# Abstract:

In this paper, we introduce a new class of variational inequality with its weak and split forms to obtain an $LU$-optimal solution to the multi-dimensional interval-valued variational problem, which is a wider class of interval-valued programming problem in operations research. Using the concept of (strict) $LU$-convexity over the involved interval-valued functionals, we establish equivalence relationships between the solutions of variational inequalities and the (strong) $LU$-optimal solutions of the multi-dimensional interval-valued variational problem. In addition, some applications are constructed to illustrate the established results.

# Keywords:

$LU$-convexity, $LU$-optimal solution, multi-dimensional inter-valued variational problem, variational inequality

# Classification:

26B25, 26D10, 49J40, 90C30

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