On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling

Two-stage stochastic equilibrium problems with equilibrium constraints: modeling and numerical schemes

“…Apart from the applications of the two stage SEPEC models discussed in this section, there are some potential applications in transportation and economics [15,21,13,28], internet service problems [36] and airline revenue management problems [16].…”

“…The concept of stationary points are important in optimization as it provides some information of optimality. This is particularly so in MPECs where obtaining a global optimal solution is often difficult and consequently various of stationary points are investigated [13,44]. The concept of Nash stationary point is relatively new: it was introduced by Hu and Ralph [15].…”

“…Zhang, Xu and Wu [44] developed a two stage SEPEC model to study generator's competition in electricity forward and spot markets and their interactions. Henrion and Römisch [13] proposed a two stage SEPEC model for studying competition in electricity markets. The authors investigated M-stationary points of the SEPEC model under the condition that the market demand in the spot market has a finite distribution.…”

Conditioning of linear-quadratic two-stage stochastic optimization problems

“…In order to apply Proposition 3.2, we have to compute explicitly the partial second-order subdifferential ∂ 2 x Ψ (explicit formulae for the other term D * N X are available from the literature, see, e.g., [5]). As a first step, we reduce the computation of ∂ 2…”

“…The assumption of strict complementarity yields that v * ∈ Ker D I and u * ∈ Im D I , where D I is the reduction of D to its active rows (see, e.g., [5,Corollary 3.7]). This entails that u * , v * = 0 which may be exploited in order to reduce the first equation in (25) to…”