# Most efficient way to find all unique solutions of q

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Alessandro Maria Laspina on 23 May 2022
Answered: arushi on 5 Oct 2023
I am designing a convex program on MATLAB whose if-else functionality is being modelled with a binary vector q that is 1xN vector. The solution of each element of the vector q is based on another variable t_ref, that is just a vector that represents a uniform step size from a certain t (current time) to a t_f (final time).
t_ref=t:(t_f-t)/(N-1):tf
t_ref is also therefore a 1xN vector.
q is then calculated by the following:
q=(t_ref<=t1) | (t_ref>=t2);
Meaning that for every value of t that is greater than or equal to a certain constant value t2 and , or less than or equal to t1 (both t1 and t2 are between 0 and tf) , q=1, and 0 every where else.
t can take on a value from anywhere between 0 and tf. The only solution I have been able to come up with is running a line search with the input being t and a small enough step size between different t values (between t0 and tf always) and then obtain the matrix of unique solutions of vector q.
I feel like there may be a more efficient trick to solving this? Or am I at the mercy of a P vs. NP problem?

arushi on 5 Oct 2023
Hi Alessandro,
I understand that you would like to efficiently calculate the binary vector q based on the vector t_ref and two constant values t1 and t2. The value of q should be 1 when t_ref is greater than or equal to t2 or less than or equal to t1, and 0 otherwise. Instead of using a line search, you can directly calculate the binary vector q using logical operations without iterating through each element of t_ref. Here's an efficient approach:
q = (t_ref <= t1)| (t_ref >= t2);
In this approach, the logical operator <= compares each element of t_ref with t1, resulting in a logical vector of the same size as t_ref with 1 where the condition is true and 0 otherwise. Similarly, the logical operator >= compares each element of t_ref with t2. Finally, the logical OR operator | combines the two conditions elementwise, resulting in the desired binary vector q .
This approach avoids the need for a line search or iterating through each element of t_ref, making it more efficient.
Note that t1 and t2 should be scalar values, not vectors, as per your description. If you have multiple values of t1 and t2, you can use element-wise comparisons and logical operations to calculate q for each pair of t1 and t2 values.

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