The New Butterfly Relaxation Method for Mathematical Programs with Complementarity Constraints

Abstract

We propose a new family of relaxation schemes for mathematical programs with complementarity constraints. We discuss the properties of the sequence of relaxed non-linear programs as well as stationary properties of limiting points. A sub-family of our relaxation schemes has the desired property of converging to an M-stationary point. A stronger convergence result is also proved in the affine case. A comprehensive numerical comparison between existing relaxation methods is performed on the library of test problems MacMPEC which shows promising results for our new method.

Appendices

Appendix

3.7 Proof of a Technical Lemma

In the proof of Theorem 3.4.4 and Theorem 3.4.5, we use the following lemma that links the gradients of G and H with the gradients of ${F_{1}(x;t)}$ and ${F_{2}(x;t)}$.

Lemma 3.7.6

Let $(I,I^c,I^-)$ be any partition of $\mathcal {I}_{GH}^{00}(x;t)$. Assume that the gradients

$$\begin{aligned} \begin{aligned}&\{ \nabla g_i(x) \ (i \in \mathcal {I}_g(x)), \ \nabla h_i(x) \ (i=1,\dots ,p), \\ {}&\nabla G_i(x) \ (i \in \mathcal {I}_G(x;\bar{t}) \cup \mathcal {I}_{GH}^{00}(x;t) \cup \mathcal {I}_{GH}^{+0}(x;t)), \\&\nabla H_i(x) \ (i \in \mathcal {I}_H(x;\bar{t}) \cup \mathcal {I}_{GH}^{00}(x;t)\cup \mathcal {I}_{GH}^{0+}(x;t)) \} \end{aligned} \end{aligned}$$

are linearly independent. Then, LICQ holds at x for (3.13).

Proof

We show that the gradients of the constraints of (3.13) are positively linearly independent. For this purpose, we prove that the trivial solution is the only solution to the equation

$$\begin{aligned} \begin{aligned} 0=&\sum _{i \in \mathcal {I}_g(x)} \eta ^g_i \nabla g_i(x) + \sum _{i=1}^{p} \eta ^h_i \nabla h_i(x)-\sum _{i \in \mathcal {I}_G(x;\bar{t})} \eta ^G_i \nabla G_i(x) - \sum _{i \in \mathcal {I}_H(x;\bar{t})} \eta ^H_i \nabla H_i(x) \\&+ \sum _{i \in \mathcal {I}_{GH}^{+0}(x;t) \cup \mathcal {I}_{GH}^{0+}(x;t)} \eta ^\Phi _i \nabla {\Phi ^B_i(G(x),H(x);t)}\\&+\sum _{i \in \mathcal {I}_{GH}^{00}(x;t)} \left( \nu ^{{F_{1}(x;t)}}_i-\mu ^{{F_{1}(x;t)}}_i+\delta ^{{F_{1}(x;t)}}_i\right) \nabla {F_{1i}(x;t)}\\&+ \left( -\nu ^{{F_{2}(x;t)}}_i+\mu ^{{F_{2}(x;t)}}_i+\delta ^{{F_{2}(x;t)}}_i\right) \nabla {F_{2i}(x;t)},\\ \end{aligned} \end{aligned}$$

where $\mathop {\mathrm {supp}}(\eta ^g)\subseteq \mathcal {I}_g(x)$, $\mathop {\mathrm {supp}}(\eta ^G)\subseteq \mathcal {I}_G(x;\bar{t})$, $\mathop {\mathrm {supp}}(\eta ^H)\subseteq \mathcal {I}_H(x;\bar{t})$, $\mathop {\mathrm {supp}}(\eta ^\Phi )\subseteq \mathcal {I}_{GH}^{+0}(x;t) \cup \mathcal {I}_{GH}^{0+}(x;t)$, $\mathop {\mathrm {supp}}(\nu ^{{F_{1}(x;t)}})\subseteq I$, $\mathop {\mathrm {supp}}(\nu ^{{F_{2}(x;t)}})\subseteq I$, $\mathop {\mathrm {supp}}(\mu ^{{F_{1}(x;t)}})\subseteq I^c$, $\mathop {\mathrm {supp}}(\mu ^{{F_{2}(x;t)}})\subseteq I^c$, $\mathop {\mathrm {supp}}(\delta ^{{F_{1}(x;t)}})\subseteq I^-$, and $\mathop {\mathrm {supp}}(\delta ^{{F_{2}(x;t)}})\subseteq I^-$ where $I \cup I^c \cup I^-=\mathcal {I}_{GH}^{00}(x;t)$ and $I,I^c,I^-$ have a two-by-two empty intersection.

By definition of ${F_{1}(x;t)}$ and ${F_{2}(x;t)}$, it holds that

$$\begin{aligned} \begin{aligned}&\nabla {F_{1i}(x;t)}=\nabla H_i(x) - {t_2}\theta '_{t_1}(G_i(x)) \nabla G_i(x), \\&\nabla {F_{2i}(x;t)}=\nabla G_i(x) - {t_2}\theta '_{t_1}(H_i(x)) \nabla H_i(x). \end{aligned} \end{aligned}$$

The gradient of ${\Phi ^B(G(x),H(x);t)}$ is given by Lemma 3.3.3.

We now replace those gradients in the equation above

$$\begin{aligned} \begin{aligned} 0= \sum _{i \in \mathcal {I}_g(x)} \lambda ^g_i \nabla g_i(x) + \sum _{i=1}^{p} \lambda ^h_i \nabla h_i(x)+\sum _{i=1}^{q} \lambda ^G_i \nabla G_i(x)+\sum _{i=1}^{q} \lambda ^H_i \nabla H_i(x), \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned}&\lambda ^G_i=-\eta ^G_i + \eta ^\Phi _i{F_{1i}(x;t)}-\left( \eta ^\Phi _i{F_{2i}(x;t)}\!+\!\nu ^{{F_{1}(x;t)}}_i\!-\!\mu ^{{F_{1}(x;t)}}_i\!+\!\delta ^{{F_{1}(x;t)}}_i\right) {t_2}\theta '_{t_1}(G_i(x)) \\& - \nu ^{{F_{2}(x;t)}}_i + \mu ^{{F_{2}(x;t)}}_i + \delta ^{{F_{2}(x;t)}}_i,\\&\lambda ^H_i=-\eta ^H_i + \eta ^\Phi _i{F_{2i}(x;t)}-\left( \eta ^\Phi _i{F_{1i}(x;t)}\!-\!\nu ^{{F_{2}(x;t)}}_i\!+\!\mu ^{{F_{2}(x;t)}}_i\!+\!\delta ^{{F_{2}(x;t)}}_i\right) {t_2}\theta '_{t_1}(H_i(x)) \\& + \nu ^{{F_{1}(x;t)}}_i -\mu ^{{F_{1}(x;t)}}_i+\delta ^{{F_{1}(x;t)}}_i. \end{aligned} \end{aligned}$$

By linear independence assumption, we obtain

$$\begin{aligned} \begin{aligned}&\eta ^g=0, \eta ^h=0, \eta ^G=0, \eta ^H=0,\eta ^\Phi _i=0 \ \forall i \in \mathcal {I}_{GH}^{0+}(x;t) \cup \mathcal {I}_{GH}^{+0}(x;t), \\&-\nu ^{{F_{1}(x;t)}}_i {t_2}\theta '_{t_1}(G_i(x)) - \nu ^{{F_{2}(x;t)}}_i=0 \text{ and } \nu ^{{F_{2}(x;t)}}_i {t_2}\theta '_{t_1}(H_i(x)) + \nu ^{{F_{1}(x;t)}}_i=0, \forall i \in I,\\&\mu ^{{F_{1}(x;t)}}_i {t_2}\theta '_{t_1}(G_i(x)) \!+\! \mu ^{{F_{2}(x;t)}}_i\!=0 \text{ and } -\!\mu ^{{F_{2}(x;t)}}_i {t_2}\theta '_{t_1}(H_i(x)) - \mu ^{{F_{1}(x;t)}}_i\!=0, \forall i \in I^c,\\&-\!\delta ^{{F_{1}(x;t)}}_i {t_2}\theta '_{t_1}(G_i(x)) \!+\! \delta ^{{F_{2}(x;t)}}_i \!\!=0 \text{ and } \!-\!\delta ^{{F_{2}(x;t)}}_i {t_2}\theta '_{t_1}(H_i(x)) \!+\! \delta ^{{F_{1}(x;t)}}_i\!=0, \forall i \in I^-. \end{aligned} \end{aligned}$$

So, it follows for $i \in I^-$ that

$$\begin{aligned} \delta ^{{F_{2}(x;t)}}_i=\delta ^{{F_{1}(x;t)}}_i {t_2}\theta '_{t_1}(G_i(x)) \text{ and } \delta ^{{F_{1}(x;t)}}_i=\delta ^{{F_{2}(x;t)}}_i {t_2}\theta '_{t_1}(H_i(x)). \end{aligned}$$

So $\delta ^{{F_{1}(x;t)}}_i=\delta ^{{F_{2}(x;t)}}_i=0$, since $i \in \mathcal {I}_{GH}^{00}(x;t)$ gives

$${t_2}\theta '_{t_1}(G_i(x)){t_2}\theta '_{t_1}(H_i(x))={t_2}\theta '_{t_1}(0){t_2}\theta '_{t_1}(0)<1$$

by properties of $\theta$ and (3.8). Similarly, we get $\mu ^{{F_{1}(x;t)}}_i=\mu ^{{F_{2}(x;t)}}_i=\nu ^{{F_{2}(x;t)}}_i=\nu ^{{F_{1}(x;t)}}_i=0$. $\square$