Enhanced intersection cutting plane and reformulation-linearization enumeration based approaches for linear complementarity problems
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In this research effort, we consider the linear complementarity problem (LCP) that arises in diverse areas including optimal control, economics, engineering, mechanics, and quadratic programming. This class of problems has posed a challenge to researchers for over three decades now. Most of the current algorithms designed to solve LCP are guaranteed to work only under some restrictive assumptions on the matrix M associated with LCP. In this research, we introduce two new algorithms based on an equivalent 0-1 mixed integer bilinear programming formulation of LCP.
In the first approach, we develop an enhanced intersection cutting plane algorithm for solving LCP. The matrix M is not assumed to possess any special structure, except that the corresponding feasible region is assumed to be bounded. A procedure is described to generate cuts that are deeper versions of Tuy's intersection cuts, based on a relaxation of the usual polar set. The proposed algorithm then attempts to find an LCP solution in the process of generating either a single or a pair of such strengthened intersection cuts. The process of generating these cuts involves a vertex ranking scheme that either finds an LCP solution, or else, these cuts eliminate the entire feasible region leading to the conclusion that no LCP solution exists. Based on the bilinear formulation, a heuristic is also proposed to front-end the algorithm, in order to possibly solve LCP.
In the second part of the dissertation, we present a global optimization algorithm based on a novel Reformulation-Linearization Technique (RLT). We do not place any restrictions on the matrix M associated with LCP in this case. This RLT scheme provides an equivalent linear, mixed integer programming formulation of LCP, that possesses a tight linear programming relaxation. The solution strategy developed is a composite impliCit enumeration-Lagrangian relaxation scheme. In addition to the bounds provided by the RLT-based relaxation, we further tighten these bounds at each node of the branch-and-bound tree through the use of strongest surrogate cuts, and strengthened intersection cuts. The heuristic developed in the previous algorithm is also invoked within this scheme.
Both algorithms have been implemented and tested on randomly generated test problems. These problems include both indefinite and negative definite defining matrices. The results of these tests indicate that both methods are effective in solving the LCP with the heuristic being quite effective in recovering LCP solutions when the matrix Mis negative definite. The Lagrangian dual approach for the implicit enumeration scheme proved to be computationally more efficient than a similar simplex based lower bounding scheme.
- Doctoral Dissertations