Abu Hassan, Malik and Monsi, Mansor and Leong, Wah June
Scaling symmetric rank one update for unconstrained optimization.
Analisis, 9 (1&2).
A basic disadvantage to the symmetric rank one (SR1) update is that the SRI update may not preserve positive definiteness when starting with a positive definite approximation. A simple remedy to this problem is to restart the update with the initial approximation mostly the identity matrix whenever this difficulty arises. However, numerical experience shows that restart with the identity matrix is not a good choice. Instead of using the identity matrix we used a positive multiple of the identity matrix. The used positive scaling factor is the optimal solution of the measure defined by the problem - maximize the determinant subject to a bound of 1 on the largest eigenvalue. This measure is motivated by considering the volume of the symmetric difference of the two ellipsoids, which arise from the current and updated quadratic models in quasi-Newton methods. A replacement in the form of positive multiple of identity matrix is provided for the SR1 when it is not positive definite. Our experiments indicate that with such simple scale, the effectiveness of the SR1 method is increased dramatically.
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