A globalized Newton method for the computation of normalized Nash equilibria

Generalized Nash Equilibrium Problems - Recent Advances and Challenges

“…In the case of GNEPs where common constraints occur, i.e., where the ν-th players' problem is given according to (19) it cannot be expected that every solution of the GNEP provides a KKT point, see [14,Example 2]. This example is stable with respect to small perturbations of the defining functions.…”

“…, N } and x * ,−ν be arbitrary but fixed. Then the FJ conditions of problem (19) with x −ν := x * ,−ν are given by…”

“…Initially, a Barzilai-Borwein step size[4] was implemented in[74] to avoid evaluations of V αβ caused by a line search. Later on, in[19], the above algorithm with the Armijo line search formula (31) is combined with a locally superlinearly convergent method (see Section 6). By standard arguments it can be shown that each accumulation point of a sequence {x k } generated by Algorithm 4 is a stationary point of V αβ .…”

“…with the identity matrix I n ∈ R n×n . Details and an explicit formula for ∇ y J α (x) can be found in [76] and [19]. Now suppose that x * is a normalized NE and CRCQ is satisfied in x * .…”

Jointly Convex Generalized Nash Equilibria and Elliptic Multiobjective Optimal Control

“…Most algorithms either are based on an equivalent variational inequality formulation, e.g., [13], or use the Nikaido-Isoda function (also called Ky-Fan function); see, e.g., [14][15][16]. We will give a precise definition of this function in the next section, and we will use it in order to extend some finite-dimensional approaches to infinite-dimensional jointly convex GNEPs.…”