I am currently using QuadProg++ for solving a dual problem. The problem also has some box constraints, i.e. constraints which limit the variable to be between two values. However, QuadProg++ has no provision which allows for incorporating such constraints. It only takes in the equality and inequality constraints. The equivalent Quadratic Programming tool in MATLAB, on the other hand, does have a provision for including box constraints.
You can take a look at the following link to see what I'm talking about:
http://www.mathworks.in/help/optim/ug/quadprog.html
Basically, I have a constraint equivalent to lb < x < ub.
I tried adding this as an inequality constraint, but it doesn't work. It results in an error, saying that the constraints are linearly dependent. However, I'm pretty sure that the constraints I'm inputting are in no way linearly dependent on each other.
Please suggest a workaround, or some other quadratic programming tool in C++, which can be of help for me. Thanks!
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I'm looking for a C++-based alternative to the SystemVerilog language.
While I doubt anything out there can match the simplicity and flexibility of the SystemVerilog constraint language, I have settled on using either Z3 or Gecode for what I'm working on, primarily because they're both under the MIT license.
What I'm looking for is:
Support for variable-sized bit vectors AND bit vector arithmetic logic operations. For example:
bit_vector a<30>;
bit_vector b<30>;
constraint {
a == (b << 2);
a == (b * 2);
b < a;
}
The problem with Gecode, as far as I can tell, is that it does not provide bit vectors right out of the box. However, its programming model seems a bit simpler, and it does provide a means for one to create their own types of variables. So I could perhaps creates some kind of wrapper around the C++ bitset, similar to how IntVar wraps around 32-bit integers. However, that would lack the ability to perform multiplication-based constrains, since C++ bitsets don't support such operations.
Z3 does provide bit vectors right out of the box, but I'm not sure how it would handle trying to set constraints on, for example, 128-bit vectors. I'm also unsure how I can specify that I want to produce a variety of randomized variables that satisfy a constraint when possible. With Gecode, it's much clearer given how thorough its documentation is.
A simplistic constraint programming model, close or similar to SystemVerilog. For example, a language where I only need to type (x == y + z) instead of something like EQ(x, y + z). As far as I can tell, both APIs provide such a simple programming model.
A means of performing constrained randomization, for the sake of producing random stimulus. As in, I can provide some random seed that, depending on the constraints, result in an answer that may differ from the previous answer. Similar to how SystemVerilog randomize calls may produce new random results. Gecode seems to support the use of random seeds. Z3, it's much less clear.
Support for weighted distribution. Gecode appears to support this via weighted sets. I imagine I can establish a relationship between certain conditions and boolean variables, and then add weights to those variables. Z3 appears to be more flexible in that you can assign weights to expressions, via the optimize class.
At the moment, I'm undecided, because Z3 lacks in documentation what Gecode lacks in out-of-the-box variable types. I'm wondering if anyone has any prior experience using either tool to achieve what SystemVerilog could. I'd like to hear any suggestions for any other API under a flexible license as well.
While z3 (or any SMT solver) can handle all of these, getting a nice sampling of satisfying assignments would be rather difficult to control. SMT solvers are optimized for just giving you a model, and they don't have much in terms of how you want to sample the solution space.
Incidentally, this is an active research area in SMT solving. Here's a paper that appeared only 6 weeks ago on this very topic: https://ieeexplore.ieee.org/document/8894251
So, I'd say if support for "good sampling" is your primary motivation, using an SMT solver is probably not the best choice. If your goal is to find satisfying assumptions for bit-vectors expressed conveniently (there are high level APIs in any language you can imagine these days), then z3 would be an extremely fine choice.
From your description, good sampling sounds like the primary motivation though, and for that SMT solvers are probably not that great. At least not for the time being.
My professor gave me a binary linear programming problem, but this problem is slightly different from optimization problems I used to solve(i.e. this is probably not maximizing or minimizing the object function.)
The problem is as follows,
Given a matrix M, for entries m_ij != 0, there are corresponding x_ijk variables.
Entries m_ij = 0 can be ignored.
x_ijk is either 0 or 1, and I want to try 5 x_ijk variables for each m_ij (that is, x_ij1, x_ij2, x_ij3, x_ij4, and x_ij5. One of them is 1 and the others are 0) are enough to satisfy some conditions(a set of inequalities).
More simply, this is to check if the set of constraints involving 5 x_ijk variables for each m_ij is a valid(or feasible) constraints.
I have solved some optimization problems, but I have never solved a problem without an objective function.
What should I set as my objective function here?
0? nothing?
I might be using lp_solve or CPLEX.
Thank you in advance for your advice!
That is correct, you can set an arbitrary constant value as an objective function.
Most of the solvers I have tried allow an empty objective function. Simply leave it out from your model.
Depending on the solver and the API you are using, it can happen that you have to set the coefficients of all variables in the objective to zero.
Don't worry, it has to work.
In response to your comment: Yes, constraint programming tools can provide better performance on feasibility problems than LP solvers (such as CPLEX). I have played with the IBM ILOG CPLEX CP Optimizer a few months ago, it is free for Academic users. Both the LP solver and the CP solver failed on my problems. Don't expect a miracle from constraint programming.
Keep in mind the that time needed to solve a constraint program grows exponentially with the size of the problem in the worse case. Sooner or later, your problems will most likely become unsolvable with either tool.
Just for your information: in the end, the constraint programming solver will call the LP solver (for example CPLEX).
My advice is: try the tool you already have / use the problem formulation that is more natural to you. Check whether the tool can solve your problem. Switch tool only if the tool fails and you cannot improve your model.
I am new to gecode and constraint programming in general.
So far, I haven't had much trouble picking up gecode, it's great. But I was wondering what is the best way to perform a "nested" cost function. Specifically, I am looking to minimize X, but within the space of solutions for which X is equal, prefer solutions which minimize Y? I could probably hack it by defining a cost function that looks like X*large_number+Y, but I'd prefer to do this properly if there's a good solution.
If anyone can point me to explain how to implement this in Gecode, that would be really helpful. Thanks!
You can define any kind of optimization criteria using the constrain member in a space in Gecode. See Section 2.5 in Modeling and Programming with Gecode for an example. In your case, the straight forward way would be to add a constrain member that adds a lexicographic constraint between the previous best solutions answer and the current space.
That being said, in general optimizing based on a lexicographic order can be wasteful (too much searching). It may often be better to first run a search optimizing the first component (X in your case). After that, re-run the search with the first components value fixed (X set to best possible value), and optimize the second value (Y in your case). Iterate as needed for all elements in the cost.
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I am looking for C++ open-source library (or just open-source Unix tool) to do: Equality test on Equations .
Equations can be build during runtime as AST Trees, string or other format.
Equations will mostly be simple algebra ones, with some assumptions about unknown functions. Domain, will be integer arithmetic (no floating point issues, as related issues are well known - Thanks #hardmath for stressing it out, I've assumed it's known).
Example: Input might contain function phi, with assumptions about it (most cases) phi(x,y)=phi(y,x) and try to solve :
equality_test( phi( (a+1)*(a+1) , a+b ) = phi( b+a, a*a + 2a + 1 )
It can be fuzzy or any equality test - what I mean is that, it does not have to always succeed (It may return "false" even if equations are equal).
If there would be problem with supporting assumptions like above about phi function, I can handle this, so just simple linear algebra equations equality testers are welcome as well.
Could you recommend some C/C++ programming libraries or Unix tools ? (open-source)
If possible, could you attach some example how such equality test, might look like in given library/tool ?
P.S. If such equality_test could (in case of success) return isomorphism - (what I mean, a kind of "mapping") - between two given equations, would be highly welcome. But tools without such capabilities are highly welcome as well.
P.S. By "fuzzy tester" I mean that in internals equation solver will be "fuzzy" in terms of looking for "isomorphism" of two functions, not in terms of testing against random inputs - I could implement this, for sure, but I try to find something with better precision.
P.P.S. There is another issue, why I need better performance solution, than brute-force "all inputs testing". Above equation is simplyfied form of my internal problem, where I do not have mapping between variables in equations. That is, I have eq1=phi( (a+1)*(a+1) , a+b ) and eq2=phi( l+k, k*k + 2k + 1 ) , and I have to find out that a==k and b==l. But this sub-problem I can handle with "brute-force" approach (even asymptotic complexity of this approach), case there is just a few variables, let it be 8. So I would need to do this equation_test for each possible mapping. If there is a tool that that whole job, I would be highly thankful, and could contribute to such project. But I don't require such functionality, simply equation_test() will be enough, I can handle rest easily.
To sum it up:
equality_test() is only one of many subproblems I have to solve, so computational complexity matters.
it does not have to be 100% reliable, but higher likelihood, than just testing equations with a few random inputs and variable mapping is highly welcome :).
output of "yes" or "no" (all additional information might be useful but in future, at this stage I need "Yes"/"No")
Your topic is one of automated theorem proving, for which a number of free/open source software packages have been developed. Many of these are meant for proof verification, but what you ask for is proof searching.
Dealing with the abstract topic of equations would be the theories mathematicians call varieties. These theories have nice properties with respect to the existence and regularity of their models.
It is possible you have in mind equations that deal specifically with real numbers or other system, which would add some axioms to the theory in which a proof is sought.
If in principle an algorithm exists to determine whether or not a logical statement can be proven in a theory, that theory is called decidable. For example, the theory of real closed fields is decidable, as Tarski showed in 1951. However a practical implementation of such an algorithm is lacking and perhaps impossible.
Here are a few open source packages that might be worth learning something about to guide your design and development:
Tac: A generic and adaptable interactive theorem prover
Prover9: An automated theorem prover for first-order and equational logic
E(quational) Theorem Prover
I am not sure for any library but how about you do it yourself by generating a random set of inputs for your equation and substituting it in both equations which have to be compared. This would give you a almost correct result given you generate considerable amount of random data.
Edit:
Also you can try http://www.wolframalpha.com/
with
(x+1)*(y+1) equals x+y+xy+2
and
(x+1)*(y+1) equals x+y+xy+1
I think you can get pretty far with using Reverse Polish Notation.
Write out your equation using RPN
Apply transformations to bring all expressions to the same form, e.g. *A+BC --> +*AB*AC (which is the RPN equivalent of A*(B+C) --> A*B+A*C), ^*BCA --> *^BA^CA (i.e. (B*C)^A --> B^A * C^A)
"Sort" the arguments of symmetric binary operator so that "lighter" operations appear on one side (e.g. A*B + C --> C + A*B)
You will have problem with dummy variables, for example sum indices. There is no other way, I think, but to try every combination of matching them on both sides of the equation.
In general, the problem is very complicated.
You can try a hack, though: use an optimizing compiler (C,Fortran) and compile both sides of the equation to optimized machine code and compare the outputs. It may work, or may not.
Opensource (GPL) project Maxima has tool simmilar to Wolfram Alpha's equals tool :
(a+b+c)+(x+y)**2 equals (x**2+b+c+a+2*x*y+y**2)
Which is is(equal()), that solves formulas :
(%i1) is(equal( (a+b+c)+(x+y)**2 , (x**2+b+c+a+2*x*y+y**2) ));
(%o1) true
For this purpose, it uses rational simplifier - ratsimp, in order to simplify the difference of two equations. When difference of two equations is simplified to zero, we know they are equal for all possible values:
(%i2) ratsimp( ((a+b+c)+(x+y)**2) - ((x**2+b+c+a+2*x*y+y**2)) );
(%o2) 0
This answer, just shows direction (like other answers). If you know about something similar, that can be used as a part of C++ Unix program - programming library ? Good C/C++ binding similar tool to this. Please, post new answer.
I need to solve a few mathematical equations in my application. Here's a typical example of such an equation:
a + b * c - d / e = a
Additional rules:
b % 10 = 0
b >= 0
b <= 100
Each number must be integer
...
I would like to get the possible solution sets for a, b, c, d and e.
Are there any libraries out there, either open source or commercial, which I can use to solve such an equation? If yes, what kind of result do they provide?
Solving linear systems can generally be solved using linear programming. I'd recommend taking a look at Boost uBLAS for starters - it has a simple triangular solver. Then you might checkout libraries targeting more domain specific approaches, perhaps QSopt.
You're venturing into the world of numerical analysis, and here be dragons. Seemingly small differences in specification can make a huge difference in what is the right approach.
I hesitate to make specific suggestions without a fairly precise description of the problem domain. It sounds superficiall like you are solving constrained linear problems that are simple enough that there are a lot of ways to do it but "..." could be a problem.
A good resource for general solvers etc. would be GAMS. Much of the software there may be a bit heavy weight for what you are asking.
You want a computer algebra system.
See https://stackoverflow.com/questions/160911/symbolic-math-lib, the answers to which are mostly as relevant to c++ as to c.
I know it is not your real question, but you can simplify the given equation to:
d = b * c * e with e != 0
Pretty sure Numerical Recipes will have something
You're looking for a computer algebra system, and that's not a trivial thing.
Lot's of them are available, though, try this list at Wikipedia:
http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems
-Adam
This looks like linear programming. Does this list help?
In addition to the other posts. Your constraint sets make this reminiscent of an integer programming problem, so you might want to check that kind of thing out as well. Perhaps your problem can be (re-)stated as one.
You must know, however that the integer programming problems tends to be one of the harder computational problems so you might end up using many clock cycles to crack it.
Looking only at the "additional rules" part it does look like linear programming, in which case LINDO or a similar program implementing the simplex algorithm should be fine.
However, if the first equation is really typical it shows yours is NOT a linear algebra problem - no 2 variables multiplying or dividing each other should appear on a linear equation!
So I'd say you definitely need either a computer algebra system or solve the problem using a genetic algorithm.
Since you have restrictions similar to those found in linear programming though you're not quite there, if you just want a solution to your specific problem I'd say pick up any of the libraries mentioned at the end of Wikipedia's article on genetic algorithms and develop an app to give you the result. If you want a more generalist approach, then you've got to simulate algebraic manipulations on your computer, no other way around.
The TI-89 Calculator has a 'solver' application.
It was built to solve problems like the one in your example.
I know its not a library. But there are several TI-89 emulators out there.