Approximation Methods for a Class of Non-Lipschitz Mathematical Programs with Equilibrium Constraints

Abstract

We consider how to solve a class of non-Lipschitz mathematical programs with equilibrium constraints (MPEC) where the objective function involves a non-Lipschitz sparsity-inducing function and other functions are smooth. Solving the non-Lipschitz MPEC is highly challenging since the standard constraint qualifications fail due to the existence of equilibrium constraints and the subdifferential of the objective function is unbounded due to the existence of the non-Lipschitz function. On the one hand, for tackling the non-Lipschitzness of the objective function, we introduce a novel class of locally Lipschitz approximation functions that consolidate and unify a diverse range of existing smoothing techniques for the non-Lipschitz function. On the other hand, we use the Kanzow and Schwartz regularization scheme to approximate the equilibrium constraints since this regularization can preserve certain perpendicular structure as in equilibrium constraints, which can induce better convergence results. Then an approximation method is proposed for solving the non-Lipschitz MPEC and its convergence is established under weak conditions. In contrast with existing results, the proposed method can converge to a better stationary point under weaker qualification conditions. Finally, a computational study on the sparse solutions of linear complementarity problems is presented. The numerical results demonstrate the effectiveness of the proposed method.

Appendix: The Convergence When Using the Regularization by Kadrani et al.

When using the regularization scheme by Kadrani et al., our approximation problem becomes

$$\begin{aligned} ({\textrm{P}}_{\sigma ,\textrm{t}}^{{\textrm{KDB}}})~~~ \min _x{} & {} F_{\sigma }(x):=f(x) + \lambda \sum _{i=1}^{r}\varphi (D_i^{\top }x;\sigma _i) \\[4pt] \mathrm{s.t.}{} & {} g(x)\le 0,~ h(x)=0,\\[4pt]{} & {} G_i(x)+t\ge 0, ~H_i(x)+t\ge 0,\\[4pt]{} & {} \Phi _i^{KDB}(x;t)\le 0\ \ i\in \{1,\ldots ,m\}, \end{aligned}$$

where $\Phi _i^{KDB}(x;t):=(H_i(x)-t)(G_i(x)-t)$ for all $i\in \{1,\ldots ,m\}$. By a straightforward calculation, we have

$$\begin{aligned} \nabla \Phi _i^{KDB}(x;t)=(H_i(x)-t)\nabla G_i(x)+(G_i(x)-t)\nabla H_i(x)\ \ i\in \{1,\ldots ,m\}. \end{aligned}$$

Theorem 4

Assume that $t^{k}\downarrow 0$ and $\sigma ^{k}\downarrow 0$ as $k\rightarrow \infty$. Let $x^k$ be a stationary solution of $\mathrm{(P_{\sigma ^k,t^k}^{KDB})}$, and $x^*$ be an accumulation point of $\{x^k\}$. If MPEC-CCQC holds at $x^*$, then $x^*$ is an M-stationary point of problem (1).

Proof

Recall that $x^k$ is a stationary point of problem $\mathrm{(P_{\sigma ^k,t^k}^{KDB})}$ if there exist multipliers $(\alpha ^k,\beta ^k,\pi ^k,\rho ^k,\gamma ^k)$ such that

$$\begin{aligned}{} & {} 0\in \nabla f(x^k) + \lambda \sum _{i=1}^{r} D_i \partial \varphi (D_i^\top x^k;\sigma ^k_i) + \sum _{i=1}^{q} \alpha _{i}^k\nabla g_{i}(x^k)+\sum _{i=1}^{s}\beta _{i}^k\nabla h_i(x^k)\nonumber \\{} & {} -\sum _{i=1}^{m}\pi _{i}^{k}\nabla G_{i}(x^k) -\sum _{i=1}^{m}\rho _{i}^{k}\nabla H_i(x^k)+\sum _{i=1}^{m}\gamma _i^{k}\nabla \Phi _{i}^{KDB}(x^k;t^k), \end{aligned}$$

(36)

$$\begin{aligned}{} & {} \alpha ^k_i\ge 0, g_i(x^k)\le 0, \alpha ^k_i g_i(x^k)=0,\ \forall i=1,\ldots ,q,\end{aligned}$$

(37)

$$\begin{aligned}{} & {} h_i(x^k)=0,\ \forall i=1,\ldots , s, \nonumber \\{} & {} \pi _i^k\ge 0, G_i(x^k)+t^k\ge 0, \pi _i^k (G_i(x^k)+t^k)=0,\ \forall i=1,\ldots ,m,\end{aligned}$$

(38)

$$\begin{aligned}{} & {} \rho _i^k\ge 0, H_i(x^k)+t^k\ge 0, \rho _i^k (H_i(x^k)+t^k)=0,\ \forall i=1,\ldots ,m,\end{aligned}$$

(39)

$$\begin{aligned}{} & {} \gamma _i^k\ge 0, \Phi _i^{KDB}(x^k;t^k)\le 0, \gamma _i^k \Phi _i^{KDB}(x^k;t^k)=0,\ \forall i=1,\ldots ,m. \end{aligned}$$

(40)

Let

$$\begin{aligned}{} & {} \mu _i^k:=\pi _i^k-\gamma _i^k(H_i(x^k)-t_k),\\{} & {} \nu _i^k:=\rho _i^k-\gamma _i^k(G_i(x^k)-t_k). \end{aligned}$$

Then (36) can be rewritten as

$$\begin{aligned} \nabla f(x^k) + \lambda \sum _{i=1}^{r} \xi ^k_i D_i + \sum _{i=1}^{q} \alpha _{i}^k\nabla g_{i}(x^k)+\sum _{i=1}^{s}\beta _{i}^k\nabla h_i(x^k)\nonumber \\ -\sum _{i=1}^{m}\mu _{i}^{k}\nabla G_{i}(x^k)-\sum _{i=1}^{m}\nu _{i}^{k}\nabla H_i(x^k)=0. \end{aligned}$$

(41)

where $\xi ^k_i \in \partial \varphi (D_i^\top x^k; \sigma ^k_i)$ for all $i=1,\ldots ,r$. By the definition of $\varphi$, it is easy to see that

$$\begin{aligned} \lim _{k\rightarrow \infty }\xi ^k_i = \partial \psi (D_i^\top x^*) = |D_i^{T}x^*|^{p-1}\textrm{sign}(D_i^{\top }x^*) \quad \forall i\in {{\mathcal {I}}}_{\ne }^{*}. \end{aligned}$$

We next show that there exists a sequence $z^k\rightarrow F(x^*)$ such that (41) holds with $\alpha _i^k\in {{\mathcal {N}}}_{(0,\infty ]}(z_i^k)$ and $(\mu _i^k,\nu _i^k)\in {{\mathcal {N}}}_{{\mathcal {C}}}(z_i^k)$. Let $z^k = (D_i^Tx^*\ i\in {{\mathcal {I}}}_0^*,g(x^k),h(x^k),R(x^*))$. It is clear that $z^k\in \Lambda$. By (37), it is easy to see that $\alpha _i^k\in {{\mathcal {N}}}_{(0,\infty ]}(z_i^k)$ for all $i=1,\ldots ,q$. One can easily observe that if $i\in {{\mathcal {I}}}_{+0}^*$, then $G_i(x^*)>0$. Thus $\pi _i^k=0$ by (38) and $\gamma _i^k(H_i(x^k)-t_k)=0$ by (40). Hence $\mu _i^k=\pi _i^k-\gamma _i^k(H_i(x^k)-t_k)=0$ for $i\in {{\mathcal {I}}}_{+0}^*$. In the same way, we have $\nu _i^k=0$ for $i\in {{\mathcal {I}}}_{0+}^*$. Moreover, it is easy to see that if $\mu _i^k<0$, then $\pi _i^k=0$ and $\gamma _i^k(H_i(x^k)-t_k)>0$. Thus, by (40), it follows $G_i(x^k)-t_k = 0$ and by (39), it follows $\rho _i^k=0$. Hence $\nu _i^k=\rho _i^k-\gamma _i^k(G_i(x^k)-t_k)=0$ and then $\mu _i^k\nu _i^k=0$ if $\mu _i^k<0$. In the same way, we can have $\mu _i^k\nu _i^k=0$ if $\nu _i^k<0$. Consequently, we have $\mu _i^k>0,\nu _i^k>0$ or $\mu _i^k\nu _i^k=0$ for all i. Thus, by the expression of the normal cone to ${{\mathcal {C}}}$, we have $(\mu _i^k,\nu _i^k)\in {{\mathcal {N}}}_{{\mathcal {C}}}(z_i^k)$.

Based on the above discussions and the expression of the normal cone to $\Lambda$, (41) can be written as

$$\begin{aligned} -\nabla f(x^k)-\lambda \sum _{i\in {{\mathcal {I}}}_{\not =}^*} \xi _{i}^{k}D_i \in \nabla F(x^k){{\mathcal {N}}}_\Lambda (z^k). \end{aligned}$$

Since MPEC-CCQC holds at $x^*$, it follows that

$$\begin{aligned} 0\in \nabla f(x^*)+\lambda \sum _{i\in {{\mathcal {I}}}_{\not =}^*} \xi _{i}^{*}D_i +\nabla F(x^*){{\mathcal {N}}}_{\Lambda }(F(x^*)), \end{aligned}$$

indicating that $x^*$ is an M-stationary point by Proposition 2. $\square$