The program carries out the Grassmann integration and performs the Lorentz trace on the involved functions, expressing the result as a compact sum of parametric integrals.
In many cases it is not possible to use derivative methods, due to the characteristics of the problem to be solved and/or its constraints, for example if the involved functions are non-smooth and/or their derivatives are not know.
If the involved functions of (P) are convex, Corollary 3.3 covers Lemma 3.1 in [5].
(2) If the involved functions of (P) are convex, Corollary 3.3 covers Lemma 3.1 in [5].
By Proposition 3.1(i) and Theorem 4.4, the necessary optimality conditions for the pessimistic semivectorial bilevel programming problem (1.6) are derived when the involved functions are strictly differentiable.
In this section we discuss sufficient optimality results under various generalized higher-order strong convexity (introduced in Section 2) hypotheses imposed on the involved functions.
Let us point out that our method clearly applies to the case where the involved functions are defined in, although we have chosen the unit interval for the sake of simplicity.
This section is devoted to investigating the relations between vector critical points and weakly quasi-efficient solution of order m for (NMP) with respect to ψ under generalized invexity (introduced in Section 2) hypotheses imposed on the involved functions.
In [22], the non-cooperative game model is called generalized Nash equilibrium problem (GNEP), which can be formulated as the quasi-equilibrium problem where the involved functions are nondifferentiable [23].
Such a situation appears in [2] where all the involved functions are defined on the set ({mathbb {N}}_{0}times {mathbb {R}}), but they can be naturally prolonged on the set ({mathbb {N}}_{0}times {mathbb {R}}^{2}) as suggested above.
