Consider two problems in a Hilbert space: A constrained convex optimization (CCO) and an equilibrium problem (EP) associated with a bifunction satisfying certain properties. It is known that CCO can be solved by the gradient-projection algorithm (GPA). It is also known that EP is equivalent to a fixed point problem. Therefore, both problems can be solved by fixed point algorithms.

Let H be a real Hilbert space and C a nonempty closed convex subset of H. A self-mapping T of C is said to be nonexpansive if ‖Tx−Ty‖≤‖x−y‖ for all x,y∈ C. A self-mapping f of C is said to be a κ-contraction if ‖f(x)−f(y)‖≤κ‖x−y‖ for all x,y ∈ C, where κ∈[0,1) is a constant. We use Fix(T) to denote the set of fixed points of T; i.e., Fix(T)={x∈C:Tx=x}.

Recall that the equilibrium problem (EP) associated to a bifunction ϕ:C×C→ℝ is to find a point u∈ C with the property

(1.1)ϕ(u,v)≥0, v∈C.

The set of solutions of EP (1.1) is denoted by EP(ϕ).

If ϕ is of the form ϕ(u,v)=⟨Au,v−u⟩ for all u,v∈ C, where A: C→ H is a mapping, then u∈EP(ϕ) if and only if u ∈ C is a solution to the variational inequality (VI):

⟨Au,v−u⟩≥0, v∈C.

Consequently, EP (1.1) includes, as special cases, numerous problems from various areas such as in physics, optimization and economics, and has been received a lot of attention; see, e.g., [7, 8, 5, 9, 12, 13, 14, 16, 17, 18, 19, 22, 28] and the references therein.

In 2010, Tian [20] considered the following iterative method:

(1.2)xn+1=αnγf(xn)+(I−μλnF)Txn, n≥0,

where F is Lipschitz and strongly monotone and γ,μ>0 are some constants. He proved that if the parameters {αn} and {λn} satisfy certain appropriate conditions, the sequence {xn} generated by (1.2) converges strongly to the unique solution x^*∈ Fix(T) of the variational inequality

⟨(γf−μF)x*,x−x*⟩≤0, for all x∈Fix(T).

Consider the constrained convex minimization problem:

(1.3)Minimize {g(x):x∈C},

where g:C→ℝ is a real-valued convex function. We denote by U the set of solutions of (1.3). It is well known that the gradient-projection algorithm (GPA) can be used to solve (1.3). If g is continuously differentiable, then GPA generates a sequence {xn} via the recursive formula:

(1.4)xn+1=PC(xn−λn∇g(xn)), n≥0,

where the initial guess x₀∈ C is chosen arbitrarily, and the parameters {λn} are positive real numbers satisfying certain conditions. The convergence of GPA (1.4) depends on the behavior of the gradient ∇g. As a matter of fact, it is known that if ∇g is α-strongly monotone and L-Lipschitzian with constants α>0 and L≥0, then the operator

(1.5)Wλ:=PC(I−λ∇g)

is a contraction for 0<λ<2α/L2. Consequently, the sequence {xn} generated by PGA (1.4) converges in norm to the unique minimizer of (1.3) provided (λn) is contained in a compact subset of (0,2α/L2), in particular, λn≡λ∈(0,2α/L2).

However, if the gradient ∇g is not strongly monotone, the operator Wλ is no longer contractive (in general). As a result, PGA (1.4) fails to converge strongly in an infinite-dimensional Hilbert space (see a counterexample in [23]). Nevertheless, if ∇g is Lipschitzian, PGA (1.4) can still converge in the weak topology.

In 2012, Tian and Liu [21] studied a composite iterative scheme by the viscosity approximation method [15, 26] for finding a common solution of an equilibrium problem and a constrained convex minimization problem:

(1.6)ϕ(un,y)+1rn⟨y−un,un−xn⟩≥0, y∈C,[3.3mm]xn+1=αnf(xn)+(1−αn)Tnun,n≥1,

where ϕ:C×C→ℝ is a bifunction, ∇g is L-Lipschitzian with L≥0 such that U∩EP(ϕ)≠∅, f is a contraction, x₁∈ C, {αn}⊂(0,1), {rn}⊂(0,∞), un=Qrnxn, T_n is determined by the relation: PC(I−λn∇g)=snI+(1−sn)Tn, sn=2−λnL4 and {λn}⊂(0,2L). They proved that the sequence {xn}, generated by (1.6), converges strongly to a point in U∩EP(ϕ) under certain conditions.

In this paper, motivated by the above results, we will propose an iterative fixed point algorithm to find a point which is a common solution to an equilibrium problem associated with a bifunction and a constrained convex minimization problem. We will prove strong convergence of the algorithm under certain conditions and identify the limit of the sequence generated by the algorithm as the unique solution of some variational inequality.

Let H be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. Weak and strong convergence are denoted by the symbols ⇀ and →, respectively. In a real Hilbert space H, we have the identity

‖λx+(1−λ)y‖2=λ‖x‖2+(1−λ)‖y‖2−λ(1−λ)‖x−y‖2

for all x,y ∈ H and λ∈ℝ. Let C be a nonempty closed convex subset of H. Then the (nearest point or metric) projection on C is defined by

PCx:=argminy∈C‖x−y‖2, x∈H.

Note that P_C is nonexpansive and characterized by the inequality (for z ∈ C)

z=PCx ⇔ ⟨x−z,z−y⟩≥0, y∈C.

The following inequality is convenient in applications. Recall that for all x,y in a Hilbert space H we have

‖x+y‖2≤‖x‖2+2⟨y,x+y⟩.

Next, given a Hilbert space H and a function φ:H→ℝ, we recal that φ is weakly lower semicontinuous l.s.c. considering H with its weak topology, that is, φ(u^)≤liminfφ(un) whenever u_n converges weakly to u^.

Lemma 2.1. ([10]) Let H be a real Hilbert space, C a closed convex subset of H and T:C → C a nonexpansive mapping with Fix(T)≠∅. If {xn} is a sequence in C such that xn⇀x and (I−T)xn→y, then (I-T)x=y. [This is known as the demiclosedness principle of nonexpansive mappings.]

Let ϕ:C×C→ℝ be a bifunction. Throughout the paper we always assume that ϕ satisfies the following (standard) conditions [2]:

(A₁) ϕ(x,x)=0 for all x∈ C;

(A₂) ϕ is monotone, i.e., ϕ(x,y)+ϕ(y,x)≤0, for all x,y∈ C;

(A₃) for each x,y,z∈C, limt↓0ϕ(tz+(1−t)x,y)≤ϕ(x,y);

(A₄) for each x∈ C, the mapping y↦ϕ(x,y) is convex, weakly lower semicontinuous (l.s.c.).

Under these conditions, it is easy to see that the solution set EP(ϕ) of the equilibrium problem (1.1) is closed and convex.

Following Combettes and Hirstoaga [6], we can define, for each fixed r>0, a mapping Qr:H→C by the equation

(2.1)Qrx=z

for x∈ H, where z∈ C is the unique solution of the inequality:

(2.2)ϕ(z,y)+1r⟨y−z,z−x⟩≥0, y∈C.

Lemma 2.2. ([6]) The mapping Q_r possesses the properties:

(i) Q_r is firmly nonexpansive, namely,

‖Qrx−Qry‖2≤⟨Qrx−Qry,x−y⟩, x,y∈H;

(ii) Fix(Qr)=EP(ϕ).

Property (ii) shows an equivalence of EP (1.1) to the fixed point problem of Qrx=x and thus EP (1.1) can be solved by fixed point methods.

Definition 2.3. ([21]) A mapping T:H→H is said to be an averaged mapping (av, for short) if T=(1−α)I+αS, where α∈(0,1), I is the identity map and S:H→ H is nonexpansive. In this case, we say that T is α-averaged (α-av, for short).

Note that firmly nonexpansive mappings (e.g., projections) are 12-av.

Proposition 2.4. ([4, 23])} The composite of finitely many averaged mappings is averaged. That is, if each of the mappings {Ti}i=1N is averaged, then so is the composite T1⋯TN. In particular, if T₁ is α₁-av, and T₂ is α₂-av, where α1,α2∈(0,1), then the composite T₁T₂ is α-av, where α=α1+α2−α1α2.

Definition 2.5. A nonlinear operator G with domain D(G) and range R(G) both in H is said to be:

(i) β-strongly monotone if there exists β>0 such that

⟨x−y,Gx−Gy⟩≥β‖x−y‖2, x,y∈D(G),

(ii) ν-inverse strongly monotone (for short, ν-ism) if there exists ν>0 such that

⟨x−y,Gx−Gy⟩≥ν‖Gx−Gy‖2, x,y∈D(G).

It can be easily seen that any projection P_C is a 1-ism.

Inverse strongly monotone (also referred to as co-coercive) operators have extensively been used in practical problems from various areas, for instance, traffic assignment problems; see, for example, [3, 11] and references therein.

Proposition 2.6. ([4, 23]) Let T be an operator of H into itself.

• (i) T is nonexpansive if and only if the complement I-T is 12-ism.

• (ii) If T is ν-ism, then for μ>0, μT is νμ-ism.

• (iii) T is averaged if and only if the complement I-T is ν-ism for some ν>12.

Indeed, for α∈(0,1), T is α-av if and only if I-T is 12α-ism.

Lemma 2.7. ([1, 25]) Assume {an} is a sequence of nonnegative real numbers such that

an+1≤(1−γn)an+γnνn+μn,

where {γn} is a sequence in [0,1], {μn} is a sequence of nonnegative real numbers and {νn} is a sequence in ℝ such that

∑ n=1∞γn=∞, limsupn→∞νn≤0, ∑ n=1∞μn<∞.

Then limn→∞an=0.

In this section we introduce an iterative fixed point algorithm for finding a point in U ∩ EP(ϕ), that is, a common solution to the equilibrium and optimization problems (1.1) and (1.3).

In the rest of this paper, we always assume that C is a nonempty closed convex subset of a Hilbert space H and g:C→ℝ is a real-valued continuously differentiable, convex function such that the gradient ∇ g is L-Lipschitz for some L≥ 0. Recall that U denotes the set of solutions of the minimization problem (1.3). Then we have that a point x^*∈ U if and only if x*∈Fix(Wλ), where W_λ is defined in (1.5); that is,

x*=PC(I−λ∇g)x*

for each λ>0. Further helpful properties of the mapping PC(I−λ∇g) are outlined in [23] and summarized below.

Lemma 3.1. Suppose a continuously differentiable, convex function g has L-Lipschitz continuous gradient ∇g

and let λ∈(0,2/L) be given.

• (i) The mapping PC(I−λ∇g) is α-av, where α=2+λL4∈(0,1); namely, one can write PC(I−λ∇g)=sI+(1−s)T, where s=2−λL4 and T=12+λL[4PC(I−λ∇g)−(2−λL)I] is nonexpansive.

• (ii) The function h(λ)x:=PC(I−λ∇g)x is uniformly Lipschitz continuous in λ∈[0,∞) over bounded x ∈ C.

Proof. (i) has been proved in [23]. To prove (ii) take λ,λ′∈[0,∞) and let r>0 be given. Setting Mr:=sup{‖∇g(x)‖:‖x‖≤r,x∈C} and noting that P_C is nonexpansive, we get

‖h(λ)−h(λ′)‖=‖PC(I−λ∇g)x−PC(I−λ′∇g)x‖      ≤|λ−λ′|‖∇g(x)‖≤Mr|λ−λ′|.

Lemma 3.2. ([27, Lemma 3.1]) Suppose F:C→H is α-Lipschitz and η-strongly monotone and let 0<μ<2ηα2. Then, for ν∈(0,1), the mapping Uν defined by

(3.1)Uνx:=x−μνFx, x∈C

is a (1−ξν)-contraction from C into H, with ξ:=1−1−μ(2η−μα2)∈(0,1].

The following is the main result of this paper.

Theorem 3.3. Assume the bifunction ϕ:C×C→ℝ satisfies the standard conditions (A₁)-(A₄), and the objective function g:C→ℝ is continuously differentiable and convex such that the gradient ∇g is L-Lipschitz with L≥0. Assume U∩EP(ϕ)≠∅. Let Q_r be defined by (2.1) for r>0. Let f be a κ-contraction of C with κ∈[0,1) and F: C→ H an α-Lipschitz and η-strongly monotone operator on C with α>0 and η>0. Assume 0<μ<2ηα2 and 0<γ<ξκ with ξ=1−1−μ(2η−μα2). Suppose {αn}, {βn} and {rn} are real sequences satisfying the following conditions:

(B₁) {αn}⊂[0,1], limn→∞αn=0 and  ∑ n=1∞αn=∞;

(B₂) {βn}⊂[0,1], limn→∞βn=0 and  ∑ n=1∞|βn+1−βn|<∞;

(B₃) {rn}⊂(0,∞), liminfn→∞rn>0 and  ∑ n=1∞|rn+1−rn|<∞;

(B₄) ∑ n=1∞|αn+1−αn|<∞ or limn→∞αnαn+1=1.

Let {x_n} be a sequence generated by the iteration process:

(3.2)un=Qrnxn,yn=PC(αnγf(xn)+(I−αnμF)Tnun),xn+1=(1−βn)yn+βnTnyn,n≥1,

where the initial point x₁ ∈ C is selected arbitrarily, {λn}⊂(0,2L), and T_n is determined by the relation PC(I−λn∇g)=snI+(1−sn)Tn with sn=2−λnL4 (cf. Lemma 3.1 (i)). Namely,

(3.3)Tn=11−sn(PC(I−λn∇g)−snI).

Suppose {λn} satisfies the condition

(B₅) 0<λ_≤λn<2L for all n, and ∑ n=1∞|λn+1−λn|<∞.

Then, the sequences {x_n} and {u_n} defined by (3.2) converge strongly to a point q∈U∩EP(ϕ), where q=PU∩EP(ϕ)(I−μF+γf)(q) is the unique solution to the following variational inequality:

⟨(μF−γf)q,q−x⟩≤0, x∈U∩EP(ϕ).

To prove Theorem 3.3 we first establish some lemmas.

Lemma 3.4. Let T_n be defined by (3.3). Let r>0 and set

M(r):=sup{4‖∇g(y)‖+L‖PCy−y‖:‖y‖≤r,y∈C}<∞.

Then, for y∈C such that ‖y‖≤r, we have

‖Tn+1y−Tny‖≤M(r)|λn+1−λn|.

Proof. By definition of T_n, we have for y∈C

Tn+1y−Tny=11−sn+1PC(y−λn+1∇g(y))−sn+11−sn+1y      −11−snPC(y−λn∇g(y))+sn1−sny=11−sn+1[PC(y−λn+1∇g(y))−PC(y−λn∇g(y))] +11−sn+1−11−snPC(y−λn∇g(y))+sn1−sn−sn+11−sn+1y=11−sn+1[PC(y−λn+1∇g(y))−PC(y−λn∇g(y))] +sn+1−sn(1−sn)(1−sn+1)(PC(y−λn∇g(y))−y).

Since P_C is nonexpansive and sn=(2−λnL)/4 (thus 1−sn=2+λn4≥12),

it turns out that (noting λnL<2)

‖Tn+1y−Tny‖≤2‖∇g(y)‖|λn+1−λn|       +L|λn+1−λn|(‖PC(y−λn∇g(y))−PCy‖+‖PCy−y‖)      ≤2‖∇g(y)‖|λn+1−λn|       +L|λn+1−λn|(λn‖∇g(y)‖+‖PCy−y‖)      ≤(4‖∇g(y)‖+L‖PCy−y‖)|λn+1−λn|.

This finishes the proof.

Lemma 3.5. For x,x′∈C and r,r′>0, we have

(3.4)‖Qrx−Q r′x‖≤1−r r′‖Q r′x−x‖

and

(3.5)‖Qrx−Q r′x′‖≤‖x−x′‖+1−r r′‖Q r′x−x‖.

Proof. Set u=Qrx and u′=Q r ′ x. By definition, we have

ϕ(u,y)+1r⟨y−u,u−x⟩≥0, for all y∈C.

In particular,

ϕ(u,u′)+1r⟨u′−u,u−x⟩≥0.

Similarly, we have

ϕ(u′,u)+1r ′ ⟨u−u′,u′−x⟩≥0.

Adding up the last two relations and using the monotonicity ϕ(u,u′)+ϕ(u′,u)≤0, we obtain

⟨u′−u,1r(u−x)−1 r ′ (u′−x)⟩≥0.

This can be rewritten as

1r‖u′−u‖2≤1r−1 r ′ ⟨u′−u,u′−x⟩    ≤1r−1 r ′ ‖u′−u‖‖u′−x‖

and (3.4) follows immediately.

Next using the nonexpansivity of Qr′, we get

‖Qrx−Q r′x′‖≤‖Q r′x′−Q r′x‖+‖Qrx−Q r′x‖      ≤‖x−x′‖+1−r r′‖Q r′x−x‖.

The proof of the lemma is complete.

Proof. Theorem 3.3First we make a convention: For the sake of convenience, we will use M>0 to stand for an appropriate constant for several estimates from various places throughout the proof which is divided into five steps.

Step 1. The sequences {x_n} and {u_n} are bounded. To see this, we take a point p∈U∩EP(ϕ) to derive from (3.2) that (noting Q_rp=p and T_np=p for all r>0 and n≥1, and P_C is nonexpansive)

‖xn+1−p‖≤(1−βn‖yn−p‖+βn‖Tnyn−p‖≤‖yn−p‖    ≤‖αnγf(xn)+(I−αnμF)Tnun−p‖    =‖αnγ[f(xn)−f(p)]+αn[γf(p)−μF(p)]     +(I−αnμF)Tnun−(I−αnμF)(p)‖.

Now since f is κ-contraction, (I−αnμF) is (1−αnξ)-contraction, T_n is nonexpansive, and ‖un−p‖=‖Qrnxn−p‖≤‖xn−p‖, it follows from the last relation that

‖xn+1−p‖≤(1−αn(ξ−γκ))‖xn−p‖+αn‖γf(p)−μF(p)‖    ≤max{‖xn−p‖,‖γf(p)−μF(p)‖ξ−γκ}.

As a result, we get, by induction,

‖xn−p‖≤max{‖x1−p‖,‖γf(p)−μF(p)‖ξ−γκ}

for all n≥1. Hence, {xn} is bounded, which implies that {un}, {yn}, {f(xn)}, {μF(Tnun)} and {Tnyn} are all bounded.

Step 2. Asymptotic regularity of {xn}: limn→∞‖xn+1−xn‖=0. To see this we use the definition of the algorithm (3.2) to derive that, after some manipulations,

(3.6)xn+2−xn+1=(1−βn+1)yn+1+βn+1Tn+1yn+1−(1−βn)yn−βnTnyn    =(1−βn+1)(yn+1−yn)+βn+1(Tn+1yn+1−Tn+1yn)     +(βn−βn+1)(yn−Tn+1yn)+βn(Tn+1yn−Tnyn).

Since T_n+1 is nonexpansive and (y_n) is bounded, and by Lemma 3.4, we obtain from (3.6),

(3.7)‖xn+2−xn+1‖≤‖yn+1−yn‖+(|βn+1−βn+1|+βn|λn+1−λn|)M,

where M>0 is a big enough constant.

To estimate ‖yn+1−yn‖, we again use (3.2) and the nonexpansiveness of P_C to get

(3.8)‖yn+1−yn‖≤‖αn+1γf(xn+1)+(I−αn+1μF)Tn+1un+1     −αnγf(xn)−(I−αnμF)Tnun‖    =‖(αn+1−αn)[γf(xn+1)−μF(Tn+1un+1)]     +αnγ[f(xn+1)−f(xn)]     +(I−αnμF)Tn+1un+1−(I−αnμF)Tnun‖.

Observe by Lemmas 3.4 and 3.5

(3.9)‖Tn+1un+1−Tnun‖≤‖Tn+1un+1−Tn+1un‖+‖Tn+1un−Tnun‖ ≤‖un+1−un‖+‖Tn+1un−Tnun‖ =‖Qrn+1xn+1−Qrnxn‖+‖Tn+1un−Tnun‖ ≤‖xn+1−xn‖+|1−rnrn+1|‖Qrn+1xn−xn‖+M|λn+1−λn| ≤‖xn+1−xn‖+|1−rnrn+1|+|λn+1−λn|M.

Now since f is κ-contraction and I−αnμF is (1−ξαn)-contraction, also using (3.9), we get from (3.8)

(3.10)‖yn+1−yn‖≤M|αn+1−αn|+κγαn‖xn+1−xn‖     +(1−ξαn)[‖xn+1−xn‖+(|1−rnrn+1|+|λn+1−λn|)M]    =[1−(ξ−κγ)αn]‖xn+1−xn‖     +(|αn+1−αn|+|1−rnrn+1|+|λn+1−λn|)M.

Substituting (3.10)) into (3.7) yields

(3.11)‖xn+2−xn+1‖≤(1−α^n)‖xn+1−xn‖+νn+δn,

where α^n=(ξ−κγ)αn, νn=M|αn+1−αn| and

δn=(|βn+1−βn|+|1−rnrn+1|+(1+βn)|λn+1−λn|)M.

By condition (B₁), we have ∑nα^ n=∞. By conditions (B₂)-(B₅), we always have ∑nδ n<∞, and moreover, ∑nν n<∞ if ∑n|αn+1−αn|<∞ or limn(ν/α^n)=0 if limn(αn/αn+1)=1. So Lemma 2.7 is applicable to (3.11), which yields ‖xn+1−xn‖→0, as claimed.

Step 3. We claim that

(3a) ‖xn−yn‖→0,

(3b) ‖xn−un‖→0,

(3c) ‖PC(I−λn∇g)xn−xn‖→0.

Indeed, since

‖xn−yn‖≤‖xn−xn+1‖+‖xn+1−yn‖    =‖xn−xn+1‖+βn‖yn−Tnyn‖    =‖xn−xn+1‖+Mβn→0 (asβn→0)

and (3a) follows. To show (3b), we first observe that the firm nonexpansivity of Qrn implies that (noting un=Qrnxn and Fix(Qrn)=EP(ϕ))

‖un−p‖2≤‖xn−p‖2−‖un−xn‖2, p∈EP(ϕ).

For the sake of brevity, we shall use the O(αn) notation in our argument below. For p∈U∩EP(ϕ), we have (noting again that P_C is nonexpansive)

‖xn+1−p‖2=‖(1−βn)(yn−p)+βn(Tnyn−p)‖2≤‖yn−p‖2    ≤‖αnγf(xn)+(I−αnμF)Tnun−p‖2    =‖αn(γf(xn)−μF(p))+(I−αnμF)Tnun−(I−αnμF)p‖2    =αn2‖γf(xn)−μF(p)‖2+‖(I−αnμF)Tnun−(I−αnμF)p‖2     +2αn⟨γf(xn)−μF(p),(I−αnμF)Tnun−(I−αnμF)p⟩    ≤αn2M+(1−ξαn)2‖un−p‖2+2αnM(1−ξαn)‖un−p‖    ≤‖un−p‖2+O(αn)    ≤‖xn−p‖2−‖un−xn‖2+O(αn).

It turns out that

‖xn−un‖2≤‖xn−p‖2−‖xn+1−p‖2+O(αn)    =O(‖xn+1−xn‖)+O(αn)→0.

This proves (3b). To verify (3c), it suffices to verify that PC(I−λn▽g)un−un→0. Since PC(I−λn▽g)=snI+(1−sn)Tn, it follows that

PC(I−λn▽g)un−un=(1−sn)Tnun−un        ≤‖Tnun−yn‖+‖yn−un‖        ≤αn‖γf(xn)−μF(Tnxn)‖+‖yn−un‖        =O(αn)+‖yn−un‖→0.

Step 4. We have the following asymptotic variational inequality:

(3.12)limsupn→∞⟨γf(q)−μF(q),yn−q⟩≤0,

where q is the unique fixed point of the contraction PU∩EP(ϕ)(I−μF+γf); namely, q=U∩EP(ϕ)(I−μF+γf)q. Alternatively, q is the unique solution of the variational inequality

(3.13)⟨γf(q)−μF(q),y−q⟩≤0, y∈U∩EP(ϕ).

To prove (3.12), take a subsequence (yni) of (y_n) weakly convergent to a point y^∈C and such that

limsupn→∞⟨γf(q)−μF(q),yn−q⟩=limi→∞⟨γf(q)−μF(q),yni−q⟩              =⟨γf(q)−μF(q),y^−q⟩.

By virtue of VI (3.13), it suffices to show y^∈U∩EP(ϕ). To see y^∈U, we use (3c) to get (noticing ‖xn−yn‖→0)

(3.14)PC(I−λni▽g)yni−yni→0.

We may assume λni→λ^; note that λ^∈(0,2/L] due to condition (B₅). It is then not hard (see Lemma 3.1(ii)) to find from (3.14) that

(3.15)PC(I−λ^▽g)yni−yni→0.

The demiclosedness principle of nonexpansive mappings then ensures that y^∈Fix(PC(I−λ^▽g)=U. It remains to show y^∈EP(ϕ). Since un=Qrnxn, it follows from the definition of Q_r and the monotonicity of ϕ that

1rn⟨y−un,un−xn⟩≥−ϕ(un,y)≥ϕ(y,un), y∈C.

This results in limsupn→∞ϕ(y,un)≤0 for each y∈C, due to the facts ‖un−xn‖→0 and infnrn>0. As ϕ(y,⋅) is l.s.c. and uni⇀y^, it turns out that ϕ(y,y^)≤0 for each y∈C. Now set yt=ty+(1−t)y^∈C with t∈(0,1). We then have by the standard conditions (A₁)-(A₄)

0=ϕ(yt,ty+(1−t)y^)≤tϕ(yt,y)+(1−t)ϕ(yt,y^)≤tϕ(yt,y).

Hence, ϕ(yt,y)≥0 and ϕ(y,yt)≤0. Letting t→0 yields ϕ(y,y^)≤0 for each y∈C. This asserts y^∈EP(ϕ) and the proof of Step 4 is complete.

Step 5. Strong convergence of {xn}: xn→q in norm, where q satisfies (3.12). Setting zn:=αnγf(xn)+(I−αnμF)Tnun (thus, yn=PCzn), we derive that

‖xn+1−q‖2=‖(1−βn)(yn−q)+βn(Tnyn−p)‖2≤‖yn−q‖2    ≤‖αnγf(xn)+(I−αnμF)Tnun−q‖2    =‖[(I−αnμF)Tnun−(I−αnμF)q]+αn(γf(xn)−μF(q))‖2    ≤‖(I−αnμF)Tnun−(I−αnμF)q‖2     +2αn⟨γf(xn)−μF(q),zn−q⟩    ≤(1−ξαn)2‖Tnun−q‖2+2γαn⟨f(xn)−f(q),xn−q⟩     +2γαn⟨f(xn)−f(q),zn−xn⟩+2αn⟨γf(q)−μF(q),zn−q⟩    ≤(1−ξαn)2‖un−q‖2+2γκαn‖xn−q‖2     +2γκαn‖xn−q‖‖zn−xn‖+2αn⟨γf(q)−μF(q),zn−q⟩    ≤(1−2(ξ−γκ)αn)‖xn−q‖2+ξ2αn2M2     +2γκαnM‖zn−xn‖+2αn⟨γf(q)−μF(q),zn−q⟩.

Here M≥‖xn−q‖ for all n. We can rewrite the last relation as

(3.16)‖xn+1−q‖2≤(1−α˜n)‖xn−q‖2+α˜nδ˜n,

where α˜n=2(ξ−γκ)αn and

δ˜n=ξ2M2αn+2γκM‖zn−xn‖+2⟨γf(q)−μF(q),zn−q⟩2(ξ−γκ).

Since yn=PCzn and Tnun∈C, we obtain

‖yn−zn‖=‖PCzn−zn‖≤‖PCzn−Tnun‖+‖Tnun−zn‖    ≤2‖zn−Tnun‖=2αn‖γf(xn)−μF(Tnun)‖→0.

It turns out from (3a) of Step 3 and (3.12) that ‖zn−xn‖→0 and

(3.17)limsupn→∞⟨γf(q)−μF(q),zn−q⟩≤0,

It is now immediately clear that ∑ n=1∞α˜n=∞ and limsupn→∞δ˜n≤0. This enables us to apply Lemma 2.7 to the relation (3.16) to arrive at ‖xn−q‖→0, that is, xn→q in norm.

Remark 3.6. The choices of the parameters (αn),(βn),(rn) are easy. For instance, any decreasing null sequence (β_n) satisfies (B₂). More precisely, the choices:

αn=1na, βn=1nb, rn=r+1nc, n≥1,

where 0<a≤1 and b,c,r>0 satisfy (B₁)-(B₄). Also, the choice λn=λ_+1nd, where λ_>0 and d>0 satisfies (B₅) for n big enough.

Remark 3.7. Theorem 3.3 is a generalization of [21, Theorem 3.2]. In addition, we used the condition liminfn→∞λn>0 for the stepsizes {λn}. However, [21, Theorem 3.2] required that λn→2L, which needs the exact value L of the Lipschitz constant of the gradient ∇g of the objective function g. In practical problems, the exact value of L would be unavailable in many circumstances; consequently, verification of the condition λn→2L turns out to be hard.