Generalized Equations

As we will see later, variational inequalities (and complementarity problems) provide a convenient and elegant tool for characterizing manifold equilibria. The aim of this chapter is to spell out how these models can be brought into the equally useful form of a generalized equation (4.1)% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgI % GiolaadoeadaqadaqaaiaadQhaaiaawIcacaGLPaaacqGHRaWkcaWG % obWaaSbaaSqaaiaadgfaaeqaaOWaaeWaaeaacaWG6baacaGLOaGaay % zkaaGaaiilaaaa!4179!$$ 0 \in C\left( z \right) + {N_Q}\left( z \right),$$ where C[ℝk fℝk] is a continuous mapping, Q a nonempty, closed, convex subset of ℝk and NQ(z) its normal cone to Q at z; cf. Definition 2.6. Q is called the feasible set of the GE (4.1). Oftentimes, the rewriting as a “nonsmooth equation” is not only possible but very helpful. While proceeding, we also collect several basic results on existence and uniqueness needed in the later chapters. Our objective is to prepare for subsequent analysis and computations.