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I know it is a vague question, but I am confused by why/when we actually want sparsity of a matrix. For example, interior-point methods work better when constraint matrix is sparse. Similarly, it is easier to invert a sparse matrix. We also make use of this property to reduce dimensions.

I would be glad if anyone could provide a general explanation regarding different applications, or point me to references. Thank you..

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I think it mainly comes down to being able to solve Ax=b quickly. For large problems we have no hope unless A is sparse (or has other structure we can exploit).

In interior point methods, at each iteration we have to solve a large linear system. If the constraint matrix is sparse, then the system we solve at each iteration may be sparse also so we could solve it quickly.

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