Necessary optimality conditions and exact penalization for non-Lipschitz nonlinear programs

Abstract

When the objective function is not locally Lipschitz, constraint qualifications are no longer sufficient for Karush-Kuhn-Tucker (KKT) conditions to hold at a local minimizer, let alone ensuring an exact penalization. In this paper, we extend quasi-normality and relaxed constant positive linear dependence (RCPLD) condition to allow the non-Lipschitzness of the objective function and show that they are sufficient for KKT conditions to be necessary for optimality. Moreover, we derive exact penalization results for the following two special cases. When the non-Lipschitz term in the objective function is the sum of a composite function of a separable lower semi-continuous function with a continuous function and an indicator function of a closed subset, we show that a local minimizer of our problem is also a local minimizer of an exact penalization problem under a local error bound condition for a restricted constraint region and a suitable assumption on the outer separable function. When the non-Lipschitz term is the sum of a continuous function and an indicator function of a closed subset, we also show that our problem admits an exact penalization under an extended quasi-normality involving the coderivative of the continuous function.Keywords Non-Lipschitz program · necessary optimality · exact penalization · error bound Mathematics Subject Classification (2010) 90C26 · 90C30 · 90C46

Optimality conditions, approximate stationarity, and applications -- a story beyond Lipschitzness

“…Furthermore, we introduce an associated qualification condition which guarantees M-stationarity of approximately stationary points. As we will show, this generalizes related considerations from Chen et al (2017); Guo and Ye (2018) which were done in a completely finite-dimensional setting. Second, we suggest an augmented Lagrangian method for the numerical solution of geometrically constrained programs and show that it computes approximately stationary points in our new sense.…”

“…The second one, given by condition (6.5), ensures in some sense that the challenging part of the objective function and the constraints of (Q) are somewhat compatible at the reference point. A similar decomposition of qualification conditions has been used in Chen et al (2017); Guo and Ye (2018) in order to ensure M-stationarity of standard nonlinear problems in finite dimensions with a composite objective function. In the latter papers, the authors referred to a condition of type (6.5) as basic qualification, and this terminology can be traced back to the works of Mordukhovich, see e.g.…”

An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints

“…image restoration or signal processing. Necessary optimality conditions and constraint qualifications addressing (5.12) can be found in [31]. In [22], the authors suggest to handle (5.12) numerically with the aid of an augmented Lagrangian method (without safeguarding) based on the (partially) augmented Lagrangian function (4.1) and the subproblems…”

Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs

“…Other than using it as a constraint qualification to ensure the KKT condition holds, RCPLD is also used in the convergence analysis of the augmented Lagrangian method to obtain a KKT point (see e.g., [1,3,15,28]). Recently, Guo and Ye [13,Corollary 3] extended RCPLD to the case where there is an extra abstract constraint set and showed that it is still a constraint qualification. In [11,Definition 4.3], a version of RCPLD called MPEC RCPLD was introduced for the mathematical programs with equilibrium constraints (MPEC) and was shown in [10,Corollary 4.1] that it is a constraint qualification for M-stationary conditions.…”

“…Note that Definition 1.1 is weaker than the one defined in Guo and Ye [13,Corollary 3] for the system containing only smooth equality and inequality constraints and one abstract constraint, in which the stronger condition {∇g i (x)} i∈I 4 ∪ {∇h i (x)} i∈I 1 is linearly dependent for every…”