Enhanced Fritz John stationarity, new constraint qualifications and local error bound for mathematical programs with vanishing constraints

Abstract

In this paper, we study the difficult class of optimization problems called the mathematical programs with vanishing constraints or MPVC. Extensive research has been done for MPVC regarding stationary conditions and constraint qualifications using geometric approaches. We use the Fritz John approach for MPVC to derive the M-stationary conditions under weak constraint qualifications. An enhanced Fritz John type stationary condition is also derived for MPVC, which provides the notion of enhanced M-stationarity under a new and weaker constraint qualification: MPVC-generalized quasinormality. We show that this new constraint qualification is even weaker than MPVC-CPLD. A local error bound result is also established under MPVC-generalized quasinormality.

On an exact penality result and new constraint qualifications for mathematical programs with vanishing constraints

“…we have following relationships in these CQ as shown in [11, Proposition 2.1] and further implication in [15].…”

“…We have used a local error bound result from [15] to establish an exact penalty result for MPVC-tailored penalty function P α under a very weak and new assumption, the MPVCgeneralized quasinormality. This CQ turns out to be strictly stronger than MPVC-ACQ, and has been illustrated by an example.…”

Optimality conditions and Mond–Weir duality for a class of differentiable semi-infinite multiobjective programming problems with vanishing constraints

“…Various stationarity concepts are widely studied in the literature and known to be important optimality conditions for optimization problems with vanishing constraints (see, for example, Dorsch et al 2012;Hoheisel and Kanzow 2007;Hoheisel et al 2012;Khare and Nath 2019;Kazemi and Kanzi 2018). Therefore, we now generalize one of such concepts of a stationary condition.…”

“…Therefore, we now generalize one of such concepts of a stationary condition. Namely, we extend the definition of a so-called KKT condition given by Achtziger and Kanzow (2008) (see also Hoheisel and Kanzow 2008;Kazemi and Kanzi 2018 ) for a scalar optimization problem with vanishing constraint and also the concept of a S-stationary point given by Khare and Nath (2019) for a scalar optimization problem with vanishing constraint to the semiinfinite vectorial case. We also introduce the concept of a "V C-S-stationary point" for the necessary optimality condition established in Theorem 15.…”

Sufficiency and Duality for Interval-Valued Optimization Problems with Vanishing Constraints Using Weak Constraint Qualifications

“…Due to the mathematical challenges and important roles in various fields, mathematical programs with vanishing constraints have attracted many mathematicians in the past decade. Mathematical programming problem with vanishing constraints is a constrained optimization problem and it is closely related to the Mathematical programs with equilibrium constraints, see for example [9,10,14]. This problem was first studied by Achtziger and Kanzow in [2] and this serves as a model for many problems from topology and structural optimization (see [2,5]).…”