α-Well-posedness for Nash Equilibria and For Optimization Problems with Nash Equilibrium Constraints

Levitin-Polyak Well-Posedness for Equilibrium Problems with Functional Constraints

“…The wellposedness of unconstrained and constrained scalar optimization problems was first introduced and studied by Tykhonov 4 and Levitin and Polyak 5 , respectively. Since then, various concepts of well-posedness have been introduced and extensively studied for scalar optimization problems 6-13 , best approximation problems [14][15][16] , vector optimization problems [17][18][19][20][21][22][23] , optimization control problems 24 , nonconvex constrained variational problems 25 , variational inequality problems 26, 27 , and Nash equilibrium problems [28][29][30][31] . The study of Levitin-Polyak well-posedness for convex scalar optimization problems with functional constraints started by Konsulova and Revalski 32 .…”

A Note on Well-posedness of Nash-type Games Problems with Set Payoff

“…In [13], the following Nash games problems with real-valued payoff is studied. A point x = (x 1 ,x 2 ) ∈ X 1 × X 2 is a Nash equilibrium if: f 1 (x 1 ,x 2 ) ≤ f 1 (z 1 ,x 2 ), ∀z 1 ∈ X 1 ; and f 2 (x 1 ,x 2 ) ≤ f 1 (x 1 , z 2 ), ∀z 2 ∈ X 2 , where f i : X 1 × X 2 → R ∪ {+∞}.…”

“…For details, we refer readers to [8,9] and the references therein. Moreover, the concept of well-posedness has been generalized to nonconvex constrained variational problems, variational inequality problems, generalized variational inequality problems and equilibrium problems, see [2,3,4,5,6,7,10,12,13,15,16,17] and the reference therein.…”

“…(ii) For each i ∈ I, if F i is a real-valued function, the problem reduces to the Nash-type game problem with real-valued payoff functions in [13].…”

“…In [13], under some closedness, convexity and lower semicontinuous of real-valued payoff functions assumptions, some well-posed results with parameter α are also obtained for the above Nash-type games problems with real-valued payoff. However, the above theorem doesn't need the convexity assumptions of real-valued payoff functions.…”