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Powell's dog leg method, also called Powell's hybrid method, is an iterative optimisation algorithm for the solution of non-linear least squares problems, introduced in 1970 by Michael J. D. Powell. [1] Similarly to the Levenberg–Marquardt algorithm, it combines the Gauss–Newton algorithm with gradient descent, but it uses an explicit trust ...
The general idea behind trust region methods is known by many names; the earliest use of the term seems to be by Sorensen (1982). [1] A popular textbook by Fletcher (1980) calls these algorithms restricted-step methods. [2]
LMA can also be viewed as Gauss–Newton using a trust region approach. The algorithm was first published in 1944 by Kenneth Levenberg, [1] while working at the Frankford Army Arsenal. It was rediscovered in 1963 by Donald Marquardt, [2] who worked as a statistician at DuPont, and independently by Girard, [3] Wynne [4] and Morrison. [5]
The Gauss–Newton method is obtained by ignoring the second-order derivative terms (the second term in this expression). That is, the Hessian is approximated by H j k ≈ 2 ∑ i = 1 m J i j J i k , {\displaystyle H_{jk}\approx 2\sum _{i=1}^{m}J_{ij}J_{ik},}
The linearizations are linear programming problems, which can be solved efficiently. As the linearizations need not be bounded, trust regions or similar techniques are needed to ensure convergence in theory. [2] SLP has been used widely in the petrochemical industry since the 1970s. [3]
Powell's method, strictly Powell's conjugate direction method, is an algorithm proposed by Michael J. D. Powell for finding a local minimum of a function. The function need not be differentiable, and no derivatives are taken. The function must be a real-valued function of a fixed number of real-valued inputs.
One such method is the famous Babylonian method, which is given by x k+1 = (x k + 2/x k)/2. Another method, called "method X", is given by x k +1 = ( x k 2 − 2) 2 + x k . [ note 1 ] A few iterations of each scheme are calculated in table form below, with initial guesses x 0 = 1.4 and x 0 = 1.42.
It addressed the instability issue of another algorithm, the Deep Q-Network (DQN), by using the trust region method to limit the KL divergence between the old and new policies. However, TRPO uses the Hessian matrix (a matrix of second derivatives) to enforce the trust region, but the Hessian is inefficient for large-scale problems. [1]