Mathematical programs with complementarity constraints and a non-Lipschitz objective: optimality and approximation

Abstract

We consider a class of mathematical programs with complementarity constraints (MPCC) where the objective function involves a non-Lipschitz sparsity-inducing term. Due to the existence of the non-Lipschitz term, existing constraint qualifications for locally Lipschitz MPCC cannot ensure that necessary optimality conditions hold at a local minimizer. In this paper, we present necessary optimality conditions and MPCC-tailored qualifications for the non-Lipschitz MPCC. The proposed qualifications are related to the constraints and the non-Lipschitz term, which ensure that local minimizers satisfy these necessary optimality conditions. Moreover, we present an approximation method for solving the non-Lipschitz MPCC and establish its convergence. Finally, we use numerical examples of sparse solutions of linear complementarity problems and the second-best road pricing problem in transportation science to illustrate the effectiveness of our approximation method for solving the non-Lipschitz MPCC.

A An iteratively reweighted \(\ell _1\) minimization method for solving problem \((P_{\epsilon ,\sigma })\)

Algorithm A.1

Choose an arbitrary initial point $y^0\in \mathbb {R}^n$ and set $\imath =0$.

• Step 1: Solve the weighted $\ell _1$ minimization problem

  $$\begin{aligned} \begin{array}{rl} \min &{} \quad f(y)+ p\sum \limits _{i=1}^r w_i^\imath |D_i^Ty|\\ {\mathrm{s.t.}} &{}\quad G(y)\ge 0,\ H(y)\ge 0, \ G(y)^TH(y)\le \sigma \end{array} \end{aligned}$$

  (A.1)

  to get $y^{\imath +1}$, where $w_i^{\imath }=(|D_i^Ty^{\imath }|+\epsilon _i)^{p-1}$ for all $i=1,\ldots ,r$.

• Step 2: Set $\imath =\imath +1$ and go to Step 1.

The proof of the following theorem uses the techniques in [31] for iteratively reweighted $\ell _1$ methods for unconstrained minimization problems. For completeness, we give a brief proof.

Theorem A.1

Any accumulation point $y^*$ of the sequence $\{y^\imath \}$ generated by Algorithm A.1 is a stationary point of problem $(P_{\epsilon ,\sigma })$.

Proof

Let q be such that $1/p + 1/q=1$, and let

$$\begin{aligned} \Upsilon _{\epsilon }(y,w):= f(y)+ p \sum _{i=1}^r \left[ w_i(|D_i^Ty|+\epsilon _i)-\frac{w_i^q}{q}\right] . \end{aligned}$$

It is easy to verify that for any $y\in \mathbb {R}^n$ and $\epsilon >0$,

$$\begin{aligned} F_{\epsilon }(y)=\min _{w\ge 0} \Upsilon _{\epsilon }(y,w), \end{aligned}$$

(A.2)

and for any $\imath \ge 0$,

$$\begin{aligned} w^{\imath } = \mathrm{Arg}\min \limits _{w\ge 0} \Upsilon _{\epsilon }(y^{\imath },w),\ \quad y^{\imath +1} \in \mathrm{Arg}\min \limits _{y\in \mathcal{F}_\sigma } \Upsilon _{\epsilon }(y, w^\imath ). \end{aligned}$$

(A.3)

It then follows that

$$\begin{aligned} F_{\epsilon }(y^{\imath +1}) = \Upsilon _{\epsilon }(y^{\imath +1},w^{\imath +1}) \le \Upsilon _{\epsilon }(y^{\imath +1},w^{\imath })\le \Upsilon _{\epsilon }(y^{\imath },w^{\imath })=F_{\epsilon }(y^{\imath }). \end{aligned}$$

(A.4)

Hence, the sequence of $\{F_{\epsilon }(y^{\imath })\}_{\imath \ge 0}$ is nonincreasing. Let $y^{\imath }\rightarrow y^*$ as $\imath \in K\rightarrow \infty$. This together with the continuity of $F_{\epsilon }$ and the monotonicity of $\{F_{\epsilon }(y^{\imath })\}_{\imath \ge 0}$ implies that $F_{\epsilon }(y^{\imath })\rightarrow F_{\epsilon }(y^*)$ as $\imath \rightarrow \infty$. Moreover, it is easy to see that $w^\imath \rightarrow (|D_i^Ty^*|+\epsilon _i)^{p-1}$ as $\imath \in K \rightarrow \infty$ for all $i=1,\ldots ,r$. Then by (A.4), we have that $\Upsilon _{\epsilon }(y^{\imath +1},w^{\imath })\rightarrow F_{\epsilon }(y^*)=\Upsilon _{\epsilon }(y^*,w^*)$. It follows from the second relation in (A.3) that $\Upsilon _{\epsilon }(y^{\imath +1}, w^\imath )\le \Upsilon _{\epsilon }(y, w^\imath )$ for all $y\in \mathcal{F}_\sigma$. Upon taking limits on both sides of this inequality as $\imath \in K\rightarrow \infty$, we have that $\Upsilon _{\epsilon }(y^*, w^*)\le \Upsilon _{\epsilon }(y, w^*)$ for all $y\in \mathcal{F}_\sigma$. This means that $y^*\in \mathrm{Arg}\min \nolimits _{y\in \mathcal{F}_\sigma } \Upsilon _{\epsilon }(y, w^*)$. By Fermat’s rule (see, e.g., [36, Theorem 10.1]), it follows that

$$\begin{aligned} 0\in \nabla f(y^*) + p\sum _{i=1}^r D_i(|D_i^Ty^*|+\epsilon _i)^{p-1} {\mathrm{sign}}\,(D_i^Ty^*) + \mathcal{N}_{\mathcal{F}_\sigma }(y^*), \end{aligned}$$

which is the stationary condition of problem $(P_{\epsilon ,\sigma })$ at $y^*$. The proof is complete. $\square$

Although any accumulation point of the sequence generated by Algorithm A.1 is a stationary point of problem $(P_{\epsilon ,\sigma })$, all iteration points may not be approximate stationary points since $\Vert y^\imath -y^{\imath +1}\Vert$ may not converge to 0 as $\imath \rightarrow \infty$. However, a weak approximate stationary point can be obtained as follows.

Theorem A.2

Assume that the sequence $\{y^\imath \}$ generated by Algorithm A.1 has a bounded subsequence. Then for any $\varepsilon >0$, there exist $\tilde{y}^\imath$ and $\zeta$ satisfying $\Vert y^\imath -\tilde{y}^\imath \Vert \le \varepsilon$ and $\Vert \zeta \Vert \le \varepsilon$ such that

$$\begin{aligned} \zeta \in \nabla f(\tilde{y}^\imath ) + p\sum _{i=1}^r D_i(|D_i^T\tilde{y}^\imath |+\epsilon _i)^{p-1} {\mathrm{sign}}\,(D_i^T\tilde{y}^\imath ) + \mathcal{N}_{\mathcal{F}_\sigma }(\tilde{y}^\imath ). \end{aligned}$$

Proof

Without loss of generality, we assume that the whole sequence $\{y^\imath \}$ is bounded. Let $\varepsilon _0>0$. Since the sequence $\{F_{\epsilon }(y^{\imath })\}_{\imath \ge 0}$ is convergent, it follows that $\Vert F_{\epsilon }(y^{\imath })-F_{\epsilon }(y^{\imath +1})\Vert \le \varepsilon _0$ when $\imath$ is sufficiently large. Then by (A.4), we have

$$\begin{aligned} 0\le \Upsilon _{\epsilon }(y^{\imath },w^{\imath })- \Upsilon _{\epsilon }(y^{\imath +1},w^{\imath }) \le \varepsilon _0. \end{aligned}$$

This means that $y^\imath$ is an $\varepsilon _0$-optimal solution to the problem of minimizing $\Upsilon _{\epsilon }(y,w^{\imath })$ on $\mathcal{F}_\sigma$. Then by Ekeland’s variational principal, there exists $\tilde{y}^{\imath }$ such that $\Vert \tilde{y}^\imath -y^\imath \Vert \le \sqrt{\varepsilon _0}$, $\Upsilon _{\epsilon }(\tilde{y}^{\imath },w^{\imath })\le \Upsilon _{\epsilon }(y^{\imath },w^{\imath })$, and $\tilde{y}^\imath$ is the unique minimizer of the problem

$$\begin{aligned} \min \ \Upsilon _{\epsilon }(y,w^{\imath }) + \sqrt{\varepsilon _0} \Vert y-\tilde{y}^l\Vert \quad \mathrm{s.t.} \ y\in \mathcal{F}_\sigma . \end{aligned}$$

Then by Fermat’s rule (see, e.g., [36, Theorem 10.1]), we have that

$$\begin{aligned} 0\in \nabla f(\tilde{y}^\imath ) + p\sum _{i=1}^r w^\imath _i D_i {\mathrm{sign}}\,(D_i^T\tilde{y}^\imath ) + \sqrt{\varepsilon _0}\mathbb {B} + \mathcal{N}_{\mathcal{F}_\sigma }(\tilde{y}^\imath ), \end{aligned}$$

where $\mathbb {B}$ stands for the closed unit ball of $\mathbb {R}^n$. This and the definition of $w^\imath$ imply that

$$\begin{aligned}&0\in \nabla f(\tilde{y}^\imath ) + p\sum _{i=1}^r D_i (|D_i^T\tilde{y}^\imath | + \epsilon _i)^{p-1} {\mathrm{sign}}\,(D_i^T\tilde{y}^\imath ) + \mathcal{N}_{\mathcal{F}_\sigma }(\tilde{y}^\imath )+ \sqrt{\varepsilon _0}\mathbb {B}\nonumber \\&\quad + p\sum _{i=1}^r D_i [(|D_i^Ty^\imath | + \epsilon _i)^{p-1}-(|D_i^T\tilde{y}^\imath | + \epsilon _i)^{p-1}]{\mathrm{sign}}\,(D_i^T\tilde{y}^\imath ). \end{aligned}$$

(A.5)

Since $\{y^{\imath }\}$ is bounded, we can find an $\varepsilon _0$ such that when $\Vert \tilde{y}^\imath -y^\imath \Vert \le \sqrt{\varepsilon _0} \le \varepsilon /2$, the norm of the term in (A.5) is bounded above by $\varepsilon /2$. Then the desired result follows immediately. $\square$