Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



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 → V and a vector q ∈ V the cone linear complementarity problem or linear complementarity problem over K, denoted as LCP(K, L, q), is to find a vector x ∈ K such thatL(x) + q ∈ 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:K∗ := {y ∈ V : hx, yi ≥ 0 ∀ x ∈ K}.Note that a subset C of V is a cone if x ∈ C ⇒ λx ∈ C for every λ ≥ 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 → V , the Cone LCPvariational inequality problem, denoted VI(K, f), is to find a x ∈ K such thathy − x, f(x)i ≥ 0, ∀ y ∈ 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, C¸ 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 ∈ Rn×n and a vector q ∈ Rn , the linear complementarity problem, denoted LCP(Rn +, M, q), is to find a x ∈ Rn + such that Mx+q ∈ Rn + and x T (Mx+q) = 0. The study of linear complementarity problem began in 1960’s for solving convex quadratic programming problems [6].


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