The linear complementarity problem and related matrix classes

Absztrakt (kivonat)

The linear complementarity problem (LCP) is one of the most important and intensively researched areas of mathematical programming. It has numerous applications in economics: for example, finding the Nash equilibria of a bimatrix game, finding the equilibria in an Arrow–Debreu market with Leontief utility functions, the optimal stopping problem of Markov chains, and the pricing of American options are all equivalent to an LCP. We examine the problem and some of the most important associated matrix classes. In particular, we prove that the matrix class P is the interior of the sets P∗ and P0; and that P0 is the closure of P∗ and P. We also study the continuity of the handicap function and show that it is not jointly continuous over the set P∗, even when the matrices are two-by-two. On the other hand, we prove that the handicap is separately lower semicontinuous for any dimension.