  This article proposes a general gH-gradient efficient-direction method and a
W-gH-gradient efficient method for the optimization problems with
interval-valued functions. The convergence analysis and the step-wise
algorithms of both the methods are presented. It is observed that the
W-gH-gradient efficient method converges linearly for a strongly convex
interval-valued objective function. To develop the proposed methods and to
study their convergence, the idea of strong convexity and sequential criteria
for gH-continuity of interval-valued function are illustrated. In the sequel, a
new definition of gH-differentiability for interval-valued functions is also
proposed. The new definition of gH-differentiability is described with the help
of a newly defined concept of linear interval-valued function. It is noticed
that the proposed gH-differentiability is superior to the existing ones. For a
gH-differentiable interval-valued function, the relation of convexity with the
gH-gradient of an interval-valued function and an optimality condition of an
interval optimization problem are derived. For the derived optimality
condition, a notion of efficient direction for interval-valued functions is
introduced. The idea of efficient direction is used to develop the proposed
gradient methods. As an application of the proposed methods, the least square
problem for interval-valued data by W-gH-gradient efficient method is solved.
The proposed method for least square problems is illustrated by a polynomial
fitting and a logistic curve fitting.
