## Doctoral Theses

### Some Geometrical Aspects of the Cone Linear Complementarity Problem.

1-28-2006

1-28-2007

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

Mohan, S. R.

#### Abstract (Summary of the Work)

Cone Linear Complementarity ProblemLet V be a finite dimensional real inner product space and K be a closed convex cone in V. Given a linear transformation L : V â†’ V and a vector q âˆˆ V the cone linear complementarity problem or linear complementarity problem over K, denoted as LCP(K, L, q), is to find a vector x âˆˆ K such thatL(x) + q âˆˆ K+ and hx, L(x) + qi = 0,where h., .i denotes an inner product on V and K is the dual cone of K defined as:Kâˆ— := {y âˆˆ V : hx, yi â‰¥ 0 âˆ€ x âˆˆ K}.Note that a subset C of V is a cone if x âˆˆ C â‡’ Î»x âˆˆ C for every Î» â‰¥ 0. The cone LCP is the special case of a variational inequality problem which is formally stated as follows:Let K be a closed convex set in V. Given a continuous function f : V â†’ V , the Cone LCPvariational inequality problem, denoted VI(K, f), is to find a x âˆˆ K such thathy âˆ’ x, f(x)i â‰¥ 0, âˆ€ y âˆˆ K.When K is a closed convex cone the VI (K, f) reduces to a (cone) complementarity problem (CP) and with the additional condition of f being affine we get a cone LCP. Volumes I and II of the recent book by Facchinei and Pang [9] provides an up to date account of finite dimensional variational inequalities and complementarity problems along with various applications and algorithmic details.Interested readers can look at the bibliography of [9] for more details.Though cone LCP is a special case of a variational inequality problem, its usefulness as a modelling framework for various practical problems and the availability of an additional structure puts it in a distinguished position. See, for example, CÂ¸ amlibel et al. [5] and Heemels et al. [29], where switched piecewise linear networks are modelled as cone LCP and [49, 50] for the reformulation of a Bilinear Matrix Inequality as a cone LCP on the cone of semidefinite matrices. Furthermore, [12] and [65] provide an excellent survey of various applications of complementarity problems in engineering and economics, and complementarity systems in optimization. The lecture notes [33] study complementarity problems in abstract spaces. Some early references related to a cone LCP (CP) include [25, 36, 37, 38]. For a recent work on cone LCP (CP) one can see [14, 15, 24, 40] and the references therein.1.1.1 Examples of a Cone LCPVarious special cases of a cone LCP (CP) are found to be of fundamental importance in the literature. We discuss briefly some of these cases in the examples Cone LCP below.Example 1.1.1 Given a real square matrix M âˆˆ RnÃ—n and a vector q âˆˆ Rn , the linear complementarity problem, denoted LCP(Rn +, M, q), is to find a x âˆˆ Rn + such that Mx+q âˆˆ Rn + and x T (Mx+q) = 0. The study of linear complementarity problem began in 1960â€™s for solving convex quadratic programming problems [6].

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