I am getting a warning in this form:
DerivativesWarning:Constraints or objectives ['traj.linkages.stage_1_grav_turn:alpha_final|coast_1:alpha_initial', 'traj.phases.stage_1_maneuver.path_constraints.path:q_alpha'] cannot be impacted by the design variables of the problem.
not sure what to make of the first one, a linkage constraint. Alpha is a parameter in the grav_turn and coast phase, and it's set to 0. The second one makes no sense, as in the the stage_1_maneuver phase alpha is a control so you can definitely control dynamic pressure * alpha. Perhaps because alpha at the end of that phase is constrained to 0?
Anyways the optimizer converges fine, and produces results that look correct and makes sense when cross checked. Just was curious about this.
In OpenMDAO V3.9.0 a feature was added that detects rows and columns of all 0 in the total derivative Jacobian. A row of all 0's means that an objective or constraint is not impacted by any of the design variables. A column of all 0's means that a design variable does not impact any constraint or objective values.
Both of these situations are potentially problematic. A 0 column means that there are less degrees of freedom than you might think, since that DV doesn't affect anything. This isn't fatal, but it is still something that is worth warning a user about.
A 0 row is much more problematic. If it the row is associated with a constraint, it means the optimizer has no ability to satisfy that constraint. You may get "lucky" and find that the constraint happens that the constraint happens to be satisfied at the initial condition anyway, and so you can technically solve the optimization problem (your specific case is likely one of these). However, mathematically the problem is singular, and unless the optimizer you using has specific code to handle this corner case it can make things difficult.
One of the primary reasons this feature was added was that the OpenMDAO dev team noticed that Dymos users were particularly prone to accidentally creating 0 rows when adding linkage and path constraints. Often these 0 rows seem to not cause harm, but we have definitely also seen cases where they give the optimizer fits.
The warning helps you identify the problem so you can correct it.
In this case, it looks like you have two separate 0 rows.
traj.linkages.stage_1_grav_turn:alpha_final|coast_1:alpha_initial means that none of the design variables you've given to the optimizer affect that constraint. Likely this means that you have specified both alpha_final and alpha_initial as fixed_final and fixed_initial respectively. You're getting away with it because the initial conditions you provided must have both alphas equal by construction.
You would still be better off either removing the constraint, or adding at least one end of the linkage or the other as a design variable.
traj.phases.stage_1_maneuver.path_constraints.path:q_alpha means that at least one of the entries in your path constraint is not affected by any DV. It is likely not the entire path constraint, but just one end of it that is fixed because its computed from fixed boundary conditions. In this case, you can simply add indices to the add_path_constraint call to exclude the first or last point from the constraint.
Since your optimization is running, the 0 rows aren't killing you. However, its good practice to clean this up. It's possible that not having them will improve performance now, or it may save you from a future situation where the optimization "mysteriously" stops working because you somehow trigger a situation where the optimizer can no longer handle the 0 rows.
Related
Such as in the 'racecar' example, could I set a lower and upper limit for the 'mass' design_parameter and then optimise the vehicle mass while solving the optimal control problem?
I see that there is an "opt" argument for phase.add_design_parameter() but when I run the problem with opt=True the value stays static. Do I need another layer to the solver that optimises this value?
This feature would be useful for allocating budgets to design decisions (e.g. purchasing a lighter chassis), and tuning parameters such as gear ratio.
It's absolutely possible, and in fact that is the intent of the opt flag on design parameters.
Just to make sure things are working as expected, when you have a design parameter with opt=True, make sure it shows up as one of the optimizer's design variables by invoking list_problem_vars on the problem instance after run_model. The documentation for list_problem_vars is here.
If it shows up as a design variable but the optimizer is refusing to change it, it could be that it sees no sensitivity wrt that variable. This could be due to
incorrectly defined derivatives in the model (wrong partials)
poor scaling (the sensitivity of the objective/constraints wrt the design parameter may be miniscule in the optimizer's units
sometimes by nature of the problem, a certain input has little to no impact on the result (this is probably the least likely here).
Things you can try:
run problem.check_totals (make sure to call problem.run_model first) and see if any of the total derivatives appear to be incorrect.
run problem.driver.scaling_report and verify that the values are not negligible in the units in which the optimizer sees them. If they're really small at the starting point, then it may be appropriate to scale the design parameter smaller (set ref to a smaller number like 0.01) so that a small change from the optimizer's perspective results in a larger change within the model.
If things don't appear to be working after trying this and I'll work with you to figure this out.
Constraint Satisfaction Problems (CSPs) are basically, you have a set of constraints with variables and the domains of values for the variables. Then given some configuration of the variables (assignment of variables to values in their domains), you check to see if the constraints are "satisfied". That is, you check to see that evaluating all of the constraints returns a Boolean "true".
What I would like to do is sort of the reverse. Instead of this Boolean "testing" if the constraints are true, I would like to instead take the constraints and enforce them on the variables. That is, set the variables to whatever values they need to be in order to satisfy the constraints. An example of this would be like in a game, you say "this box's right side is always to the left of its containing box's right side," or, box.right < container.right. Then the constraint solving engine (like Cassowary for the game example) would take the box and set its "right" property to whatever number value it resolved to. So instead of the constraint solver giving you a Boolean value "yes the variable configuration satisfies the constraints", it instead updates the variables' configuration with appropriate values, "you have updated the variables". I think Cassowary uses the Simplex Algorithm for solving its constraints.
I am a bit confused because Wikipedia says:
constraint satisfaction is the process of finding a solution to a set of constraints that impose conditions that the variables must satisfy. A solution is therefore a set of values for the variables that satisfies all constraints—that is, a point in the feasible region.
That seems different than the constraint satisfaction problem, of which it says:
An evaluation is consistent if it does not violate any of the constraints.
That's why it seems CSPs are to return Boolean values, while in CS you can set the values. Not quite clear the distinction.
Anyways, I am looking for general techniques on Constraint Solving, in the sense of setting variables like in the simplex algorithm. However, I would like to apply it to any situation, not just linear programming. Some standard and simple example constraints are:
All variables are different.
box.right < container.right
The sum of all variables < 10
Variable a goes before variable b in evaluation.
etc.
For the first case, seeing if the constraints are satisfied (Boolean true) is pretty easy: iterate through the pairs of variables, and if any pair is not equal to each other, return false, otherwise return true after processing all variables.
However, doing the equivalent of setting the variables doesn't seem possible at first glance: iterate through the pairs of variables, and if they are not equal, perhaps you set the first one to the second one. You might have to do some fixed point thing, processing some of them more than once. And then figuring out what value to set them to seems arbitrary how I just did it. Maybe instead you need some further (nested) constraints defining how set the values (e.g. "set a to b if a > b, otherwise set b to a"). The possibilities are customizable.
In addition, for simpler cases like box.right < container.right, it is even complicated. You could say at first that if box.right >= container.right then set box.right = container.right. But maybe actually you don't want that, but instead you want some iPhone-like physics "bounce" where it overextends and then bounces back with momentum. So again, the possibilities are large, and you should probably have additional constraints.
So my question is, similar to how for testing the constraints (for Boolean value) is standardized to CSP, I am wondering if there are any references or standardizations in terms of setting the values used by the constraints.
The only thing I have seen so far is that Cassowary simplex algorithm example which works well for an array of linear inequalities on real-numbered variables. I would like to see something that can handle the "All variables are different" case, and the other cases listed, as well as the standard CSP example problems like for scheduling, box packing, etc. I am not sure why I haven't encountered more on setting/updating constraint variables instead of the Boolean "yes constraints are satisfied" problem.
The only limits I have are that the constraints work on finite domains.
If it turns out there is no standardization at all and that every different constraint listed requires its own entire field of research, that would be good to know. Then I at least know what the situation is and why I haven't really seen much about it.
CSP is a research fields with many publications each year. I suggest you to read one of the books on the subject, like Rina Dechter's.
For standardized CSP languages, check MiniZinc on one hand, and XCSP3 on the other.
There are two main approaches to CSP solving: systematic and stochastic (also known as local search). I have worked on three different CSP solvers, one of them stochastic, but I understand systematic solvers better.
There are many different approaches to systematic solvers. It is possible to fill a whole book covering all the possible approaches, so I will explain only the two approaches I believe the most in:
(G)AC3 which propagates constraints, until all global constraints (hyper-arcs) are consistent.
Reducing the problem to SAT, and letting the SAT solver do the hard work. There is a great algorithm that creates the CNF lazily, on demand when the solver is already working. In a sence, this is a hybrid SAT/CSP algorithm.
To get the AC3 approach going you need to maintain a domain for each variable. A domain is basically a set of possible assignments.
For example, consider the domains of a and b: D(a)={1,2}, D(b)={0,1} and the constraint a <= b. The algorithm checks one constraint at a time, and when it reaches a <= b, it sees that a=2 is impossible, and also b=0 is impossible, so it removes them from the domains. The new domains are D'(a)={1}, D'(b)={1}.
This process is called domain propagation. Using a queue of "dirty" constraints, or "dirty" variables, the solver knows which constraint to propagate next. When the queue is empty, then all constraints (hyper arcs) are consistent (this is where the name AC3 comes from).
When all arcs are consistent, then the solver picks a free variable (with more than one value in the domain), and restricts it to a single value. In SAT, this is called a decision It adds it to the queue and propagates the constraints. If it gets to a conflict (a constraint can't be satisfied), it goes back and undos an earlier decision.
There are a lot of things going on here:
First, how the domains are represented. Some solvers only hold a pair of bounds for each domain. Others, have a set of integers. My solver holds an interval set, or a bit vector.
Then, how the solver knows to propagate a constraint? Some solvers such as SAT solvers, Minion, and HaifaCSP, use watches to avoid propagating irrelevant constraints. This has a significant performance impact on clauses.
Then there is the issue of making decisions. Usually, it is good to choose a variable that has a small domain and high connectivity. There are many papers comparing many different strategies. I prefer a dynamic strategy that resembles the VSIDS of SAT solvers. This strategy is auto-tuned according to conflicts.
Making decision on the value is also important. Many simply take the smallest value in the domain. Sometimes this can be suboptimal if there is a constraint that limits a sum from below. Another option is to randomly choose between max and min values. I tune it further, and use the last assigned value.
After everything, there is the matter of backtracking. This is a whole can of worms. The problem with simple backtracking is that sometimes the cause for conflicts happened at the first decision, but it is detected only at the 100'th. The best thing is to analyze the conflict, and realize where the cause of the conflict is. SAT solvers have been doing this for decades. But CSP representation is not as trivial as CNF. So not many solvers could do it efficiently enough.
This is a nontrivial subject that can fill at least two university courses. Just the subject of conflict analysis can take half of a course.
In Applying Arc-Consistency (AC3) algorithms on one Constraint Satisfaction Problem, if domain of one variable be empty, what is the next step?
1) halt.
2) do backtrack.
3) start from another initial state.
4) it depends on that we are in which step.
Solution (4). I think (1) is true because it mean we cannot find any consistent assignment and halt. anyone can describe why (4) is true?
With the particular algorithm you're using, if the domain of a variable shrinks until it is empty, then it means that the constraint problem has no solutions. Therefore the algorithm should halt in the failure state.
I want to model an HTTP interaction, i.e. a sequence of HTTPRequest/HTTPResponse, and I am trying to model this as a transition system.
I defined an ordering on a class State by using:
open util/ordering[State]
where a State is simply a set of Messages:
sig State {
msgSet: set Message
}
Each pair of (HTTPRequest->HTTPResponse) and (HTTPResponse->HTTPRequest) is represented as a rule in my transition system.
The rules are expressed in Alloy as predicates that let one move from one state to another.
E.g., this is a rule generating an HTTPResponse after a particular HTTPRequest is received:
pred rsp1 [s, s': State] {
one msg: Request, msg':Response | (
// Preconditions (previous Request)
msg.method=get &&
msg.address.url=sample_com &&
// Postconditions (next Response)
msg'.status=OK_200 &&
// previous Request has to be in previous state
msg in s.msgSet &&
// Response generated is added to next state
s'.msgSet = s.msgSet + msg'
}
Unfortunately, the model created seems to be too complex: we have a dozen of rules (more complex than the one above but following the same pattern) and the execution is very slow.
EDIT: In particular, the CNF generation is extremely slow, while the solving takes a reasonable amount of time.
Do you have any suggestion on how to model a similar transition system?
Thank you very much!
This is a model with an impressive level of detail; thank you for sharing it!
None of the various forms of honestAction by itself takes more than two or three minutes to find an instance (or in some cases to fail to find any instance), except for rsp8, which takes quite a while by itself (it ran for fifteen minutes or so before I stopped it).
So the long CNF preparation times you are observing are apparently caused by either (a) just predicate rsp8 that's causing your time issues, or (b) the size of the disjunction in the honestAction predicate, or (c) both.
I suspect but have not proved that the time issue is caused by combinatorial explosion in the number of individuals required to populate a model and the number of constraints in the model.
My first instinct (it's not more than that) would be to cut back on the level of detail in the model, in particular the large number of singleton signatures which instantiate your abstract signatures. These seem (I could be wrong) to be present either for bookkeeping purposes (so you can identify which rule licenses the transition from one state to another), or because the modeler doesn't trust Alloy to generate concrete instances of signatures like UserName, Password, Code, etc.
As the model now is, it looks as if you're doing a lot of work to define all the individuals involved in a particular example, instead of defining constraints and letting Alloy do the work of finding examples. (Using Alloy to check the properties a particular concrete example can be useful, but there are other ways to do that.)
Since so many of the concrete signatures in the model are constrained to singleton cardinality, I don't actually know that defining them makes the task of finding models more complex; for all I know, it makes it simpler. But my instinct is to think that it would be more useful to know (as well as possibly easier for Alloy to establish) that state transitions have a particular property in general, no matter what hosts, users, and URIs are involved, than to know that property rsp1 applies in all the cases where the host is named examplecom and the address URI is example_url_https and whatnot.
I conjecture that reducing the number of individuals whose existence and properties are prescribed, and the constraints on which individuals can be involved in which state transitions, will reduce the CNF generation time.
If your long-term goal is to test long sequences of state transitions to test whether from a given starting point it's possible or impossible to arrive at a particular state (or kind of state), you may need to re-think the approach to enable shorter sequences of state transitions to do the job.
A second conjecture would involve less restructuring of the model. For reasons I don't think I understand fully, sometimes quantification with one seems to hurt rather than help performance, as in this example, where explicitly quantifying some variables with some instead of one turned out to make a problem tractable instead of intractable.
That question involves quantification in a predicate, not in the model overall, and the quantification with one wasn't intended in the first place, so it may not be relevant here. But we can test the effect of the one keyword on this model in a simple way: I commented out everything in honestAction except rsp8 and ran the predicate first != last in a scope of 8, once with most of the occurrences of one commented out and once with those keywords intact. With the one keywords commented out, the Analyser ran the problem in 24 seconds or so; with the one keywords in place, it ran for 500 seconds so far before I decided the point was made and terminated it.
So I'd try removing the keyword one from all of the signatures with instance-specific individuals, leaving it only on get, post, OK_200, etc., and appData. I would also try doing without the various subtypes of Key, SessionID, URL, Host, UserName, and Password, or at least constraining their cardinality in the run command.
I am working on a structure from motion application and I am tracking a number of markers placed on the object to determine the rigid structure of the object.
The app is essentially using standard Levenberg-Marquardt optimization over multiple camera views and minimizing the differences between expected marker points and the marker points obtained in 2D from each view.
For each marker point and each view the following function is minimised:
double diff = calculatedXY[index] - observedXY[index]
Where calculatedXY value depends on a number of unknown parameters that need to be found via the optimization and observedXY is the marker point position in 2D. In total I have (marker points * views) number of functions like the one above that I am aiming to minimise.
I have coded up a simulation of the camera seeing all the marker points but I was wondering how to handle the cases when during running the points are not visible due to lighting, occlusion or just not being in the camera view. In the real running of the app I will be using a web cam to view the object so it is likely that not all markers will be visible at once and depending on how robust my computer vision algorithm is, I might not be able to detect a marker all the time.
I thought of setting the diff value to be 0 (sigma squared difference = 0) in the case where the marker point could not be observed, could this skew the results however?
Another thing I noticed is that the algorithm is not as good when presented with too many views. It is more likely to estimate a bad solution when presented with too many views. Is this a common problem with bundle adjustment due to the increased likeliness of hitting a local minimum when presented with too many views?
It is common practice to just leave out terms corresponding to missing markers. Ie. don't try to minimise calculateXY-observedXY if there is no observedXY term. There's no need to set anything to zero, you shouldn't even be considering this term in the first place - just skip it (or, I guess in your code, it's equivalent to set the error to zero).
Bundle adjustment can fail terribly if you simply throw a large number of observations at it. Build your solution up incrementally by solving with a few views first and then keep on adding.
You might want to try some kind of 'robust' approach. Instead of using least squares, use a "loss function"1. These allow your optimisation to survive even if there are a handful of observations that are incorrect. You can still do this in a Levenberg-Marquardt framework, you just need to incorporate the derivative of your loss function into the Jacobian.