Abstract
Error bounds play an important role in the research of mathematical programming. Using some techniques of nonsmooth analysis, we establish some results on the existence of higher-order error bounds for difference functions with set constraints.
Similar content being viewed by others
1 Introduction
Let X be a Banach space, and let Ω be a nonempty closed convex subset of X. Let \(f: X\rightarrow\mathbb{R}\cup\{+\infty\}\) be a proper lower semicontinuous function. We assume that
Let \(a\in S\), \(\tau>0\), and \(\lambda>0\). We say that f has a λ-order local error bound τ at a if there exists \(\delta>0\) such that
where \(B_{X}\) denotes the closed unit ball of X, \(d(x,S)=\inf\{\| x-y\| | y\in S\}\), and \([f(x)]_{+}=\max\{f(x), 0\}\). We say that f has a λ-order global error bound τ if \(a+\delta B_{X}\) in (1.1) can be replaced by the whole space X.
Error bounds play an important role in convergence and perturbation analysis of some algorithms and mathematical programming [1–3]. In the last twenty years, many researchers studied error bounds and obtained a lot of interesting results; see the survey papers [2, 3] and the references therein. However, these results are mainly concerned with error bounds in the case \(\lambda=1\). Recently, Huang [4] considered higher-order error bounds for strongly convex multifunctions. Huang [5] and Huang and Li [6] also considered mixed-order error bounds for gamma paraconvex multifunctions. Zheng and Ng [7] studied Hölder weak sharp minimizers, which closely relate to error bounds for lower semicontinuous functions.
Recall that a function \(f: X\rightarrow\mathbb{R}\cup\{+\infty\}\) is said to be a difference (DC) function if f is the difference of two (convex) functions. Many nonconvex optimization problems are difference structure optimization problems. In 2012, Le Thi, Pham Dinh, and Ngai [8] studied error bounds for DC functions in \(\mathbb{R}^{n}\). In 2014, Huang and Li [9] studied error bounds for DC multifunctions. In 2016, Van Hang and Yao [10] established sufficient conditions for the existence of error bounds for difference functions and applications. However, all results mentioned in this paragraph are concerned with \(\lambda =1\). It is natural for us to consider higher-order error bounds for difference functions.
The rest of this paper is organized as follows. In Sect. 2, we give some notions. In Sect. 3, we establish sufficient and necessary conditions for the existence of higher-order error bounds for difference functions with set constraints in terms of nonsmooth analysis tools.
2 Preliminaries
Let X be a real Banach space, and let \(\tau, \lambda>0\). We say that \(f: X\rightarrow \mathbb{R}\cup\{+\infty\}\) is a λ-order strongly convex function with modulus τ if for all \(x, y\in X\) and \(t\in(0,1)\),
In the particular case \(\tau=0\), strongly convex functions reduce to convex functions. We say that f is proper if its domain \(\operatorname{dom}(f):=\{x\in X | f(x)<+\infty\}\neq\emptyset\). Let \(x_{0}\in\operatorname{dom}(f)\). We say that f is lower semicontinuous at \(x_{0}\) if
Let \(u\in X\). We define
It is known that
if f is a convex function. For a convex function f and \(\bar{x}\in \operatorname{dom}(f)\), the subdifferential of f at x̄ is defined as
Let Ω be a closed convex subset of X, and let \(a\in\varOmega\). The tangent cone of Ω at a is defined as
Let \(A\subset\mathbb{R}^{n}\) and \(x\in\mathbb{R}^{n}\). The projection of x on A is defined as
The limiting normal cone of A at \(a\in A\) [11] is defined as
Let \(\bar{x}\in X\) and \(\delta>0\). By \(B(\bar{x}, \delta)\) we denote the open ball with center at x̄ and radius δ and by \(S_{X}\) the unit sphere of X. By \(\operatorname{bdry}(S)\) we denote the boundary of a set S.
3 Main results
In this section, we assume that X is a real Banach space unless stated otherwise, \(\varOmega\subset X\) is a nonempty closed convex set, \(g, h: X\rightarrow\mathbb{R}\cup\{+\infty\}\) are two proper functions, and \(S=\{x\in\varOmega | g(x)-h(x)\leq 0\}\neq\emptyset\). We first establish sufficient conditions for the existence of higher-order error bounds for difference functions.
Theorem 3.1
Let \(g: X\rightarrow\mathbb{R}\cup\{ +\infty\}\) be a lower semicontinuous function, and let \(h: X\rightarrow\mathbb{R}\) be a continuous function. Let \(\tau>0\), \(\delta>0\), \(\lambda>0\), and \(\bar{x}\in S\). Suppose that, for each \(x\in\varOmega\setminus S\) such that \(\|x-\bar{x}\|<\delta\), there exists \(u\in T(\varOmega; x)\cap S_{X}\) such that
Then
Proof
Suppose on the contrary that (3.2) is not true. Then there exists \(\tilde{x}\in\varOmega\) such that \(\|\tilde{x}-\bar{x}\|<\frac{2}{3}\delta\) and
Since \(\inf_{x\in\varOmega} [g(x)-h(x)]_{+}=0\), it follows from (3.3) that
By the completeness of X and the closedness of Ω we know that Ω is a complete metric space with respect to the metric induced by the norm of X. By the Ekeland variational principle [12] there exists \(x_{0}\in\varOmega\) such that
and
From (3.4) we have that \(x_{0}\notin S\), and so \(g(x_{0})-h(x_{0})>0\). Let \(u\in T(\varOmega; x_{0})\cap S_{X}\). Since Ω is a convex set, there exists \(t_{0}>0\) such that \(x_{0}+tu\in\varOmega\) for all \(t\in(0, t_{0})\). Since \(g-h\) is lower semicontinuous at \(x_{0}\), there exists \(t_{1}\in(0, t_{0})\) such that
for all \(t\in(0, t_{1})\). It follows from (3.5) that
for all \(t\in(0, t_{1})\), that is,
since \(\|u\|=1\). Therefore
Taking lim inf as \(t\rightarrow0^{+}\), we get
By (3.4) we have
and so \(d(\tilde{x},S)\leq2d(x_{0},S)\). This inequality and (3.6) imply that
By (3.4) we get
Inequalities (3.7) and (3.8) are a contradiction to (3.1). The proof is completed. □
We now give necessary conditions for the existence of higher-order error bounds for difference functions.
Theorem 3.2
Let \(g, h: X\rightarrow\mathbb{R}\) be two continuous convex functions. Let \(\tau>0\), \(\delta>0\), \(\lambda >0\), and \(\bar{x}\in S\). Suppose that
Then, for each \(x\in\varOmega\setminus S\) such that \(\|x-\bar{x}\|<\frac {\delta}{2}\), there exist \(a\in S\) and \(u\in T(\varOmega; x)\cap S_{X}\) such that
Proof
Let \(x\in\varOmega\setminus S\) be such that \(\|x-\bar{x}\| <\frac{\delta}{2}\). Take \(a\in\operatorname{bdry}(S)\) such that\(\|x-a\| <2d(x,S)\). Then
As \(a\in\operatorname{bdry}(S)\), we have \(g(a)-h(a)=0\). By (3.9),
that is,
Let \(x^{*}\in\partial g(x)\) and \(a^{*}\in\partial h(a)\). Then
It follows from (3.11) that
Denote \(u:=\frac{a-x}{\|a-x\|}\). Then \(u\in T(\varOmega; x)\cap S_{X}\) since Ω is a convex set. The last inequality implies that
Since \(h'(a; u)=\max_{a^{*}\in\partial h(a)} \langle a^{*}, u\rangle\) and \(g'(x; u)=\max_{x^{*}\in\partial g(x)} \langle x^{*}, u\rangle \), from this inequality it follows that
Therefore (3.10) is verified. □
Corollary 3.1
Let \(g: X\rightarrow\mathbb{R}\) be a continuous convex function, and let \(\lambda>0\). Then the following two statements are equivalent:
-
(i)
There exist \(\tau>0\) and \(\delta>0\) such that
$$\tau d(x,S)^{\frac{1+\lambda}{\lambda}}\leq\bigl[g(x)\bigr]_{+}, \quad \forall x \in\varOmega\cap B(\bar{x}, \delta); $$ -
(ii)
There exist \(\tau'>0\) and \(\delta'>0\) such that, for each \(x\in\varOmega\setminus S\) with \(\|x-\bar{x}\|<\delta'\), there exists \(u\in T(\varOmega; x)\cap S_{X}\) such that
$$g'(x;u)\leq-\tau' d(x,S)^{\frac{1}{\lambda}}. $$
Proof
The conclusion directly follows from Theorems 3.1 and 3.2 by taking \(h=0\). □
Theorem 3.3
Let \(g: X\rightarrow\mathbb{R}\) be a λ-order strongly convex function with modulus τ, and let \(h: X\rightarrow \mathbb{R}\) be a convex function. If for each \(x\in\varOmega\setminus S\), there exists \(y\in\operatorname{bdry}(S)\) such that
then
Proof
Let \(x\in\varOmega\). Without loss of generality, we may assume that \(x\in\varOmega\setminus S\). Take \(y\in\operatorname{bdry}(S)\) such that (3.12) holds. Let \(t\in(0, 1)\). Since g is a λ-order strongly convex function with modulus η, we have
that is,
Letting \(t\rightarrow0^{+}\), we get
Since h is a convex function, we have
Adding (3.13) and (3.14), we have
due to assumption (3.12) and the equality \(g(y)-h(y)=0\). Since \(y\in\operatorname{bdry}(S)\), it follows from (3.15) that
□
We now give an example to illustrate Theorem 3.3.
Example 3.1
Let \(g, h: \mathbb{R}\rightarrow\mathbb {R}\) be defined as
Clearly, g is a second-order strongly convex function with modulus \(\tau=1\). It is easy to calculate that \(S=\{x\in\mathbb{R} | g(x)-h(x)\leq0\}=[0,1]\). Take \(\varOmega=(-\infty, 1]\). Let \(x\in\varOmega\setminus S=(-\infty, 0)\). There exists \(y=0\in\operatorname{bdry}(S)\) such that
All conditions of Theorem 3.3 are satisfied. By Theorem 3.3 we have
Theorem 3.4
Take \(X=\mathbb{R}^{n}\), and let \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}\) be an mth-order smooth function (m is a positive integer number), \(\delta>0\), \(S:=\{x\in\mathbb{R}^{n} | f(x)\leq0\}\neq\emptyset\), and \(\bar {x}\in\operatorname{bdry}(S)\). Suppose that, for each \(x\in B(\bar{x},\delta)\setminus S\), there exists \(u\in P_{S}(x)\) such that
and
Then there exist τ> and \(\eta>0\) such that
Proof
Suppose on the contrary that there exists a sequence \(\{ x_{k}\}\subset B(\bar{x},\delta)\) with \(x_{k}\rightarrow\bar{x}\) such that
Clearly, \(x_{k}\notin S\) for all k. By the assumption we can take \(u_{k}\in P_{S}(x_{k})\) such that
Clearly, \(\|x_{k}-u_{k}\|=d(x_{k},S)\). Letting \(k\rightarrow\infty\), we have \(u_{k}\rightarrow\bar{x}\). By (3.17) we get
since \(f(u_{k})=0\). By the Taylor theorem,
for some \(\theta_{k}\in(0, 1)\), and (3.18)–(3.20) imply that
that is,
Since \(u_{k}\in P_{S}(x_{k})\) and \(\{\frac{x_{k}-u_{k}}{\|x_{k}-u_{k}\| }\}\) is bounded, without loss of generality, we may assume that \(\frac{x_{k}-u_{k}}{\|x_{k}-u_{k}\|}\rightarrow v\in N(S, \bar{x})\). Letting \(k\rightarrow\infty\) in (3.21), we have
This contradicts (3.16). The proof is completed. □
We now give an example to illustrate Theorem 3.4.
Example 3.2
Let \(f: \mathbb{R}^{2}\rightarrow\mathbb {R}\) be defined by
Clearly, \(S=\{(0,0)\}\). Take \(\bar{x}=(0,0)\in\operatorname{bdry}(S)\). Then \(N(S,\bar{x})=\mathbb{R}^{2}\). Let \(x\in\mathbb{R}^{2}\setminus S\). There exists \(u=(0,0)\in P_{S}(x)\) such that \(f'(u)(x-u)=0\) and
All conditions of Theorem 3.4 are satisfied. By Theorem 3.4 there exists \(\tau>0\) (\(\tau=1\)) such that
4 Conclusion
In this paper, we establish two existence theorems of higher-order error bounds for difference functions and an existence theorem of higher-order error bounds for mth-order smooth functions. Moreover, the coefficients in Theorems 3.1–3.4 can be calculated. It is interesting for us to consider higher-order error bounds for DC multifunctions.
References
Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. Natl. Bur. Stand. 49, 263–265 (1952)
Azé, D.: A survey on error bounds for lower semicontinuous functions. In: Proceedings of MODE-SMAI Conference. ESAIM Proc., vol. 13, pp. 1–17 (2003)
Pang, J.S.: Error bounds in mathematical programming. Math. Program., Ser. B 79(1–3), 299–332 (1997)
Huang, H.: Global error bounds with exponents for multifunctions with set constraints. Commun. Contemp. Math. 12(3), 417–435 (2010)
Huang, H.: Coderivative conditions for error bounds of gamma-paraconvex multifunctions. Set-Valued Var. Anal. 20(4), 567–579 (2012)
Huang, H., Li, R.X.: Global error bounds for gamma-multifunctions. Set-Valued Var. Anal. 19(3), 487–504 (2011)
Zheng, X.Y., Ng, K.F.: Hölder weak sharp minimizers and Hölder tilt-stability. Nonlinear Anal. Theory Methods Appl. 120, 186–201 (2015)
Le Thi, H.A., Pham Dinh, T., Ngai, H.V.: Exact penalty and error bounds in DC programming. J. Glob. Optim. 52(3), 509–535 (2012)
Huang, H., Li, R.X.: Error bounds for the difference of two convex multifunctions. Set-Valued Var. Anal. 22(2), 447–465 (2014)
Van Hang, N.Y., Yao, J.C.: Sufficient conditions for error bounds of difference functions and applications. J. Glob. Optim. 66(3), 439–456 (2016)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47(2), 324–353 (1974)
Acknowledgements
The authors are greatly indebted to the reviewers and the Editor for their valuable comments.
Funding
The first author was supported by the National Natural Science Foundation of China (Grant No. 11461080).
Author information
Authors and Affiliations
Contributions
The authors have made the same contribution. Both authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Huang, H., Xia, M. Higher-order error bound for the difference of two functions. J Inequal Appl 2018, 290 (2018). https://doi.org/10.1186/s13660-018-1883-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-018-1883-8