A Sequential Optimality Condition Related to the Quasi-normality Constraint Qualification and Its Algorithmic Consequences

Abstract

In the present paper, we prove that the augmented Lagrangian method converges to KKT points under the quasinormality constraint qualification, which is associated with the external penalty theory. An interesting consequence is that the Lagrange multipliers estimates computed by the method remain bounded in the presence of the quasinormality condition. In order to establish a more general convergence result, a new sequential optimality condition for smooth constrained optimization, called PAKKT, is defined. The new condition takes into account the sign of the dual sequence, constituting an adequate sequential counterpart to the (enhanced) Fritz-John necessary optimality conditions proposed by Hestenes, and later extensively treated by Bertsekas. PAKKT points are substantially better than points obtained by the classical Approximate KKT (AKKT) condition, which has been used to establish theoretical convergence results for several methods. In particular, we present a simple problem with complementarity constraints such that all its feasible points are AKKT, while only the solutions and a pathological point are PAKKT. This shows the efficiency of the methods that reach PAKKT points, particularly the augmented Lagrangian algorithm, in such problems. We also provided the appropriate strict constraint qualification associated with the PAKKT sequential optimality condition, called PAKKT-regular, and we prove that it is strictly weaker than both quasinormality and cone continuity property. PAKKT-regular connects both branches of these independent constraint qualifications, generalizing all previous theoretical convergence results for the augmented Lagrangian method in the literature.where f : R n → R and X is the feasible set composed of equality and inequality constraints of the form X = {x | h(x) = 0, g(x) ≤ 0}, * This work has been partially supported by CEPID-CeMEAI (FAPESP 2013/07375-0), FAPESP (Grant 2013/05475-7) and CNPq (Grant 303013/2013-3).

Optimality Conditions and Constraint Qualifications for Generalized Nash Equilibrium Problems and Their Practical Implications

“…The recent paper [2] shows global convergence under QN. Since QN is independent of CCP, the results from [6] and from [2] give different global convergence results, while [2] and [6] generalize the original global convergence proof under CPLD in [1]. In a similar fashion, we will prove global convergence of an augmented Lagrangian method for GNEPs under CCP-GNEP (in Corollary 6.1) or QN-GNEP (in Theorem 6.3).…”

“…In the augmented Lagrangian literature of optimization, global convergence has been proved under CPLD in [1], with improvements in [4,5], and more recently, under CCP in [6]. The recent paper [2] shows global convergence under QN. Since QN is independent of CCP, the results from [6] and from [2] give different global convergence results, while [2] and [6] generalize the original global convergence proof under CPLD in [1].…”

“…The main differences of our approaches are that we focus on optimality conditions and CQs that are associated with the proposed method, in particular, our global convergence proof is based on the CCP constraint qualification, while the one in [28] is based on the (stronger) CPLD. We also present a convergence result based on the QN constraint qualification, which extends [2] from optimization to GNEPs, but proving in addition that the dual sequence is bounded. A main contribution of this paper is that AKKT is not an optimality condition for a general GNEP.…”

Condições de otimalidade para otimização cônica

“…Esta propriedade torna as condições sequenciais de otimalidade ferramentas úteis para fornecer naturalmente uma condição de otimalidade perturbada, que é adequada para a definição de critérios de parada e análise de complexidade para vários algoritmos. Além disso, um estudo cuidadoso da relação das condições sequenciais de otimalidade com as medidas clássicas de estacionaridade sob uma condição de qualificação, pode produzir resultados de convergência global sob hipóteses fracas, como podemos ver em [AFSS19,AMRS16,1.3 AMRS18].…”

“…This property makes sequential optimality conditions useful tools for naturally providing a perturbed optimality condition, which is suitable for the definition of stopping criteria and complexity analysis for several algorithms. Also, a careful study of the relation of sequential optimality conditions with classical stationarity measures under a constraint qualification, yields global convergence results under weak assumptions [AFSS19,AMRS16,AMRS18].…”