Local Convergence Analysis of a Primal–Dual Method for Bound-Constrained Optimization Without SOSC

Abstract

We propose a local convergence analysis of a primal–dual interior point algorithm for the solution of a bound-constrained optimization problem. The algorithm includes a regularization technique to prevent singularity of the matrix of the linear system at each iteration, when the second-order sufficient conditions do not hold at the solution. These conditions are replaced by a milder assumption related to a local error-bound condition. This new condition is a generalization of the one used in unconstrained optimization. We show that by an appropriate updating strategy of the barrier parameter and of the regularization parameter, the proposed algorithm owns a superlinear rate of convergence. The analysis is made thanks to a boundedness property of the inverse of the Jacobian matrix arising in interior point algorithms. An illustrative example is given to show the behavior and the gain obtained by this regularization strategy.

A Appendix: Properties of the Regularized Jacobian Matrix

The matrix of the linear system (3) is of the form

$$\begin{aligned} J_\theta (w)= \begin{pmatrix} H(w) + \theta I &{} -I \\ Z &{} X\end{pmatrix}, \end{aligned}$$

(42)

where $w=(x,z)\in {\mathbb {R}}^{2n}$, $H(w)\in {\mathbb {R}}^{n\times n}$ is symmetric, and $\theta \ge 0$ is a regularization parameter. Two properties of this matrix are stated here. The statements of the following lemmas can be understood in a general framework of linear algebra. They are stated without proof. Detailed proofs can be found in [22, Lemma 5.8 and Lemma 5.9].

The first lemma is related to an upper bound of the inverse of $J_\theta (w)$, for points on the boundary of the nonnegative orthant. This lemma is used in the proof of Lemma 5.1.

Lemma A.1

Let us define ${{{\mathcal {C}}}}=\{w=(x,z)\in {\mathbb {R}}^{2n}: 0\le x\perp z\ge 0\}$. Let $w^*\in {{{\mathcal {C}}}}$ be such that

$$\begin{aligned} a:=\min \{x_i^*+z_i^*:i=1,\ldots ,n\}>0. \end{aligned}$$

Let ${\mathcal {A}}=\{i: x^*_i = 0\}$ be the active set at $x^*$ related to the bounds $x\ge 0$. Let $H: {\mathbb {R}}^{2n}\rightarrow {\mathbb {R}}^{n\times n}$ be a continuous function such that $H(w)=H(w)^\top$. Assume that, for all $u\in {\mathbb {R}}^n$ such that $u_i=0$ for all $i\in {\mathcal {A}}$, $u^\top H(w^*) u\ge 0$. Then, for all $\varepsilon \in ]0,a[$, there exists $C>0$ such that for all $w\in {B}(w^*,\varepsilon )\cap {{{\mathcal {C}}}}$ and $\theta >0$, the matrix $J_\theta (w)$ is nonsingular and

$$\begin{aligned} \Vert J_\theta (w)^{-1}\Vert \le C \max \{\theta ,1/\theta \} \end{aligned}$$

The second lemma also gives an upper bound of the inverse of $J_\theta (w)$, but for points that are located in the interior of the positive orthant. This property is used in the proof of Lemma 5.2.

Lemma A.2

Let $w^*=(x^*,z^*)\in {\mathbb {R}}^{2n}$ be such that

$$\begin{aligned} 0\le x^*\perp z^*\ge 0\quad \text{ and }\quad a:=\min \{x_i^*+z_i^*|i=1,\ldots n\}>0. \end{aligned}$$

Let $H: {\mathbb {R}}^{2n}\rightarrow {\mathbb {R}}^{n\times n}$ be a continuous function such that for all $w\in {\mathbb {R}}^{2n}$, $H(w)=H(w)^\top$. Let $\theta :{\mathbb {R}}^{2n} \rightarrow {\mathbb {R}}_{++}$ be a continuous function such that $\theta (w)\ge \gamma \Vert x\circ z\Vert ^t$, for all $w\in {\mathbb {R}}^{2n}$, for some constants $\gamma >0$ and $t\in ]0,1[$. Assume that for all $w\in {\mathbb {R}}^{2n}_{++}$, $H(w)+X^{-1}Z\succeq 0$. For all $w\in {\mathbb {R}}^{2n}_{++}$, the matrix $J_{\theta (w)}(w)$ is nonsingular and for all $\varepsilon \in ]0,a[$, there exists $C>0$ such that for all $w\in B(w^*,\varepsilon )\cap {\mathbb {R}}^{2n}_{++}$,

$$\begin{aligned} \Vert J_{\theta (w)}(w)^{-1}\Vert \le \dfrac{C}{\theta (w)}. \end{aligned}$$