Linesearch 1. Given x∈domf2, σ>0, θ∈(0,1) and δ>0. |
Input α=σ. |
While α‖∇f1(FBα(x))−∇f1(x)‖>δ‖FBα(x)−x‖, do |
α=θα. |
End |
Output α. |
We study and investigate a convex minimization problem of the sum of two convex functions in the setting of a Hilbert space. By assuming the Lipschitz continuity of the gradient of the function, many optimization methods have been invented, where the stepsizes of those algorithms depend on the Lipschitz constant. However, finding such a Lipschitz constant is not an easy task in general practice. In this work, by using a new modification of the linesearches of Cruz and Nghia [
Citation: Adisak Hanjing, Pachara Jailoka, Suthep Suantai. An accelerated forward-backward algorithm with a new linesearch for convex minimization problems and its applications[J]. AIMS Mathematics, 2021, 6(6): 6180-6200. doi: 10.3934/math.2021363
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We study and investigate a convex minimization problem of the sum of two convex functions in the setting of a Hilbert space. By assuming the Lipschitz continuity of the gradient of the function, many optimization methods have been invented, where the stepsizes of those algorithms depend on the Lipschitz constant. However, finding such a Lipschitz constant is not an easy task in general practice. In this work, by using a new modification of the linesearches of Cruz and Nghia [
Throughout this article, we suppose that H is a real Hilbert space with an inner product ⟨⋅,⋅⟩ and the induced norm ‖⋅‖. We are interested in studying the following unconstrained convex minimization problem:
minimizef1(x)+f2(x),subject tox∈H | (1.1) |
where f1,f2:H→R∪{∞} are two proper, lower semi-continuous and convex functions such that f1 is differentiable on an open set containing the domain of f2. Problem (1.1) has been widely studied due to its applications which can be used in various real-world applications such as in signal and image processing, in regression problems, and in classification problems, etc., see [3,5,8,10,11,13] and the references therein. One of the important topics of studying Problem (1.1) is to invent some efficient procedures for approximating minimizers of f1+f2. Various optimization methods were introduced and developed by many researchers, see [3,5,6,7,8,12,14,15,16,25,26,27,32], for instance.
If a minimizer x∗ of f1+f2 exists, it is known that x∗ is characterized by the fixed point equation of the forward-backward operator
x∗=FBα(x∗):=proxαf2⏟backward step(x∗−α∇f1(x∗))⏟forward step, | (1.2) |
where α>0, proxf2 is the proximity operator of f2 and ∇f1 stands for the gradient of f1. The above equation leads to the following iterative method:
Method 1. Let x1∈domf2. For k≥1, let
xk+1=proxαkf2(xk−αk∇f1(xk)), |
where 0<αk<2L and L is a Lipschitz constant of ∇f1.
This method is well known as the forward-backward splitting algorithm [8,15], which includes the proximal point algorithm [17,24], the gradient method [4,9] and the CQ algorithm [1] as special cases. It is observed from Method 1 that we need to assume the Lipschitz continuity condition on the gradient of f1 and the stepsize αk depends on the Lipschitz constant L, which is not an easy task to find in general practice (see [3,5,8,12] for other relevant methods).
In the sequel, we set the standing hypotheses on Problem (1.1) as follows:
(H1) f1,f2:H→R∪{∞} are two proper, lower semi-continuous and convex functions with domf2⊆domf1, and domf2 is nonempty, closed and convex;
(H2) f1 is differentiable on an open set containing domf2. The gradient ∇f1 is uniformly continuous on any bounded subset of domf2 and maps any bounded subset of domf2 to a bounded set in H.
We note that the second part of (H2) is a weaker assumption than the Lipschitz continuity assumption on ∇f1.
Cruz and Nghia [7] proposed a technique for selecting the stepsize αk which is independent of the Lipschitz constant L by using the following linesearch process.
Linesearch 1. Given x∈domf2, σ>0, θ∈(0,1) and δ>0. |
Input α=σ. |
While α‖∇f1(FBα(x))−∇f1(x)‖>δ‖FBα(x)−x‖, do |
α=θα. |
End |
Output α. |
Linesearch 1 is a particular case of the linesearch proposed in [29] for inclusion problems and it was shown that this linesearch is well defined, that is, it stops after finitely many steps, see [7, Lemma 3.1] and [29, Theorem 3.4(a)]. Cruz and Nghia [7] employed the forward-backward iteration where the stepsize αk is generated by Linesearch 1.
Method 2. Let x1∈domf2, σ>0, θ∈(0,1) and δ∈(0,12). For k≥1, let
xk+1=proxαkf2(xk−αk∇f1(xk)), |
where αk:= Linesearch 1(xk,σ,θ,δ).
In optimization theory, to speed up the convergence of iterative methods, many mathematicians often use the inertial-type extrapolation [20,22] by adding the technical term βk(xk−xk−1). The control parameter βk is called an inertial parameter, which controls the momentum xk−xk−1. Using Linesearch 1, Cruz and Nghia [7] also introduced an accelerated algorithm with the inertial technical term as follows.
Method 3. Let x0=x1∈domf2, α0=σ>0, θ∈(0,1), δ∈(0,12), and t1=1. For k≥1, let
tk+1=1+√1+4t2k2,βk=tk−1tk+1, yk=Pdomf2(xk+βk(xk−xk−1)),xk+1=proxαkf2(yk−αk∇f1(yk)), |
where αk:= Linesearch 1(yk,αk−1,θ,δ).
The technique of choosing βk in Method 3 was first mentioned in the fast iterative shrinkage-thresholding algorithm (FISTA) by Beck and Teboulle [5]. Weak convergence results of Methods 2 and 3 were obtained for solving Problem (1.1) with (H1) and (H2).
Recently, Kankam et al. [14] proposed a modification of Linesearch 1 as follows.
Linesearch 2. Given x∈domf2, σ>0, θ∈(0,1) and δ>0. |
Input α=σ. |
While αmax{‖∇f1(FB2α(x))−∇f1(FBα(x))‖,‖∇f1(FBα(x))−∇f1(x)‖} |
>δ(‖FB2α(x)−FBα(x)‖+‖FBα(x)−x‖), do |
α=θα. |
End |
Output α, |
where FB2α(x):=FBα(FBα(x)). |
They showed the well-definedness of Linesearch 2 and introduced the following double forward-backward algorithm whose stepsize is generated by Linesearch 2.
Method 4. Let x1∈domf2, σ>0, θ∈(0,1) and δ∈(0,18). For k≥1, let
yk=proxαkf2(xk−αk∇f1(xk)),xk+1=proxαkf2(yk−αk∇f1(yk)), |
where αk:= Linesearch 2(xk,σ,θ,δ).
A weak convergence theorem of Method 4 was proved and an application in signal recovery was illustrated, see [14].
In this paper, inspired and motivated by the results of Cruz and Nghia [7] and Kankam et al. [14], and other related researches, we aim to improve Linesearches 1 and 2 and introduce a new accelerated algorithm using our proposed linesearch for the convex minimization problem of the sum of two convex functions. The paper is organized as follows. Basic definitions, notations and some useful tools for proving our convergence results are given in Section 2. Our main result is in Section 3. In this section, we introduce a new modification of Linesearches 1 and 2 and present a double forward-backward algorithm by using an inertial technique for solving Problem (1.1) with Hypotheses (H1) and (H2). After that, a weak convergence theorem of the proposed method is proved. The complexity of our reduced algorithm is also discussed. In Section 4, we apply the convex minimization problem to an image restoration problem and a regression problem. We analyze and illustrate the convergence behavior of our method, and also compare its efficiency with the well-known methods in the literature.
The mathematical symbols used throughout this paper are as follows. R, R+ and N are the set of real numbers, the set of nonnegative real numbers, and the set of positive integers, respectively. Id stands for the identity operator on H. Denote weak and strong convergence of a sequence {xk}⊂H to x∈H by xk⇀x and xk→x, respectively. The set of all weak-cluster points of {xk} is denoted by ωw(xk). If C is a nonempty closed convex subset of H, then PC stands for the metric projection from H onto C, i.e., for each x∈H, PCx is the unique element in C such that ‖x−PCx‖=dist(x,C):=infy∈C‖x−y‖.
Let us recall the concept of the proximity operator which extends the notion of the metric projection. Let f:H→R∪{∞} be a proper, lower semi-continuous and convex function. The proximity (or proximal) operator [2,18] of f, denoted by proxf is defined for each x∈H, proxfx is the unique solution of the minimization problem:
minimizey∈Hf(y)+12‖x−y‖2. | (2.1) |
If f:=iC is an indicator function on C (defined by iC(x)=0 if x∈C; otherwise iC(x)=∞), then proxf=PC.
The proximity operator can be formulated in the equivalent form
proxf=(Id+∂f)−1:H→domf, | (2.2) |
where ∂f is the subdifferential of f defined by
∂f(x):={u∈H:f(x)+⟨u,y−x⟩≤f(y),∀y∈H},∀x∈H. |
Moreover, we have the following useful fact:
x−proxαf(x)α∈∂f(proxαf(x)),∀x∈H,α>0. | (2.3) |
The following is a property of the subdifferential operator.
Lemma 2.1 ([23]). If f:H→R∪{∞} is a proper, lower semi-continuous and convex function, then the graph of ∂f defined by Gph(∂f):={(x,y)∈H×H:y∈∂f(x)} is demiclosed, i.e., if the sequence {(xk,yk)}⊂Gph(∂f) satisfies that xk⇀x and yk→y, then (x,y)∈Gph(∂f).
We end this section by providing useful tools for proving our main results.
Fact. Let x,y∈H. The following inequalities hold on H:
‖x±y‖2=‖x‖2±2⟨x,y⟩+‖y‖2. | (2.4) |
Lemma 2.2 ([12]). Let {ak} and {tk} be two sequences of nonnegative real numbers such that
ak+1≤(1+tk)ak+tkak−1,∀k∈N. |
Then the following holds
ak+1≤K⋅k∏j=1(1+2tj),whereK=max{a1,a2}. |
Moreover, if ∑∞k=1tk<∞, then {ak} is bounded.
Lemma 2.3 ([31]). Let {ak} and {bk} be two sequences of nonnegative real numbers such that ak+1≤ak+bk for all k∈N. If ∑∞k=1bk<∞, then limk→∞ak exists.
Lemma 2.4 (Opial [19]). Let {xk} be a sequence in H such that there exists a nonempty set Ω⊂H satisfying:
(i) For every p∈Ω, limk→∞‖xk−p‖ exists;
(ii) ωw(xk)⊂Ω.
Then, {xk} converges weakly to a point in Ω.
In this section, using the idea of Linesearches 1 and 2, we introduce a new linesearch and present an inertial double forward-backward algorithm with the proposed linesearch for solving the convex minimization problem of the sum of two convex functions without any Lipschitz continuity assumption on the gradient. A weak convergence result of our proposed method is analyzed and established.
We now focus on Problem (1.1) and assume that f1 and f2 satisfy Hypotheses (H1) and (H2). The solution set of (1.1) is denoted by Ω. Also, suppose that Ω≠∅. For simplicity, we let F:=f1+f2 and denote by FBα the forward-backward operator of f1 and f2 with respect to α, that is, FBα:=proxαf2(Id−α∇f1). Here, our linesearch is designed as follows.
Linesearch 3. Given x∈domf2, σ>0, θ∈(0,1), μ∈(0,1) and δ>0. |
Input α=σ. |
While α{(1−μ)‖∇f1(FB2α(x))−∇f1(FBα(x))‖+μ‖∇f1(FBα(x))−∇f1(x)‖} |
>δ(‖FB2α(x)−FBα(x)‖+‖FBα(x)−x‖), do |
α=θα. |
End |
Output α. |
Remark 3.1. The loop termination condition of Linesearch 3 is weaker than that of Linesearch 2. Indeed, since
α{(1−μ)‖∇f1(FB2α(x))−∇f1(FBα(x))‖+μ‖∇f1(FBα(x))−∇f1(x)‖}≤αmax{‖∇f1(FB2α(x))−∇f1(FBα(x))‖,‖∇f1(FBα(x))−∇f1(x)‖}, |
it follows from the well-definedness of Linesearch 2 that our linesearch also stops after finitely many steps, see [14, Lemma 3.2] for more details.
Using Linesearch 3, we propose the following iterative method with the inertial technical term.
Method 5. |
Initialization: Take x1=y0∈domf2, σ>0, θ∈(0,1), μ∈(0,12] and δ∈(0,μ4). Let {βk}⊂R+. |
Iterative steps: For k≥1, calculate xk+1 as follows: |
Step 1. Compute |
zk=FBαk(xk)=proxαkf2(xk−αk∇f1(xk)),(3.1) |
yk=FBαk(zk)=proxαkf2(zk−αk∇f1(zk)),(3.2) |
where αk:= Linesearch 3(xk,σ,θ,μ,δ). |
Step 2. Compute |
xk+1=Pdomf2(yk+βk(yk−yk−1)).(3.3) |
Set k:=k+1 and return to Step 1. |
To verify the convergence of Method 5, the following result is needed.
Lemma 3.2. Let {xk} be the sequence generated by Method 5. For each k≥1 and p∈domf2, we have
‖xk−p‖2−‖yk−p‖2≥2αk[F(yk)+F(zk)−2F(p)]+(1−4δμ)(‖xk−zk‖2+‖zk−yk‖2). |
Proof. Let k≥1 and p∈domf2. By (2.3), (3.1) and (3.2), we get
xk−zkαk−∇f1(xk)∈∂f2(zk)andzk−ykαk−∇f1(zk)∈∂f2(yk). |
Then, by the definition of the subdifferential of f2, we have
f2(p)−f2(zk)≥⟨xk−zkαk−∇f1(xk),p−zk⟩=1αk⟨xk−zk,p−zk⟩+⟨∇f1(xk),zk−p⟩ | (3.4) |
and
f2(p)−f2(yk)≥⟨zk−ykαk−∇f1(zk),p−yk⟩=1αk⟨zk−yk,p−yk⟩+⟨∇f1(zk),yk−p⟩. | (3.5) |
By the assumptions on f1 and f2, we have the following fact:
f1(x)−f1(y)≥⟨∇f1(y),x−y⟩,∀x∈domf1,∀y∈domf2. | (3.6) |
From (3.6), we get
f1(p)−f1(xk)≥⟨∇f1(xk),p−xk⟩, | (3.7) |
and
f1(p)−f1(zk)≥⟨∇f1(zk),p−zk⟩. | (3.8) |
Summing (3.4), (3.5), (3.7) and (3.8) yields
2F(p)−F(zk)−f2(yk)−f1(xk)≥⟨∇f1(xk),zk−p⟩+⟨∇f1(zk),yk−p⟩+⟨∇f1(xk),p−xk⟩+⟨∇f1(zk),p−zk⟩+1αk[⟨xk−zk,p−zk⟩+⟨zk−yk,p−yk⟩]=⟨∇f1(xk),zk−xk⟩+⟨∇f1(zk),yk−zk⟩+1αk[⟨xk−zk,p−zk⟩+⟨zk−yk,p−yk⟩]=⟨∇f1(xk)−∇f1(zk),zk−xk⟩+⟨∇f1(zk),zk−xk⟩+⟨∇f1(yk),yk−zk⟩+⟨∇f1(zk)−∇f1(yk),yk−zk⟩+1αk[⟨xk−zk,p−zk⟩+⟨zk−yk,p−yk⟩]≥⟨∇f1(zk),zk−xk⟩+⟨∇f1(yk),yk−zk⟩−‖∇f1(xk)−∇f1(zk)‖‖zk−xk‖−‖∇f1(zk)−∇f1(yk)‖‖yk−zk‖+1αk[⟨xk−zk,p−zk⟩+⟨zk−yk,p−yk⟩]. |
Using (3.6) again, the above inequality becomes
2F(p)−F(zk)−f2(yk)−f1(xk)≥f1(yk)−f1(xk)−‖∇f1(xk)−∇f1(zk)‖‖zk−xk‖−‖∇f1(zk)−∇f1(yk)‖‖yk−zk‖+1αk[⟨xk−zk,p−zk⟩+⟨zk−yk,p−yk⟩]≥f1(yk)−f1(xk)−‖∇f1(xk)−∇f1(zk)‖(‖yk−zk‖+‖zk−xk‖)−‖∇f1(zk)−∇f1(yk)‖(‖yk−zk‖+‖zk−xk‖)+1αk[⟨xk−zk,p−zk⟩+⟨zk−yk,p−yk⟩]=f1(yk)−f1(xk)+1αk[⟨xk−zk,p−zk⟩+⟨zk−yk,p−yk⟩]−(‖∇f1(xk)−∇f1(zk)‖+‖∇f1(zk)−∇f1(yk)‖)(‖yk−zk‖+‖zk−xk‖). | (3.9) |
Since αk:= Linesearch 3(xk,σ,θ,μ,δ), we have
αk{(1−μ)‖∇f1(yk)−∇f1(zk)‖+μ‖∇f1(zk)−∇f1(xk)‖}≤δ(‖yk−zk‖+‖zk−xk‖). | (3.10) |
It follows from (3.9) and (3.10) that
1αk[⟨xk−zk,zk−p⟩+⟨zk−yk,yk−p⟩]≥F(yk)+F(zk)−2F(p)−(‖∇f1(xk)−∇f1(zk)‖+‖∇f1(zk)−∇f1(yk)‖)(‖yk−zk‖+‖zk−xk‖)≥F(yk)+F(zk)−2F(p)−1μ{(1−μ)‖∇f1(yk)−∇f1(zk)‖+μ‖∇f1(zk)−∇f1(xk)‖}(‖yk−zk‖+‖zk−xk‖)≥F(yk)+F(zk)−2F(p)−δαkμ(‖yk−zk‖+‖zk−xk‖)2≥F(yk)+F(zk)−2F(p)−2δαkμ(‖yk−zk‖2+‖zk−xk‖2). | (3.11) |
From (2.4), we get
⟨xk−zk,zk−p⟩=12(‖xk−p‖2−‖xk−zk‖2−‖zk−p‖2), | (3.12) |
and
⟨zk−yk,yk−p⟩=12(‖zk−p‖2−‖zk−yk‖2−‖yk−p‖2). | (3.13) |
Therefore, we conclude from (3.11)-(3.13) that
‖xk−p‖2−‖yk−p‖2≥2αk[F(yk)+F(zk)−2F(p)]+(1−4δμ)(‖xk−zk‖2+‖zk−yk‖2). |
Now we are ready to prove a convergence result of Method 5, which is the main theorem of this paper.
Theorem 3.3. Let {xk} be the sequence generated by Method 5. If αk≥α>0, βk≥0 for all k∈N and ∑∞k=1βk<∞, then {xk} converges weakly to a point in Ω.
Proof. Let p∗∈Ω, αk≥α>0, βk≥0 for all k∈N and ∑∞k=1βk<∞. By applying Lemma 3.2, we have
‖xk−p∗‖2−‖yk−p∗‖2≥2αk[F(yk)−F(p∗)+F(zk)−F(p∗)]+(1−4δμ)(‖xk−zk‖2+‖zk−yk‖2)≥(1−4δμ)‖xk−zk‖2≥0, | (3.14) |
which implies
‖yk−p∗‖≤‖xk−p∗‖. | (3.15) |
Let x0=x1. By (3.3) and (3.15), we have
‖xk+1−p∗‖=‖Pdomf2(yk+βk(yk−yk−1))−p∗‖≤‖yk+βk(yk−yk−1)−p∗‖≤‖yk−p∗‖+βk‖yk−yk−1‖≤(1+βk)‖yk−p∗‖+βk‖yk−1−p∗‖≤(1+βk)‖xk−p∗‖+βk‖xk−1−p∗‖,∀k∈N. | (3.16) |
By Lemma 2.2, we have ‖xk+1−p∗‖≤K⋅∏kj=1(1+2βj), where K:=max{‖x1−p∗‖,‖x2−p∗‖}, and {xk} is bounded. Thus, ∑∞k=1βk(‖xk−p∗‖+‖xk−1−p∗‖)<∞. In view of (3.16), it follows from Lemma 2.3 that limk→∞‖xk−p∗‖ exists. Also, we have limk→∞‖xk−p∗‖=limk→∞‖yk−p∗‖. Then, by (3.14), we get
limk→∞‖xk−zk‖=0. | (3.17) |
Next, we show that ωw(xk)⊂Ω. Pick ˉp∈ωw(xk), then there is a subsequence {xki} of {xk} such that xki⇀ˉp. Thus, zki⇀ˉp. By (H2), we have
limk→∞‖∇f1(xk)−∇f1(zk)‖=0. | (3.18) |
From (2.3), we get
xki−zkiαki+∇f1(zki)−∇f1(xki)∈∂f2(zki)+∇f1(zki)=∂F(zki). | (3.19) |
Here, by zki⇀ˉp and (3.17)-(3.19), it follows from Lemma 2.1 that 0∈∂F(ˉp), that is, ˉp∈Ω. So, ωw(xk)⊂Ω. Using Lemma 2.4, we obtain that xk⇀p for some p∈Ω. This completes the proof.
The following method is obtained by reducing the inertial step in Method 5.
Method 6. |
Initialization: Take x1∈domf2, σ>0, θ∈(0,1), μ∈(0,12] and δ∈(0,μ4). |
Iterative steps: For k≥1, compute |
zk=FBαk(xk)=proxαkf2(xk−αk∇f1(xk)), (3.20) |
xk+1=FBαk(zk)=proxαkf2(zk−αk∇f1(zk)), (3.21) |
where αk:= Linesearch 3(xk,σ,θ,μ,δ). |
We also analyze the convergence and the complexity of Method 6.
Theorem 3.4. Let {xk} be the sequence generated by Method 6. If αk≥α>0 for all k∈N, then the following statements hold:
(i) {xk} converges weakly to a point in Ω;
(ii) If δ∈(0,μ8), then {F(xk)} is decreasing and
F(xk)−minx∈HF(x)≤[dist(x1,Ω)]22αk,∀k∈N. | (3.22) |
Proof. Setting βk:=0 in Method 5, (i) is obtained directly from Theorem 3.3. To prove (ii), let δ∈(0,μ8). We first show that {F(xk)} is decreasing. By Lemma 3.2, we get
‖xk−p‖2−‖xk+1−p‖2≥2αk[F(xk+1)+F(zk)−2F(p)]+(1−4δμ)(‖xk−zk‖2+‖zk−xk+1‖2),∀p∈domf2. | (3.23) |
Taking p:=zk in (3.23), we have
‖xk−zk‖2−‖xk+1−zk‖2≥2αk[F(xk+1)−F(zk)]+(1−4δμ)(‖xk−zk‖2+‖zk−xk+1‖2). | (3.24) |
Also, taking p:=xk in (3.23), we have
−‖xk+1−xk‖2≥2αk[F(xk+1)+F(zk)−2F(xk)]+(1−4δμ)(‖xk−zk‖2+‖zk−xk+1‖2). | (3.25) |
Combining (3.24) and (3.25), we get
‖xk−zk‖2−‖xk+1−zk‖2−‖xk+1−xk‖2≥4αk[F(xk+1)−F(xk)]+2(1−4δμ)(‖xk−zk‖2+‖zk−xk+1‖2), |
which implies
F(xk+1)−F(xk)≤−14αk{(1−8δμ)‖xk−zk‖2+(3−8δμ)‖zk−xk+1‖2+‖xk+1−xk‖2}. |
This together with δ∈(0,μ8) yields F(xk+1)≤F(xk), that is, {F(xk)} is decreasing. Next, we will show that (3.22) is true. Pick any p∗∈Ω. From (3.23), we have
‖xk−p∗‖2−‖xk+1−p∗‖2≥2αk[F(xk+1)−F(p∗)+F(zk)−F(p∗)]+(1−4δμ)(‖xk−zk‖2+‖zk−xk+1‖2)≥2α[F(xk+1)−F(p∗)]. |
Hence
F(xi+1)−F(p∗)≤12α(‖xi−p∗‖2−‖xi+1−p∗‖2),∀i∈N. |
It follows that
k∑i=1F(xi+1)−kF(p∗)≤12αk∑i=1(‖xi−p∗‖2−‖xi+1−p∗‖2)=12α(‖x1−p∗‖2−‖xk+1−p∗‖2)≤‖x1−p∗‖22α. |
By the decreasing property of {F(xk)}, we get
k[F(xk)−F(p∗)]≤k∑i=1F(xi+1)−kF(p∗)≤‖x1−p∗‖22α,∀p∗∈Ω. | (3.26) |
It follows from (3.26) that
F(xk)−minx∈HF(x)≤infp∗∈Ω‖x1−p∗‖22αk=[dist(x1,Ω)]22αk. |
We note that the stepsize condition on {αk} in Theorems 3.3 and 3.4 needs the boundedness from below by a positive real number. Next, we show that this condition can be guaranteed by the Lipschitz continuity assumption on the gradient.
Let C⊂H be a nonempty closed convex set. Recall that an operator T:C→H is said to be Lipschitz continuous if there exists L>0 such that ‖Tx−Ty‖≤L‖x−y‖ for all x,y∈C.
Proposition 3.5. Let {αk} be the sequence generated by Linesearch 3 of Method 5 (or Method 6). If ∇f1 is Lipschitz continuous on domf2 with a coefficient L>0, then αk≥min{σ,δθL} for all k∈N.
Proof. Let ∇f1 be L-Lipschitz continuous on domf2. From Linesearch 3, we know that αk≤σ for all k∈N. If αk<σ, then αk=σθmk where mk is the smallest positive integer such that
αk{(1−μ)‖∇f1(FB2αk(xk))−∇f1(FBαk(xk))‖+μ‖∇f1(FBαk(xk))−∇f1(xk)‖}≤δ(‖FB2αk(xk)−FBαk(xk)‖+‖FBαk(xk)−xk‖). |
Let ¯αk:=αkθ>0. By the Lipschitz continuity of ∇f1 and the above expression, we have
¯αkL(‖FB2¯αk(xk)−FB¯αk(xk)‖+‖FB¯αk(xk)−xk‖)≥¯αk(‖∇f1(FB2¯αk(xk))−∇f1(FB¯αk(xk))‖+‖∇f1(FB¯αk(xk))−∇f1(xk)‖)≥¯αk{(1−μ)‖∇f1(FB2¯αk(xk))−∇f1(FB¯αk(xk))‖+μ‖∇f1(FB¯αk(xk))−∇f1(xk)‖}>δ(‖FB2¯αk(xk)−FB¯αk(xk)‖+‖FB¯αk(xk)−xk‖), |
which implies that αk>δθL. Consequently, αk≥min{σ,δθL} for all k∈N.
Remark 3.6. It is worth mentioning again that the Lipschitz continuity assumption on the gradient of f1 is sufficient for Hypothesis (H2). However, if we assume this assumption further, the calculation of the stepsize αk generated by Linesearch 3 is still independent of the Lipschitz constant.
In this section, we apply the convex minimization problem, Problem (1.1), to an image restoration problem and a regression problem. To slove these problems, we analyze and illustrate the convergence behavior of our method (Method 5) and compare its efficiency with Methods 1-4 and 6. We also consider the following method with another linesearch strategy [21]:
Linesearch 4. Given x∈domf2, ˉσ>0, and ˉθ∈(0,1). |
Input α=ˉσ. |
While F(FBα(x))>f1(x)+⟨FBα(x)−x,∇f1(x)⟩+12α‖FBα(x)−x‖2+f2(FBα(x)), do |
α=ˉθα. |
End |
Output α. |
Method 7. Let x1=y0∈domf2, ˉσ>0, ˉθ∈(0,1), ρk>0, and t1=1. For k≥1, let
yk=proxαkf2(xk−αk∇f1(xk)),tk+1=1+√1+4ρkt2k2,βk=tk−1tk+1, xk+1=yk+βk(yk−yk−1), |
where αk:= Linesearch 4(xk,ˉσ,ˉθ).
Note that Method 7 is well known as the FISTA with backtracking, see [5,25].
All experiments and visualizations are done with MATLAB and are performed on a laptop computer (Intel Core-i5/4.00 GB RAM/Windows 8/64-bit).
Many problems rising in image and signal processing, especially the image restoration problem, can be formulated as the following equation:
y=Ax+ε, | (4.1) |
where x∈RN is an original image, y∈RM is the observed image, ε is an additive noise and A∈RM×N is the blurring operation. To approximate the original image, we need to minimize the value of ε by using the LASSO model [28]:
minx∈RN{12‖y−Ax‖22+λ‖x‖1}, | (4.2) |
where λ>0 is a regularization parameter, ‖⋅‖1 is the l1-norm, and ‖⋅‖2 is the Euclidean norm. In this situation, we apply Problem (1.1) to the LASSO model (4.2) by setting
f1(x)=12‖y−Ax‖22andf2(x)=λ‖x‖1, |
where A=RW, R is the matrix representing the blur operator and W is the inverse of a three stage Haar wavelet transform. We use the peak signal-to-noise ratio (PSNR) in decibel (dB) [30] and the structural similarity index metric (SSIM) [33] as the image quality measures, which are formulated as follows:
PSNR(xk)=10log10(2552MSE(k)), |
where MSE(k)=1M‖xk−x‖22, M is the number of image samples, and x is the original image and
SSIM(xk,x)=(2νxkνx+C1)(2ξxkx+C2)(ν2xk+ν2x+C1)(ξ2xk+ξ2x+C2), |
where {νxk,ξxk} and {νx,ξx} denote the mean intensity and standard deviation set of the deblerring image xk and the original image x, respectively, ξxnx denotes its cross correlation, and C1, C2 are small constants value to avoid instability problem when the denominator is too close to zero.
Example 4.1. Consider two grayscale images, Cameraman (see Figure 1 (a)) and Boy (see Figure 2 (a)), with size of 256×256, which are contaminated by Gaussian blur of filter size 9×9 with standard deviation ˆσ=4, noise 10−5 (see Figure 1 (b) and Figure 2 (b)).
Firstly, we test the efficiency of Method 5 for recovering the Caneraman image at the 300th iteration by choosing various linesearch parameters and different starting points as shown in Table 1.
Initial points x1 | Linesearch parameters | PSNR | SSIM | |||
σ | θ | μ | δ | |||
Blurred image | 5 | 0.1 | 0.1 | 0.02 | 31.554 | 0.6193 |
1 | 0.5 | 0.1 | 0.02 | 31.096 | 0.6081 | |
0.1 | 0.9 | 0.5 | 0.12 | 28.721 | 0.5717 | |
10 | 0.9 | 0.5 | 0.12 | 33.051 | 0.6708 | |
Random selection | 5 | 0.1 | 0.1 | 0.02 | 30.554 | 0.6043 |
1 | 0.5 | 0.1 | 0.02 | 29.660 | 0.5934 | |
0.1 | 0.9 | 0.5 | 0.12 | 28.694 | 0.5640 | |
10 | 0.9 | 0.5 | 0.12 | 32.798 | 0.6554 |
It is observed from Table 1 that the linesearch parameter σ influences the image recovery performance of Method 5, while choosing the starting point x1 has no significant impact on the convergence behavior of our method.
Next, we test and analyze the image recovery performance of Methods 1-7 for recovering the images (Cameraman and Boy) by setting the parameters as in Table 2 and by choosing the blurred images as the starting points. The maximum iteration number for all methods is fixed at 500. The comparative experiments for recovering the Cameraman and Boy images are as follows. Original images, blurry image contaminated by Gaussian blur and restored images by Methods 1-7 are shown in Figures 1 and 2. The PSNR and SSIM results are shown in Figures 3 and 4. It can be seen that Method 5 gives the higher values of PSNR and SSIM than the others, meanwhile Method 6 gives the better results than the other methods without the inertial step. Therefore, our method has the highest image recovery efficiency comparing with the studied methods.
Methods | |||||||
Parameters | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
λ=10−5 | √ | √ | √ | √ | √ | √ | √ |
αk=k(k+1)L, L=λmax(A⊤A) | √ | - | - | - | - | - | - |
σ=10, θ=0.9, δ=0.1 | - | √ | √ | √ | √ | - | - |
μ=0.5 | - | - | - | - | √ | √ | - |
βk={kk+1if1≤k≤50012kotherwise | - | - | - | - | √ | - | - |
ˉσ=10, ˉθ=0.9, ρk=1 | - | - | - | - | - | - | √ |
We first introduce a brief concept of extreme learning machine (ELM) for regression problems. Let S={(xk,tk):xk∈Rn,tk∈Rm,k=1,2,...,N} be a training set of N distinct samples where xk is an input data and tk is a target. For any single hidden layer of ELM, the output of the i-th hidden node is
hi(x)=G(ai,bi,x), |
where G is an activation function, ai and bi are parameters of the i-th hidden node. The output function of ELM for a single-hidden layer feedforward neural networks (SLFNs) with M hidden nodes is
Ok=M∑i=1ξihi(xk), |
where ξi is the output weight of the i-th hidden node. The hidden layer output matrix H is defined by
H=[G(a1,b1,x1)⋯G(aM,bM,x1)⋮⋱⋮G(a1,b1,xN)⋯G(aM,bM,xN)]N×M. |
The main goal of ELM is to find ξ=[ξ⊤1,...,ξ⊤M]⊤ such that
Hξ=T, | (4.3) |
where T=[t⊤1,...,t⊤N]⊤. From (4.3), we can estimate the weight ξ by ξ=H†T where H†=(H⊤H)−1H⊤ is the pseudo-inverse matrix of H.
Example 4.2. This example aims to predict the sine function by using the ELM method defined as follows:
∙ Create randomly 10 distinct points y1,y2,...,y10∈[−4,4];
∙ Create a training matrix R=[y1y2...y10]⊤ and a training target matrix S=[sin(y1)sin(y2)...sin(y10)]⊤;
∙ Create a testing matrix V=[−4−3.99−3.98...4] and a target matrix T=[sin(−4)sin(−3.99)...sin(y10)]⊤;
∙ Using sigmoid as an activation function generates the hidden layer output matrix H1 of the training matrix R with the hidden node M=100;
∙ Choose a regularization parameter λ=10−5. Formulate a convex minimization problem via the LASSO model that
f1(ξ)=12‖H1ξ−S‖22andf2(ξ)=λ‖ξ‖1, |
and find the optimal weight ξk of this problem by employing Methods 1-5 with certain number of iterations;
∙ Using sigmoid as an activation function generates the hidden layer output matrix H2 of the testing matrix V with the hidden node M=100;
∙ Calculate the output Ok=H2ξk and calculate the mean squared error (MSE) at k-th iteration by
MSE(k)=1N‖Ok−T‖22, |
where N is the number of distinct samples.
The parameters for each prediction method are set as in Table 3.
Methods | |||||||
Parameters | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
αk=k(k+1)L, L=λmax(H⊤H) | √ | - | - | - | - | - | - |
σ=0.1, θ=0.49, δ=0.1 | - | √ | √ | √ | √ | √ | - |
μ=0.5 | - | - | - | - | √ | √ | - |
βk={kk+1if1≤k≤1000012kotherwise | - | - | - | - | √ | - | - |
ˉσ=10−5, ˉθ=0.49, ρk=1 | - | - | - | - | - | - | √ |
Now a graph of predicting the sine function by Methods 1-7 at the 500th iteration is shown in Figure 5. We show a comparative result of iteration numbers, mean squared error values and times in Table 4 when the stopping criterion is set as: MSE≤10−3 or at the 10000th iteration. Also, a graph of the mean squared error is shown in Figure 6. It can be observed that Method 5 gives a better performance for predicting the sine function than the other tested methods.
Methods | Iteration No. | Time (s) | MSE |
1 | 10000 | 0.7986 | 0.0408 |
2 | 10000 | 12.1314 | 0.0032 |
3 | 3999 | 1.3463 | 9.9995×10−4 |
4 | 10000 | 17.7834 | 0.0014 |
5 | 338 | 0.6531 | 9.9239×10−4 |
6 | 10000 | 18.3197 | 0.0014 |
7 | 10000 | 3.0340 | 0.0067 |
In this paper, we discuss the convex minimization problem of the sum of two convex functions in a Hilbert space. The challenge of removing the Lipschitz continuity assumption on the gradient of the function attracts us to study the concept of the linesearch process. We introduce a new linesearch and propose an inertial forward-backward algorithm whose stepsize does not depend on any Lipschitz constant for solving the considered problem without any Lipschitz continuity condition on the gradient. It is shown that the sequence generated by our proposed method converges weakly to a minimizer of the sum of those two convex functions under some mild control conditions. As applications, we employ our method to recover blurry images in an image restoration problem and predict the sine function in a regression problem. The results of the experiments show that our method has a higher efficiency than the well-known methods in [5,7,14,15,25].
This research was supported by Thailand Science Research and Innovation under the project IRN62W0007 and Chiang Mai University.
The authors declare no conflict of interest.
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1. | José A. Vásquez-Coronel, Marco Mora, Karina Vilches, A Review of multilayer extreme learning machine neural networks, 2023, 56, 0269-2821, 13691, 10.1007/s10462-023-10478-4 | |
2. | Adisak Hanjing, Panadda Thongpaen, Suthep Suantai, A new accelerated algorithm with a linesearch technique for convex bilevel optimization problems with applications, 2024, 9, 2473-6988, 22366, 10.3934/math.20241088 |
Linesearch 1. Given x∈domf2, σ>0, θ∈(0,1) and δ>0. |
Input α=σ. |
While α‖∇f1(FBα(x))−∇f1(x)‖>δ‖FBα(x)−x‖, do |
α=θα. |
End |
Output α. |
Linesearch 2. Given x∈domf2, σ>0, θ∈(0,1) and δ>0. |
Input α=σ. |
While αmax{‖∇f1(FB2α(x))−∇f1(FBα(x))‖,‖∇f1(FBα(x))−∇f1(x)‖} |
>δ(‖FB2α(x)−FBα(x)‖+‖FBα(x)−x‖), do |
α=θα. |
End |
Output α, |
where FB2α(x):=FBα(FBα(x)). |
Linesearch 3. Given x∈domf2, σ>0, θ∈(0,1), μ∈(0,1) and δ>0. |
Input α=σ. |
While α{(1−μ)‖∇f1(FB2α(x))−∇f1(FBα(x))‖+μ‖∇f1(FBα(x))−∇f1(x)‖} |
>δ(‖FB2α(x)−FBα(x)‖+‖FBα(x)−x‖), do |
α=θα. |
End |
Output α. |
Method 5. |
Initialization: Take x1=y0∈domf2, σ>0, θ∈(0,1), μ∈(0,12] and δ∈(0,μ4). Let {βk}⊂R+. |
Iterative steps: For k≥1, calculate xk+1 as follows: |
Step 1. Compute |
zk=FBαk(xk)=proxαkf2(xk−αk∇f1(xk)),(3.1) |
yk=FBαk(zk)=proxαkf2(zk−αk∇f1(zk)),(3.2) |
where αk:= Linesearch 3(xk,σ,θ,μ,δ). |
Step 2. Compute |
xk+1=Pdomf2(yk+βk(yk−yk−1)).(3.3) |
Set k:=k+1 and return to Step 1. |
Method 6. |
Initialization: Take x1∈domf2, σ>0, θ∈(0,1), μ∈(0,12] and δ∈(0,μ4). |
Iterative steps: For k≥1, compute |
zk=FBαk(xk)=proxαkf2(xk−αk∇f1(xk)), (3.20) |
xk+1=FBαk(zk)=proxαkf2(zk−αk∇f1(zk)), (3.21) |
where αk:= Linesearch 3(xk,σ,θ,μ,δ). |
Linesearch 4. Given x∈domf2, ˉσ>0, and ˉθ∈(0,1). |
Input α=ˉσ. |
While F(FBα(x))>f1(x)+⟨FBα(x)−x,∇f1(x)⟩+12α‖FBα(x)−x‖2+f2(FBα(x)), do |
α=ˉθα. |
End |
Output α. |
Initial points x1 | Linesearch parameters | PSNR | SSIM | |||
σ | θ | μ | δ | |||
Blurred image | 5 | 0.1 | 0.1 | 0.02 | 31.554 | 0.6193 |
1 | 0.5 | 0.1 | 0.02 | 31.096 | 0.6081 | |
0.1 | 0.9 | 0.5 | 0.12 | 28.721 | 0.5717 | |
10 | 0.9 | 0.5 | 0.12 | 33.051 | 0.6708 | |
Random selection | 5 | 0.1 | 0.1 | 0.02 | 30.554 | 0.6043 |
1 | 0.5 | 0.1 | 0.02 | 29.660 | 0.5934 | |
0.1 | 0.9 | 0.5 | 0.12 | 28.694 | 0.5640 | |
10 | 0.9 | 0.5 | 0.12 | 32.798 | 0.6554 |
Methods | |||||||
Parameters | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
λ=10−5 | √ | √ | √ | √ | √ | √ | √ |
αk=k(k+1)L, L=λmax(A⊤A) | √ | - | - | - | - | - | - |
σ=10, θ=0.9, δ=0.1 | - | √ | √ | √ | √ | - | - |
μ=0.5 | - | - | - | - | √ | √ | - |
βk={kk+1if1≤k≤50012kotherwise | - | - | - | - | √ | - | - |
ˉσ=10, ˉθ=0.9, ρk=1 | - | - | - | - | - | - | √ |
Methods | |||||||
Parameters | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
αk=k(k+1)L, L=λmax(H⊤H) | √ | - | - | - | - | - | - |
σ=0.1, θ=0.49, δ=0.1 | - | √ | √ | √ | √ | √ | - |
μ=0.5 | - | - | - | - | √ | √ | - |
βk={kk+1if1≤k≤1000012kotherwise | - | - | - | - | √ | - | - |
ˉσ=10−5, ˉθ=0.49, ρk=1 | - | - | - | - | - | - | √ |
Methods | Iteration No. | Time (s) | MSE |
1 | 10000 | 0.7986 | 0.0408 |
2 | 10000 | 12.1314 | 0.0032 |
3 | 3999 | 1.3463 | 9.9995×10−4 |
4 | 10000 | 17.7834 | 0.0014 |
5 | 338 | 0.6531 | 9.9239×10−4 |
6 | 10000 | 18.3197 | 0.0014 |
7 | 10000 | 3.0340 | 0.0067 |
Linesearch 1. Given x∈domf2, σ>0, θ∈(0,1) and δ>0. |
Input α=σ. |
While α‖∇f1(FBα(x))−∇f1(x)‖>δ‖FBα(x)−x‖, do |
α=θα. |
End |
Output α. |
Linesearch 2. Given x∈domf2, σ>0, θ∈(0,1) and δ>0. |
Input α=σ. |
While αmax{‖∇f1(FB2α(x))−∇f1(FBα(x))‖,‖∇f1(FBα(x))−∇f1(x)‖} |
>δ(‖FB2α(x)−FBα(x)‖+‖FBα(x)−x‖), do |
α=θα. |
End |
Output α, |
where FB2α(x):=FBα(FBα(x)). |
Linesearch 3. Given x∈domf2, σ>0, θ∈(0,1), μ∈(0,1) and δ>0. |
Input α=σ. |
While α{(1−μ)‖∇f1(FB2α(x))−∇f1(FBα(x))‖+μ‖∇f1(FBα(x))−∇f1(x)‖} |
>δ(‖FB2α(x)−FBα(x)‖+‖FBα(x)−x‖), do |
α=θα. |
End |
Output α. |
Method 5. |
Initialization: Take x1=y0∈domf2, σ>0, θ∈(0,1), μ∈(0,12] and δ∈(0,μ4). Let {βk}⊂R+. |
Iterative steps: For k≥1, calculate xk+1 as follows: |
Step 1. Compute |
zk=FBαk(xk)=proxαkf2(xk−αk∇f1(xk)),(3.1) |
yk=FBαk(zk)=proxαkf2(zk−αk∇f1(zk)),(3.2) |
where αk:= Linesearch 3(xk,σ,θ,μ,δ). |
Step 2. Compute |
xk+1=Pdomf2(yk+βk(yk−yk−1)).(3.3) |
Set k:=k+1 and return to Step 1. |
Method 6. |
Initialization: Take x1∈domf2, σ>0, θ∈(0,1), μ∈(0,12] and δ∈(0,μ4). |
Iterative steps: For k≥1, compute |
zk=FBαk(xk)=proxαkf2(xk−αk∇f1(xk)), (3.20) |
xk+1=FBαk(zk)=proxαkf2(zk−αk∇f1(zk)), (3.21) |
where αk:= Linesearch 3(xk,σ,θ,μ,δ). |
Linesearch 4. Given x∈domf2, ˉσ>0, and ˉθ∈(0,1). |
Input α=ˉσ. |
While F(FBα(x))>f1(x)+⟨FBα(x)−x,∇f1(x)⟩+12α‖FBα(x)−x‖2+f2(FBα(x)), do |
α=ˉθα. |
End |
Output α. |
Initial points x1 | Linesearch parameters | PSNR | SSIM | |||
σ | θ | μ | δ | |||
Blurred image | 5 | 0.1 | 0.1 | 0.02 | 31.554 | 0.6193 |
1 | 0.5 | 0.1 | 0.02 | 31.096 | 0.6081 | |
0.1 | 0.9 | 0.5 | 0.12 | 28.721 | 0.5717 | |
10 | 0.9 | 0.5 | 0.12 | 33.051 | 0.6708 | |
Random selection | 5 | 0.1 | 0.1 | 0.02 | 30.554 | 0.6043 |
1 | 0.5 | 0.1 | 0.02 | 29.660 | 0.5934 | |
0.1 | 0.9 | 0.5 | 0.12 | 28.694 | 0.5640 | |
10 | 0.9 | 0.5 | 0.12 | 32.798 | 0.6554 |
Methods | |||||||
Parameters | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
λ=10−5 | √ | √ | √ | √ | √ | √ | √ |
αk=k(k+1)L, L=λmax(A⊤A) | √ | - | - | - | - | - | - |
σ=10, θ=0.9, δ=0.1 | - | √ | √ | √ | √ | - | - |
μ=0.5 | - | - | - | - | √ | √ | - |
βk={kk+1if1≤k≤50012kotherwise | - | - | - | - | √ | - | - |
ˉσ=10, ˉθ=0.9, ρk=1 | - | - | - | - | - | - | √ |
Methods | |||||||
Parameters | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
αk=k(k+1)L, L=λmax(H⊤H) | √ | - | - | - | - | - | - |
σ=0.1, θ=0.49, δ=0.1 | - | √ | √ | √ | √ | √ | - |
μ=0.5 | - | - | - | - | √ | √ | - |
βk={kk+1if1≤k≤1000012kotherwise | - | - | - | - | √ | - | - |
ˉσ=10−5, ˉθ=0.49, ρk=1 | - | - | - | - | - | - | √ |
Methods | Iteration No. | Time (s) | MSE |
1 | 10000 | 0.7986 | 0.0408 |
2 | 10000 | 12.1314 | 0.0032 |
3 | 3999 | 1.3463 | 9.9995×10−4 |
4 | 10000 | 17.7834 | 0.0014 |
5 | 338 | 0.6531 | 9.9239×10−4 |
6 | 10000 | 18.3197 | 0.0014 |
7 | 10000 | 3.0340 | 0.0067 |