Single-iterative saddle-point algorithms for solving constrained optimization problems via augmented Lagrangians

Abstract

The theory of augmented Lagrange functions makes it possible to unify most of known constrained optimization algorithms as special cases of a general Newton-like algorithms of saddle-point seeking. Penalty methods, gradient projection methods, multiplier methods, quadratic approximation methods etc can be treated uniformly in this framework. Moreover, a new class of single-iterative saddlepoint algorithm generalising the previously mentioned methods results from this approach. The particular algorithms are described in detail : a double-variable metric algorithm with variable metrics used for the approximation of the primal and the dual hessian, and a single-variable metric algorithm with variable metrics used for the approximation of the primal hessian and with an inversion of the dual hessian. Both algorithms have finite termination in quadratic case and quadratic convergence in more general case; both are effective as well for nonlinear as for linear constraints. Examples of application and results of preliminary tests are given at the end.