Davidon-Broyden Rank-one Minimization Methods in Hilbert Space with Application to Optimal Control Problems
National Aeronautics and Space Administration, 1972 - Control theory - 31 pages
The Davidon-Broyden class of rank one, quasi-Newton minimization methods is extended from Euclidean spaces to infinite-dimensional, real Hilbert spaces. For several techniques of choosing the step size, conditions are found which assure convergence of the associated iterates to the location of the minimum of a positive definite quadratic functional. For those techniques, convergence is achieved without the problem of the computation of a one-dimensional minimum at each iteration. The application of this class of minimization methods for the direct computation of the solution of an optimal control problem is outlined. The performance of various members of the class are compared by solving a sample optimal control problem. Finally, the sample problem is solved by other known gradient methods, and the results are compared with those obtained with the rank one quasi-Newton methods.
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ansn APPLICATION TO OPTIMAL applied basic algorithm bounded set Broyden ref choice choosing the step chosen by method class of minimization Class of Rank-One conjugate gradient method convex hull Corollary Davidon-Broyden Class denote DFP algorithm DFP method element of H equation 12 Euclidean spaces example problem Figure finite number form a Schauder functional evaluations given by equation Goldfarb ref Hence Horwitz and Sarachik ieN(n infinite-dimensional real Hilbert J(un J(xn Langley Research Center lemma mathematical induction methods of choosing minimum monotone sequence Moreover Number of functional number of steps one-dimensional minimization operator from H optimal control problem positive self-adjoint linear Proof quadratic functional quasi-Newton methods Quasi-Newton Minimization Methods rank-one algorithms rate of convergence real Hilbert space real numbers sample optimal control Sarachik ref Schauder basis self-adjoint operator SPACE WITH APPLICATION steepest descent strongly positive self-adjoint theorem tion v(n+l variables x e H x,Ax