Again, none. Reddit users have discussed the etymology of the term, linking it to songs by Soulja Boy, among other sources. DM: Direct message. Each week, users post lists of people that they think others should follow using the FF or FollowFriday hashtag. MM: Music Monday. Another recurring Twitter topic. In this case, users post a song or two that will get your week off to a better start.
NSFW: Not safe for work. NSFL: Not safe for life. Usually a humorous disclaimer that something formerly innocent is going to be irreparably sullied if you click the link. QQ: Crying. Rather than an abbreviation, this is an emoticon, a picture created in text. The tails of the capital Q form tears, while the circles are the eyes. We emphasize that this is just one example of the various appropriate choices.
But this particular algorithm seems reasonable for our purposes. Linesearch SNM is known to be quite efficient, and at the same time, it is easy to implement in its basic form. We note that we do not use any enhancements, such as crashing and nonmonotone linesearch see, e. The reason is that these are intended to improve global behavior of the algorithm, while we are concerned with local behavior. Thus a simple implementation of the global scheme is sufficient for our purposes, as our principal conclusions refer to the local convergence properties.
Assumptions A1 and A2 are evidently satisfied for the adopted choices. In order to guarantee A3 , we need to assume the error bound. Recall that this bound is equivalent to semistability. Thus, assumption A3 is satisfied. We proceed with the formal statement of the algorithm. Variable ''Alg'' below is used to select between the two variants of the algorithm that would be compared to each other.
Note however that switching to an active-set step is forbidden on the first iteration, and also in the case when the identified index sets differ from the corresponding index sets at the previous iteration.
This is done in order to prevent the algorithm from switching to the active-set strategy too early, when the sets are not yet stabilized and are likely to give incorrect identification.
Preliminary step. Initialization step. If this point is well-defined and 3. If is well-defined but 3. Gradient step. Linesearch step. Note that 3. This modification does not affect the rate of convergence. To see this, note that 2. The well-known formula. The stopping criterion is. The cases when an algorithm did not terminate according to this criterion after iterations are referred to as failures.
The algorithm was implemented in Matlab, making use of the standard option for treating sparse matrices. Next to the latter number, in the brackets, stands the number of active-set iterations at the ''tail'' of the process right before convergence had been declared. Finally, the number of gradient steps and the number of evaluations of F are reported recall that for both algorithms, the Jacobian of F is evaluated once per iteration.
Failures are marked by ''-''. Recall that by failure we mean that 3. In particular, for billups both variants of the algorithm converged to a local minimizer of the merit function, which is not an MCP solution.
But for this problem, this is typical for most MCP algorithms. We note that the problems mentioned above are considered among the difficult ones in MCP literature. Since we were able to solve essentially all the other problems, this means that our implementation of the global scheme, though simple, is sufficiently robust.
The rest of this discussion focuses on the cases for which at least one algorithm did not fail. For 3 problems, namely duopoly, shubik, and ne-hard, switching to our step at some early stage prevented failure although for ne-hard, the accuracy obtained by the algorithm without switching was of order 10 -9 , i. The opposite situation was observed for games only, though the obtained accuracy in the latter case was also of order 10 We do not have an explanation for the behavior on those 4 problems and regard it as probably ''accidental''.
Moreover, the possible negative global effect of switching to the active-set step too early can be avoided by the following simple trick. When an active-set step is accepted, we can store the previous iterate as a back-up, and restart the algorithm from that point if an active-set step is rejected on some subsequent iteration which indicates that the switch occurred too early. For 20 test problems the active-set step was never accepted, but trying it never harmed drastically.
The number of evaluations of F for the algorithm without the switching option is typically not much less than for the complete variant, especially when the number of iterations is relatively large. As mentioned above, for duopoly and shubik this actually prevented failure, but for games this caused it. For freebert this resulted in some extra linesearch steps. This points to the obvious fact that switching to the active-set strategy too early should be avoided.
For 19 test problems our step was used ''properly'' on the final stage of the process. For badfree and handskoop, this was evidently rather advantageous. In the other cases apparently either BD -regular or not regular even in our sense , the conclusions are overall similar to the situation when our step was never used. In those cases, we do not win or lose much. On the one hand, we typically have to pay the price of a few more evaluations of F but not always. Finally, the ''proper'' use of our step never increased and sometimes decreased the overall number of iterations.
General conclusions are as follows. The option of switching to our step never harms too much, though we certainly have to pay some extra price for computing it at some iterations at those where the index sets do not change , even if this step is eventually rejected.
But this is consistent with the main goal of the presented approach. We also point out that the efficiency of the identification procedure is of crucial importance for the methods presented in this paper. According to our numerical experience, the switching to our local method usually occurs exactly one iteration after the correct identification is obtained. Additional tuning of the identification procedure i. Actually, identification techniques other than the one described above could also be tried.
Also, note that we have not been using any heuristic considerations to decide whether or not the active-set step should be computed. Developing such heuristics can certainly save some computational work as well. For example, even if the active sets have not changed from one iteration to the next, we may decide not to compute the active-set step if the residual is relatively large i.
Other important issues are feasible versions of the method, and different globalization schemes which would better fit the structure of the method. This will be the subjects of future research. Example 3. At the final step, x 13 - » 3. The next four problems, taken from [14, Example ] Example 3. Specifically, det L 3 » 4. At the final step, x 7 - » 1. The rate of convergence is superlinear. In particular, solution of a given degenerate linear system depends on the linear solver, and can affect the overall convergence.
For Y S , 2-regularity holds, but weak regularity does not. Master of Science. Origin of S. State Militia. What else does SM mean? How is SM pronounced? Where does SM come from?
How is SM used in real life? The Cradle of Mankind W. Opuscula Robert Gordon Latham. San Marino. The symbol for the element samarium. Published by Houghton Mifflin Company.
0コメント