# Smoothing sample average approximation method for solving stochastic second-order-cone complementarity problems

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## Abstract

In this paper, we consider stochastic second-order-cone complementarity problems (SSOCCP). We first use the so-called second-order-cone complementarity function to present an expected residual minimization (ERM) model for giving reasonable solutions of SSOCCP. Then, we introduce a smoothing function, by which we obtain a smoothing approximate ERM model. We further show that the global solution sequence and weak stationary point sequence of this smoothing approximate ERM model converge to the global solution and the weak stationary point of the original ERM model as the smoothing parameter tends to zero respectively. Moreover, since the ERM formulation contains an expectation, we employ a sample average approximate method for solving the smoothing ERM model. As the convergence analysis, we first show that the global optimal solution of this smoothing sample average approximate problem converges to the global optimal solution of the ERM problem with probability one. Subsequently, we consider the weak stationary points’ convergence results of this smoothing sample average approximate problem of ERM model. Finally, some numerical examples are given to explain that the proposed methods are feasible.

## Keywords

Second-order-cone Stochastic complementarity problems Sample average approximation Smoothing function Convergence## MSC

90C33 65K05 90C15## 1 Introduction

*i*, \(l=0\), and the mapping

*F*has the form

In fact, many engineering and practical problems, such as the three-dimensional frictional contact problems [1] and the robust Nash equilibria [2], can be changed into SOCCP directly. In addition, as the special case of the SOCCP, some basic theory, effective algorithms, and important applications of complementarity problems in engineering, economics have been developed. However, in practice, several types of uncertain data such as weather, supply, demand, cost, etc. may be involved in SOCCP. The stochastic problems are aimed at a practical treatment of such problems under uncertainty. It is well known that stochastic second-order-cone complementarity problems (SSOCCP) are more complicated than SOCCP and they have found applications in more fields. Therefore, it is meaningful and interesting to study the general SSOCCP.

*i*, \(l=0\), and the mapping

*F*has the form

Note that problem (1.3) may not have a solution in general. There have been proposed three ways to deal with (1.3). One way was suggested by Gürkan et al. [3], who used an expectation of \(F_{0}\) instead of \(F_{0}\) for giving a simple nonlinear complementarity problems reformulation. Another way was presented by Chen and Fukushima [4], who made use of the so-called NCP function to present the expected residual minimization formulation for SNCP. The last was proposed by Lin and Fukushima [5]. They formulated SNCP as a special here-and-now model of stochastic mathematical program with equilibrium constraints. Moreover, Luo and Wang [6] presented the ERM and CVaR reformulation for solving stochastic generalized complementarity problem.

Motivated by the above work, we will propose a reformulation of SSOCCP for giving reasonable solutions of SSOCCP. In the rest of this paper, we assume that Ω is a nonempty compact set. We will often write \(x=(x_{1},x_{2})\) for \((x_{1}^{T},x_{2}^{T})^{T}\) and \((x,y,z)\) for \((x^{T},y^{T},z^{T})^{T}\) for simplicity. In addition, for any \(x=(x_{1},x_{2})\in R\times R^{n-1}\) and \(y=(y_{1},y_{2})\in R\times R^{n-1}\), we define their Jordan product as \(x\circ y=(x^{T}y, y_{1}x_{2}+x_{1}y_{2})\). We will write \(x^{2}\) to mean \(x\circ x\) and write \(x+y\) to mean the usual componentwise addition of vectors. Moreover, if \(x\in\mathcal{K}^{n}\), then there exists a unique vector in \(\mathcal{K}^{n}\), which we denote by \(x^{1/2}\) such that \((x^{1/2})^{2}=x^{1/2}\circ x^{1/2}=x\). \(F(\cdot,\cdot,\cdot,\xi)\) is twice continuous differentiable with respect to \((x,y,z)\) and \(F(x,y,z,\cdot)\) is continuous differentiable with respect to \(\xi\in \Omega\). For each \(t=1,\ldots, n+l\), \(\nabla_{(x,y,z)}F_{t}(x,y,z,\xi )\) denotes the gradient of \(F_{t}\) on \((x,y,z)\), \(\nabla ^{2}_{(x,y,z)}F_{t}(x,y,z,\xi)\) denotes the Hessian matrix of \(F_{t}\) on \((x,y,z)\), and \(\frac{\partial F_{t}}{\partial x_{j}}\) denotes a partial derivative on \(x_{j}\). \(\nabla_{(x,y,z)}F(x,y,z,\xi)=[\nabla _{(x,y,z)}F_{1}(x,y,z,\xi),\ldots,\nabla_{(x,y,z)}F_{n+l}(x,y,z,\xi )]^{T}\). For an \(m\times n\) matrix *A*, \(\|A\|_{\mathcal{F}}:=\sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}|a_{ij}|^{2}}\) denotes its Frobenius norm. Moreover, *I* and *O* denote the identity matrix and null matrix with suitable dimensions, respectively.

The remainder of the paper is organized as follows. In Sect. 2, some preliminaries are given and a smoothing sample average approximation problem is presented. The convergence analysis of global solution and weak stationary point of this approximation problem are established in Sect. 3 and Sect. 4, respectively. Moreover, the conclusions are given in Sect. 5.

## 2 Deterministic reformulation and its approximation problems

Because of the existence of a random element *ξ*, we cannot generally expect that there exists a vector \((x,y,z)\) satisfying (1.1) or (1.2) for almost all \(\xi\in\Omega\), that is, both (1.1) and (1.2) may not have a solution in general. Therefore, an important issue in the study of SSOCCP is to present an appropriate deterministic formulation of the considered problem. Before giving the reformulation, we first introduce some related functions and their properties.

*x*onto the second-order-cone \(\mathcal{K}^{n}\), that is, \([x]_{+}=\operatorname{argmin}_{{x'}\in\mathcal {K}^{n}}\|x'-x\|\). For simplicity, set \(\phi_{0}(x,y):=\phi_{\mathrm{NR}}(x,y)\) in this paper. Note that this function is nondifferentiable but strongly semismooth.

*k*, a smoothing sample average approximation to problem (2.2) as follows:

For the sample average approximation method, the strong law of large numbers guarantees that the following result holds with probability one (abbreviated by “w.p.1.”).

## Lemma 2.1

It is generally known that the boundedness of an iteration sequence is the basic requirement in the iteration methods for solving various optimization problems. To ensure the boundedness of the iteration sequence of problem (2.1), it is essential to take the level sets of \(\theta(x,y,z)\) into account. And it is worth noting that if \(\theta(x,y,z)\) is coercive, i.e., \(\lim_{\|x\|\rightarrow \infty}{\theta(x,y,z)}=+\infty\), then the level sets of \(\theta (x,y,z)\) are bounded. Next, we study the coerciveness of \(\theta (x,y,z)\) under appropriate conditions.

## Theorem 2.1

*If the sequence*\(\{(x^{k},y^{k})\}\subseteq R^{n}\times R^{n}\)*satisfies*\(\liminf_{k\rightarrow+\infty}\lambda_{1}(x^{k})=-\infty\)*or*\(\liminf_{k\rightarrow+\infty}\lambda_{1}(y^{k})=-\infty\), *then*\(\|\theta(x^{k},y^{k},z^{k})\|\rightarrow+\infty\)*as*\(k\rightarrow +\infty\). *Here*, \(\lambda_{1}(x)\)*is the spectral value of*\(x=(x_{1}, x_{2})\in R\times R^{n-1}\), *given by*\(\lambda_{1}(x)=x_{1}-\|x_{2}\|\).

## 3 Convergence of global optimal solution

In this section, we will show that the global solution sequence of smoothing approximation problem (2.2) converges to the global solution of the true ERM problem (2.1). Then we will consider that the global solution sequence of smoothing SAA approximation problem (2.3) converges to the global solution of the so-called true problem (2.1) when the smoothing parameter tends to 0^{+} as sample size increases to +∞.

From now on, we denote by \(\bar{S}(\mu)\) the set of global optimal solutions of problem (2.2).

## Theorem 3.1

*Suppose that for each**μ*, \((x(\mu),y(\mu),z(\mu))\)*is a global optimal solution of problem* (2.2) *and*\(\lim_{\mu\to0^{+}} x(\mu)=\hat{x}\), \(\lim_{\mu\to0^{+}} y(\mu )=\hat{y}\), \(\lim_{\mu\to0^{+}} z(\mu)=\hat{z}\). *Then we have*\((\hat{x},\hat{y},\hat{z})\)*is a global optimal solution of problem* (2.1).

## Proof

*D*be a compact convex set containing the sequence \(\{(x(\mu), y(\mu), z(\mu))\}\). By the continuous differentiability of

*F*on the compact set \(D\times\Omega\) and Theorem 16.8 of [12], we have

Let *μ* vary as *k* increases, that is, let \(\mu=\mu_{k}\rightarrow 0^{+}\) as \(k\rightarrow+\infty\). In the following, we discuss the convergence for this case.

## Theorem 3.2

*Suppose that for each**k*, \((x^{k}(\mu_{k}),y^{k}(\mu_{k}),z^{k}(\mu_{k}))\)*is a global optimal solution of problem* (2.3) *and*\(\lim_{k\to+\infty} x^{k}(\mu_{k})=\bar{x}\), \(\lim_{k\to+\infty} y^{k}(\mu_{k})=\bar{y}\), \(\lim_{k\to+\infty} z^{k}(\mu_{k})=\bar{z}\), *w*.*p*.1. *Then we have*\((\bar{x},\bar{y},\bar{z})\)*is a global optimal solution of problem* (2.1) *w*.*p*.1.

## Proof

*A*be a compact convex set containing the sequence \(\{(x^{k}(\mu_{k}), y^{k}(\mu_{k}), z^{k}(\mu_{k}))\}\). By the continuous differentiability of

*F*on the compact set \(A\times\Omega\), there exist constants \(C_{1}>0\), \(C_{2}>0\) such that

*k*and each \(\xi^{i}\), there exist \(\gamma^{ki}_{x}(\mu_{k})=\alpha_{ki}x^{k}(\mu_{k})+(1-\alpha_{ki}) \bar {x}\), \(\gamma^{ki}_{y}(\mu_{k})=\alpha_{ki}y^{k}(\mu_{k})+(1-\alpha_{ki}) \bar{y}\), \(\gamma^{ki}_{z}(\mu_{k})=\alpha_{ki}z^{k}(\mu_{k})+(1-\alpha _{ki}) \bar{z}\), and \((\gamma^{ki}_{x}(\mu_{k}), \gamma^{ki}_{y}(\mu _{k}), \gamma^{ki}_{z}(\mu_{k}))\in A\) with \(\alpha_{ki}\in[0,1]\) such that

## 4 Convergence of weak stationary point

We first give some definitions associated with a weak stationary point of problem (2.1), then we show that the sequence of weak stationary point for (2.2) converges to a weak stationary point for (2.1) as \(\mu\rightarrow0^{+}\). Then we will consider the convergence of the stationary point of (2.3) while \(\mu=\mu_{k}\rightarrow0^{+}\) as \(k\rightarrow+\infty\).

## Definition 4.1

*g*at \(x_{0}\) is defined as

*X*.

## Definition 4.2

## Definition 4.3

*μ*, a point \((x(\mu),y(\mu),z(\mu))\) satisfying the following equation is called a weak stationary point of (2.2):

## Definition 4.4

*μ*and

*k*, a point \((x^{k}(\mu),y^{k}(\mu),z^{k}(\mu ))\) satisfying the following equation is called a stationary point of (2.3):

Especially, more details about weak stationary points can be seen in [13].

## Theorem 4.1

*Let*\((x(\mu),y(\mu),z(\mu))\)*be a weak stationary point of* (2.2) *for each**μ**and*\(\lim_{\mu\to0^{+}} x(\mu)=\tilde{x}\), \(\lim_{\mu\to0^{+}} y(\mu )=\tilde{y}\), \(\lim_{\mu\to0^{+}} z(\mu)=\tilde{z}\). *Suppose that*\(\|\phi_{\mu}(x,y)\|^{2}\)*satisfies gradient consistency*. *Then*\((\tilde {x},\tilde{y},\tilde{z})\)*is a weak stationary point of problem* (2.1).

## Proof

**D**be a compact convex set containing the sequence \(\{(x(\mu), y(\mu), z(\mu))\}\). By the twice continuous differentiability of

*F*on the compact set \(\mathbf{D}\times\Omega\) and Theorem 16.8 of [12], we have

**D**, there exists \(C_{3}>0\) such that

## Remark 4.1

The condition \(\|\phi_{\mu}(x,y)\|^{2}\) satisfying gradient consistency has been proved in [14]. In fact, the smoothing function \(\phi_{\mu}\) used in this paper is a special form of smoothing function considered in [14].

## Lemma 4.1

*Suppose that*\(\lim_{k\rightarrow+\infty}x^{k}(\mu_{k})=\tilde{x}\), \(\lim_{k\rightarrow+\infty}y^{k}(\mu_{k})=\tilde{y}\), \(\lim_{k\rightarrow+\infty}z^{k}(\mu_{k})=\tilde{z}\),

*w*.

*p*.1.

*Then we have that the following limit holds with probability one*:

## Proof

**A**be a compact convex set containing the sequence \(\{(x^{k}(\mu_{k}), y^{k}(\mu_{k}), z^{k}(\mu_{k}))\}\). By the twice continuous differentiability of

*F*on the compact set

**A**and the continuity of

*F*with respect to

*ξ*on a compact set Ω, there exist constants \(C_{4}>0\), \(C_{5}>0\), \(C_{6}>0\) such that

*F*and (4.9) into account, we have

## Theorem 4.2

*Let*\((x^{k}(\mu_{k}),y^{k}(\mu_{k}),z^{k}(\mu_{k}))\)*be a stationary point to* (2.3) *for each**k**and*\(\lim_{k\rightarrow+\infty}x^{k}(\mu_{k})=\tilde{x}\), \(\lim_{k\rightarrow+\infty}y^{k}(\mu_{k})=\tilde{y}\), \(\lim_{k\rightarrow +\infty}z^{k}(\mu_{k})=\tilde{z}\)*w*.*p*.1. *Then*\((\tilde{x},\tilde {y},\tilde{z})\)*is a weak stationary point of problem* (2.1) *w*.*p*.1.

## Proof

## 5 Numerical examples

In this section, we use random generator *rand* to generate a random sequence \(\{\xi^{1},\ldots,\xi^{N_{k}}\}\) of *ξ* and employ the Matlab unconstrained optimization command *fminunc* to solve the corresponding problems (2.1) and (2.3).

## Example 5.1

*ξ*be the uniformly distributed random variable in \((0,1)\), and \(F:R^{2}\times R^{2}\times\Omega\rightarrow R^{2}\) is defined as follows:

Computational results for Example 5.1

\(N^{k}\) | \(x^{*}=(0.0105,0.0030)^{T}\), \(y^{*}=(0.0089, 0.0094)^{T}\) | ||
---|---|---|---|

\(x^{k}\) | \(y^{k}\) | | |

10 | \((0.0115,0.0030)^{T}\) | \((0.0091,0.0093)^{T}\) | 0.0010 |

10 | \((0.0114,0.0030)^{T}\) | \((0.0090,0.0093)^{T}\) | 0.0009 |

10 | \((0.0112,0.0029)^{T}\) | \((0.0089,0.0094)^{T}\) | 0.0007 |

## Example 5.2

*ξ*be the uniformly distributed random variable in \((0,1)\), and \(F:R^{2}\times R^{2}\times\Omega\rightarrow R^{2}\) is defined as follows:

Computational results for Example 5.2

\(N^{k}\) | \(x^{*}=(1.8420, -0.4211)^{T}\), \(y^{*}=(3.1497, 2.3370)^{T}\) | |||
---|---|---|---|---|

\(\mu^{k}\) | \(x^{k}\) | \(y^{k}\) | | |

10 | 0.01 | \((1.9321, -0.5234)^{T}\) | \((3.2850, 2.5873)^{T}\) | 0.3155 |

10 | 0.008 | \((1.8564, -0.4328)^{T}\) | \((3.1589, 2.4692)^{T}\) | 0.1338 |

10 | 0.005 | \((1.8462, -0.4281)^{T}\) | \((3.1503, 2.3981)^{T}\) | 0.0616 |

## 6 Results and discussion

In this paper, we give a reasonable reformulation for solving SSOCCP, which is called ERM model. To solve this ERM model, we pay attention to two problems: one is how to handle the property of non-differentiability, the other is how to obtain the value of expectation for the objective function. We employ a smoothing technique and a sample average approximation method to deal with the problems. Further, we show that the global solution and weak stationary point of this smoothing sample average approximation ERM problem converge to those of the true ERM model correspondingly.

## Notes

### Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant No. 11501275 and Scientific Research Fund of Liaoning Provincial Education Department under Grant No. L2015199.

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

## Competing interests

The authors declare that they have no competing interests.

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