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-Bangalore)


Parthasarathy, Thiruvenkatachari (TSMU-Bangalore; ISI)

Abstract (Summary of the Work)

This thesis is composed of chapters 1 to 5. In Chapter 1, we formally define SDLCP and give some examples of SDLCP and show that SDLCP is a special case of variational inequality problem. Also we show that LCP is indeed a special case of SDLCP. Later section of this chapter deals with definitions and notations.In Chapter 2, we are concerned with the results which have been obtained in an effort to generalize the P-matrix condition of LCP to SDLCP. It is known that a matrix M is a P-matrix if and only if M does not re verse the sign of any nonzero vector. This notion of the P-matrix property has been extended to the SDLCP setup through the P-property or the Pa- property. If M is a P-matrix, then LCP(M,4) has a unique solution for every g E Rn. But, if L has the P-property, it need not imply that, L has the GUS-property.It is known that the P-property may not imply the GUS-property al- though it is true in the LCP situation. It has been shown that the P2- property implies the GUS-property. We will also give an example to show the GUS-property need not imply the P2-property. Also from this example, it becomes clear that the P2-property is stronger than the P-property.If L has the strong monotonicity property, then it has the GUS-property and the Pa-property also implies the GUS-property. In view of this we would like to know whether there is any relationship between the strong monotonicity property and the P2-property. Could we say the strong monotonicity property = the P2 property? We answer this question affirmatively. If a linear transformation L: Sn + Sn has the strong monotonicity property, then it has the P2-property.We provide an example to show that the P2-property need not imply the strong monotonicity property in general. Finally, in this chapter we derive a set of necessary and sufficient conditions for the linear map MA to have the strong monotonicity property when AE R2x2.


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Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


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