Elsevier

Operations Research Letters

Volume 45, Issue 5, September 2017, Pages 456-460
Operations Research Letters

Error bounds for affine variational inequalities with second-order cone constraints

https://doi.org/10.1016/j.orl.2017.06.006Get rights and content

Abstract

In this paper, error bounds for affine variational inequalities with second-order cone constraints are considered. Examples are given to show that, in general, Lipschitz error bounds may be invalid for affine second-order cone inclusion problems. We provide a sufficient condition (not stronger than Mangasarian–Fromovitz constraint qualification), under which a local Lipschitz error bound is valid for the variational inequality problem. Moreover, under a full row rank assumption, a local Hölder error bound is established for the variational inequality problem and the Hölder exponent is bounded by a function of problem dimensions.

Introduction

Error bound conditions play an important role in the convergence analysis of numerical algorithms for optimization problems and variational inequalities. For example, the convergence analysis of matrix splitting methods, can be established under an error bound condition, for symmetric and monotone affine variational inequalities [19]. Another example is the convergence analysis of the modified Levenberg–Marquardt methods. The quadratic convergence rate can be obtained under the local error bound condition, see [33] and [4].

Error bounds for nonlinear systems, such as systems of convex functions, polynomial functions and lower semi-continuous functions, have been investigated extensively and readers are referred to [8], [10], [11], [12], [14], [16], [17], [18], [24], [28], [29], [30] and the references therein.

There are many important results on error bounds for variational inequalities (VIs) over convex polyhedral sets. Motivated by the convergence analysis of matrix splitting algorithms, Luo and Tseng (1992) [19] established a local Lipschitz-type error bound for affine VIs using projection-type residual functions. The conclusion can also be obtained from error bounds for piecewise quadratic systems (see [16]). Pang (1987) [23] proved that global Lipschitz error bound holds for VIs with polyhedral constraints under the condition of strong monotonicity and Lipschitz continuity. Xiu and Zhang (2002) [31] extended these results to general VIs with convex constraints. An error bound for a system or an optimization problem is usually described in terms of residual functions. For variational inequality problems, gap functions are served as residual functions. For instance, under strong monotonicity and Lipschitz continuity, Lipschitz error bounds in terms of gap functions of VI problems can be established, see [5], [25], [26], [27], [32]. Moreover, these error bound results are extended to nonsmooth VIs and subanalytic VIs, see [20] and [15], respectively.

In this paper, we mainly consider error bounds for the affine variational inequalities (AVIs) with second-order cone (SOC) constraints. This variational inequality problem can be stated as: find an xX satisfying xx,Mx+q0,xX,where X={x|AxbQ} and Q=i=1pQmi+1 and Qmi+1 is a second-order cone in Rmi+1 as i=1,2,,p. A second-order cone LCP problem is a special case of Problem (1), which has found many applications in optimizations. For example, the system of KKT conditions for a quadratic programming problem with second-order cone constraints can be formulated as a second-order cone LCP problem. Lipschitz error bound condition has already been used in the convergence analysis of algorithms for solving Problem (1). For example, the convergence results for the matrix-splitting method by [6] and the proximal gradient descent method by [22] are established under the Lipschitz error bound. Since affine variational inequalities are not necessarily strictly monotone or strongly monotone, the known results are not applicable to get a Lipschitz error bound for Problem (1). To overcome this difficulty, in this paper, we propose a sufficient condition, under which a local Lipschitz-type error bound holds for Problem (1). Moreover, based on fractional error bounds for polynomial systems proposed by Li et.  al. (2015,2016) [13], [14], we establish a local Hölder error bound for Problem (1) under a full row rank condition.

The rest of this paper will be organized as follows. In the next section, examples are presented for illustrating that in general, Lipschitz error bounds may be invalid for second-order cone inclusion problems or semidefinite inequalities. Hence it also may be invalid for VIs with SOC constraints. In Section 3, we provide a sufficient condition, under which Problem (1) has a local Lipschitz-type error bound with projection-type residual functions. A local Hölder error bound for AVIs with SOC constraints is established in Section 4. It is proved that the Hölder exponent can be bounded by a function of problem dimensions. The final section makes a conclusion of the paper.

Section snippets

Two examples for second-order cone and semidefinite inequalities

It has been proved by Luo and Tseng (1992) [19] and Li [16] that affine variational inequalities have Lipschitz error bounds without any regularity assumptions. But for second-order cone inclusion problems and semidefinite inequalities, Lipschitz error bounds are only satisfied under some regularity conditions, see [7], [9], [11], [21]. The following example shows that Lipschitz error bounds may be invalid for SOC problems when these regularity conditions fail.

Example 2.1

Consider a second-order cone

Preliminary

It is well known (from the convexity of X) that the solution set of Problem (1) is equivalent to X={xX|x=[xMxq]X},where [D]X denotes the orthogonal projection of D onto X. Denote the projection [xMxq]X by z. Then z is the optimal solution of the following second-order cone constrained quadratic program, min12zx+Mx+q2s. t.AzbQ.The KKT conditions of the problem can be written as follows.

KKT conditions: xzMxq+ATλ=0,AizbiQmi+1,λiQmi+1,λi(Aizbi)=0,i=1,,p.

A local Lipschitz error bound

In view of Example 2.1,

Results on Hölder error bounds

In this section, local Hölder error bounds for AVI with SOC constraints will be proved based on the recent error bound results on polynomials given by Li, Mordukhovich and Pham (2015) [14]. By the nonsmooth Łojasiewicz’s inequality, they established local Hölder error bounds for polynomial systems as follows.

Proposition 4.1

Li, Mordukhovich and Pham [14], Theorem 3.5

Let gi as i=1,,r and hj as j=1,,s be real polynomials on Rn with degree at most d. Denote the solution set by S, i.  e., S={x|gi(x)0,i=1,,r,hj(x)=0,j=1,,s}. Let x0S be a given

Conclusion

It is well known that linear VIs have Lipschitz error bounds. However for AVIs with SOC constraints, Lipschitz error bounds may be invalid unless some constraint qualifications are satisfied. In this paper, through converting the problem into a nonconvex quadratic system via the KKT condition, we proposed a condition under which the variational inequality problem has a local Lipschitz error bound with a projection-type residual function. It is worth noting that here no strongly monotone

Acknowledgments

We are greatly grateful for referees’ constructive suggestions and comments for improving the paper. This paper is supported by the National Science Foundation of China (11601061, 11301347 and 11301246), the Fundamental Research Funds for the Central Universities (DUT16LK07) and the General Project of Scientific Research of the Education Department of Liaoning Province (L2015291).

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