.5.9) where η∈ 0,L1
with Lbeing the spectral radius of T∗T and T∗ is the adjoint ofT. The identity operator is denoted byIandPC, PQ are metric projections ontoC,Q,respectively.
They obtained a weak convergence of the sequence{xn}generated by (2.5.9) to a solution of (1.2.4)-(1.2.5) under the condition that the solution set of problem (1.2.4)-(1.2.5) is nonempty, A, F are L1, L2-co-coercive operators respectively, λ ∈ [0,2α], where α :=
min{L1, L2}, and for allx which are solutions of (1.2.4), Ay, PC(I−λA)(y)−x
≥0, ∀ y∈ H. (2.5.10)
Observe that Algorithm (2.5.9) does not require the product space formation, thus it fully exploits the attractive splitting structure of the SVIP (1.2.4)-(1.2.5). However, the authors obtained a weak convergence of this method under some strong assumptions which are the fact that both mappings are required to be co-coercive and (2.5.10). Many authors have studied several methods which do not rely on assumption (2.5.10) for solving SVIP and other related problems (see for example [131, 145, 183]), but their methods also relied on the co-coercivity of the cost operators.
In a quest to overcome these limitations, Tian and Jiang [243] proposed an iterative method and they defined it as follows:
Algorithm 2.5.8.
yn=PC(xn−τnA∗(I−S′)T xn) vn=PC(yn−λnAyn)
xn+1 =PC(yn−λnAvn)
(2.5.11)
where {τn} ⊂ [a, b], {λn} ⊂ [c, d] for some c, d∈ 0,L1
, T : H1 → H2 is a bounded linear operator, S′ : H2 → H2 is a nonexpansive mapping and A : C → H1 is a mono- tone and Lipschitz continuous mapping. They obtained a weak convergence result of the sequence generated by Algorithm (2.5.11) to the following problem; Find
x∗ ∈ C such that ⟨Ax∗, x−x∗⟩, ∀ x∈ C such that T x∗ ∈F(S′) (2.5.12) where F(S′) is the set of fixed points of S′. Since strong convergence results are more desirable and more applicable than the weak convergence results in infinite dimensional spaces, there is need to develop algorithms that generate strong convergence results.
Tian and Jiang [244] modified Algorithm (2.5.11) into the following viscosity method and they defined it as follows:
Algorithm 2.5.9.
yn=PC(xn−τnA∗(I−S′)T xn) vn=PC(yn−λnAyn)
tn=PC(yn−λnAvn)
xn+1 =αnh(xn) + (1−αn)tn
(2.5.13)
where{τn} ⊂[a, b], {λn} ⊂[c, d] for somec, d∈ 0,L1
, {αn} ⊂(0,1), T :H1 → H2 is a bounded linear operator, S′ : H2 → H2 is a nonexpansive mapping, h is a contrac- tion mapping and A : C → H1 is a monotone and Lipschitz continuous mapping. We observe that the conditions on the underlying operators in Algorithms (2.5.11)-(2.5.13) do not require the strong co-coercive assumption but involve computation of many pro- jections which makes them computationally expensive and may affect the efficiency of Algorithms (2.5.11)-(2.5.13). Algorithms (2.5.11)-(2.5.13) can be used to solve the SVIP (1.2.4)-(1.2.5) if we set S′ = PQ(I − λA) and let A be co-coercive. This implies that when solving the SVIP (1.2.4)-(1.2.5), these methods (Algorithm (2.5.11)-(2.5.13)) still relies on the co-coercive assumption on the underlying operator A. To weaken the condi- tion on the underlying operators, Pham et al. [206] combined the Halpern method with the subgradient extragradient method for solving the SVIP (1.2.4)-(1.2.5) in real Hilbert spaces when the underlying operators A and F are pseudomonotone and Lipschitz con- tinuous. The authors obtained a strong convergence result of their proposed method (see Appendix (4.3.16)) to a solution of the SVIP (1.2.4)-(1.2.5) under the following condi- tions: lim sup
n→∞
⟨A(xn), y−yn⟩ ≤ ⟨A(¯x), y −y⟩,¯ for every sequences {xn} and {yn} in H1
converging weakly to ¯x and ¯y respectively, and lim sup
n→∞
⟨F(cn), d−dn⟩ ≤ ⟨F(¯c), d−d⟩¯ for every sequences {cn} and {dn} inH2 converging weakly to ¯cand ¯d respectively.
To accelerate the convergence of iterative methods for solving the SVIP (1.2.4)-(1.2.5), the inertial and relaxation techniques were employed. This dynamical approach leads to the following Relaxed Inertial Tseng’s Forward-Backward-Forward (RITFBF) with parameter ϕn=θh2n, ∀n≥1 (see, for example [25]):
Algorithm 2.5.10.
wn =xn+αn(xn−xn−1) yn =PC(wn−λnAwn)
xn+1 = (1−ϕn)wn+ϕn(yn+λn(Awn−Ayn)).
(2.5.14)
When ϕn = 1 and αn = 0 for all n ≥ 1, Algorithm (2.5.14) reduces to the well-known Tseng’s forward-backward-forward method [253], which is known to converge weakly to a solution of the classical VIP.
Given the importance of these two techniques (inertial and relaxation), it is of interest to consider their combination for solving optimization problems. Attouch and Cabot [26,27]
employed both techniques into proximal algorithms (resulting to Relaxed Inertial Proxi- mal Algorithms (RIPA)) for solving convex minimization and null point problems. Also, Iutzeler and Hendrickx [129] studied the influence of inertial and relaxation techniques on the numerical performance of iterative schemes.
Remark 2.5.3. Several of the existing methods for solving the SVIP require that the problem be transformed into a product space problem which does not fully exploit the splitting structure of the SVIP. Therefore, there is need to develop efficient algorithms for solving the SVIP which does not require the product space formulation and converges to the minimum-norm solution of the SVIP. Note that in many practical problems with physical and engineering backgrounds, it is very important if the minimum-norm solutions of such problems can be found (see for example, [121, Section 5] and [54,261]).