add constraint rule pyomo - constraints

I have been trying to create a constraint based on a certain condition that is the binary output of my function ('Check_Condition'). The type of constraint is just a basic "conflicting" constraint to avoid that two conflicting variables are chosen. The constraint seems to be ignored (I included the print statement as a check), although I do not get any error. Here is the code. I do not want to use nested loops as they considerably slow down the code. Thanks!
model.n = RangeSet(n_operations)
model.triangular = Set(within=model.n * model.n, filter=lambda i, j: j < i)
def con(model, i, j):
if Check_Condition(operations[i],operations[j])==1:
print('added constraint')
return model.x[i] + model.x[j] <= 1
else:
return Constraint.Skip
model.OverlapConst = Constraint(model.triangular, rule=con)

Related

Recursion without executing all methods on each recursive call

What's the term for when a recursive algorithm doesn't always reach all functions on each level? Consider this code:
function f(value):
if val < -10 return g(value + 1)
if var > 10 return h(value - 1)
else return 0
function g(value):
if value % 2 == 0 return f(value / 2)
else return h(value)
function h(value):
if value % 2 == 1 return g(value - 1)
else return h(value - 1)
Each recursive call may call a different function such as when we start with 15:
f(15)
h(14)
h(13)
g(12)
f(6)
return 0
I've never heard of a specific term for what you describe, it's simply referred to as recursion.
The fact that it executes potentially different pathways for certain levels of recursion doesn't really make it special, any more than the same feature in a loop:
for i = 1 to 10:
if i is even:
print i, " is even"
else:
print i, " is odd"
That's still called a loop (not something like "alternating loop" or "iteration dependent behavioural loop"). It's similar to your recursion example.
In fact, recursion always has different behaviour at at least one level (the last one) since, without that, it would recur forever and you would most likely overflow the stack.

How to use predicate exactly in MiniZinc

New MiniZinc user here ... I'm having a problem understanding the syntax of the counting constraint:
predicate exactly(int: n, array[int] of var int: x, int: v)
"Requires exactly n variables in x to take the value v."
I want to make sure each column in my 10r x 30c array has at least one each of 1,2 and 3, with the remaining 7 rows equal to zero.
If i declare my array as
array[1..10,1..30] of var 0..3: s;
how can I use predicate exactly to populate it as I need? Thanks!
Well, the "exactly" constraint is not so useful here since you want at least one occurrence of 1, 2, and 3. It's better to use for example the count function:
include "globals.mzn";
array[1..10,1..30] of var 0..3: s;
solve satisfy;
constraint
forall(j in 1..30) (
forall(c in 1..3) (
count([s[i,j] | i in 1..10],c) >= 1
)
)
;
output [
if j = 1 then "\n" else " " endif ++
show(s[i,j])
| i in 1..10, j in 1..30
];
You don't have do to anything about 0 since the domain is 0..3 and all values that are not 1, 2, or 3 must be 0.
Another constraint is "at_least", see https://www.minizinc.org/2.0/doc-lib/doc-globals-counting.html .
If you don't have read the MiniZinc tutorial (https://www.minizinc.org/downloads/doc-latest/minizinc-tute.pdf), I strongly advice you to. The tutorial teaches you how to think Constraint Programming and - of course - MiniZinc.

Why do i get this error - MATLAB

I have the image and the vector
a = imread('Lena.tiff');
v = [0,2,5,8,10,12,15,20,25];
and this M-file
function y = Funks(I, gama, c)
[m n] = size(I);
for i=1:m
for j=1:n
J(i, j) = (I(i, j) ^ gama) * c;
end
end
y = J;
imshow(y);
when I'm trying to do this:
f = Funks(a,v,2)
I am getting this error:
??? Error using ==> mpower
Integers can only be combined with integers of the same class, or scalar doubles.
Error in ==> Funks at 5
J(i, j) = (I(i, j) ^ gama) * c;
Can anybody help me, with this please?
The error is caused because you're trying to raise a number to a vector power. Translated (i.e. replacing formal arguments with actual arguments in the function call), it would be something like:
J(i, j) = (a(i, j) ^ [0,2,5,8,10,12,15,20,25]) * 2
Element-wise power .^ won't work either, because you'll try to "stuck" a vector into a scalar container.
Later edit: If you want to apply each gamma to your image, maybe this loop is more intuitive (though not the most efficient):
a = imread('Lena.tiff'); % Pics or GTFO
v = [0,2,5,8,10,12,15,20,25]; % Gamma (ar)ray -- this will burn any picture
f = cell(1, numel(v)); % Prepare container for your results
for k=1:numel(v)
f{k} = Funks(a, v(k), 2); % Save result from your function
end;
% (Afterwards you use cell array f for further processing)
Or you may take a look at the other (more efficient if maybe not clearer) solutions posted here.
Later(er?) edit: If your tiff file is CYMK, then the result of imread is a MxNx4 color matrix, which must be handled differently than usual (because it 3-dimensional).
There are two ways I would follow:
1) arrayfun
results = arrayfun(#(i) I(:).^gama(i)*c,1:numel(gama),'UniformOutput',false);
J = cellfun(#(x) reshape(x,size(I)),results,'UniformOutput',false);
2) bsxfun
results = bsxfun(#power,I(:),gama)*c;
results = num2cell(results,1);
J = cellfun(#(x) reshape(x,size(I)),results,'UniformOutput',false);
What you're trying to do makes no sense mathematically. You're trying to assign a vector to a number. Your problem is not the MATLAB programming, it's in the definition of what you're trying to do.
If you're trying to produce several images J, each of which corresponds to a certain gamma applied to the image, you should do it as follows:
function J = Funks(I, gama, c)
[m n] = size(I);
% get the number of images to produce
k = length(gama);
% Pre-allocate the output
J = zeros(m,n,k);
for i=1:m
for j=1:n
J(i, j, :) = (I(i, j) .^ gama) * c;
end
end
In the end you will get images J(:,:,1), J(:,:,2), etc.
If this is not what you want to do, then figure out your equations first.

How to define a parameter recursively in GAMS?

I need to define a set of parameters that have a natural recursive relation.
Here is a MWE where I try to define the factorial function over a set of (nine) parameters S:
$title TitleOfProblem
set S / s1*s9 /;
alias(S, S1, S2);
set delta1(S1,S2);
delta1(S1,S2) = yes$(ord(S1) + 1 = ord(S2));
parameter f(S);
f(S) = 1$(ord(S) = 1) + (ord(S) * sum(S1$(delta1(S1, S)), f(S1)))$(ord(S) > 1);
display f;
"delta1" is a relation containing pairs of elements in sorted order that differ by 1. Logically, the definition of f matches the definition of the factorial function (for inputs 1 to 9), but GAMS doesn't seem to like that f is defined recursively. The output of GAMS compilation looks something like this:
f(S) = 1$(ord(S) = 1) + (ord(S) * sum(S1$(delta1(S1, S)), f(S1)))$(ord(S) > 1);
$141
141 Symbol neither initialized nor assigned
A wild shot: You may have spurious commas in the explanatory
text of a declaration. Check symbol reference list.
Question:
Is it possible to recursively define a parameter in GAMS? If not, what is a work-around?
(P.S. Someone with enough rep should create a tag "GAMS" and add it to this question.)
Someone showed me a solution for my example using a while loop. However, this solution is specific to factorial and does not generalize to an arbitrary recursive function.
$title factorial
set S / s1*s9 /;
parameter f(S);
parameter temp;
Loop(S,
temp=ord(s);
f(S)=ord(s);
While(temp > 1,
f(S) = f(S) * (temp-1);
temp = temp - 1;
);
);
display f;

iterative version of easy recursive algorithm

I have a quite simple question, I think.
I've got this problem, which can be solved very easily with a recursive function, but which I wasn't able to solve iteratively.
Suppose you have any boolean matrix, like:
M:
111011111110
110111111100
001111111101
100111111101
110011111001
111111110011
111111100111
111110001111
I know this is not an ordinary boolean matrix, but it is useful for my example.
You can note there is sort of zero-paths in there...
I want to make a function that receives this matrix and a point where a zero is stored and that transforms every zero in the same area into a 2 (suppose the matrix can store any integer even it is initially boolean)
(just like when you paint a zone in Paint or any image editor)
suppose I call the function with this matrix M and the coordinate of the upper right corner zero, the result would be:
111011111112
110111111122
001111111121
100111111121
110011111221
111111112211
111111122111
111112221111
well, my question is how to do this iteratively...
hope I didn't mess it up too much
Thanks in advance!
Manuel
ps: I'd appreciate if you could show the function in C, S, python, or pseudo-code, please :D
There is a standard technique for converting particular types of recursive algorithms into iterative ones. It is called tail-recursion.
The recursive version of this code would look like (pseudo code - without bounds checking):
paint(cells, i, j) {
if(cells[i][j] == 0) {
cells[i][j] = 2;
paint(cells, i+1, j);
paint(cells, i-1, j);
paint(cells, i, j+1);
paint(cells, i, j-1);
}
}
This is not simple tail recursive (more than one recursive call) so you have to add some sort of stack structure to handle the intermediate memory. One version would look like this (pseudo code, java-esque, again, no bounds checking):
paint(cells, i, j) {
Stack todo = new Stack();
todo.push((i,j))
while(!todo.isEmpty()) {
(r, c) = todo.pop();
if(cells[r][c] == 0) {
cells[r][c] = 2;
todo.push((r+1, c));
todo.push((r-1, c));
todo.push((r, c+1));
todo.push((r, c-1));
}
}
}
Pseudo-code:
Input: Startpoint (x,y), Array[w][h], Fillcolor f
Array[x][y] = f
bool hasChanged = false;
repeat
for every Array[x][y] with value f:
check if the surrounding pixels are 0, if so:
Change them from 0 to f
hasChanged = true
until (not hasChanged)
For this I would use a Stack ou Queue object. This is my pseudo-code (python-like):
stack.push(p0)
while stack.size() > 0:
p = stack.pop()
matrix[p] = 2
for each point in Arround(p):
if matrix[point]==0:
stack.push(point)
The easiest way to convert a recursive function into an iterative function is to utilize the stack data structure to store the data instead of storing it on the call stack by calling recursively.
Pseudo code:
var s = new Stack();
s.Push( /*upper right point*/ );
while not s.Empty:
var p = s.Pop()
m[ p.x ][ p.y ] = 2
s.Push ( /*all surrounding 0 pixels*/ )
Not all recursive algorithms can be translated to an iterative algorithm. Normally only linear algorithms with a single branch can. This means that tree algorithm which have two or more branches and 2d algorithms with more paths are extremely hard to transfer into recursive without using a stack (which is basically cheating).
Example:
Recursive:
listsum: N* -> N
listsum(n) ==
if n=[] then 0
else hd n + listsum(tl n)
Iteration:
listsum: N* -> N
listsum(n) ==
res = 0;
forall i in n do
res = res + i
return res
Recursion:
treesum: Tree -> N
treesum(t) ==
if t=nil then 0
else let (left, node, right) = t in
treesum(left) + node + treesum(right)
Partial iteration (try):
treesum: Tree -> N
treesum(t) ==
res = 0
while t<>nil
let (left, node, right) = t in
res = res + node + treesum(right)
t = left
return res
As you see, there are two paths (left and right). It is possible to turn one of these paths into iteration, but to translate the other into iteration you need to preserve the state which can be done using a stack:
Iteration (with stack):
treesum: Tree -> N
treesum(t) ==
res = 0
stack.push(t)
while not stack.isempty()
t = stack.pop()
while t<>nil
let (left, node, right) = t in
stack.pop(right)
res = res + node + treesum(right)
t = left
return res
This works, but a recursive algorithm is much easier to understand.
If doing it iteratively is more important than performance, I would use the following algorithm:
Set the initial 2
Scan the matrix for finding a 0 near a 2
If such a 0 is found, change it to 2 and restart the scan in step 2.
This is easy to understand and needs no stack, but is very time consuming.
A simple way to do this iteratively is using a queue.
insert starting point into queue
get first element from queue
set to 2
put all neighbors that are still 0 into queue
if queue is not empty jump to 2.

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