A smoothing quasi-Newton method for solving general second-order cone complementarity problems

Abstract

Recently, there are much interests in studying smoothing Newton method for solving montone second-order cone complementarity problem (SOCCP) or SOCCPs with Cartesian $P/P_0$-property. In this paper, we propose a smoothing quasi-Newon method for solving general SOCCP. We show that the proposed method is well-defined without any additional assumption and has global convergence under standard conditions. Moreover, under the Jacobian nonsingularity assumption, the method is shown to have local superlinear or quadratic convergence rate. Our preliminary numerical experiments show the method could be very effective for solving SOCCPs.

Appendix: The proof of the result (c) given in Remark 4.1

We only need to prove that the result (c) holds when the SOCCP has Cartesian $P_0$-property. For any $z=(\mu ,x)\in \mathbb {R}_{++}\times \mathbb {R}^n$, let $\mathrm {d}x=(\mathrm {d}x_{1},\ldots ,\mathrm {d}x_{r})\in \mathbb {R}^{n_1}\times \cdots \times \mathbb {R}^{n_r}$ satisfy ${\Phi }'_{x}(z)\mathrm {d}x=0.$ Then, by (2.13) and (2.15), we have

$$\begin{aligned} (I_{n_i}-L_{w_i}^{-1}L_{x_i})\mathrm {d}x_i+(I_{n_i} -L_{w_i}^{-1}L_{F_i(x)})F_i'(x)\mathrm {d}x_i=0,~~~\forall ~i=1,\ldots ,r, \end{aligned}$$

i.e.,

$$\begin{aligned} (L_{w_i}-L_{x_i})\mathrm {d}x_i+(L_{w_i}-L_{F_i(x)}) F_i'(x)\mathrm {d}x_i=0,~~~\forall ~i=1,\ldots ,r. \end{aligned}$$

(7.1)

Since $w_i^2-(x_i^2+F_i(x)^2)=2\mu ^2\mathrm {e}_i\succ _{\mathbb {K}^{n_i}}0,$ it follows from [14, Proposition 2.2] that

$$\begin{aligned}&L_{w_i}-L_{x_i}\succ 0,~~L_{w_i}-L_{F_i(x)}\succ 0, \end{aligned}$$

(7.2)

$$\begin{aligned}&(L_{w_i}-L_{x_i})(L_{w_i}-L_{F_i(x)})\succ 0,~~\forall ~i=1,\ldots ,r. \end{aligned}$$

(7.3)

So, by (7.1) and (7.2) we have

$$\begin{aligned} F_i'(x)\mathrm {d}x_i=-(L_{w_i}-L_{F_i(x)})^{-1} (L_{w_i}-L_{x_i})\mathrm {d}x_i,~~~\forall ~i=1,\ldots ,r. \end{aligned}$$

(7.4)

Now we suppose that $\mathrm {d}x\ne 0$. Since F has the Cartesian $P_0$-property, $F'$ has the Cartesian $P_0$-property. Hence, there exists an index $v\in \{1,\ldots ,r\}$ such that

$$\begin{aligned} \mathrm {d}x_v\ne 0,~~~\mathrm {d}x_v^T(F'(x)\mathrm {d}x)_v\ge 0, \end{aligned}$$

(7.5)

which together with (7.4) give

$$\begin{aligned} \mathrm {d}x_v^T(L_{w_v}-L_{F_v(x)})^{-1}(L_{w_v}-L_{x_v})\mathrm {d}x_v\le 0. \end{aligned}$$

(7.6)

Let ${\overline{\mathrm {d}x_v}}:=(L_{w_v}-L_{F_v(x)})^{-1}\mathrm {d}x_v$. Then it follows from (7.3) that

$$\begin{aligned} \mathrm {d}x_v^T(L_{w_v}-L_{F_v(x)})^{-1}(L_{w_v}-L_{x_v})\mathrm {d}x_v=\overline{\mathrm {d}x_v}^T (L_{w_v}-L_{x_v})(L_{w_v}-L_{F_v(x)}){\overline{\mathrm {d}x_v}}^T\ge 0. \end{aligned}$$

(7.7)

By (7.6) and (7.7), we have

$$\begin{aligned} \overline{\mathrm {d}x_v}^T(L_{w_v}-L_{x_v}) (L_{w_v}-L_{F_v(x)}){\overline{\mathrm {d}x_v}}^T=0, \end{aligned}$$

which together with (7.3) give $\overline{\mathrm {d}x_v}=0$ and hence ${\mathrm {d}x_v}=0$ This contradicts with ${\mathrm {d}x_v}\ne 0$ in (7.5). Thus, we have $\mathrm {d}x=0$ and complete the proof. $\square$