{ "_id": "6ad324cf2c5c76d3dd0c9a99", "Package": "GNE", "Type": "Package", "Title": "Computation of Generalized Nash Equilibria", "Version": "0.99-7", "Authors@R": "person(\"Christophe\", \"Dutang\", role = c(\"aut\", \"cre\"),\nemail = \"dutangc@gmail.com\", comment = c(ORCID = \"0000-0001-6732-1501\"))", "Description": "Compute standard and generalized Nash Equilibria of\nnon-cooperative games. Optimization methods available are\nnonsmooth reformulation, fixed-point formulation, minimization\nproblem and constrained-equation reformulation. 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"engine": "knitr::rmarkdown", "headings": [ "Introduction", "GNEP as a nonsmooth equation", "Notation and definitions", "A classic example", "Usage example", "Localization of the GNEs", "Benchmark of the complementarity functions and the computation methods", "Initial point $z_0 = (4, -4, 1, 1)$", "Initial point $z_0 = (-4, 4, 1, 1)$", "Initial point $z_0 = (3, 0, 1, 1)$", "Initial point $z_0 = (0, 3, 1, 1)$", "Initial point $z_0 = (-1, -1, 1, 1)$", "Initial point $z_0 = (0, 0, 1, 1)$", "Conclusions", "Special case of shared constraints with common multipliers", "Constrained-equation reformulation of the KKT system", "GNEP as a fixed point equation or a minimization problem", "NI reformulation", "QVI reformulation", "The jointly convex case", "In this subsection, we present reformulations for a subclass of GNEP called jointly convex case.Firstly, the jointly convex setting requires that the constraint function is common to all players $g^1=\\dots =g^N= g$.Then, we assume, there exists a closed convex subset $X \\subset \\R^n$ such that for all player $i$,$$", "NIF formulation for the jointly convex case", "QVI formulation for the jointly convex case", "List of examples", "Tables for the nonsmooth reformulation", "Appendix for the nonsmooth reformulation", "\\subsection{Semismooth reformulation -- Shared constraint case\\label{app:ceq:jointcase}}The Jacobian is given by$$\\Jac \\widetilde H(x, \\tilde \\lambda, \\tilde w) =\\left(\\begin{matrix}\\Jac_x \\bar L (x, \\tilde \\lambda) & \\Jac_{\\tilde \\lambda} \\bar L (x, \\tilde \\lambda) & 0 \\\\Jac_x \\tilde g(x) & 0 & I \\0 & \\diag[\\tilde w] & \\diag[\\tilde \\lambda]\\end{matrix}\\right),$$where$$\\Jac_{\\tilde \\lambda} \\bar L (x, \\tilde \\lambda)" ], "created": "2023-03-29 19:59:07", "modified": "2024-10-17 14:21:24", "commits": 3 } ], "_score": 5.181843587944773, "_indexed": true, "_nocasepkg": "gne", "_universes": [ "r-forge", "dutangc" ], "_binaries": [ { "r": "4.7.0", "os": "linux", "version": "0.99-7", 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