Download or read book Algorithms and Interfaces for Structured Variational Inequalities and Their Extensions written by Youngdae Kim and published by . This book was released on 2017 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Variational inequalities (VIs) are a generalization of nonlinear system of equations, so-called generalized equations. In addition to the system of equations, they subsume geometric first-order optimality conditions, nonlinear (linear) complementarity problems, and mixed complementarity problems. Representative applications are equilibrium problems such as generalized Nash equilibrium problems (GNEPs) and multiple optimization problems with equilibrium constraints (MOPECs). This thesis is concerned with algorithms and interfaces for structured variational inequalities and their extensions. Algorithms and interfaces are closely related to each other in a way that interfaces helping identify problem structures can lead to more robust and efficient algorithms, and structure-exploiting algorithms can guide us to design better structure-exposing interfaces. Interfaces exposing problem structures are described based on an extended mathematical programming (EMP) framework, where the framework allows us to formulate equilibrium problems in a natural and intuitive way in modeling languages, for example AMPL, GAMS, or Julia, without requiring the modeler to supply derivatives. Extensions to support some complicated structures such as shared constraints, shared variables, and quasi-variational inequalities (QVIs) are presented. Our interfaces generate a human-readable file from which we can easily identify high-level structure of the problem. We present an extension to PATH and two general-purpose solvers, PATHAVI and SELKIE, each of which utilizes problem structures, such as implicitly defined variables, polyhedral constraints, and groups of interacting agents, respectively. These structures are identified through our interfaces. An extension to PATH exploits implicitly defined variables by restoring their feasibility via projection using the implicit function theorem. Projection is performed in both their primal and dual spaces. PATHAVI is a structure-preserving solver for affine variational inequalities such that it follows a piecewise-linear (PL) path on a PL-manifold constructed using given polyhedral constraints without applying any reduction. This is a key contrast to the existing solver PATH which is oblivious of those constraints except for preprocessing purposes and a QR decomposition-based method that performs a reduction, thus destroying the structure, if there is nontrivial lineality space. SELKIE is a solver for equilibrium problems which enables various decomposition schemes based on groups of agents information to be instantiated in a flexible and adaptable way. Parallelism can be achieved either whenever independent groups of agents are detected or per user's request. A sub-solver for each sub-model can be chosen so that a highly efficient solver can be employed tailored to a certain problem type. Examples illustrating the efficiency and effectiveness of our extension and solvers are given. All our interfaces and solvers have been implemented and are available within GAMS/EMP.