Inexact variable metric method for convex-constrained optimization problems

Inexact gradient projection method with relative error tolerance

“…Here, • F denotes the Frobenius matrix norm A F = A, A , where the inner product is given by A, B = tr(A T B). Problem (41) and its variants appear in applications in different areas such as statistics, physics and economics [15,18,26,50], and were considered, for example, in the numerical tests of [8,24,31,32].…”

“…Note that if rank(W * ) ≤ p, then W p coincides with W * . As far as we know, this approach was first proposed in [2] and was also used in [24,23].…”

“…This means that we stopped the execution of the main algorithms when the above convergence metric is satisfied for two consecutive iterations. We used a similar strategy as in [24] to generate the test instances of (41). Given the dimensions n and m, with m ≥ n, we randomly generate A (a sparse matrix with density 10 −4 ) with elements between (−1, 1).…”

On the inexact scaled gradient projection method

“…We randomly generated 100 instances of each problem as described in Section 6.1. The dimension of the problems and the parameter c in (42) were also taken arbitrarily. For Problem I, we choose 100 ≤ n ≤ 800 and 10 ≤ c ≤ 50, whereas for Problem II, we choose 10 ≤ n ≤ 200 and 100 ≤ c ≤ 1000.…”

“…For this reason, it may not be justified to carry out exact projections when the iterates are far from the solution of the problem. In order to reduce the computational effort spent on projections, inexact procedures that become more and more accurate when approaching the solution, have been proposed, resulting in more efficient methods; see for exemple [13,16,38,42,60,64,57]. On the other hand, non-monotonous searches can improve the probability of finding an optimal global solution, in addition to potentially improving the speed of convergence of the method as a whole, see for example [24,55,63].…”