QPALM: a proximal augmented lagrangian method for nonconvex quadratic programs

Abstract

We propose QPALM, a nonconvex quadratic programming (QP) solver based on the proximal augmented Lagrangian method. This method solves a sequence of inner subproblems which can be enforced to be strongly convex and which therefore admit a unique solution. The resulting steps are shown to be equivalent to inexact proximal point iterations on the extended-real-valued cost function, which allows for a fairly simple analysis where convergence to a stationary point at an $R$-linear rate is shown. The QPALM algorithm solves the subproblems iteratively using semismooth Newton directions and an exact linesearch. The former can be computed efficiently in most iterations by making use of suitable factorization update routines, while the latter requires the zero of a monotone, one-dimensional, piecewise affine function. QPALM is implemented in open-source C code, with tailored linear algebra routines for the factorization in a self-written package LADEL. The resulting implementation is shown to be extremely robust in numerical simulations, solving all of the Maros-Meszaros problems and finding a stationary point for most of the nonconvex QPs in the Cutest test set. Furthermore, it is shown to be competitive against state-of-the-art convex QP solvers in typical QPs arising from application domains such as portfolio optimization and model predictive control. As such, QPALM strikes a unique balance between solving both easy and hard problems efficiently.

Proof of Theorem 2.2

Proof of Theorem 2.2 (Inexact nonconvex PP [53, §4.1]) The proximal inequality

$$\begin{aligned} \varphi (x^{k+1}) {}+{} \tfrac{1}{2}\Vert x^{k+1}-x^k-e^k\Vert _{\varSigma _\textsf { x}^{-1}}^2 {}\le {} \varphi (x^k) {}+{} \tfrac{1}{2}\Vert e^k\Vert _{\varSigma _\textsf { x}^{-1}}^2, \end{aligned}$$

cf. (1.3), yields

$$\begin{aligned} \varphi (x^{k+1})+ & {} \tfrac{1}{4}\Vert x^{k+1}-x^k\Vert _{\varSigma _\textsf { x}^{-1}}^2 \le \varphi (x^{k+1}) {}+{} \tfrac{1}{2}\Vert x^{k+1}-x^k-e^k\Vert _{\varSigma _\textsf { x}^{-1}}^2\nonumber \\&+\tfrac{1}{2}\Vert e^k\Vert _{\varSigma _\textsf { x}^{-1}}^2 {}\le {} \varphi (x^k) {}+{} \Vert e^k\Vert _{\varSigma _\textsf { x}^{-1}}^2, \end{aligned}$$

(A.1)

proving assertions 2.2(ii) and 2.2(iv), and similarly 2.2(i) follows by invoking [47, Lem. 2.2.2]. Next, let $(x^k)_{k\in K}$ be a subsequence converging to a point $x^\star$; then, it also holds that $(x^{k+1})_{k\in K}$ converges to $x^\star$ owing to assertion 2.2(ii). From the proximal inequality (1.3) we have

$$\begin{aligned} \varphi (x^{k+1}) {}+{} \tfrac{1}{2} \Vert x^{k+1}-x^k-e^k\Vert _{\varSigma _\textsf { x}^{-1}}^2 {}\le {} \varphi (x^\star ) {}+{} \tfrac{1}{2} \Vert x^\star -x^k-e^k\Vert _{\varSigma _\textsf { x}^{-1}}^2, \end{aligned}$$

so that passing to the limit for $K\ni k\rightarrow \infty$ we obtain that $\limsup _{k\in K}\varphi (x^{k+1})\le \varphi (x^\star )$. In fact, equality holds since $\varphi$ is lsc, hence from assertion 2.2(i) we conclude that $\varphi (x^{k+1})\rightarrow \varphi (x^\star )$ as $k\rightarrow \infty$, and in turn from the arbitrarity of $x^\star$ it follows that $\varphi$ is constantly equal to this limit on the whole set of cluster points. To conclude the proof of assertion 2.2(iii), observe that the inclusion $\varSigma _\textsf { x}^{-1}(x^k+e^k-x^{k+1}) {}\in {} {\hat{\partial \varphi }}(x^{k+1})$, cf. (1.4), implies that

$$\begin{aligned} {{\,\mathrm{dist}\,}}\bigl (0,\partial \varphi (x^{k+1})\bigr ) {}\le {} {{\,\mathrm{dist}\,}}\bigl (0,{\hat{\partial \varphi }}(x^{k+1})\bigr ) {}\le {} \Vert \varSigma _\textsf { x}^{-1}\Vert \left( \Vert x^k-x^{k+1}\Vert {}+{} \Vert e^k\Vert \right) ,\nonumber \\ \end{aligned}$$

(A.2)

and with limiting arguments (recall that $\lim _{k\in K}\varphi (x^k)=\varphi (\lim _{k\in K}x^k)$) the claimed stationarity of the cluster points is obtained.