Correction to: J Optim Theory Appl https://doi.org/10.1023/A:1014853129484

In this article the below author names are included.

Rimpi and C.S. Lalitha

1 Introduction

We correct an error in the proof of Theorem 3.5 in [1].

2 Mathematical Details

Theorem 3.5 in [1] says, if \(\bar{x}\in S\) is a local minimum of (P) and f is locally Lipschitz which admits an USRC \(\partial ^*f(\bar{x})\) at \(\bar{x}\), then

$$\begin{aligned} 0\in {{\,\textrm{cl}\,}}(\overline{{{\,\textrm{co}\,}}}(\partial ^*f(\bar{x}))+T^\circ (S,\bar{x})). \end{aligned}$$
(1)

The following example justifies that the above theorem fails to hold.

Example 2.1

Let \(f:\mathbb {R}^2\rightarrow \mathbb {R}\) and the feasible set S be defined as \(f(x_1,x_2)=-x_1+|x_2|\) and \(S=\{(x_1,x_2)\in \mathbb {R}^2:\ 0\le x_1\le |x_2|\}.\) Clearly, \(\bar{x}=(0,0)\) is a global minimum, \(T(S,\bar{x})=S\) and \({{\,\textrm{co}\,}}(T(S,\bar{x}))={{\,\textrm{co}\,}}(S)=\{(x_1,x_2)\in \mathbb {R}^2: x_1\ge 0\}.\) Now, for any \(v=(v_1,v_2)\in \mathbb {R}^2\), \(f^{+}_d(\bar{x},v)=-v_1+|v_2|.\) Also it can be seen that \(\partial ^*f(\bar{x})=\{(-1,1),(-1,-1)\}\) is an upper semi-regular convexificator of f at \(\bar{x}\). As \(T(S,\bar{x})=S\), it follows that \( f^{+}_d(\bar{x},v)\ge 0,\) for all \(v\in T(S,\bar{x}).\) Thus \(\sup \limits _{\zeta \in \partial ^*f(\bar{x})}\langle \zeta , v\rangle \ge 0,\) for all \(v\in T(S,\bar{x}).\) Clearly, \(\sup \limits _{\zeta \in \partial ^*f(\bar{x})}\langle \zeta , \overline{v}\rangle =f^{+}_d(\bar{x},\overline{v})=-1<0\) for \(\overline{v}=(2,1)\in {{\,\textrm{co}\,}}(T(S,\bar{x}))\). Moreover, \(T^\circ (S,\bar{x})=\{(x,0)\in \mathbb {R}^2:x\le 0\}\) and \(\overline{{{\,\textrm{co}\,}}}(\partial ^*f(\bar{x}))=\{(-1,t)\in \mathbb {R}^2:-1\le t\le 1\}\). Hence \(0\notin {{\,\textrm{cl}\,}}(\overline{{{\,\textrm{co}\,}}}(\partial ^*f(\bar{x}))+ T^\circ (S,\bar{x})).\) Thus Theorem 3.5 in [1] fails to hold for f at \(\bar{x}\).

We rectify the error in the above theorem by assuming the tangent cone to be convex. For this we first recall the notion of support functions from [2]. The support function \(\sigma _A(x):\mathbb {R}^n\rightarrow \mathbb {R}\cup \{+\infty \}\) of a nonempty set \(A\subseteq \mathbb {R}^n\) is defined as \( \sigma _A(x):=\sup \limits _{a\in A}\langle x,a\rangle .\)

A correct statement of Theorem 3.5 in [1] should be as follows.

Theorem 2.1

If \(\bar{x}\in S\) is a local minimum of (P), \(T(S,\bar{x})\) is a convex cone and f is locally Lipschitz which admits an USRC \(\partial ^*f(\bar{x})\) at \(\bar{x}\), then (1) holds. Further, if \(\partial ^*f(\bar{x})\) is bounded, then

$$\begin{aligned} 0\in {{\,\textrm{co}\,}}(\partial ^*f(\bar{x}))+T^\circ (S,\bar{x}). \end{aligned}$$
(2)

Proof

As \(\bar{x}\) is a local minimum of (P), there exists \(\epsilon >0\) such that \(f(\bar{x})\le f(x)\) for all \(x\in B(\bar{x},\epsilon )\cap S\). For \(v\in T(S,\bar{x})\), there exist sequences \((t_k)_{k\in \mathbb {N}}\) and \((v_k)_{k\in \mathbb {N}}\) with \(t_k\downarrow 0\) and \(v_k\rightarrow v\) such that \(\bar{x}+t_kv_k\in S\). Thus there exists \(k_0\in \mathbb {N}\) such that \(\bar{x}+t_kv_k\in B(\bar{x},\epsilon )\cap S\) for all \(k\ge k_0\), which implies \(f(\bar{x})\le f(\bar{x}+t_kv_k)\) for all \(k\ge k_0\). As f is locally Lipschitz with Lipschitz constant say, L, hence for every \(v\in T(S,\bar{x})\) we have

$$\begin{aligned} f^+_d(\bar{x},v)&=\limsup \limits _{t\downarrow 0}\dfrac{f(\bar{x}+tv)-f(\bar{x})}{t}\\&\ge \limsup \limits _{k\rightarrow \infty } \left[ \dfrac{f(\bar{x}+t_kv)-f(\bar{x}+t_kv_k)}{t_k}+\dfrac{f(\bar{x}+t_kv_k)-f(\bar{x})}{t_k}\right] \\&\ge \lim \limits _{k\rightarrow \infty }\left[ -L\Vert v_k-v\Vert \right] +\limsup \limits _{k\rightarrow \infty }\left[ \dfrac{f(\bar{x}+t_kv_k)-f(\bar{x})}{t_k}\right] \ge 0. \end{aligned}$$

As \(\partial ^*f(\bar{x})\) is an USRC of f at \(\bar{x}\), it follows from [2, Proposition 2.2.1 (p. 211)] that

$$\begin{aligned} \sigma _{{{\,\textrm{co}\,}}(\partial ^*f(\bar{x}))}(v)=\sigma _{\partial ^*f(\bar{x})}(v) =\sup _{\zeta \in \partial ^*f(\bar{x})}\langle \zeta ,v\rangle \ge 0,\ \text {for all}\ v\in T(S,\bar{x}). \end{aligned}$$
(3)

As \(T(S,\bar{x})\) is convex, hence by applying [2, Example 2.3.1 (p. 215)] for \(K=T^\circ (S,\bar{x})\) we deduce that

$$\begin{aligned} \sigma _{T^\circ (S,\bar{x})}(v)={\left\{ \begin{array}{ll} 0,\quad &{}\text {if}\ v\in T(S,\bar{x}),\\ +\infty ,\quad &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$
(4)

In view of [2, Theorem 3.3.3(i) (p. 226)] the support function of the set \(U={{\,\textrm{cl}\,}}({{\,\textrm{co}\,}}(\partial ^*f(\bar{x}))+T^\circ (S,\bar{x}))\) is

$$\begin{aligned} \sigma _U(v)=\sigma _{{{\,\textrm{co}\,}}(\partial ^*f(\bar{x}))}(v)+\sigma _{T^\circ (S,\bar{x})}(v), \ \text {for all}\ v\in T(S,\bar{x}). \end{aligned}$$
(5)

Using (3)–(5) and the fact that \(\sigma _K(v)\) is infinite for \(v\notin T(S,\bar{x})\), we conclude that \(\sigma _U(v)\ge 0 \ \text {for all}\ v\in \mathbb {R}^n.\) Thus, by [2, Theorem 2.2.2 (p. 211)], \(0\in U={{\,\textrm{cl}\,}}({{\,\textrm{co}\,}}(\partial ^*f(\bar{x}))+T^\circ (S,\bar{x})).\)

If \(\partial ^*f(\bar{x})\) is a bounded set then \({{\,\textrm{co}\,}}(\partial ^*f(\bar{x}))\) is compact as \(\partial ^*f(\bar{x})\) is a closed set. Hence (1) reduces to (2) as \({{\,\textrm{co}\,}}(\partial ^*f(\bar{x}))+T^\circ (S,\bar{x})\) is a closed set. \(\square \)