Conclusion

Throughout this study, we discuss second-order cone programming theoretically as well as practically. Both the long-step path-following and predictor-corrector algorithms are studied. Existence of the step length $\alpha$ is shown for the long-step method. Furthermore, a specific algorithm for choosing the centering parameter $\sigma$ and the size of the neighborhood $\gamma$ is developed and tested successfully on our test cases. Next, several variants of predictor-corrector algorithms are also compared. We analyze the numerical behavior of the linear and quadratic combinations of predictor and corrector directions. The idea of inexact Newton's method and of repeating the predictor direction one more time are also integrated into the algorithm. According to our numerical results, the inexact method for SOCP can be expected to behave similarly as in LP. Overall, Algorithm 4.4 using the normal equations works best for both full and sparse cases. The linear constraints in Example 5 lead to a full matrix. The inexact method works competitively in every case except Example 5.

Figure 2: The condition number of the coefficient matrix for full linear system and normal equations
Figure 1: Comparison of $\mu$ from Algorithm 4.3, H2,H3,H4