Projected dynamical systems and variational inequalities with applications
The variational inequality problem is defined as follows: for a closed convex set \(K\subset R^k\) and a function \(F:K\to R^k\) determine \(x^*\in K\) such that \(\langle F(x^*)^t,x-x^*\rangle\geq 0\;\forall x\in K\), where \(\langle.,.\rangle\) denotes the inner product in \(R^k\). Under appropriate conditions, the above may be shown to be equivalent to solving a system of nonlinear equations, solving a complementarity problem, solving an optimization exercise or solving a fixed point problem for some projection map. These relationships between a wide variety of problems provides the setting for the enquiry into solutions to the variational inequality problem described above and referred to as VI\((F,K)\). These discussions are also used to introduce the notion of a projected dynamical system. Let \(K\) be as above; for any \(v\in R^k\) and \(x\in K\) define the projection of the vector \(v\) at \(x\) (with respect to \(K\)), \(\Pi_K(x,v)\) by \[ \Pi_K(x,v)=\lim_{\delta\to 0} {P_K(x+\delta v)-x\over \delta}, \] where \(P_K(x)=\text{argmin}_{z\in K}|x-z|\), and \(|\cdot|\) denotes the Euclidean norm. Consider the system of ordinary differential equations defined by \(\dot x=\Pi_K(x,-F(x))\), where \(F(x)\) is a vector field on \(K\). Since the right-hand side is discontinuous on the boundary of \(K\), the notion of solution to the differential equation needs to be defined: \(x:[0,\infty)\to K\) is a solution if it is absolutely continuous and \(\dot x=\Pi_K(x,-F(x))\) save on a set of Lebesgue measure zero. Given an ordinary differential equation on the above type, the projected dynamic system PDS\((F,K)\) is defined to be the map \(\Phi:K\times R\to K\), where \(\Phi(x,t)=\phi_x(t)\) satisfies \(\dot\phi_x(t)=\Pi_K(\phi_x(t),-F(x(t)))\), \(\phi_x(0)=x\). The connection between VI\((F,K)\) and PDS\((F,K)\) rests on the result that if \(K\) is a convex polyhedron then the equilibrium points of PDS\((F,K)\) coincide with VI\((F,K)\). The book is based on the links mentioned above and consists of two parts: Part I (Chapters 1-4) presents the PDS and establishes the links mentioned above between PDS and VI; the analysis of the introduced dynamical system is done through the Skhorokhod Problem and discusses existence and uniqueness of the solution path (Chapter 2), develops tools for stability analysis (Chapter 3) and provides algorithms for the computation of equilibrium patterns (Chapter 4). Part II (Chapters 5 - 8) contains applications of the methods developed in Part I to a wide range of problems. In Chapter 5 an application is made to oligopolistic market equilibria. Two types of such equilibria are considered; first, the classical Cournot-Nash equilibrium and then the spatial network oligopoly models. In each case, the variational inequality formulation is first presented together with the corresponding projected dynamical system. Stability analysis is conducted using the tools developed in Chapter 3 and algorithms for computation of the solution is also provided. Chapter 6 discusses the spatial price equilibrium models along lines described above. Chapters 7 and 8 discuss traffic network equilibrium models. Wardropian equilibrium conditions are cast into variational inequality formulations and the corresponding projected dynamical systems are derived in the case of elastic demands (Chapter 7) and for fixed demands (Chapter 8). And as for the earlier applications, stability analysis is conducted and algorithms are provided. The approach adopted by the authors is claimed to allow the study of equilibrium problems (which have been mainly studied in static settings) within a dynamic and behavioral framework.
scientific article; zbMATH DE number 920061