Solving a class of linear projection equations
The author proposes four new methods to find search directions when solving by the author's projection and contraction method [Appl. Math. Optimization 25, No. 3, 247-262 (1992; Zbl 0767.90086)] the linear projection equations of the class \(u= P_ \Omega [u- (Mu+ q)]\), where \(M\in \mathbb{R}^{n\times n}\) is a positive semidefinite (not necessarily symmetric) matrix, \(q\in \mathbb{R}^ n\), \(\Omega\subset \mathbb{R}^ n\) is a closed convex set and \(P_ \Omega(\cdot)\) denotes the projection on the set \(\Omega\). He concentrates on the cases when the solution set of such an equation is nonempty and the projection onto \(\Omega\) is simple to carry out (e.g. when \(\Omega\) is a general orthant, a box, a sphere, a cylinder or a subspace). The previous variant of the projection and contraction method has many advantages, but the search direction applied in it may lead to a very slow convergence for ill-conditioned problems. The proposed methods can be viewed as extensions of well-known methods for unconstrained optimization. The first method seems namely to be an extension of the steepest descent method, the second one an extension of the Newton method, the third one can be regarded as combination of these two and the fourth one an extension of the Levenberg-Marquardt method. The author proves their linear convergence only, but he is convinced that the proposed directions are better than his original one and that the use of Newton-like direction will lead to a substantial improvement in computational efficiency. He announces also the developing of these methods onto nonlinear problems.
scientific article; zbMATH DE number 691559