Convergence analysis

4.3 On split variational inequality problem beyond monotonicity

4.3.2 Convergence analysis

Then, we can see that

η_n ≤ ∥T w_n−z_n∥²−ϵ 1 +∥T^∗(T w_n−z_n)∥² 1 +∥T^∗(T wn−zn)∥² , which implies that

ηn+ηn∥T^∗(T wn−zn)∥² ≤ ∥T wn−zn∥²−ϵ−ϵ∥T^∗(T wn−zn)∥². This further gives (4.3.15) and consequently (4.3.16).

As we shall see in our convergence analysis, (4.3.15) and (4.3.16) play crucial role in our proofs. Therefore, one can either choose the stepsize in Step 3 or that in (4.3.17), to ensure the convergence of Algorithms 4.3.2 and 4.3.3.

Combining (4.3.21) and (4.3.22), we have

⟨T w_n−y_n−λ(F T w_n−F y_n), y_n−T p⟩ ≥0. (4.3.23) From Step 2 and (4.3.23), we obtain

⟨T w_n−T p, r_n⟩ = ⟨T w_n−y_n, r_n⟩+⟨y_n−T p, r_n⟩

= ⟨T wn−yn, rn⟩+⟨yn−T p, T wn−yn−λ(F T wn−F yn)⟩

≥ ⟨T w_n−y_n, r_n⟩, which implies that

−⟨T w_n−T p, r_n⟩ ≤ −⟨T w_n−y_n, r_n⟩. (4.3.24) Since z_n =T w_n−γ₂β_nr_n, we have that ||β_n·r_n||² =γ₂⁻²||z_n−T w_n||².

Thus, from Lemma 2.1.1, (4.3.5) and (4.3.24) (nothing that β_n and γ₂ are nonnegative), we obtain

∥z_n−T p∥² =∥T w_n−γ₂β_nr_n−T p∥²

=∥T w_n−T p∥²+γ₂²β_n²∥r_n∥²−2γ₂β_n⟨T w_n−T p, r_n⟩

≤ ∥T w_n−T p∥²+γ₂²β_n²∥r_n∥²−2γ₂β_n⟨T w_n−y_n, r_n⟩

=∥T wn−T p∥²+γ₂²β_n²∥rn∥²−2γ2βn·βn∥rn∥²

=∥T w_n−T p∥²−γ₂(2−γ₂)∥β_n·r_n∥²

=∥T w_n−T p∥²−γ₂⁻¹(2−γ₂)||z_n−T w_n||². (4.3.25) Also, we obtain from Step 3, Lemma 2.1.1 and (4.3.25) that

∥b_n−p∥² =∥w_n−p∥²+η²_n∥T^∗(z_n−T w_n)∥²+ 2η_n⟨w_n−p, T^∗(z_n−T w_n)⟩

=∥w_n−p∥²+η²_n∥T^∗(z_n−T w_n)∥²+ 2η_n⟨T w_n−T p, z_n−T w_n⟩

=∥w_n−p∥²+η²_n∥T^∗(z_n−T w_n)∥² +η_n

∥z_n−T p∥²− ∥T w_n−T p∥²− ∥z_n−T w_n∥²

≤ ∥w_n−p∥²+η_n²∥T^∗(z_n−T w_n)∥²−η_n∥z_n−T w_n∥² (4.3.26)

=∥w_n−p∥²−η_n

∥z_n−T w_n∥²−η_n∥T^∗(z_n−T w_n)∥² . Thus, by (4.3.15), we obtain that

∥b_n−p∥² ≤ ∥w_n−p∥², which implies from (4.3.20) that

∥b_n−p∥ ≤ ∥w_n−p∥ ≤ ∥x_n−p∥+δ_nM₁, ∀n≥1. (4.3.27) Following similar argument used in obtaining (4.3.25), we get

∥t_n−p∥² =∥b_n−γ₁γ_nv_n−p∥²

≤ ∥b_n−p∥² −γ₁⁻¹(2−γ₁)∥t_n−b_n∥². (4.3.28)

Also, by Step 5, we get

∥x_n+1−p∥=∥(1−θ_n−δ_n)(b_n−p) +θ_n(t_n−p)−δ_np∥

≤ ∥(1−θ_n−δ_n))(b_n−p) +θ_n(t_n−p)∥+δ_n∥p∥. (4.3.29) But from (4.3.27) and (4.3.28), we obtain that

∥(1−θ_n−δ_n)(b_n−p) +θ_n(t_n−p)∥² = (1−θ_n−δ_n)²∥b_n−p∥²+θ_n²∥t_n−p∥² + 2(1−θ_n−δ_n)θ_n⟨b_n−p, t_n−p⟩

≤(1−θ_n−δ_n)²∥w_n−p∥²+θ²_n∥b_n−p∥² + 2(1−θ_n−δ_n)θ_n∥b_n−p∥²

≤(1−δ_n)²∥w_n−p∥². (4.3.30) Hence, (4.3.29) becomes

∥x_n+1−p∥ ≤(1−δ_n)∥w_n−p∥+δ_n∥p∥

≤(1−δ_n) [∥x_n−p∥+δ_nM₁] +δ_n∥p∥

≤(1−δ_n)∥x_n−p∥+δ_n(M₁+∥p∥), which implies from Lemma 2.5.35 that {x_n} is bounded.

Lemma 4.3.5. Let {x_n} be a sequence generated by Algorithm 4.3.2 under Assumption 4.3.1. Suppose that, there exists a subsequence {x_nk} of {x_n} which converges weakly to a point z ∈ H1 and lim

k→∞∥bn_k−wn_k∥= 0 = lim

k→∞∥bn_k −tn_k∥, then z ∈Γ.

Proof. Without loss of generality, we may assume that zn ̸=T wn. Then, η_n ∈

ϵ, ∥z_n−T w_n∥²

∥T^∗(z_n−T w_n)∥² −ϵ

. Thus, putting (4.3.16) into (4.3.26), we obtain that

∥b_nk−p∥² ≤ ∥w_nk −p∥²−η_nkϵ∥T^∗(z_nk −T w_nk)∥²

≤ ∥w_nk −p∥²−ϵ²∥T^∗(z_nk −T w_nk)∥², (4.3.31) which implies that

ϵ²∥T^∗(zn_k−T wn_k)∥² ≤ ∥wn_k−p∥²− ∥bn_k −p∥²

≤ ∥w_nk−b_nk∥²+ 2∥w_nk−b_nk∥∥b_nk −p∥.

Thus, by our hypothesis, we obtain

k→∞lim ∥T^∗(z_nk −T w_nk)∥= 0. (4.3.32) From (4.3.26) and (4.3.32), we obtain

η_nk∥z_nk−T w_nk∥² ≤ ∥w_nk −p∥² − ∥b_nk−p∥²+η_n²

k∥T^∗(z_nk −T w_nk)∥²

≤ ∥w_nk −b_nk∥²+ 2∥w_nk −b_nk∥∥b_nk−p∥

+η_n²

k∥T^∗(z_nk −T w_nk)∥² →0, as k→ ∞.

Since η_nk > ϵ >0, we obtain that

k→∞lim ∥z_nk −T w_nk∥= 0. (4.3.33) From (4.3.12), we obtain

⟨T w_nk−y_nk, r_nk⟩ ≥(1−λL₂)∥T w_nk−y_nk∥². (4.3.34) Since λ ∈ (0,_L¹

2), we have that 1−λL₂ > 0. Thus, we obtain from (4.3.34) and (4.3.5) that

∥T w_nk−y_nk∥² ≤

1 1−λL2

⟨T w_nk −y_nk, r_nk⟩

=

1 1−λL2

β_nk∥r_nk∥²

=

1 1−λL2

β_nk∥r_nk∥∥T w_nk −y_nk −λ(F T w_nk −F y_nk)∥

≤

1 1−λL2

β_nk∥r_nk∥(∥T w_nk −y_nk∥+λ∥F T w_nk −F y_nk∥)

≤

1 1−λL2

β_nk∥r_nk∥(∥T w_nk −y_nk∥+λL₂∥T w_nk−y_nk∥)

=

1 +λL₂ 1−λL2

∥T w_nk−y_nk∥β_nk∥r_nk∥

=γ₂⁻¹

1 +λL₂ 1−λL₂

∥T w_nk−y_nk∥∥z_nk −T w_nk∥,

where the last equality follows from z_nk =T w_nk−γ₂β_nkr_nk, that is,β_nk||r_nk||=γ₂⁻¹||z_nk− T w_nk||.

Therefore, we obtain from (4.3.33) that

∥T w_nk−y_nk∥ ≤γ₂⁻¹

1 +λL₂ 1−λL₂

∥T w_nk−z_nk∥ →0, as k → ∞. (4.3.35) By the characteristic property of PQ, we obtain for allx∈ Q that

⟨T w_nk −λF T w_nk −y_nk, x−y_nk⟩ ≤0.

This implies that 1

λ⟨T w_nk −y_nk, x−y_nk⟩+⟨F T w_nk, y_nk −T w_nk⟩ ≤ ⟨F T w_nk, x−T w_nk⟩. (4.3.36) Thus, we obtain from (4.3.35) that

0≤lim inf

k→∞ ⟨F T w_nk, x−T w_nk⟩, ∀x∈ Q. (4.3.37) Now, note that

⟨F y_nk, x−y_nk⟩ = ⟨F y_nk−F T w_nk, x−T w_nk⟩+⟨F T w_nk, x−T w_nk⟩

+ ⟨F y_nk, T w_nk −y_nk⟩. (4.3.38)

Moreover, since F is Lipschitz continuous on H₂, we obtain from (4.3.35) that

k→∞lim ||F T w_nk −F y_nk||= 0.

Hence, we obtain from (4.3.35), (4.3.37) and (4.3.38) that 0≤lim inf

k→∞ ⟨F y_nk, x−y_nk⟩, ∀x∈ Q. (4.3.39) We now show that T z ∈VI(F,Q). Let {c_k} be a sequence of positive numbers such that c_k+1 ≤c_k, ∀k ≥1 and c_k →0 as k→ ∞. For each k ≥1, denote by N_k, the smallest positive integer such that

⟨F y_nj, x−y_nj⟩+c_k ≥0, ∀j ≥N_k, (4.3.40) where the existence of N_k follows from (4.3.39). Note that {N_k} is increasing since {c_k} is decreasing. Also, since {yN_k} ⊂ Q for all k ≥ 1, we have that F yN_k ̸= 0. Hence, we can set q_Nk = _{∥F y}^{F yNk}

Nk∥², for each k ≥ 1. Then, ⟨F y_Nk, q_Nk⟩ = 1, for each k ≥ 1. Thus by (4.3.40), we have ⟨F y_Nk, x+c_kq_Nk−y_Nk⟩ ≥0, which by the pseudomonotonicity of F on H2, yields

⟨F(x+c_kq_Nk), x+c_kq_Nk −y_Nk⟩ ≥0. (4.3.41) This further implies that

⟨F x, x−y_Nk⟩ ≥ ⟨F x−F(x+c_kq_Nk), x+c_kq_Nk −y_Nk⟩ −c_k⟨F x, q_Nk⟩. (4.3.42) Now, from Step 2 and (4.3.18), we get that

n→∞lim ∥w_n−x_n∥= lim

n→∞α_n∥x_n−x_n−1∥= lim

n→∞δ_n· α_n

δ_n∥x_n−x_n−1∥= 0. (4.3.43) Since the subsequence {x_nk} of {x_n} is weakly convergent to a point z ∈ H₁, it follows that the subsequence {w_nk} of {w_n} is also weakly convergent to z ∈ H₁. Again, since T is a bounded linear operator, we obtain that {T w_nk} converges weakly to T z. Hence, by (4.3.35), we have that {y_nk} also converges weakly to T z. Note that T z ∈ Q since {y_nk} ⊂ Q. Thus, by the sequentially weakly continuity ofF onQ, we obtain that{F y_nk} converges weakly to F T z. Now, ifF T z= 0,then T z ∈V I(F,Q).So, let F T z̸= 0. Then, by the weakly lower semicontinuity of ∥ · ∥, we obtain that

0<∥F T z∥ ≤lim inf

k→∞ ∥F y_Nk∥.

Since {y_nk} ⊂ {y_Nk}, we obtain that 0≤lim sup

k→∞

∥c_kq_Nk∥= lim sup

k→∞

ck

∥F y_nk∥

≤

lim sup

k→∞

c_k lim inf

k→∞ ∥F y_nk∥ ≤ 0

∥F T z∥ = 0.

Therefore, lim

k→∞c_kq_Nk = 0. Thus, letting k→ ∞ in (4.3.42), we have

⟨F x, x−T z⟩ ≥0, ∀x∈ Q, (4.3.44)

which implies by Lemma 2.5.8 that T z ∈V I(F,Q).

Next, we show that z ∈V I(A,C). Now, observe that, if we follow similar argument used in getting (4.3.35), and noting (by our hypothesis) that limk→∞∥b_nk −t_nk∥ = 0, we can also get

∥b_nk−u_nk∥ ≤r₁⁻¹

1 +µL₁ 1 +µL1

∥b_nk−t_nk∥ →0, as k→ ∞. (4.3.45) Thus, using (4.3.45), the characteristics property of PC and the Lipschitz continuity of A on H₁, we obtain (by following similar argument used in getting (4.3.39)) that

0≤lim inf

k→∞ ⟨Au_nk, y−u_nk⟩, ∀y∈ C. (4.3.46) Now, since {w_nk} converges weakly to z, we obtain by our hypothesis and (4.3.45) that the subsequences {b_nk} and {u_nk} of {b_n} and {u_n} respectively, converge weakly to z.

Again, since {u_nk} ⊂ C, we have thatz ∈ C. Thus, by the sequential weakly continuity of A on C, we obtain that {Au_nk} converges weakly to Az.

Thus, we can follow similar proof used in obtaining (4.3.44), to get

⟨Ay, y−z⟩ ≥0, ∀y ∈ C, (4.3.47) which by Lemma 2.5.8, implies z ∈VI(A,C). Hence, we conclude that z ∈Γ.

We are now in position to give the main theorem for Algorithm 4.3.2.

Theorem 4.3.6. Let {xn} be a sequence generated by Algorithm 4.3.2 under Assumption 4.3.1. Then, {x_n} converges strongly to p∈Γ, where ∥p∥= min{∥z∥:z ∈Γ}.

Proof. Let p∈Γ. Then, from Step 2 and (4.3.19), we obtain

∥w_n−p∥² = ∥x_n−p∥²+α²_n∥x_n−xn−1∥²+ 2α_n⟨x_n−p, x_n−xn−1⟩

≤ ∥x_n−p∥²+α²_n∥x_n−xn−1∥²+ 2α_n||x_n−p||||x_n−xn−1||

= ∥x_n−p∥²+αn∥x_n−xn−1∥

δn· α_n

δ_n∥x_n−xn−1∥+ 2∥x_n−p∥

≤ ∥xn−p∥²+αn∥xn−xn−1∥[δnM1+ 2∥xn−p∥]. (4.3.48) Since by Lemma 4.3.4, {x_n} is bounded, then, there exists M₂ >0 such that

(δ_nM₁+∥x_n−p∥)≤M₂, ∀n≥1.

Hence, we obtain from (4.3.48) that

∥w_n−p∥² ≤ ∥x_n−p∥²+ 3α_n∥x_n−xn−1∥M₂. (4.3.49)

Thus, from (4.3.27) and (4.3.28), we get

∥(1−θ_n)b_n+θ_nt_n−p∥² =∥(1−θ_n)(b_n−p) +θ_n(t_n−p)∥²

= (1−θ_n)²∥b_n−p∥²+θ_n²∥t_n−p∥² + 2(1−θ_n)θ_n⟨b_n−p, t_n−p⟩

≤(1−θ_n)²∥w_n−p∥²+θ²_n∥w_n−p∥² + 2(1−θn)θn⟨bn−p, tn−p⟩

≤(1−θ_n)²∥w_n−p∥²+θ²_n∥w_n−p∥²+ 2(1−θ_n)θ_n∥w_n−p∥²

≤ ∥x_n−p∥²+ 3α_n∥x_n−xn−1∥M₂. From Step 5, the previous inequality and (4.3.18), we get

∥x_n+1−p∥² =∥(1−δ_n) [(1−θ_n)b_n+θ_nt_n−p]−[δ_nθ_n(b_n−t_n) +δ_np]∥²

≤(1−δ_n)²∥(1−θ_n)b_n+θ_nt_n−p∥²−2⟨δ_nθ_n(b_n−t_n) +δ_np, x_n+1−p⟩

≤(1−δ_n)∥(1−θ_n)b_n+θ_nt_n−p∥²+ 2δ_nθ_n⟨b_n−t_n, p−x_n+1⟩ + 2δn⟨p, p−xn−1⟩

≤(1−δ_n)∥(1−θ_n)b_n+θ_nt_n−p∥²+ 2δ_nθ_n∥b_n−t_n∥∥p−x_n+1∥ + 2δ_n⟨p, p−x_n+1⟩

≤(1−δn) ∥xn−p∥²+ 3αn∥xn−xn−1∥M2

+ 2δnθn∥bn−tn∥∥p−xn+1∥ + 2δ_n⟨p, p−x_n+1⟩

≤(1−δ_n)∥x_n−p∥²+ 3α_n∥x_n−xn−1∥M₂+ 2δ_nθ_n∥b_n−t_n∥∥p−x_n+1∥ + 2δ_n⟨p, p−x_n+1⟩

= (1−δ_n)∥x_n−p∥²+δ_nd_n, n ≥1, where d_n = 3^α_δⁿ

n∥x_n−xn−1∥M₂+ 2 (θ_n∥b_n−t_n∥∥p−x_n+1∥+⟨p, p−x_n+1⟩). According to Lemma2.5.36, to conclude the proof, it sufficies to show that lim sup

k→∞

dn_k ≤0 for every subsequence {∥x_nk −p∥}of {∥x_n−p∥} satisfying the condition;

lim inf

k→∞ (∥xn_k+1−p∥ − ∥xn_k −p∥)≥0. (4.3.50) To show that lim sup

k→∞

dn_k ≤ 0, suppose that {∥xn_k −p∥} is a subsequence of {∥xn−p∥}

such that (4.3.50) holds. Then, lim inf

k→∞ ∥xn_k+1−p∥²− ∥xn_k −p∥²

= lim inf

k→∞

h

(∥xn_k+1−p∥ − ∥xn_k −p∥)

×(∥x_nk+1−p∥+∥x_nk −p∥)i

≥0. (4.3.51)

Now, observe that

∥(1−θn−δn)(bn−p) +θn(tn−p)∥² = (1−θn−δn)²∥bn−p∥²

+ 2(1−θ_n−δ_n)θ_n⟨b_n−p, t_n−p⟩+θ_n²∥t_n−p∥²

≤(1−θ_n−δ_n)²∥b_n−p∥²+ (1−θ_n−δ_n)θ_n∥b_n−p∥² + (1−θ_n−δ_n)θ_n∥t_n−p∥²+θ²_n∥t_n−p∥²

≤(1−θ_n−δ_n)(1−δ_n)∥b_n−p∥² (4.3.52) +θ_n(1−δ_n)∥t_n−p∥².

Also, since {x_n} and {δ_n} are bounded, there exists M₃ >0 such that (δn∥p∥+ 2||xn+1−(1−δn)p||)||p|| ≤M3

, for all n ≥1. Hence, from Step 5, (4.3.28), (4.3.49) and (4.3.52), we get

∥x_n+1−p∥² =∥(1−θ_n−δ_n)(b_n−p) +θ_n(t_n−p)−δ_np∥²

=∥(1−θ_n−δ_n)(b_n−p) +θ_n(t_n−p)∥²+δ_n²∥p∥²

−2δ_n⟨(1−θ_n−δ_n)(b_n−p) +θ_n(t_n−p), p⟩

≤ ∥(1−θ_n−δ_n)(b_n−p) +θ_n(t_n−p)∥²

+δ_n δ_n∥p∥²+ 2||(1−θ_n−δ_n)(b_n−p) +θ_n(t_n−p)|| ||p||

=∥(1−θ_n−δ_n)(b_n−p) +θ_n(t_n−p)∥² +δ_n(δ_n∥p∥+ 2||x_n+1−(1−δ_n)p||)||p||

≤(1−θ_n−δ_n)(1−δ_n)∥b_n−p∥²+θ_n(1−δ_n)∥t_n−p∥²+δ_nM₃

≤(1−θn−δn)(1−δn)∥bn−p∥²+θn(1−δn)(∥bn−p∥² (4.3.53)

−γ₁⁻¹(2−γ₁)∥t_n−b_n∥²) +δ_nM₃

≤(1−θ_n−δ_n)(1−δ_n)∥w_n−p∥² +θ_n(1−δ_n)(∥w_n−p∥²

−γ₁⁻¹(2−γ₁)∥t_n−b_n∥²) +δ_nM₃

≤ ∥w_n−p∥²−θ_n(1−δ_n)γ₁⁻¹(2−γ₁)∥t_n−b_n∥²+δ_nM₃

≤ ∥x_n−p∥²−θ_n(1−δ_n)γ₁⁻¹(2−γ₁)∥t_n−b_n∥² +δ_n

3αn

δ_n∥x_n−xn−1∥M₂+M₃

, Thus, from (4.3.51) and (4.3.19), we obtain

lim sup

k→∞

(1−δ_nk)θ_nkγ₁⁻¹(2−γ₁)∥t_nk −b_nk∥²

≤lim sup

k→∞

∥x_nk −p∥²− ∥x_nk+1−p∥² + lim sup

k→∞

(δ_nk(3M₁M₂+M₃))

=−lim inf

k→∞ ∥x_nk+1−p∥²− ∥x_nk −p∥²

≤0, which implies that

k→∞lim ∥t_nk −b_nk∥= 0. (4.3.54)

From (4.3.53) and (4.3.31), we obtain

∥x_nk+1−p∥² ≤(1−θ_nk −δ_nk)(1−δ_nk)∥b_nk −p∥²+θ_nk(1−δ_nk)∥b_nk −p∥²+δ_nkM₃

≤ ∥b_nk−p∥²+δ_nkM₃

≤ ∥w_nk −p∥²−ϵ²∥T^∗(z_nk −T w_nk)∥²+δ_nkM₃, which implies from (4.3.49) that

lim sup

k→∞

∥T^∗(z_nk −T w_nk)∥² ≤ 1

ϵ² lim sup

k→∞

∥x_nk−p∥²− ∥x_nk+1−p∥² + 1

ϵ² lim sup

k→∞

δ_nk

3α_nk

δ_nk∥x_nk−x_nk−1∥M₂+M₃

≤ −1

ϵ² lim inf

k→∞ ∥x_n+1−p∥²− ∥x_nk−p∥²

≤0.

Thus,

k→∞lim ∥T^∗(z_nk −T w_nk)∥= 0. (4.3.55) Hence, by Step 3, we obtain

∥bn_k −wn_k∥=ηn_k∥T^∗(zn_k −T wn_k)∥ →0, as k→ ∞. (4.3.56) From Step 5 and (4.3.54), we get

∥x_nk+1−b_nk∥ ≤θ_nk∥b_nk −t_nk∥+δ_nk∥b_nk∥ →0, as k→ ∞. (4.3.57) From (4.3.56) and (4.3.57), we get

k→∞lim ∥xn_k+1−wn_k∥= 0. (4.3.58) From (4.3.43) and (4.3.58), we get

k→∞lim ∥x_nk+1−x_nk∥= 0. (4.3.59) By Lemma 4.3.4, there exists a subsequence {x_{nk j}}of {x_nk}which converges weakly to z, such that

lim sup

k→∞

⟨p, p−x_nk⟩= lim

j→∞⟨p, p−x_{nk j}⟩=⟨p, p−z⟩.

Also, since we have established (4.3.54) and (4.3.56), then we can apply Lemma 4.3.5 to get that z∈Γ.

Thus, since p=P_Γ0, we obtain from the previous inequality that lim sup

k→∞

⟨p, p−xn_k⟩=⟨p, p−z⟩ ≤0, which implies by (4.3.59) that

k→∞lim⟨p, p−x_nk+1⟩ ≤0. (4.3.60) Now, recall thatdn_k = 3^α_δ^nk

nk∥xn_k−xn_k−1∥M2+2 (θn_k∥bn_k −tn_k∥∥p−xn_k+1∥+⟨p, p−xn_k+1⟩). Thus, by (4.3.18), (4.3.54) and (4.3.60), we obtain that lim sup

k→∞

d_nk ≤ 0. Hence, get that

n→∞lim ∥x_n−p∥= 0. Therefore,{x_n} converges strongly to p=P_Γ0.

Remark 4.3.7.

ˆ Observe that by settingH1 =H2 =H, F = 0 andT =IH (the identity operator on H) in Theorem4.3.6, we obtain as a corollary, an inertial projection and contraction method requiring only one projection onto the feasible setC per iteration, with fixed stepsize for solving the classical VIP (1.2.1) whenAis pseudomonotone and Lipschitz continuous.

ˆ The conclusions of Lemma 4.3.4, Lemma 4.3.5 and Theorem 4.3.6 still hold even if µ ∈ (0,_L¹

1) and λ ∈ (0,_L¹

2) in Algorithm 4.3.2 are replaced with variable stepsizes µ_n and λ_n, respectively such that

0< inf

n≥1µ_n ≤sup

n≥1

µ_n< 1

L₁ and 0< inf

n≥1λ_n≤sup

n≥1

λ_n< 1 L₂.

We now turn to establish the strong convergence of Algorithm 4.3.3. We begin with the following useful results.

Remark 4.3.8.

(i) From (4.3.3) in Algorithm 4.3.3, it is clear thatλ_n+1 ≤λ_n, ∀n≥1.Also, since F is L₂-Lipschitz continuous, we get in the case ofF T w_n ̸=F y_n in Algorithm4.3.3, that

λn+1 = min

a₂||T w_n−y_n||

||F T w_n−F y_n||, λn

≥min a₂

L₂, λn

, which by induction, implies that {λ_n} is bounded below by minn

a2

L2, λ₁o

. Since {λ_n} is also monotone nonincreasing, we have that the limit exists, and lim

n→∞λ_n ≥ min{_L^a2

2, λ₁}>0.

(ii) Similar to (i), we have that the limit of the stepsize {µ_n} exists and lim

n→∞µ_n ≥ min{_L^a1

1, µ₁}>0.

Lemma 4.3.9. Let {x_n} be a sequence generated by Algorithm 4.3.3 under Assumption 4.3.1. Then,

||T w_n−y_n|| ≤γ₂⁻¹

λ_n+1+λ_na₂ λ_n+1−λ_na₂

||T w_n−z_n||, ∀n≥1 (4.3.61) and

||b_n−u_n|| ≤γ₁⁻¹

µ_n+1+µ_na₁ µ_n+1−µ_na₁

||b_n−t_n||, ∀n≥1. (4.3.62) Proof. From (4.3.3), we know that

||F T w_n−F y_n|| ≤ a₂

λ_n+1||T w_n−y_n||, ∀n≥1, (4.3.63)

holds for both F T w_n=F y_n and F T w_n̸=F y_n. Thus, similar to (4.3.12), we obtain that

⟨T w_n−y_n, r_n⟩ ≥ ∥T w_n−y_n∥²−λ_n∥T w_n−y_n∥∥F T w_n−F y_n∥

≥(1−λ_n a₂ λn+1

)∥T w_n−y_n∥², which implies that

∥T w_n−y_n∥² ≤ 1

1−λn a2

λn+1

⟨T w_n−y_n, r_n⟩

≤ 1

1−λ_nλ^a2

n+1

β_n∥r_n∥

∥T w_n−y_n∥+λ_n a₂

λ_n+1∥T w_n−y_n∥

=

1 + _λ^λna2

n+1

1− ^λ_λ^na2

n+1

!

βn∥rn∥∥T wn−yn∥

=γ₂⁻¹

λ_n+1+λ_na₂ λ_n+1−λ_na₂

∥T w_n−z_n∥∥T w_n−y_n∥.

This further gives (4.3.61). In a similar manner, we get (4.3.62).

In the light of Remark 4.3.8 and Lemma 4.3.9, we have the following result.

Theorem 4.3.10. Let{x_n}be a sequence generated by Algorithm4.3.3under Assumption 4.3.1. Then, {x_n} converges strongly to p∈Γ, where ∥p∥= min{∥z∥:z ∈Γ}.

Proof. Let p ∈ Γ. Then, by replacing µ and λ with µn and λn, respectively in Lemma 4.3.4, we can easily get that {x_n} is bounded.

Also, from Remark 4.3.8 (i), we obtain that

n→∞lim

λn+1+λna2

λ_n+1−λ_na₂

= 1 +a2

1−a₂. (4.3.64)

Similarly, from Remark 4.3.8 (ii), we obtain that

n→∞lim

µ_n+1+µ_na₁ µ_n+1−µ_na₁

= 1 +a₁

1−a₁. (4.3.65)

Furthermore, we can follow the same argument as in the proof of Theorem 4.3.6to obtain (4.3.54), (4.3.55) and (4.3.56). Also, using (4.3.55) and (4.3.56), we can get (4.3.33).

Hence, from (4.3.33), (4.3.61) and (4.3.64), we obtain that

k→∞lim ∥T w_nk −y_nk∥= 0.

Again, from (4.3.54), (4.3.47) and (4.3.65), we obtain that

k→∞lim ∥b_nk−u_nk∥= 0.

The rest of the proof follows the same arguments as in Lemma4.3.5and Theorem4.3.6.

Remark 4.3.11.

ˆ Similar to Remark 4.3.7, we can obtain as corollary, an inertial projection and con- traction method with adaptive stepsize, requiring only one projection onto the feasi- ble setC per iteration, for solving the classical VIP (1.2.4) whenAis pseudomonotone and Lipschitz continuous.

ˆ As seen in our convergence analysis, we did not follow the usual “Two Cases Ap- proach” (i.e., Case 1 and Case 2) used in many works to guarantee strong conver- gence of iterative methods. Thus, the techniques and ideas employed in our strong convergence analysis are new for solving the SVIP (1.2.4)-(1.2.5).

ˆ If the operators A and F are monotone and Lipschitz continuous, then we do not need them to be sequentially weakly continuous. This is because the sequential weakly continuity assumption was only used after (4.3.36) to get the conclusion of Lemma 4.3.5. But, from (4.3.36), we obtain

0≤

F T w_nk, x−T w_nk + 1

λ

y_nk−T w_nk, x−y_nk +

F T w_nk, T w_nk −y_nk

≤

F T w_nk −F x, x−T w_nk +

F x, x−T w_nk + 1

λ∥y_nk −T w_nk∥∥x−y_nk∥ +∥F T w_nk∥∥T w_nk−y_nk∥

≤

F x, x−T w_nk + 1

λ∥y_nk−T w_nk∥∥x−y_nk∥+∥F T w_nk∥∥T w_nk −y_nk∥, ∀x∈ Q, where the last inequality follows from the monotoncity of F. Thus, by passing limit as k → ∞ and noting that {T w_nk} converges weakly to z, we obtain from (4.3.35) that

⟨F x, x−T z⟩ ≥0, ∀x∈ Q,

which follows from Lemma 2.5.8 that T z ∈ V I(F,Q). Similarly, we also get that z ∈V I(A,C). Hence, the conclusion of Lemma 4.3.5 holds.

ˆ In finite dimensional spaces, Theorem 4.3.6 and Theorem4.3.10 are still true if the operatorsAandF are only required to be pseudomonotone and Lipschitz continuous.

This is an improvement over the result of He et al. [121] since no product space formulation is required even with the relaxed pseudomonotonicity assumption.

ˆ The conclusions of Theorem 4.3.6 and Theorem 4.3.10 are still valid if Step 1 of Algorithm 4.3.2 and Algorithm 4.3.3 is replaced with the following: 0 ≤ αn ≤ α¯n, where

¯ α_n:=

(minn

α,_∥x ^τn

n−xn−1∥

o

, if x_n̸=xn−1

α, otherwise,

with α∈[0,1).