Gianni Di Pillo

An Augmented Lagrangian Function with Improved Exactness Properties

In this paper we introduce a new exact augmented Lagrangian function for the solution of general nonlinear programming problems. For this Lagrangian function a complete equivalence between its unconstrained minimization on an open set and the solution of the original constrained problem can be established under mild assumptions and without requiring the boundedness of the feasible set of the constrained problem. Moreover we describe an unconstrained algorithmic model which is globally convergent toward KKT pairs of the original constrained problem. The algorithmic model can be endowed with a superlinear rate of convergence by a proper choice of the search direction in the unconstrained minimization, without requiring strict complementarity.

On Exact Augmented Lagrangian Functions in Nonlinear Programming

The problem considered here is the nonlinear programming problem:

A truncated Newton method in an augmented Lagrangian framework for nonlinear programming

{\text{minimize }}f(x){\text{ s}}{\text{.t}}{\text{. }}g(x){\mkern 1mu} \leqslant {\mkern 1mu} 0,

Convergence to Second-Order Stationary Points of a Primal-Dual Algorithm Model for Nonlinear Programming

(NLP) where f : ℝ n → ℝ and g : ℝ n → ℝ m are twice continuously differentiable functions.

An Exact Augmented Lagrangian Function for Nonlinear Programming with Two-Sided Constraints

In this paper we propose a recursive quadratic programming algorithm for nonlinear programming problems with inequality constraints that uses as merit function a differentiable exact penalty function. The algorithm incorporates an automatic adjustment rule for the selection of the penalty parameter and makes use of an Armijo-type line search procedure that avoids the need to evaluate second order derivatives of the problem functions. We prove that the algorithm possesses global and superlinear convergence properties. Numerical results are reported.

Optimal Location of a Measurement Point in a Diffusion Process

In this paper we propose a primal-dual algorithm for the solution of general nonlinear programming problems. The core of the method is a local algorithm which relies on a truncated procedure for the computation of a search direction, and is thus suitable for large scale problems. The truncated direction produces a sequence of points which locally converges to a KKT pair with superlinear convergence rate.The local algorithm is globalized by means of a suitable merit function which is able to measure and to enforce progress of the iterates towards a KKT pair, without deteriorating the local efficiency. In particular, we adopt the exact augmented Lagrangian function introduced in Pillo and Lucidi (SIAM J. Optim. 12:376–406, 2001), which allows us to guarantee the boundedness of the sequence produced by the algorithm and which has strong connections with the above mentioned truncated direction.The resulting overall algorithm is globally and superlinearly convergent under mild assumptions.

Support vector machines for surrogate modeling of electronic circuits

Constrained global optimization problems can be tackled by using exact penalty approaches. In a preceding paper, we proposed an exact penalty algorithm for constrained problems which combines an unconstrained global minimization technique for minimizing a non-differentiable exact penalty function for given values of the penalty parameter, and an automatic updating of the penalty parameter that occurs only a finite number of times. However, in the updating of the penalty parameter, the method requires the evaluation of the derivatives of the problem functions. In this work, we show that an efficient updating can be implemented also without using the problem derivatives, in this way making the approach suitable for globally solving constrained problems where the derivatives are not available. In the algorithm, any efficient derivative-free unconstrained global minimization technique can be used. In particular, we adopt an improved version of the DIRECT algorithm. In addition, to improve the performances, the approach is enriched by resorting to derivative-free local searches, in a multistart framework. In this context, we prove that, under suitable assumptions, for every global minimum point there exists a neighborhood of attraction for the local search. An extensive numerical experience is reported.

Exact augmented lagrangian approach to multilevel optimization of large-scale systems

We define a primal-dual algorithm model (second-order Lagrangian algorithm, SOLA) for inequality constrained optimization problems that generates a sequence converging to points satisfying the second-order necessary conditions for optimality. This property can be enforced by combining the equivalence between the original constrained problem and the unconstrained minimization of an exact augmented Lagrangian function and the use of a curvilinear line search technique that exploits information on the nonconvexity of the augmented Lagrangian function.