  In this paper, we attempt to propose Ekeland's variational principle for
interval-valued functions (IVFs). To develop the variational principle, we
study the concept of sequence of intervals. In the sequel, the idea of
gH-semicontinuity for IVFs is explored. A necessary and sufficient condition
for an IVF to be gH-continuous in terms of gH-lower and upper semicontinuity is
given. Moreover, we prove a characterization for gH-lower semicontinuity by the
level sets of the IVF. With the help of this characterization result, we ensure
the existence of a minimum for an extended gH-lower semicontinuous,
level-bounded and proper IVF. To find an approximate minima of a gH-lower
semicontinuous and gH-Gateaux differentiable IVF, the proposed Ekeland's
variational principle is used.
