The Investigation of Some Essential Concepts of Extended Fuzzy-Valued Convex Functions and Their Applications

In this paper, we are thus motivated to define and introduce the extended fuzzy-valued convex functions that can take the singleton fuzzy values <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mo>−</a:mo><a:mover accent="true"><a:mi>∞</a:mi><a:mrow><a:mo>˜</a:mo></a:mrow></a:mover></a:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" id="M2"><d:mo>+</d:mo><d:mover accent="true"><d:mi>∞</d:mi><d:mrow><d:mo>˜</d:mo></d:mrow></d:mover></d:math> at some points. Such functions can be characterized using the notions of effective domain and epigraph. In this way, we study important concepts such as fuzzy indicator function and fuzzy infimal convolution for extended fuzzy-valued functions. Finally, we introduce the concept of directional generalized derivative for extended above functions and its properties. Eventually, we give a practical example that will illustrate well the directional <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M3"><g:mi>g</g:mi></g:math> -derivative for the extended fuzzy-valued convex function.