×

Penalty and barrier methods: A unified framework. (English) Zbl 0953.90045

Summary: It is established that many optimization problems may be formulated in terms of minimizing a function \(x\rightarrow f_0 (x) + H_\infty(f_1 (x), f_2 (x),\ldots,f_m (x)) + L_\infty(Ax-b)\), where the \(f_i\) are closed functions defined on \(\mathbb{R}^N\), and where \(H_\infty\) and \(L_\infty\) are the recession functions of closed, proper, convex functions \(H\) and \(L\). \(A\) is a linear transformation from \(\mathbb{R}^N\) to a finite dimensional vector space \(Y\) with \(b\in Y\). A generic algorithm, based on the properties of recession functions, is proposed. This algorithm not only encompasses almost all penalty and barrier methods in nonlinear programming and in semidefinite programming, but also generates new types of methods. Primal and dual convergence theorems are given.

MSC:

90C25 Convex programming
90C31 Sensitivity, stability, parametric optimization
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
PDFBibTeX XMLCite
Full Text: DOI