On the velocity projection map for polyhedral Skorokhod problems. (English) Zbl 1073.90050

Day, Martin V.

Appl. Math. E-Notes 5, 52-59 (2005).

Each facet of a convex polyhedron \(G\subset\mathbb{R}^n\) defined by a finite system of linear constraints \(x\cdot n_i\geq c_i\) is associated with a restoration vector \(d_i\) such that \(n_i\cdot d_i=1\). For each \(x\in\partial G\), \(d(x)\) is a set of all combinations \(\sum\alpha_i d_i\) with positive coefficients, where the summation is over active indices at \(x\) and \(\left| d(x)\right| =1\). A point \(y\in G\) is the {discrete projection} \(\Pi(x)\), \(x\in\mathbb{R}^n\), if \(r(y-x)\in d(y)\) for some positive \(r\). The main result of the paper is in establishment of equivalence between existence of \(\Pi(y)\) and solvability of the complementarity problem, both, locally and in the entire \(\mathbb{R}^n\). The complementarity problem is formulated as follows: Given \(x\in\partial G\) and \(v\in\mathbb{R}^n\), find \(w=v+\sum \beta_i d_i\) such that \(\beta_i\) and \(n_i\cdot w\) are non-negative and not simultaneously positive, and the summation is over active indices at \(x\).

Reviewer: Dmitry Silin (Berkeley)

MSC:

90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
93A30	Mathematical modelling of systems (MSC2010)

Keywords:

convex polyhedron; Skorokhod problem; discrete projection map; complementarity problem