1 Erratum to: Optim Lett (2011) 5:549–556 DOI 10.1007/s11590-011-0323-1

The proof of Theorem 3 in the original publication of the article contains an incorrect statement that we fix below.

Theorem 3

Let \(\mathbf {K}\) in (1.2) be compact and let Assumption 1 hold true. For every fixed \(\mu >0,\) choose \(\mathbf {x}_\mu \in \mathbf {K}\) to be an arbitrary stationary point of \(\phi _\mu \) in \(\mathbf {K}\).

Then every accumulation point \(\mathbf {x}^*\in \mathbf {K}\) of such a sequence \((\mathbf {x}_\mu )\subset \mathbf {K}\) with \(\mu \rightarrow 0,\) is a global minimizer of \(f\) on \(\mathbf {K},\) and if \(\nabla f(\mathbf {x}^*)\ne 0,\) \(\mathbf {x}^*\) is a KKT point of \(\mathbf {P}.\)

Proof

Let \(\mathbf {x}_\mu \in \mathbf {K}\) be a stationary point of \(\phi _\mu \), which by Lemma 2 is guaranteed to exist. So

$$\begin{aligned} \nabla \phi _\mu (\mathbf {x}_\mu )=\nabla f(\mathbf {x}_\mu )-\sum _{j=1}^m\frac{\mu }{g_j(\mathbf {x}_\mu )} \nabla g_j(\mathbf {x}_\mu )=0. \end{aligned}$$
(0.1)

As \(\mu \rightarrow 0\) and \(\mathbf {K}\) is compact, there exists \(\mathbf {x}^*\in \mathbf {K}\) and a subsequence \((\mu _\ell )\subset \mathbb {R}_+\) such that \(\mathbf {x}_{\mu _\ell }\rightarrow \mathbf {x}^*\) as \(\ell \rightarrow \infty \). We need consider two cases:

Case when \(g_j(\mathbf {x}^*)>0,\,\forall j=1,\ldots ,m\). Then as \(f\) and \(g_j\) are continuously differentiable, \(j=1,\ldots ,m\), taking limit in (0.1) for the subsequence \((\mu _\ell )\), yields \(\nabla f(\mathbf {x}^*)=0\) which, as \(f\) is convex, implies that \(\mathbf {x}^*\) is a global minimizer of \(f\) on \(\mathbb {R}^n\), hence on \(\mathbf {K}\).

Case when \(g_j(\mathbf {x}^*)=0\) for some \(j\in \{1,\ldots ,m\}\). Let \(J:=\{j\,:\,g_j(\mathbf {x}^*)=0\}\ne \emptyset \). We next show that for every \(j\in J\), the sequence of ratios \((\mu _\ell /g_j(\mathbf {x}_{\mu _\ell })\), \(\ell =1,\ldots \), is bounded. Indeed let \(j\in J\) be fixed arbitrary. As Slater’s condition holds, let \(\mathbf {x}_0\in \mathbf {K}\) be such that \(g_j(\mathbf {x}_0)>0\) for all \(j=1,\ldots ,m\); then \(\langle \nabla g_j(\mathbf {x}^*),\mathbf {x}_0-\mathbf {x}^*\rangle >0\). Indeed, as \(\mathbf {K}\) is convex, \(\langle \nabla g_j(\mathbf {x}^*),\mathbf {x}_0+\mathbf {v}-\mathbf {x}^*\rangle \ge 0\) for all \(\mathbf {v}\) in some small enough ball \(\mathbf {B}(0,\rho )\) around the origin. So if \(\langle \nabla g_j(\mathbf {x}^*),\mathbf {x}_0-\mathbf {x}^*\rangle =0\) then \(\langle \nabla g_j(\mathbf {x}^*),\mathbf {v}\rangle \ge 0\) for all \(\mathbf {v}\in \mathbf {B}(0,\rho )\), in contradiction with \(\nabla g_j(\mathbf {x}^*)\ne 0\). Next,

$$\begin{aligned} \langle \nabla f(\mathbf {x}_{\mu _\ell }),\mathbf {x}_0-\mathbf {x}^*\rangle&= \underbrace{\sum _{k\not \in J}^m\frac{\mu _\ell }{g_k(\mathbf {x}_{\mu _\ell })} \langle \nabla g_k(\mathbf {x}_{\mu _\ell }),\mathbf {x}_0-\mathbf {x}^*\rangle }_{A_\ell }\nonumber \\&\quad +\underbrace{\sum _{k\in J}^m\frac{\mu _\ell }{g_k(\mathbf {x}_{\mu _\ell })} \langle \nabla g_k(\mathbf {x}_{\mu _\ell }),\mathbf {x}_0-\mathbf {x}^*\rangle }_{B_\ell } \end{aligned}$$
(0.2)

Observe that

  • Every term of the sum \(B_\ell \) is nonnegative for sufficiently large \(\ell \), say \(\ell \ge \ell _0\), because \(\mathbf {x}_{\mu _\ell }\rightarrow \mathbf {x}^*\) and \(\langle \nabla g_k(\mathbf {x}^*),\mathbf {x}_0-\mathbf {x}^*\rangle >0\) for all \(k\in J\).

  • \(A_\ell \rightarrow 0\) as \(\ell \rightarrow \infty \) because \(\mu _\ell \rightarrow 0\) and \(g_k(\mathbf {x}_{\mu _\ell })\rightarrow g_k(\mathbf {x}^*)>0\) for all \(k\not \in J\).

Therefore \(\vert A_\ell \vert \le A\) for all sufficiently large \(\ell \), say \(\ell \ge \ell _1\), and so for every \(j\in J\):

$$\begin{aligned} \langle \nabla f(\mathbf {x}_{\mu _\ell }),\mathbf {x}_0-\mathbf {x}^*\rangle +A \ge \frac{\mu _\ell }{g_j(\mathbf {x}_{\mu _\ell })}\langle \nabla g_j(\mathbf {x}_{\mu _\ell }),\mathbf {x}_0-\mathbf {x}^*\rangle ,\quad \ell \ge \ell _2:=\max [\ell _0,\ell _1], \end{aligned}$$

which shows that for every \(j\in J\), the nonnegative sequence \((\mu _\ell /g_j(\mathbf {x}_{\mu _\ell }))\), \(\ell \ge \ell _2\), is bounded from above.

So take a subsequence (still denoted \((\mu _\ell )\), \(\ell \in \mathbb {N}\), for convenience) such that the ratios \(\mu _\ell /g_j(\mathbf {x}_{\mu _\ell })\) converge for all \(j\in J\), that is,

$$\begin{aligned} \lim _{\ell \rightarrow \infty }\,\frac{\mu _\ell }{g_j(\mathbf {x}_{\mu _\ell })}= \lambda _j\ge 0,\quad \forall \,j\in J, \end{aligned}$$

and let \(\lambda _j:=0\) for every \(j\not \in J\), so that \(\lambda _jg_j(\mathbf {x}^*)=0\) for every \(j=1,\ldots ,m\). Taking limit in (0.1) as \(\ell \rightarrow \infty \), yields:

$$\begin{aligned} \nabla f(\mathbf {x}^*)=\sum _{j=1}^m\lambda _j\,\nabla g_j(\mathbf {x}^*), \end{aligned}$$
(0.3)

which shows that \((\mathbf {x}^*,\lambda )\in \mathbf {K}\times \mathbb {R}^m_+\) is a KKT point for \(\mathbf {P}\). Finally, invoking Theorem 1, \(\mathbf {x}^*\) is also a global minimizer of \(\mathbf {P}\). \(\square \)