Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 6489190, 7 pages
https://doi.org/10.1155/2020/6489190

Research Article

A Gradient Projection Algorithm with a New Stepsize for Nonnegative Sparsity-Constrained Optimization

Ye Li,1,2 Jun Sun,³ and Biao Qu¹

¹Qufu Normal University, Rizhao 276826, Shandong, China
²Shandong Women’s University, Jinan 250300, Shandong, China
³Beijing Jiaotong University, Beijing 100044, China

Correspondence should be addressed to Ye Li; 35030@sdwu.edu.cn

Received 26 May 2020; Revised 30 July 2020; Accepted 8 August 2020; Published 26 August 2020

Guest Editor: Chuanjun Chen

Copyright © 2020 Ye Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Nonnegative sparsity-constrained optimization problem arises in many fields, such as the linear compressing sensing problem and the regularized logistic regression cost function. In this paper, we introduce a new stepsize rule and establish a gradient projection algorithm. We also obtain some convergence results under milder conditions.

1. Introduction

In this paper, we are mainly concerned with the nonnegative sparsity-constrained optimization problem (NN-SCO):

where is a continuously differential function with a lower bound. is a sparse set, where is a given integer regulating the sparsity level in and is the nonnegative orthant in . is the norm of , counting the number of nonzero elements in . Many application problems can be translated into problem (1), such as the widely studied linear compressing sensing problem of with being a sensing matrix, is the observation vector, and is the Euclidean norm in [1]. Problem (1) has also used to the regularized logistic regression cost function [2].

Recently, a great deal of work has been devoted to algorithms for sparsity-constrained optimization problem. Beck and Eldar [3] established the IHT algorithm which converges to L-stationary under the Lipchitz continuity of the gradient of objective function. Beck and Hallak [4] generalized these results to sparse symmetric sets. Lu [5] designed a nonmonotone algorithm for symmetric set constraint problems. Pan, Xiu, and Zhou [6, 7] established the B-stationary, C-stationary, and -stationary based on the Bouligand tangent cone and Clarke tangent. Recently, Pan, Zhou, and Xiu [8] established the improved IHT algorithm (IIHT) for problem (1) by using Armijo line search. They proved that any accumulation point converged to an -stationary point under the restricted strong smoothness of objective function which is weaker than the Lipchitz continuity of the gradient.

Inspired by the above literature studies, in this paper, we establish a gradient projection algorithm with a new stepsize. The new algorithm removes the condition of the restricted strong smoothness of objective function which makes it more applicable. Meanwhile, we prove the convergence of the algorithm.

The rest of this paper is organized as follows. In Section 2, we present some notations, definitions, and lemmas. In Section 3, we give the algorithm of (1) and prove the convergence properties.

2. Preliminaries

2.1. Notations

To make it easier to read, we give some used notations as follows:

2.2. Definitions

Definition 1 (see [8]). Let be a given feasible point of (1). We say that is an -stationary point, if there exists such that

Definition 2 (see [9]). A function is called 2s-restricted strongly smooth (2s-RSS) with parameter , and if for any satisfying , it holds that

Definition 3 (see [9]). A function is called 2s-restricted strongly convex (2s-RSC) with parameter , and if for any satisfying , it holds that

If and only if for any and , we have

In particular, in (5), if , the function is called 2s-restricted convex (2s-RC).

Definition 4 (see [10]). The projected gradient of is defined by

2.3. Lemmas

Lemma 1 (see [8]). For, vector is an -stationary point if and only if

In particular, when , .

When , .

Lemma 2 (see [8]). .

Lemma 3 (see [8]). For any , we have

where is a vector whose th component is one and others are zeros.

3. Main Results

In this section, we establish a new algorithm which improves the IIHT algorithm for (1) and then we analyze its convergence properties. At first, let us develop the gradient projection algorithm with a new stepsize rule.

Algorithm 1.  

 Step 1. Initialize , , and , and set. Step 2. Compute , where

 Step 3. Compute where satisfies .  Step 4. If , then stop; otherwise, set and go to Step 2.

Next, let us list the following assumptions for convenience:

(1)For any , (2) is bounded below on

Lemma 4. Let the sequence be generated by Algorithm 1, and set . Then, we have

where .

Proof. Let

Then,

Thus,

Then, (11) is tenable.

Lemma 5. We suppose . For and , we have

where .

Proof. Since , by the definition of projection, we get

Moreover,

Because is a constant independent of , we can get

Therefore,

By Lemma 4, we get

Hence,

Let . We get

Lemma 6. Let the sequence be generated by Algorithm 1. Then,

(1)(2) is an increasing sequence, and when , converges(3)(4)for any , if , we have

Proof.  

(1)Since , we get

 Setting in (15), formula (1) can be obtained.(2)We can easily get that is an increasing sequence by (15). Moreover, by the assumptions , we can get that converges.(3)Let in (1). We can get

 Summing over both sides of this inequality, we get

 Since is bounded below, we get

(4)It easily can be got by (2).

Lemma 7. Let the sequence be generated by Algorithm 1. Suppose that the function is 2s-RC. We have

Proof. Because the sequence be generated by Algorithm 1, we get . By Lemma 4 and Lemma 5 in reference [8], we can get

Theorem 1. Let the sequence be generated by Algorithm 1. Then, the following results hold:

(1)Any accumulation of sequence is an -stationary point.(2)If is 2s-RC, the projected gradient sequence converges to zero, i.e.,

Proof.  

(1)Suppose that is an accumulation point of sequence . Then, there exists a subsequence converges to . Because

 we get

 Moreover,

 We consider the next two cases:

Case 1. For , there must exist a sufficiently large index and a constant such that

 By and (33), we can get

 Since

 without loss of generality, we can suppose . Let . We get

 i.e.,

Case 2. For , we consider two subcases.

Subcase 1. When , we get

 Due to the property of the projections and , we have

 Thus,

 Taking limits on both sides, we obtain

Subcase 2. When , suppose , and we have

 For all sufficiently large , we have

 Since , for all sufficiently large , we have

 which contradicts with . Thus, . Summarizing the two cases, we obtain

 Thus, is an -stationary point of (1).(2)Set . By Lemma 3, we have

By Definition 4, we have

Moreover, the maximum value is taken at . For any , there exists and satisfies

Because and , we get

i.e.,

Thus, for any , we get

Taking , we get

By the Cauchy–Schwartz inequality, we get

i.e.,

By Lemma 7, we get

Taking limits on both sides and using Lemma 6, we have

By (32), we get

Theorem 2. Let the sequence be generated by Algorithm 1. is an accumulation point of the sequence . Suppose is , then the following results hold:

(1)If then is a global minimizer of (1)(2)If , then is a local minimizer of (1)

Proof.  

(1)For , we have . Since is , by Definition 3, we have

 Because is an accumulation point of the sequence . By Theorem 1, is an -stationary. By Lemma 1, we can get

 Thus, is a global minimizer of (1).(2)If , then . In fact, for all sufficiently large , taking , we get

For any , we have

Thus,

By and , we have

For any satisfying , we have . Since is , by Definition 3, Theorem 1, and Lemma 1, we have

Thus, is a local minimizer of (1).

Theorem 3. Let the sequence be generated by Algorithm 1. is a limit of the sequence . Suppose is with parameter and for all sufficiently large , and we have

where .

Proof. By Theorem 2, we get . As is with parameter , for any and , we have

Set . By Theorem 2, we get . For all sufficiently large , we have

For all sufficiently large , we have

Because , we get

Since and , we have . Thus,

Setting , we get

Thus,

By and we get .
From and , we have .
Thus,

where . Thus, the sequence is Q-linear convergence to .

4. Conclusions

In this paper, we are mainly concerned with the nonnegative sparsity-constrained optimization problem. We introduce a new stepsize rule and propose a new gradient projection algorithm to solve this problem. The new algorithm removes the condition of the restricted strong smoothness of objective function which makes the new algorithm more applicable. Meanwhile, we prove the convergence of the algorithm.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This project was supported by the National Science Foundation of Shandong Province (no. ZR2018MA019).

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