Constant Positive Linear Dependence Research Articles

Derivative-free methods for nonlinear programming with general lower-level constraints

Augmented Lagrangian methods for derivative-free continuous optimization with constraints are introduced in this paper. The algorithms inherit the convergence results obtained by Andreani, Birgin, Martinez and Schuverdt for the case in which analytic derivatives exist and are available. In particular, feasible limit points satisfy KKT conditions under the Constant Positive Linear Dependence (CPLD) constraint qualification. The form of our main algorithm allows us to employ well established derivative-free subalgorithms for solving lower-level constrained subproblems. Numerical experiments are presented.

On Augmented Lagrangian Methods with General Lower-Level Constraints

Augmented Lagrangian methods with general lower-level constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems in which the constraints are only of the lower-level type. Inexact resolution of the lower-level constrained subproblems is considered. Global convergence is proved using the constant positive linear dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The resolution of location problems in which many constraints of the lower-level set are nonlinear is addressed, employing the spectral projected gradient method for solving the subproblems. Problems of this type with more than $3 \times 10^6$ variables and $ 14 \times 10^6$ constraints are solved in this way, using moderate computer time. All the codes are available at http://www.ime.usp.br/$\sim$egbirgin/tango/.

Augmented Lagrangian methods under the constant positive linear dependence constraint qualification

Two Augmented Lagrangian algorithms for solving KKT systems are introduced. The algorithms differ in the way in which penalty parameters are updated. Possibly infeasible accumulation points are characterized. It is proved that feasible limit points that satisfy the Constant Positive Linear Dependence constraint qualification are KKT solutions. Boundedness of the penalty parameters is proved under suitable assumptions. Numerical experiments are presented.

On the Relation between Constant Positive Linear Dependence Condition and Quasinormality Constraint Qualification

The constant positive linear dependence (CPLD) condition for feasible points of nonlinear programming problems was introduced by Qi and Wei (Ref. 1) and used in the analysis of SQP methods. In that paper, the authors conjectured that the CPLD could be a constraint qualification. This conjecture is proven in the present paper. Moreover, it is shown that the CPLD condition implies the quasinormality constraint qualification, but that the reciprocal is not true. Relations with other constraint qualifications are given.

Corrigendum: On the Constant Positive Linear Dependence Condition and Its Application to SQP Methods

We correct an assertion in the quoted paper [Qi and Wei, SIAM J. Optim., 10 (2000), pp. 963--981].

On the Constant Positive Linear Dependence Condition and Its Application to SQP Methods

In this paper, we introduce a constant positive linear dependence condition (CPLD), which is weaker than the Mangasarian--Fromovitz constraint qualification (MFCQ) and the constant rank constraint qualification (CRCQ). We show that a limit point of a sequence of approximating Karush--Kuhn--Tucker (KKT) points is a KKT point if the CPLD holds there. We show that a KKT point satisfying the CPLD and the strong second-order sufficiency conditions (SSOSC) is an isolated KKT point. We then establish convergence of a general sequential quadratical programming (SQP) method under the CPLD and the SSOSC. Finally, we apply these results to analyze the feasible SQP method proposed by Panier and Tits in 1993 for inequality constrained optimization problems. We establish its global convergence under the SSOSC and a condition slightly weaker than the Mangasarian--Fromovitz constraint qualification, and we prove superlinear convergence of a modified version of this algorithm under the SSOSC and a condition slightly weaker than the linear independence constraint qualification.