Weak sharp minima for interval-valued functions and its primal-dual characterizations using generalized Hukuhara subdifferentiability

Abstract

This article introduces the concept of weak sharp minima for convex interval-valued functions. To solve constrained and unconstrained convex IOPs by WSM, we provide primal and dual characterizations of the set of WSM. The primal characterization is given in terms of gH-directional derivatives. On the other hand, to derive dual characterizations, we propose the notions of the support function of a subset of $I({\mathbb {R}})^{n}$ and gH-subdifferentiability for convex IVFs. Further, we develop the required gH-subdifferential calculus for convex IVFs. Thereafter, by using the proposed gH-subdifferential calculus, we provide dual characterizations for the set of WSM of objective IVFs of convex constrained and unconstrained IOPs. Two applications of the proposed theory are presented. The first one determines the set of WSM of a minimum risk portfolio interval optimization problem. In the second application, we propose a way to find weak efficient solutions of linear and nonlinear IOPs using WSM.

A Proof of Lemma 1

(i). Let ${\textbf {A}}=[\underline{a},\overline{a}], {\textbf {B}}=[\underline{b},\overline{b}], \text { and } {\textbf {C}}=[\underline{c},\overline{c}]$. We have

$$\begin{aligned} r\le \underline{a} \text { and } r\le \overline{a}. \end{aligned}$$

(57)

Similarly, by ${\textbf {A}}\preceq {\textbf {B}}\ominus _{gH}{} {\textbf {C}}$, we have

$$\begin{aligned} \underline{a}\le \min \left\{ \underline{b}-\underline{c}, \overline{b}-\overline{c}\right\} \text {and } \overline{a}\le \max \left\{ \underline{b} -\underline{c},\overline{b}-\overline{c}\right\} . \end{aligned}$$

(58)

• Case 1. Let $\min \left\{ \underline{b} -\underline{c},\overline{b}-\overline{c}\right\} =\overline{b}-\overline{c}$ and $\max \left\{ \underline{b} -\underline{c},\overline{b}-\overline{c}\right\} =\underline{b}-\underline{c}$. Then, from (57) and (58), we get

  $$\begin{aligned} \overline{c}+r\le \overline{b} \text { and } \underline{c}+r\le \underline{b}. \end{aligned}$$

  Hence, ${\textbf {C}}\oplus [r,r]\preceq {\textbf {B}}.$

• Case 2. When $\min \left\{ \underline{b}-\underline{c}, \overline{b}-\overline{c}\right\} =\underline{b}-\underline{c}$ and $\max \left\{ \underline{b}-\underline{c},\overline{b} -\overline{c}\right\} =\overline{b}-\overline{c}$. Proof contains similar steps as in Case 1.

(ii). Let ${\textbf {A}}=[\underline{a},\overline{a}]$ and ${\textbf {B}}=[\underline{b},\overline{b}].$ Then,

$$\begin{aligned}&\Big \{(1-\lambda )\odot {\textbf {A}} \oplus \lambda \odot {\textbf {B}}\Big \}\ominus _{gH}{} {\textbf {A}}\\&\quad =\Big \{(1-\lambda )\odot [\underline{a},\overline{a}] \oplus \lambda \odot [\underline{b},\overline{b}]\Big \} \ominus _{gH}[\underline{a},\overline{a}]\\&\quad =\left[ (1-\lambda )\underline{a}+\lambda \underline{b}, (1-\lambda )\overline{a}+\lambda \overline{b}\right] \ominus _{gH}[\underline{a},\overline{a}\Big ]\\&\qquad \text {because }\lambda \in [0,1]\\&\quad =\Big [\min \Big \{\lambda \underline{b} -\lambda \underline{a},\lambda \overline{b} -\lambda \overline{a}\Big \},\max \Big \{\lambda \underline{b} -\lambda \underline{a},\lambda \overline{b} -\lambda \overline{a}\Big \}\Big ]\\&\quad =\lambda \odot \Big \{{{\textbf {A}}} \ominus _{gH}{} {\textbf {B}}\Big \}. \end{aligned}$$