Thank you very much. This problem is very similar to Can this be solved in CVX?, with some additional requirements. I have demand vector \bf d. The matrix \bf L has N(=64) rows and P(=72) columns. Each row in \bf L has some supply. My target is to find M rows of \bf L such that the summation of these M vectors satisfies the demand. The vector \alpha_i is an integer variable gives the the index as well as the number of times a particular row in \bf L is chosen. If the row 2 is one of the M rows and is chosen just once then \alpha_2=1. If it was chosen twice then \alpha_2=2.
Now, the supply provided by the each row in \bf L is defined for unit time. The time unit can be second, millisecond or even minute. In this problem , we want to minimize the norm of the difference vector \bf d-s over W(=16) units of time. There are total M opportunities to select the rows. The scheduler selects the rows (M times) within the given cycle of W units of time.
The supply from any selected row (on one of the M different occasions) should last for at least one unit of time. Therefore, the supply from any selected row can last for maximum 13 units of time.
So, in this problem, I need to find the index of the selected rows (\alpha_i)'s and the duration of each selected row so that the norm is minimized.