Hybrid proximal linearized algorithm for the split DC program in infinite-dimensional real Hilbert spaces.

To be the best of our knowledge, the convergence theorem for the DC program and split DC program are proposed in finite-dimensional real Hilbert spaces or Euclidean spaces. In this paper, to study the split DC program, we give a hybrid proximal linearized algorithm and propose related convergence theorems in the settings of finite- and infinite-dimensional real Hilbert spaces, respectively.

Introduction

Let H be a real Hilbert space, and let be a proper lower semicontinuous and convex function. Define a sequence by taking arbitrarily and Then converges weakly to a minimizer of f under suitable conditions, and this is called the proximal point algorithm (PPA). This algorithm is useful, however, only for convex problems, because the idea for this algorithm is based on the monotonicity of subdifferential operators of convex functions. So, it is important to consider the relation between nonconvex functions and proximal point algorithm.

The DC program is the well-known nonconvex problem of the form where are proper lower semicontinuous convex functions. Here, the function f is called a DC function, and the functions g and h are called the DC components of f. (In the DC program, the convention is adopted to avoid the ambiguity that does not present any interest.) It is well known that a necessary condition for to be a local minimizer of f is . However, this condition is hard to be reached. So, many researchers focus their attentions on finding points such that , where x is called a critical point of f [1].

It is worth mentioning the richness of the class of DC functions that is a subspace containing the class of lower- functions. In particular, contains the space of functions with locally Lipschitz continuous gradients. Further, is closed under the operations usually considered in optimization. For example, a linear combination, a finite supremum, or the product of two DC functions remain DC. It is also known that the set of DC functions defined on a compact convex set of is dense in the set of continuous functions on this set.

The interest in the theory of DC functions has much increased in the last years. Some interesting optimality conditions and duality theorems related to the DC program are given. For more details, we refer to [2-9].

In 2003, Sun, Sampaio, and Candido [10] proposed a proximal point algorithm to study problem (DCP).

Algorithm 1.1

(Proximal point algorithm for (DCP) [10])

Let be a sequence in , and let be proper lower semicontinuous and convex functions. Let be generated as follows:

In 2016, Souza, Oliveira, and Soubeyran [11] proposed a proximal linearized algorithm to study the DC program.

Algorithm 1.2

(Proximal linearized algorithm [11])

Let be a sequence in , and let be proper lower semicontinuous and convex functions. Let be generated as follows:

Besides, some algorithms for the DC program are proposed to analyze and solve a variety of highly structured and practical problems (see, for example, [12]).

On the other hand, Chuang [13] introduced the following split DC program (split minimization problems for DC functions): where and are real Hilbert spaces, is a linear bounded mapping with adjoint , and are proper lower semicontinuous and convex functions, and and for all and . Further, to study problem (SDCP), Chuang [13] gave the following split proximal linearized algorithm and related convergence theorem in finite-dimensional real Hilbert spaces.

Algorithm 1.3

(Split proximal linearized algorithm)

Let be generated as follows:

Besides, there are also some important algorithms for the related problems in the literature; see, for example, [14-17].

In this paper, motivated by the works mentioned, we first give an hybrid proximal linearized algorithm and then propose a related convergence theorem in finite-dimensional real Hilbert spaces. Next, we propose related convergence theorems in infinite-dimensional real Hilbert space.

Preliminaries

Let H be a real Hilbert space with inner product and norm . We denote the strong and weak convergence of to by and , respectively. For all and , we have

Definition 2.1

Let H be a real Hilbert space, let , and let . Then,

B is monotone if for all .

B is β-strongly monotone if for all .

Definition 2.2

Let H be a real Hilbert space, and let be a set-valued mapping with domain . Then,

B is monotone if for any and .

B is maximal monotone if its graph is not properly contained in the graph of any other monotone mapping.

B is ρ-strongly monotone () if for all , , and .

Definition 2.3

Let H be a real Hilbert space, and let . Then,

f is proper if .

f is lower semicontinuous if is closed for each .

f is convex if for every and .

f is ρ-strongly convex () if for all and .

f is Gâteaux differentiable at if there is such that for each .

f is Fréchet differentiable at x if there is such that

Example 2.1

Let H be a real Hilbert space. Then is a 2-strongly convex function.

Example 2.2

Let , where is a real symmetric positive definite matrix, and . Then g is a strongly convex function.

Definition 2.4

Let be a proper lower semicontinuous and convex function. Then the subdifferential ∂f of f is defined by for each .

Lemma 2.1

([18, 19])

Let be a proper lower semicontinuous and convex function. Then:

∂f is a set-valued maximal monotone mapping;

f is Gâteaux differentiable at if and only if consists of a single element, that is, [18, Prop. 1.1.10];

A Fréchet differentiable function f is convex if and only if ∇f is a monotone mapping.

Lemma 2.2

([19, Example 22.3(iv)])

Let , let H be a real Hilbert space, and let be a proper lower semicontinuous and convex function. If f is ρ-strongly convex, then ∂f is ρ-strongly monotone.

Lemma 2.3

([19, Prop. 16.26])

Let H be a real Hilbert space, and let be a proper lower semicontinuous and convex function. Let and be sequences in H such that for all . Then if and , then .

Lemma 2.4

([20])

Let H be a real Hilbert space, let be a set-valued maximal monotone mapping, and let . The mapping defined by for is a single-valued mapping.

Main results in finite-dimensional real Hilbert space

Let ρ and L be real numbers with . Let and be finite-dimensional real Hilbert spaces, and let be a nonzero linear and bounded mapping with adjoint . Let be proper lower semicontinuous and convex functions, let be proper lower semicontinuous and convex functions, and let for and for . Further, we assume that and are bounded from below, and are Fréchet differentiable, and are L-Lipschitz continuous, and and are ρ-strongly convex.

Choose , let β be a real number, and let be a sequence in such that Since and , we have , and then Besides, we know that which implies that Let be a sequence in , and let r be a real number with and Thus we have and So, we have and then Let be defined by We further assume that . The following result of Chuang [13] plays an important role in this paper.

Lemma 3.1

([13])

Under the assumptions in this section, let Then if and only if .

Proposition 3.1

([13])

If and , then the set is a singleton.

In this section, we propose the following algorithm to study the split DC program.

Algorithm 3.1

Let be arbitrary, and let be defined as follows:

Remark 3.1

The stop criteria in Algorithm 3.1 is given by Lemma 3.1.

Theorem 3.1

Let be generated by Algorithm 3.1. Then converges to x̄, where .

Proof

Take any and , and let w and n be fixed. First, we know that By (3.2) and Lemma 2.4 we have By (3.2) again, there exists such that Since , we have that . By Lemma 2.2, is ρ-strongly monotone, and this implies that By (3.4) and (3.5) we have Hence, by (3.6), Similarly to (3.2), we have and Similarly to (3.3), we have and Similarly to (3.7), we have and Next, we set By (3.10) and (3.11) we have By (3.15) we have and By (3.18) we know that for each . Besides, we have By (3.19) we have Next, we have On the other hand, we have which implies that By (3.12), (3.13), (3.21), and (3.23) we have We also have By (3.24) and (3.25) we have By (3.14) we have By (3.7), (3.26), and (3.27) we have By (3.28), exists, is a bounded sequence, and By the assumptions we have Since is bounded, there exists a subsequence of such that . Thus, , , , and . By (3.8), (3.9), and Lemma 2.3 we get that . By Proposition 3.1, . Further, . Therefore the proof is completed. □

Main results in infinite-dimensional real Hilbert space

Let and be infinite-dimensional real Hilbert spaces. Let δ, ρ, L, A, , , , , , , , , and be the same as in Sect. 3.

Definition 4.1

Let C be a nonempty closed convex subset of a real Hilbert space H, and let . Let . Then:

T is a nonexpansive mapping if for all ;

T is a firmly nonexpansive mapping if for all , that is, for all .

Lemma 4.1

([21])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a nonexpansive mapping, and let be a sequence in C. If and , then .

Definition 4.2

Let , let H be a real Hilbert space, and let be a proper lower-semicontinuous and convex function. Then the proximal operator of g of order β is defined by for each . In fact, we know that and is a firmly nonexpansive mapping.

Lemma 4.2

([22, Lemma 2.3])

Let H be a real Hilbert space, and let be a proper lower-semicontinuous and convex function. For , we have

The following result plays an important role when we study our convergence theorem in an infinite-dimensional real Hilbert space.

Lemma 4.3

Let H be a real Hilbert space, let be proper lower-semicontinuous and convex functions, and suppose that h is Fréchet differentiable. Then for all and , we have

By Lemma 4.2 we have Thus, and then Therefore the proof is completed. □

Lemma 4.4

Let , let H be a real Hilbert space, and let be a proper lower semicontinuous and ρ-strongly convex function. Then is a contraction mapping. In fact, .

Lemma 4.5

Let , let H be a real Hilbert space, and let be proper lower semicontinuous and convex functions. Further, we assume that h is Fréchet differentiable, ∇h is L-Lipschitz continuous, and g is ρ-strongly convex. Let be defined by for each . Then the following are satisfied.

If , then T is a contraction mapping.

If , then T is a nonexpansive mapping.

For , we have Thus the proof is completed. □

Theorem 4.1

In Theorem 3.1, let and be an infinite-dimensional real Hilbert space and assume that . Then the sequence generated by Algorithm 3.1 converges weakly to the unique solution x̄ of problem (SDCP).

By Proposition 3.1 we know that . Since , we may assume that there exists a real number such that . By (3.11) we have Similarly, we have By (3.30) we know that By (4.2) and (4.3) we have and By (4.4), (4.5), and Lemma 4.3 we have and Besides, we have to show that is a bounded sequence. Since is infinite dimensional, there exist and a subsequence of such that . By (4.3) we know that and . Hence, by (4.6), Lemma 4.1, and Lemma 4.5 we have that , which implies that . Since A is linear, we have . Hence, by (4.7), Lemma 4.1 and Lemma 4.5, we have , which implies that . So, , and thus exists. So, by Opial’s condition, we get . Therefore the proof is completed. □

Remark 4.1

To the best of our knowledge, the convergence theorems for the DC program and split DC program are proposed in finite-dimensional Hilbert spaces. Here, Theorem 4.1 is a convergence theorem for the split DC program in infinite-dimensional real Hilbert spaces.

Following the same argument as in the proof of Theorem 4.1, we get the following convergence theorem in infinite-dimensional real Hilbert spaces.

Theorem 4.2

Let and be infinite-dimensional real Hilbert spaces. Let A, , , , , , , and be the same as in Sect. 3. Let . Let be a sequence in . Let be a sequence in such that . Then the sequence generated by Algorithm 1.3 converges weakly to some .

Application to DC program

Let ρ, L, δ, be the same as in Sect. 3. Let H be an infinite-dimensional Hilbert space, and let be proper lower semicontinuous and convex functions. Besides, we also assume that h is Fréchet differentiable, ∇h is L-Lipschitz continuous, and g is ρ-strongly convex. Let for all and assume that f is bounded from below.

Let be a sequence in with and Let be defined by and assume that .

The following algorithm and convergence theorem are given by Algorithm 3.1 and Theorem 4.1, respectively.

Algorithm 5.1

Let be arbitrary, and let be generated as follows:

Theorem 5.1

Assume that . Then the sequence generated by Algorithm 5.1 converges weakly to the unique solution x̄ of problem (SDCP).

The following algorithm is a particular case of Algorithm 1.3.

Algorithm 5.2

([13])

Let be arbitrary, and let be generated as follows:

By Theorem 4.2 we get the following result, which it is a generalization of [13, Thm. 4.1].

Theorem 5.2

Let . Let be a sequence in . Let be a sequence in such that . Let be generated by Algorithm 5.2. Then converges weakly to some .

Next, we can get the following algorithm and convergence theorem by Algorithm 5.2 and Theorem 5.2, respectively. Further, Theorem 5.3 is a generalization of [13, Thm. 4.2].

Algorithm 5.3

([13])

Let be arbitrary, and let be generated as follows:

Theorem 5.3

Let . Let be a sequence in . Let be a sequence in such that . Let be generated by Algorithm 5.3. Then converges weakly to some .

If for all , then we have the following result.

Theorem 5.4

Let . Let be a sequence in . Let be arbitrary, and let be generated by Then converges weakly to some .

Following similar argument as in the proof of Theorem 4.1, we get the statement of Theorem 5.4. □