G. Di Pillo

Exact penalty functions in constrained optimization

In this paper formal definitions of exactness for penalty functions are introduced and sufficient conditions for a penalty function to be exact according to these definitions are stated, thus providing a unified framework for the study of both nondifferentiable and continuously differentiable penalty functions. In this framework the best-known classes of exact penalty functions are analyzed, and new results are established concerning the correspondence between the solutions of the constrained problem and the unconstrained minimizers of the penalty functions.

A New Class of Augmented Lagrangians in Nonlinear Programming

In this paper a new class of augmented Lagrangians is introduced, for solving equality constrained problems via unconstrained minimization techniques. It is proved that a solution of the constrained problem and the corresponding values of the Lagrange multipliers can be found by performing a single unconstrained minimization of the augmented Lagrangian. In particular, in the linear quadratic case, the solution is obtained by minimizing a quadratic function. Numerical examples are reported.

Exact Penalty Methods

Exact penalty methods for the solution of constrained optimization problems are based on the construction of a function whose unconstrained minilnizing points are also solution of the constrained problem. In the first part of this paper we recall some definitions concerning exactness properties of penalty functions, of barrier functions, of augmented Lagrangian functions, and discuss under which assumptions on the constrained problem these properties can be ensured. In the second part of the paper we consider algorithmic aspects of exact penalty methods; in particular we show that, by making use of continuously differentiable functions that possess exactness properties, it is possible to define implementable algorithms that are globally convergent with superlinear convergence rate towards KKT points of the constrained problem.

A smooth method for the finite minimax problem

We consider unconstrained minimax problems where the objective function is the maximum of a finite number of smooth functions. We prove that, under usual assumptions, it is possible to construct a continuously differentiable function, whose minimizers yield the minimizers of the max function and the corresponding minimum values. On this basis, we can define implementable algorithms for the solution of the minimax problem, which are globally convergent at a superlinear convergence rate. Preliminary numerical results are reported.

A New Version of the Price‘s Algorithm for Global Optimization

We present an algorithm for finding a global minimum of a multimodal,multivariate function whose evaluation is very expensive, affected by noise andwhose derivatives are not available. The proposed algorithm is a new version ofthe well known Prices algorithm and its distinguishing feature is that ittries to employ as much as possible the information about the objectivefunction obtained at previous iterates. The algorithm has been tested on alarge set of standard test problems and it has shown a satisfactorycomputational behaviour. The proposed algorithm has been used to solveefficiently some difficult optimization problems deriving from the study ofeclipsing binary star light curves.

An exact penalty function method with global convergence properties for nonlinear programming problems

In this paper a new continuously differentiable exact penalty function is introduced for the solution of nonlinear programming problems with compact feasible set. A distinguishing feature of the penalty function is that it is defined on a suitable bounded open set containing the feasible region and that it goes to infinity on the boundary of this set. This allows the construction of an implementable unconstrained minimization algorithm, whose global convergence towards Kuhn-Tucker points of the constrained problem can be established.

A Continuously Differentiable Exact Penalty Function for Nonlinear Programming Problems with Inequality Constraints

In this paper it is shown that, given a nonlinear programming problem with inequality constraints, it is possible to construct a continuously differentiable exact penalty function whose global or local unconstrained minimizers correspond to global or local solutions of the constrained problem.

A new augmented Lagrangian function for inequality constraints in nonlinear programming problems

In this paper, a new augmented Lagrangian function is introduced for solving nonlinear programming problems with inequality constraints. The relevant feature of the proposed approach is that, under suitable assumptions, it enables one to obtain the solution of the constrained problem by a single unconstrained minimization of a continuously differentiable function, so that standard unconstrained minimization techniques can be employed. Numerical examples are reported.

Exact penalization via dini and hadamard conditional derivatives

Exact penalty functions for nonsmooth constrained optimization problems are analyzed tfy using the notion of (Dini) Hadamard directional derivative with respect to the constraint set. Weak conditions are given guaranteeing equivalence of the sets of stationary, global minimum, local minimum points of the constrained problem and of the penalty function

An approach to constrained global optimization based on exact penalty functions

In the field of global optimization many efforts have been devoted to solve unconstrained global optimization problems. The aim of this paper is to show that unconstrained global optimization methods can be used also for solving constrained optimization problems, by resorting to an exact penalty approach. In particular, we make use of a non-differentiable exact penalty function