- Research
- Open Access
Self-adaptive iterative method for solving boundedly Lipschitz continuous and strongly monotone variational inequalities
- Songnian He^{1, 2},
- Lili Liu^{2} and
- Aviv Gibali^{3, 4}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-018-1941-2
© The Author(s) 2018
- Received: 2 August 2018
- Accepted: 10 December 2018
- Published: 18 December 2018
Abstract
In this paper we introduce a new self-adaptive iterative algorithm for solving the variational inequalities in real Hilbert spaces, denoted by \(\operatorname{VI}(C, F)\). Here \(C\subseteq \mathcal{H}\) is a nonempty, closed and convex set and \(F: C\rightarrow \mathcal{H}\) is boundedly Lipschitz continuous (i.e., Lipschitz continuous on any bounded subset of C) and strongly monotone operator. One of the advantages of our algorithm is that it does not require the knowledge of the Lipschitz constant of F on any bounded subset of C or the strong monotonicity coefficient a priori. Moreover, the proposed self-adaptive step size rule only adds a small amount of computational effort and hence guarantees fast convergence rate. Strong convergence of the method is proved and a posteriori error estimate of the convergence rate is obtained.
Primary numerical results illustrate the behavior of our proposed scheme and also suggest that the convergence rate of the method is comparable with the classical gradient projection method for solving variational inequalities.
Keywords
- Variational inequalities
- Self-adaptive iterative methods
- Boundedly Lipschitz continuous
- Strongly monotone
MSC
- 47J20
- 90C25
- 90C30
- 90C52
1 Introduction
The variational inequality problem (VIP) was introduced and studied by Fichera [9, 10] (see also [22]). Since then VIPs have been studied and applied in a wide variety of problems arising in different fields, for example, engineering science, structural analysis, economics, optimization, operations research, see [1, 2, 6–11, 14, 16–20, 22, 24–26] and the references therein.
We point out that most of the algorithms mentioned above use variable parameter sequences satisfying (1.6), this might be essential when the feasible set C is more complex and thus the relaxation projection technique has to be used. On the other hand, when C is easy to project onto and the constants η and L are unknown, the usage of the parameter sequence \(\{\lambda_{n}\}_{n=0}^{\infty }\) satisfying (1.6) is not a good choice due to the computational effort of doing so. So, our main motivation of this paper is to propose a new simple and fast converging iterative algorithm with self-adaptive parameter selection.
One of the main advantages of our new proposed method is that it does not require a priori the knowledge of the Lipschitz constant of F on any bounded subset of C or the strong monotonicity coefficient. Moreover, the proposed self-adaptive step size rule only adds a small amount of computational effort and hence guarantees fast convergence rate. Strong convergence of the method is proved and a posteriori error estimate of the convergence rate is obtained. Primary numerical results demonstrate the applicability and efficiency of the algorithm.
The outline of the paper is as follows. In Sect. 2, we recall some basic definitions and results which are useful for our analysis. Our self-adaptive iterative algorithm is presented and analyzed in Sect. 3. Then, in Sect. 4, three numerical experiments which demonstrate and compare our algorithm’s performance with two related methods are presented. Final conclusions are given in Sect. 5.
2 Preliminaries
- (i)
→ denotes strong convergence.
- (ii)
⇀ denotes weak convergence.
- (iii)
\(\omega_{w}(x_{n}) =\{x\mid \exists\ \{x_{n_{k}}\}_{k=1} ^{\infty }\subset \{x_{n}\}_{n=1}^{\infty }\text{ such that } x_{n_{k}} \rightharpoonup x\}\) denotes the weak ω-limit set of \(\{x_{n}\}_{n=1}^{\infty }\).
- (iv)
\(S(x,r)\) denotes the closed ball with center \(x\in \mathcal{H}\) and radius \(r>0\).
Lemma 2.1
Lemma 2.2
([21])
Let \(T: C\rightarrow C\) be a nonexpansive mapping. Then \(I-T\) is demiclosed at 0 in the sense that if \(\{x_{n}\}_{n=1}^{\infty }\) is a sequence in C such that \(x_{n}\rightharpoonup x\) and \(\Vert x_{n}-Tx _{n}\Vert \rightarrow 0\) as \(n\rightarrow \infty \), it follows that \(x-Tx=0\), i.e., \(x\in \operatorname {Fix}(T)\). Here \(\operatorname {Fix}(T)=\{x\in \mathcal{H}\mid Tx=x \}\) is the set of fixed points of T.
Definition 2.3
Lemma 2.4
([23])
- (i)
\(\sum_{n=0}^{\infty }\gamma_{n} = \infty \),
- (ii)
\(\sum_{n=1}^{\infty }\vert \gamma_{n}\sigma_{n}\vert = \infty \), or \(\limsup_{n\rightarrow \infty }\sigma_{n}\leq 0\).
Theorem 2.5
([18])
Let C be a nonempty closed convex subset of a real Hilbert space \(\mathcal{H}\). If \(F: C\rightarrow \mathcal{H}\) is a strongly monotone and boundedly Lipschitz continuous operator, then the variational inequality \(\operatorname {VI}(C, F)\) has a unique solution.
3 The self-adaptive iterative algorithm
Let \(\mathcal{H}\) be a real Hilbert space with inner product \(\langle \cdot ,\cdot \rangle \) and induced norm \(\Vert \cdot \Vert \), and let C be a nonempty closed convex subset of \(\mathcal{H}\). Let \(F: C\rightarrow \mathcal{H}\) be a strongly monotone and boundedly Lipschitz continuous operator. Throughout this section, we always assume that we do not need to know or to estimate its strong monotonicity coefficient η and the Lipschitz constant \(L_{B}\) on any bounded subset B of C. Also, we always assume that the projection operator \(P_{C}\) is easy to calculate. Using Theorem 2.5, \(\operatorname {VI}(C, F)\) has a unique solution, denoted by \(x^{*}\).
Now we are ready to present our self-adaptive iterative algorithm for solving \(\operatorname {VI}(C, F)\).
Algorithm 3.1
(Self-adaptive iterative algorithm)
- Step 1.Choose \(x_{0}\in C\) arbitrarily and set \(n:=1\). Calculate \(x_{1}\) byIf \(x_{1}=x_{0}\), then \(x_{0}\) is the unique solution of \(\operatorname {VI}(C, F)\) and stop the algorithm.$$ x_{1}=P_{C}\bigl(x_{0}-F(x_{0}) \bigr). $$Otherwise, set$$ \eta_{0}=\frac{\langle F(x_{1})-F(x_{0}),x_{1}-x_{0}\rangle }{ \Vert x_{1}-x _{0} \Vert ^{2}},\quad\quad L_{0}= \frac{ \Vert F(x_{1})-F(x_{0}) \Vert }{ \Vert x_{1}-x_{0} \Vert }, \quad \text{and} \quad \mu_{0}=\frac{\eta_{0}}{L_{0}^{2}}. $$
- Step 2.Given the current iterate \(x_{n}\), computeand$$\begin{aligned}& \eta_{n}= \textstyle\begin{cases} \min \{\eta_{n-1}, \frac{\langle F(x_{n})-F(x_{n-1}),x_{n}-x_{n-1} \rangle }{ \Vert x_{n}-x_{n-1} \Vert ^{2}}, \frac{\langle F(x_{n})-F(x_{0}),x _{n}-x_{0}\rangle }{ \Vert x_{n}-x_{0} \Vert ^{2}}\} , &\text{if } x_{n}\neq x_{0}, \\ \min \{\eta_{n-1}, \frac{\langle F(x_{n})-F(x_{n-1}),x_{n}-x_{n-1} \rangle }{ \Vert x_{n}-x_{n-1} \Vert ^{2}}\}, & \text{if } x_{n}= x_{0}, \end{cases}\displaystyle \\& L_{n}= \textstyle\begin{cases} \max \{L_{n-1}, \frac{ \Vert F(x_{n})-F(x_{n-1}) \Vert }{ \Vert x_{n}-x_{n-1} \Vert }, \frac{ \Vert F(x_{n})-F(x_{0}) \Vert }{ \Vert x_{n}-x_{0} \Vert }\} ,& \text{if } x_{n}\neq x_{0}, \\ \max \{L_{n-1}, \frac{ \Vert F(x_{n})-F(x_{n-1}) \Vert }{ \Vert x_{n}-x_{n-1} \Vert }\}, & \text{if } x_{n}= x_{0}, \end{cases}\displaystyle \end{aligned}$$$$ \mu_{n}=\frac{\eta_{n}}{L_{n}^{2}}. $$
- Step 3.Update the next iterate asIf \(x_{n+1}=x_{n}\), STOP, \(x_{n}\) is the unique solution of \(\operatorname {VI}(C, F)\).$$ x_{n+1}=P_{C}\bigl(x_{n}- \mu_{n}F(x_{n})\bigr),\quad n\geq 1. $$(3.1)
Otherwise, set \(n:=n+1\) and return to Step 2.
Remark 3.2
- (1)
It is easy to see by a simple induction that the sequences \(\{\eta_{n}\}_{n=0}^{\infty }\), \(\{L_{n}\}_{n=0}^{\infty }\), and \(\{\mu_{n}\}_{n=0}^{\infty }\) are well defined. Also the calculations of \(\eta_{n}\), \(L_{n}\), and \(\mu_{n}\) only add a small amount of computational load. Indeed, for any \(n\geq 1\), the values of \(\{F(x_{k})\}_{k=0}^{n}\) have been obtained in the previous calculations.
- (2)Let η be the strong monotonicity coefficient of F. Then the following properties directly follow from the definitions of \(\eta_{n}\), \(L_{n}\) and \(\mu_{n}\):
- (i)
\(\{\eta_{n}\}_{n=0}^{\infty }\) is monotone nonincreasing and \(\eta_{n}\geq \eta \) for all \(n\geq 0\).
- (ii)
\(\{L_{n}\}_{n=0}^{\infty }\) is monotone nondecreasing and \(L_{n}\geq \eta_{n}\) holds for all \(n\geq 0\). Particularly, if F is L-Lipschitz continuous, then \(L_{n}\leq L\) holds for all \(n\geq 0\).
- (iii)
\(\{\mu_{n}\}_{n=0}^{\infty }\) is monotone nonincreasing and \(\mu_{n}=\frac{\eta_{n}}{L_{n}^{2}}\leq \frac{1}{\eta_{n}}\leq \frac{1}{ \eta }\) holds for all \(n\geq 0\). In particular, if F is L-Lipschitz continuous, then \(\mu_{n}\geq \frac{\eta }{L^{2}}\) holds for all \(n\geq 0\).
- (i)
Next we present a strong convergence theorem of Algorithm 3.1.
Theorem 3.3
Assume that F is boundedly Lipschitz continuous and strongly monotone on the feasible set, then any sequence \(\{x_{n}\}_{n=0}^{\infty }\) generated by Algorithm 3.1 converges strongly to the unique solution \(x^{*}\) of problem \(\operatorname {VI}(C,F)\).
Proof
To present a complete convergence analysis of Algorithm 3.1, the next theorem establishes the algorithm’s convergence rate.
Theorem 3.4
Proof
Observe that this estimate can be easily obtained by letting \(m\rightarrow \infty \) in (3.5). □
Since a Lipschitz continuous operator is obviously boundedly Lipschitz continuous, the following results are straightforward.
Corollary 3.5
Assume that F is Lipschitz continuous and strongly monotone on the feasible set, then the sequence \(\{x_{n}\}_{n=0}^{\infty }\) generated by Algorithm 3.1 converges strongly to the unique solution \(x^{*}\) of problem \(\operatorname {VI}(C,F)\).
Corollary 3.6
4 Numerical results
In this section, we present three numerical examples which demonstrate the performance of the self-adaptive iterative algorithm (Algorithm 3.1). All implementations and testing are preformed with Matlab R2014b on an HP Pavilion notebook with Intel(R) Core(TM) i5-3230M CPU@2.60 GHz and 4 GB RAM running on Windows 10 Home Premium operating system.
Example 1
Consider the variational inequality problem \(\operatorname {VI}(C, F)\) (1.1) with the set \(C :=\{(x,y)\mid x^{2}+y^{2}\leq 1\}\) and \(F:C\rightarrow R^{2}\) defined by \(F(x,y)=(2x+2y+\sin (x),-2x+2y+\sin (y))^{ \top }\), \(\forall\ (x,y)^{\top }\in C\).
Comparison of Algorithm 3.1 with GPM and VPGPM
\(E_{n}\) | Iter. | CPU (in s) | ||||
---|---|---|---|---|---|---|
VPGPM | GPM | Algorithm 3.1 | VPGPM | GPM | Algorithm 3.1 | |
1⋅10^{−1} | 7 | 11 | 5 | 0.000064 | 0.000488 | 0.001099 |
1⋅10^{−2} | 15 | 22 | 9 | 0.000126 | 0.000554 | 0.001709 |
1⋅10^{−3} | 31 | 33 | 13 | 0.000213 | 0.000621 | 0.002412 |
1⋅10^{−4} | 66 | 44 | 17 | 0.000420 | 0.000684 | 0.003223 |
1⋅10^{−5} | 141 | 54 | 21 | 0.000855 | 0.000746 | 0.003691 |
1⋅10^{−6} | 302 | 65 | 25 | 0.001784 | 0.000820 | 0.004325 |
1⋅10^{−7} | 649 | 76 | 29 | 0.003787 | 0.000900 | 0.004826 |
1⋅10^{−8} | 1398 | 87 | 33 | 0.008293 | 0.001003 | 0.005116 |
From Table 1 and Fig. 1, we conclude that the VPGPM performs the worst, regardless of the number of iterations or the computing time, and Algorithm 3.1 and the GPM are roughly the same since Algorithm 3.1 needs the least number of iterations, while the GPM needs the shortest computing time. Although Algorithm 3.1 requires a little longer computing time than GPM due to parameter self-adaptive selection, Algorithm 3.1 still shows obvious superiority, not only because it requires the least number of iterations, but also because it does not need to know the constants L and η.
Example 2
Consider the variational inequality problem \(\operatorname {VI}(C, F)\) (1.1) with the set \(C :=\{(x,y)\mid x\geq 0,y\geq 0\}\) and \(F:C\rightarrow R^{2}\) defined by \(F(x,y)=(2x+2y+\exp (x),-2x+2y+\exp (y))^{\top }\), \(\forall\ (x,y)^{\top }\in C\).
It is easy to see that F is strongly monotone and boundedly Lipschitz continuous on C and \(x^{*}=(0,0)^{\top }\) is the unique solution. Since F is not Lipschitz continuous on C, so GPM and VPGPM are not applicable for this example. Choosing the starting point \(x_{0}=(1,1)^{ \top }\) and using Algorithm 3.1 to solve this example, we find that the exact solution \(x^{*}=(0,0)^{\top }\) can be obtained by only one iteration.
Example 3
Consider the variational inequality problem \(\operatorname {VI}(C, F)\) (1.1) with the set \(C :=\{(x,y)\mid x\geq 0\}\) and \(F:C\rightarrow R^{2}\) defined by \(F(x,y)=(2x+2y+\exp (x),-2x+2y+\exp (y))^{\top }\), \(\forall\ (x,y)^{\top }\in C\).
Numerical results of Algorithm 3.1
\(E_{n}\) | Iter. | CPU (in s) |
---|---|---|
1⋅10^{−1} | 3 | 0.000094 |
1⋅10^{−2} | 11 | 0.000282 |
1⋅10^{−3} | 19 | 0.000479 |
1⋅10^{−4} | 28 | 0.000686 |
1⋅10^{−5} | 36 | 0.000867 |
1⋅10^{−6} | 45 | 0.001082 |
1⋅10^{−7} | 54 | 0.001207 |
1⋅10^{−8} | 62 | 0.001425 |
The numerical results in Tables 1 and 2 show that the convergence rate of Algorithm 3.1 for solving boundedly Lipschitz continuous variational inequalities is almost the same as that of GPM for solving Lipschitz continuous variational inequalities.
5 Conclusions
In this paper, in the setting of Hilbert spaces, a new self-adaptive iterative algorithm is proposed for solving \(\operatorname {VI}(C, F)\) governed by boundedly Lipschitz continuous and strongly monotone operator \(F: C\rightarrow \mathcal{H}\) under the assumption that \(P_{C}\) has a closed-form formula. The advantages of our algorithm are not only having no need to know or estimate the strong monotonicity coefficient and Lipschitz constant on any bounded subset of the feasible set, but also having a fast convergence rate because the parameter self-adaptive selection process only adds a small amount of computational effort. Currently, as far as we know, such algorithms for solving strongly monotone and boundedly Lipschitz continuous variational inequalities have not been considered before.
Declarations
Availability of data and materials
Not applicable.
Funding
The first author is supported by Open Fund of Tianjin Key Lab for Advanced Signal Processing (No. 2017ASP-TJ03).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Baiocchi, C., Capelo, A.: Variational and Quasi Variational Inequalities. Wiley, New York (1984) MATHGoogle Scholar
- Cai, G., Gibali, A., Iyiola, O.S., Shehu, Y.: A new double-projection method for solving variational inequalities in Banach spaces. J. Optim. Theory Appl. 178, 219–239 (2018) MathSciNetView ArticleGoogle Scholar
- Cegielski, A., Gibali, A., Reich, S., Zalas, R.: An algorithm for solving the variational inequality problem over the fixed point set of a quasi-nonexpansive operator in Euclidean space. Numer. Funct. Anal. Optim. 34, 1067–1096 (2013) MathSciNetView ArticleGoogle Scholar
- Cegielski, A., Zalas, R.: Methods for variational inequality problem over the intersection of fixed point sets of quasi-nonexpansive operators. Numer. Funct. Anal. Optim. 34, 255–283 (2013) MathSciNetView ArticleGoogle Scholar
- Censor, Y., Gibali, A.: Projections onto super-half-spaces for monotone variational inequality problems in finite dimensional space. J. Nonlinear Convex Anal. 9, 461–475 (2008) MathSciNetMATHGoogle Scholar
- Cottle, R.W., Giannessi, F., Lions, J.L.: Variational Inequalities and Complementarity Problems. Theory and Applications. Wiley, New York (1980) Google Scholar
- Dong, Q.L., Lu, Y.Y., Yang, J.: The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 65, 2217–2226 (2016) MathSciNetView ArticleGoogle Scholar
- Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problem. Vol. I and II. Springer Series in Operations Research. Springer, New York (2003) MATHGoogle Scholar
- Fichera, G.: Sul problema elastostatico di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 34, 138–142 (1963) MathSciNetMATHGoogle Scholar
- Fichera, G.: Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, Mem., Cl. Sci. Fis. Mat. Nat., Sez. I, VIII. Ser. 7, 91–140 (1964) MathSciNetMATHGoogle Scholar
- Fukushima, M.: A relaxed projection method for variational inequalities. Math. Program. 35, 58–70 (1986) MathSciNetView ArticleGoogle Scholar
- Gibali, A., Reich, S., Zalas, R.: Iterative methods for solving variational inequalities in Euclidean space. J. Fixed Point Theory Appl. 17, 775–811 (2015) MathSciNetView ArticleGoogle Scholar
- Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities in Hilbert space. Optimization 66, 417–437 (2017) MathSciNetView ArticleGoogle Scholar
- Glowinski, R., Lions, J.L., Tremoliers, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981) Google Scholar
- Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Dekker, New York (1984) MATHGoogle Scholar
- Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990) MathSciNetView ArticleGoogle Scholar
- He, S.N., Tian, H.L.: Selective projection methods for solving a class of variational inequalities. Numer. Algorithms (2018). https://doi.org/10.1007/s11075-018-0499-X View ArticleGoogle Scholar
- He, S.N., Xu, H.K.: Variational inequalities governed by boundedly Lipschitzian and strongly monotone operators. Fixed Point Theory 10, 245–258 (2009) MathSciNetMATHGoogle Scholar
- He, S.N., Yang, C.P.: Solving the variational inequality problem defined on intersection of finite level sets. Abstr. Appl. Anal. 2013, 942315 (2013) MathSciNetMATHGoogle Scholar
- Lions, J.L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–512 (1967) View ArticleGoogle Scholar
- Opial, Z.: Weak convergence of the sequence of successive approximations of nonexpansive mappings. Bull. Am. Math. Soc. 73, 595–597 (1967) MathSciNetView ArticleGoogle Scholar
- Stampacchia, G.: Formes bilineaires coercivites sur les ensembles convexes. C. R. Acad. Sci. 258, 4413–4416 (1964) MATHGoogle Scholar
- Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002) MathSciNetView ArticleGoogle Scholar
- Xu, H.K., Kim, T.H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185–201 (2003) MathSciNetView ArticleGoogle Scholar
- Yamada, I.: The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473–504. North-Holland, Amsterdam (2001) View ArticleGoogle Scholar
- Yang, H.M., Bell, G.H.: Traffic restraint, road pricing and network equilibrium. Transp. Res., Part B, Methodol. 31, 303–314 (1997) View ArticleGoogle Scholar