Local convergence of the method of multipliers for variational and optimization problems under the noncriticality assumption

Abstract

We present a local convergence analysis of the method of multipliers for equality-constrained variational problems (in the special case of optimization, also called the augmented Lagrangian method) under the sole assumption that the dual starting point is close to a noncritical Lagrange multiplier (which is weaker than second-order sufficiency). Local $Q$-superlinear convergence is established under the appropriate control of the penalty parameter values. For optimization problems, we demonstrate in addition local $Q$-linear convergence for sufficiently large fixed penalty parameters. Both exact and inexact versions of the method are considered. Contributions with respect to previous state-of-the-art analyses for equality-constrained problems consist in the extension to the variational setting, in using the weaker noncriticality assumption instead of the usual second-order sufficient optimality condition (SOSC), and in relaxing the smoothness requirements on the problem data. In the context of optimization problems, this gives the first local convergence results for the augmented Lagrangian method under the assumptions that do not include any constraint qualifications and are weaker than the SOSC. We also show that the analysis under the noncriticality assumption cannot be extended to the case with inequality constraints, unless the strict complementarity condition is added (this, however, still gives a new result).

Appendix

This appendix contains lemmas concerning nonsingularity of matrices of certain structure, used in the analysis above. The first one is a refined version of [23, Lemma 1].

Lemma 3

Let $H$ be an $n\times n$-matrix, $B$ be an $l\times n$-matrix, and assume that

$$\begin{aligned} H\xi \not \in \mathrm{im}\, B^{\mathrm{T}}\quad \forall \,\xi \in \ker B\setminus \{0\}. \end{aligned}$$

(64)

Then for any $M > 0$ there exists $\gamma >0$ such that

$$\begin{aligned} \left\| \left( \tilde{H} + t(B+\varOmega )^{\mathrm{T}}\tilde{B}\right) \xi \right\| \ge \gamma \Vert \xi \Vert \quad \forall \,\xi \in \mathrm{I\! \mathrm R}^n\end{aligned}$$

for every $n\times n$-matrix $\tilde{H}$ close enough to $H$, every $l\times n$-matrix $\tilde{B}$ close enough to $B$, every $t\in \mathrm{I\! \mathrm R}$ such that $|t|$ is sufficiently large, and for every $l\times n$-matrix $\varOmega$ satisfying $\Vert \varOmega \Vert \le M/|t|$.

Proof

Suppose the contrary, i.e., that for some $M>0$ there exist sequences $\{H_k\}$ of $n\times n$-matrices, $\{B_k\}$ and $\{\varOmega _k\}$ of $l\times n$-matrices, $\{t_k\}\subset \mathrm{I\! \mathrm R}$, and $\{\xi ^k\}\subset \mathrm{I\! \mathrm R}^n\setminus \{0\}$, such that $\{H_k\}\rightarrow H, \{B_k\}\rightarrow B, |t_k|\rightarrow \infty , \Vert \varOmega _k\Vert \le M/|t_k|$ for all $k$, and

$$\begin{aligned} H_k\xi ^k + t_k(B+\varOmega _k)^{\mathrm{T}}B_k\xi ^k=o(\Vert \xi ^k\Vert ) \end{aligned}$$

(65)

as $k\rightarrow \infty$. Without loss of generality we may assume that $\Vert \xi ^k\Vert =1$ for all $k$ and that $\{\xi ^k\}\rightarrow \xi \ne 0$. Then (65) means the existence of a sequence $\{ w^k\} \subset \mathrm{I\! \mathrm R}^n$ such that $\{ w_k\} \rightarrow 0$ and

$$\begin{aligned} H_k\xi ^k + t_k(B+\varOmega _k)^{\mathrm{T}}B_k\xi ^k =w_k \end{aligned}$$

(66)

for all $k$. Therefore, it must hold that $B^{\mathrm{T}}B\xi =0$, since

$$\begin{aligned} B^{\mathrm{T}}B_k\xi ^k = -\frac{1}{t_k}H_k\xi ^k - \varOmega _k^{\mathrm{T}}B_k\xi ^k +\frac{1}{t_k} w^k \end{aligned}$$

tends to $0$ as $k\rightarrow \infty$. Consequently, $\xi \in \ker B$.

On the other hand, (66) implies that

$$\begin{aligned} H_k\xi ^k + t_k\varOmega _k^{\mathrm{T}}B_k\xi ^k - w^k = -t_k B^{\mathrm{T}}B_k\xi ^k \in \mathop {\hbox {im}}B^{\mathrm{T}}\end{aligned}$$

for all $k$, where the second term in the left-hand side tends to zero as $k\rightarrow \infty$ because $\{t_k\varOmega _k\}$ is bounded and $\{B_k\xi ^k\}\rightarrow B\xi =0$. Hence, $H\xi \in \mathop {\hbox {im}}B^{\mathrm{T}}$ by the closedness of $\mathop {\hbox {im}}B^{\mathrm{T}}$. This completes a contradiction with (64). $\square$

Lemma 4

Under the assumptions of Lemma 3, for any $M>0$ and any $\varepsilon >0$ it holds that for every $n\times n$-matrix $\tilde{H}$ close enough to $H$, every $l\times n$-matrix $\tilde{B}$ close enough to $B$, every real $t$ such that $|t|$ is sufficiently large, and for all $l\times n$-matrices $\varOmega$ satisfying $\Vert \varOmega \Vert \le M/|t|$, the matrix $\tilde{H} + t(B+\varOmega )^{\mathrm{T}}\tilde{B}$ is nonsingular and

$$\begin{aligned} \left\| \left( \tilde{H} + t(B+\varOmega )^{\mathrm{T}}\tilde{B}\right) ^{-1} (B+\varOmega )^{\mathrm{T}}\right\| \le \varepsilon . \end{aligned}$$

(67)

Proof

Fix arbitrary $M>0$ and $\varepsilon >0$. The assertion regarding nonsingularity of the matrix $\tilde{H} + t(B+\varOmega )^{\mathrm{T}}\tilde{B}$ follows directly from Lemma 3. Therefore, we only have to prove that (possibly by making $\tilde{H}$ closer to $H, \tilde{B}$ closer to $B$, and $|t|$ larger) one can additionally ensure (67).

By contradiction, suppose first that there exist sequences $\{H_k\}$ of $n\times n$-matrices, $\{B_k\}$ and $\{\varOmega _k\}$ of $l\times n$-matrices, $\{t_k\}$ of reals, and $\{\eta ^k\}\subset \mathrm{I\! \mathrm R}^n$, such that $\{H_k\}\rightarrow H, \{B_k\}\rightarrow B, |t_k|\rightarrow \infty , \Vert \varOmega _k\Vert \le M/|t_k|, \Vert \eta ^k\Vert =1$ and $\det (H_k + t_k(B+\varOmega _k)^{\mathrm{T}}B_k)\ne 0$ for all $k$, and for

$$\begin{aligned} \xi ^k=(H_k + t_k(B+\varOmega _k)^{\mathrm{T}}B_k)^{-1}(B+\varOmega _k)^{\mathrm{T}}\eta ^k \end{aligned}$$

(68)

it holds that

$$\begin{aligned} \Vert \xi ^k\Vert > \varepsilon \end{aligned}$$

(69)

for all $k$. By (68) we have that

$$\begin{aligned} (B+\varOmega _k)^{\mathrm{T}}\eta ^k = H_k\xi ^k + t_k(B+\varOmega _k)^{\mathrm{T}}B_k\xi ^k. \end{aligned}$$

(70)

Due to (69), the sequence $\{\eta ^k/\Vert \xi ^k\Vert \}$ is bounded. Without loss of generality we may assume that the sequence $\{\xi ^k/\Vert \xi ^k\Vert \}$ converges to some $\xi \in \mathrm{I\! \mathrm R}^n$ such that $\Vert \xi \Vert =1$. Then dividing both sides of (70) by $t_k\Vert \xi ^k\Vert$ and passing onto the limit as $k\rightarrow \infty$, we obtain that $B^{\mathrm{T}}B\xi =0$, and hence, $\xi \in \ker B$.

Furthermore, by (70), it holds that

$$\begin{aligned} H_k\frac{\xi ^k}{\Vert \xi ^k\Vert } -\varOmega _k^{\mathrm{T}}\frac{\eta ^k}{\Vert \xi ^k\Vert }+t_k\varOmega _k^{\mathrm{T}}B_k\frac{\xi ^k}{\Vert \xi ^k\Vert } = \frac{1}{\Vert \xi ^k\Vert }B^{\mathrm{T}}(\eta ^k-t_k B_k\xi ^k)\in \mathop {\hbox {im}}B^{\mathrm{T}}\end{aligned}$$

for all $k$. The second term in the left-hand side tends to zero because $\{\Vert \varOmega _k\Vert \}\rightarrow 0$ while the sequence $\{\eta ^k/\Vert \xi ^k\Vert \}$ is bounded. Moreover, the third term in the left-hand side tends to zero as well, because $\{t_k\varOmega _k\}$ is bounded while $\{B_k\xi ^k/\Vert \xi ^k\Vert \}\rightarrow B\xi =0$. Therefore, by closedness of $\mathop {\hbox {im}}B^{\mathrm{T}}$, it follows that $H\xi \in \mathop {\hbox {im}}B^{\mathrm{T}}$, which contradicts (64). $\square$

Lemma 5

In addition to the assumptions of Lemma 3, let $H$ be symmetric.

Then for any $M>0$ and any $\varepsilon >0$ it holds that for every symmetric $n\times n$-matrix $\tilde{H}$ close enough to $H$, every real $t$ such that $|t|$ is sufficiently large, and for all $l\times n$-matrices $\varOmega$ satisfying $\Vert \varOmega \Vert \le M/|t|$, the matrix $\tilde{H} + t(B+\varOmega )^{\mathrm{T}}(B+\varOmega )$ is nonsingular and the following estimate is valid

$$\begin{aligned} \left\| t(B+\varOmega )\left( \tilde{H} + t(B+\varOmega )^{\mathrm{T}}(B+\varOmega )\right) ^{-1}(B+\varOmega )^{\mathrm{T}}\right\| \le 1 + \varepsilon . \end{aligned}$$

(71)

Proof

Again, nonsingularity of $\tilde{H} + t(B+\varOmega )^{\mathrm{T}}(B+\varOmega )$ is given by Lemma 3. If at the same time the estimate (71) does not hold, there must exist sequences $\{H_k\}$ of symmetric $n\times n$-matrices, $\{\varOmega _k\}$ of $l\times n$-matrices, $\{t_k\}$ of reals, and $\{\eta ^k\}\subset \mathrm{I\! \mathrm R}^n$, such that $\{H_k\}\rightarrow H, |t_k|\rightarrow \infty$, and for all $k$ it holds that $\Vert \varOmega _k\Vert \le M/|t_k|, \Vert \eta ^k\Vert =1, \det (H_k + t_k(B+\varOmega _k)^{\mathrm{T}}(B+\varOmega _k))\ne 0$, and

$$\begin{aligned} \left\| t_k(B+\varOmega _k)\left( H_k + t_k(B+\varOmega _k)^{\mathrm{T}}(B+\varOmega _k)\right) ^{-1}(B+\varOmega _k)^{\mathrm{T}}\eta ^k\right\| > 1 + \varepsilon . \end{aligned}$$

(72)

For each $k$ set

$$\begin{aligned} W_k&= (B+\varOmega _k)\left( H_k + t_k(B+\varOmega _k)^{\mathrm{T}}(B+\varOmega _k)\right) ^{-1}\nonumber \\&= \left( \left( H_k + t_k(B+\varOmega _k)^{\mathrm{T}}(B+\varOmega _k)\right) ^{-1}(B+\varOmega _k)^{\mathrm{T}}\right) ^{\mathrm{T}}, \end{aligned}$$

(73)

where the symmetry of $H_k$ was taken into account. Due to Lemma 4 we have that $\{ W_k\} \rightarrow 0$.

Furthermore, for each $k$ the vector $\eta ^k$ can be decomposed into the sum

$$\begin{aligned} \eta ^k = \eta ^k_1 + \eta ^k_2, \end{aligned}$$

where $\eta ^k_1\in \ker B^{\mathrm{T}}=(\mathop {\hbox {im}}B)^\bot$ and $\eta ^k_2\in \mathop {\hbox {im}}B$. Observe that $t_kW_k(B+\varOmega _k)^{\mathrm{T}}\eta ^k_1=W_k(t_k\varOmega _k^{\mathrm{T}})\times \eta ^k_1$, and since the sequences $\{\eta ^k_1\}$ and $\{t_k\varOmega _k\}$ are bounded, and $\{ W_k\} \rightarrow 0$, we conclude that $\{ t_kW_k(B+\varOmega _k)^{\mathrm{T}}\eta ^k_1\} \rightarrow 0$. On the other hand, as $\eta ^k_2\in \mathop {\hbox {im}}B$, there exists $\xi ^k_2\in \mathrm{I\! \mathrm R}^n$ such that $B\xi ^k_2=\eta ^k_2$ and the sequence $\{\xi ^k_2\}$ is bounded. Therefore, employing (73),

$$\begin{aligned} \left\| t_kW_k(B+\varOmega _k)^{\mathrm{T}}\eta ^k_2\right\|&= \left\| W_k(t_k(B+\varOmega _k)^{\mathrm{T}}) B\xi ^k_2\right\| \\&\le \left\| W_k(H_k + t_k(B+\varOmega _k)^{\mathrm{T}}(B+\varOmega _k))\xi ^k_2\right\| \\&+ \left\| W_k(H_k +t_k(B+\varOmega _k)^{\mathrm{T}}\varOmega _k)\xi ^k_2\right\| \\&= \left\| (B+\varOmega _k)\xi ^k_2\right\| + \left\| W_k(H_k +t_k(B+\varOmega _k)^{\mathrm{T}}\varOmega _k)\xi ^k_2\right\| \\&\le \Vert \eta ^k_2\Vert +\Vert \varOmega _k\xi ^k_2\Vert + \left\| W_k(H_k +t_k(B+\varOmega _k)^{\mathrm{T}}\varOmega _k)\xi ^k_2\right\| \\&\le 1 +\Vert \varOmega _k\xi ^k_2\Vert + \left\| W_k(H_k +t_k(B+\varOmega _k)^{\mathrm{T}}\varOmega _k)\xi ^k_2\right\| . \end{aligned}$$

The last two terms in the right-hand side tend to zero because the sequences $\{\xi ^k_2\}$ and $\{H_k +t_k(B+\varOmega _k)^{\mathrm{T}}\varOmega _k\}$ are bounded, while $\{\varOmega _k\} \rightarrow 0$ and $\{ W_k\} \rightarrow 0$. Therefore,

$$\begin{aligned} \limsup _{k\rightarrow \infty }\left\| t_kW_k(B+\varOmega _k)^{\mathrm{T}}\eta ^k\right\|&\le \lim _{k\rightarrow \infty }\left\| t_kW_k(B+\varOmega _k)^{\mathrm{T}}\eta _1^k\right\| \\&+\,\, \limsup _{k\rightarrow \infty }\left\| t_kW_k(B+\varOmega _k)^{\mathrm{T}}\eta _2^k\right\| \\&\le 1, \end{aligned}$$

which contradicts (72). $\square$