### PRACTICAL POLYNOMIAL TIME ALGORITHMS FOR LINEAR COMPLEMENTARITY PROBLEMS

Shinji Mizuno, Akiko Yoshise, Takeshi Kikuchi
1989 Journal of the Operations Research Society of Japan
In this paper, we first propose three practical algorithms for linear complementarity problems, which are based on the polynomial time method of Kojima, Mizuno and Yoshise (5), and compare them by showing the computational complexities. Then we modify two of the algorithms in order to accelerate them. Through the computational experiments for three types of linear complementarity problems, we compare the proposed algorithms in practice and see the efficiency of the modified algorithms. We also
more » ... lgorithms. We also estimate the practical computational complexity of each algorithm for each type of problems. Chapter 2 describes the outline of the method of [5] . The method traces a path of centers from an initial point to a solution by generating a sequence of feasible points. In Chapter 3, we construct three algorithms, Algorithm A, Algorithm B and Algorithm C, which are based on the method of Chapter 2. It is shown that Algorithms A, Band C require at most O(n4Ld, O(n 3 . 5 Ld and O(n 3 Ll) arithmetic operations, respectively, where Ll is a number which depends on the initial point and the final point. Chapter 4 describes two modified algorithms, Algorithm A' and Algorithm B', which accelerate Algorithm A and Algorithm B, respectively. In Chapter 5, we show some computational results for three types of LCPs, Problem 1, Problem 2 and Problem 3. Using the computational results, we estimate the practical computational complexity of each algorithm for each type of LCPs. We also compare Algorithms A, Band C in practice and see the differences between Algorithms A and A' and between Algorithms Band B'. Chapter 6 gives the conclusions. will be in proportional to some exponential of the problem size m for each problem and each algorithm, i.e., there are c and d such that