Nonlinear rescaling vs. smoothing technique in convex optimization

Abstract.

We introduce an alternative to the smoothing technique approach for constrained optimization. As it turns out for any given smoothing function there exists a modification with particular properties. We use the modification for Nonlinear Rescaling (NR) the constraints of a given constrained optimization problem into an equivalent set of constraints.¶The constraints transformation is scaled by a vector of positive parameters. The Lagrangian for the equivalent problems is to the correspondent Smoothing Penalty functions as Augmented Lagrangian to the Classical Penalty function or MBFs to the Barrier Functions. Moreover the Lagrangians for the equivalent problems combine the best properties of Quadratic and Nonquadratic Augmented Lagrangians and at the same time are free from their main drawbacks.¶Sequential unconstrained minimization of the Lagrangian for the equivalent problem in primal space followed by both Lagrange multipliers and scaling parameters update leads to a new class of NR multipliers methods, which are equivalent to the Interior Quadratic Prox methods for the dual problem.¶We proved convergence and estimate the rate of convergence of the NR multipliers method under very mild assumptions on the input data. We also estimate the rate of convergence under various assumptions on the input data.¶In particular, under the standard second order optimality conditions the NR method converges with Q-linear rate without unbounded increase of the scaling parameters, which correspond to the active constraints.¶We also established global quadratic convergence of the NR methods for Linear Programming with unique dual solution.¶We provide numerical results, which strongly support the theory.