[For the entire collection see Zbl 0668.00024.]
Variable metric gradient algorithms for unconstrained minimization amount to generalized steepest-descent iterations, (1a) \(u\to u+\sigma v\), (1b) \(v=-S\nabla J(u)\), where u is a vector in a real Hilbert space \({\mathcal U}\) with inner product \(<u,v>\), S is a positive-definite bounded linear transformation on \({\mathcal U}\), \(\sigma\) is a positive real number, and S and \(\sigma\) are determined at each stage by auxiliary rules designed to decrease the continuously differentiable real-valued function J, to drive \(\| \nabla J(u)\|\) to zero, and to ensure rapid convergence to any nearby nonsingular relative minimizer for J. The quantity \(S\nabla J(u)\) appearing in (1b) may be viewed as the gradient of J in the metric derived from the weighted inner product \(<u,v>_ S=<u,S^{-1}v>\); hence the name “variable metric” gradient method for (1).
A related family of constrained minimization algorithms is obtained from (1) by adding a projection step, i.e., by implementating the recursion (2a) \(u\to P_{\Omega}(u+\sigma v)\), (2b) \(v=-S\nabla J(u)\), where \(\Omega\) is a given nonempty set in which the minimizers of J are sought, \(P_{\Omega}\) denotes nearest point projection into \(\Omega\) relative to the ground metric \(\| u\|^ 2=<u,u>\) in \({\mathcal U}\), and S and \(\sigma\) are chosen to decrease J, to force some appropriate constrained minimization counterpart of \(\| \nabla J(u)\|\) to zero, and to induce rapid local convergence to nonsingular relative minimizers of J in \(\Omega\).
We begin with a brief examination of the unconstrained variable metric gradient scheme (1), since the general factors that govern the selection of S and \(\sigma\) are well understood and readily accessible in this special case. A modified step length rule suitable for unscaled gradient projection \((S=I)\) is described next, and the resulting iteration (2) is seen to behave much like the ordinary steepest-descent method for unconstrained problems. These considerations provide a paradigm for the subsequent extension to scaled gradient projection; however, certain new considerations enter at this level of generality. Furthermore, it is possible to achieve major simplifications in the calculation of -S\(\nabla J(u)\) for the kind of structured objective functions commonly seen in optimal control problems. Specialized variable metric gradient projection methods that capitalize on this structure are formulated for discrete- time optimal control problems and their continuous-time limits.