Second-order lower radial tangent derivatives and applications to set-valued optimization.

Area of Science:

• Optimization Theory
• Set-Valued Analysis
• Mathematical Analysis

Background:

• Set-valued optimization problems involve complex functions where outputs can be sets.
• Understanding optimality conditions is crucial for solving these problems efficiently.
• Existing methods may not fully capture the nuanced behavior of set-valued maps.

Purpose of the Study:

• To introduce and define four new second-order tangent derivatives for set-valued maps.
• To explore the fundamental properties of these newly defined second-order tangent derivatives.
• To establish second-order necessary optimality conditions for Henig efficient elements in set-valued optimization.

Main Methods:

• Utilizing concepts of radial tangent cones and second-order radial tangent sets.
• Defining second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative.
• Analyzing the properties of these derivatives to derive optimality conditions.

Main Results:

• Successfully introduced four distinct second-order tangent derivative concepts for set-valued maps.
• Established second-order necessary optimality conditions for identifying Henig efficient elements.
• Separated the second-order tangent derivatives of objective and constraint functions in the optimality conditions.

Conclusions:

• The newly introduced second-order tangent derivatives provide a powerful tool for analyzing set-valued optimization problems.
• The established optimality conditions offer a refined approach to determining efficiency in these complex problems.
• This work contributes to the theoretical framework of optimization with set-valued maps.