equilibrium problems with equilibrium constraints - Convex ...

18 Chapter 2 Mathematical Program with Equilibrium Constraints
B. β (¯x) with ¯d β satisfying
B. β (¯x) ¯d β = ¯b
and the rest of the components in ¯d being 0.
Similarly, for every sufficiently large ν, the vector d ν is a basic solution of
(2.23) associated with the basis B. β (x ν ) with (d ν ) β satisfying
B. β (x ν )(d ν ) β = b ν
and the rest of the components in d ν being 0.
From (2.21), it is clear that H j (x ν ; a H t ν
)/G j (x ν ; a G t ν
) → ¯λ G /¯λ H , and hence,
b ν → ¯b as ν → ∞. With B(x ν ) converging to B(¯x), it follows that the sequence
{d ν } is bounded and d ν → ¯d as ν → ∞.
It is easy to see that d ν is a critical direction of Reg(t ν ) (2.13) at x ν for ν large
enough. If the constraint G j (x; a G t ν
)H j (x; a H t ν
) ≤ t ν is active at x ν , we examine the
term associated with this constraint in the Lagrangian function of Reg(t ν ) for the
second-order necessary optimality conditions. In particular,
λ GH
j,ν d T ν ∇ 2 ( G j (x ν ; a G t ν
)H j (x ν ; a H t ν
) − t ν
)
dν
= λ GH
j,ν H j(x ν ; a H t ν
) d T ν ∇2 G j (x ν ; a G t ν
) d ν + λ GH
j,ν G j(x ν ; a G t ν
) d T ν ∇2 H j (x ν ; a H t ν
) d ν
− λ GH
j,ν H j(x ν ; a H 2
t ν
)
G j (x ν ; a G t ν
) .
While the first two terms in the above equation are bounded, the third term
j,ν H j(x ν ; a H 2
t ν
)
G j (x ν ; a G t ν
)
−λ GH
→ −∞, as ν → ∞,
since λ GH
j,ν H j(x ν ; a H t ν
) → −¯λ j > 0 and G j (x ν ; a G t ν
) → 0 + . It is easy to check that
all other terms in d T ν ∇2 L(x ν , λ ν )d ν are bounded, and hence, the second-order
necessary optimality condition of Reg(t ν ) (2.13) fails at x ν for sufficiently large ν.
(iii) Since, from (ii), ¯x is an M-stationary point and the ULSC holds at ¯x, it
follows that ¯x is a strongly stationary point, and hence, a B-stationary point.