In this paper, we consider a sufficiently broad class of non-linear mathematical programs with disjunctive constraints, which, e.g. include complemetarity/vanishing constraints. We present an extension the concept [Formula: see text]-stationarity which can be easily combined well-known notion M-stationarity to obtain stronger property so-called text]-stationarity. show how (and thus also M-stationarity) efficiently verified for considered problem by computing text]-stationary solutions...

The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This encompasses whole range standard generalized differential (non-directional) notions relies on very weak (non-restrictive) qualification conditions having also character. derived rules facilitate application tools exploiting difficult problems variational analysis including, instance, various stability sensitivity issues....

As a starting point of our research, we show that, for fixed order γ≥1 , each local minimizer rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary the classical sense (corresponding to stationarity 1), satisfies conditions terms coderivative construction γ or asymptotically stationary with respect critical direction as well certain sense. By ruling out latter case constraint qualification not stronger than directional metric subregularity, end up new...

.In this paper, we propose second-order sufficient optimality conditions for a very general nonconvex constrained optimization problem, which covers many prominent mathematical programs. Unlike the existing results in literature, our prove to be sufficient, an essential local minimizer of second order, under merely basic smoothness and closedness assumptions on data defining problem. In part, comprehensive first- variational analysis disjunctive systems demonstrate how objects appearing can...

In the paper we provide new conditions ensuring isolated calmness property and Aubin of parameterized variational systems with constraints depending, apart from parameter, also on solution itself. Such include, e.g., quasi-variational inequalities implicit complementarity problems. Concerning property, possible restrictions imposed parameter are admitted. Throughout paper, tools directional limiting generalized differential calculus employed enabling us to impose only rather weak (non-...

Estimating the regular normal cone to constraint systems plays an important role for derivation of sharp necessary optimality conditions. We present two novel approaches and introduce a new stationarity concept which is stronger than M-stationarity. apply our theory three classes mathematical programs frequently arising in literature.

Abstract In this paper, we characterize Lipschitzian properties of different multiplier-free and multiplier-dependent perturbation mappings associated with the stationarity system a so-called generalized nonlinear program popularized by Rockafellar. Special emphasis is put on investigation isolated calmness property at around point. The latter decisive for locally fast convergence semismooth* Newton-type method Gfrerer Outrata. Our central result characterization point mapping via...

This paper is devoted to the study of tilt stability local minimizers, which plays an important role in both theoretical and numerical aspects optimization. notion has been comprehensively investigated unconstrained framework as well for problems nonlinear programming with $C^2$-smooth data. Available results nonpolyhedral conic programs were obtained only under strong constraint nondegeneracy assumptions. Here we develop approach second-order variational analysis, allows us establish...

Abstract We establish two types of estimates for generalized derivatives set-valued mappings which carry the essence basic patterns observed throughout pile calculus rules. These also illustrate role essential assumptions that accompany these patters, namely calmness on one hand and (fuzzy) inner calmness* other. Afterwards, we study relationship between sufficient conditions various notions (inner) calmness. The aforementioned are applied in order to recover several prominent rules tangents...

We propose an SQP algorithm for mathematical programs with vanishing constraints which solves at each iteration a quadratic program linear constraints. The is based on the newly developed concept of $${\mathcal {Q}}$$ -stationarity (Benko and Gfrerer in Optimization 66(1):61–92, 2017). demonstrate how {Q}}_M$$ -stationary solutions can be obtained. show that all limit points sequence iterates generated by basic method are least M-stationary some extension we also guarantee stronger property points.

In this paper, we study continuity and Lipschitzian properties of set-valued mappings, focusing on inner-type conditions. We introduce new notions inner calmness* and, its relaxation, fuzzy calmness*. show that polyhedral maps enjoy examine (fuzzy) a multiplier mapping associated with constraint systems in depth. Then utilize these to develop some rules generalized differential calculus, mainly for the primal objects (e.g. tangent cones). particular, propose an exact chain rule graphical...

We propose an SQP algorithm for mathematical programs with complementarity constraints which solves at each iteration a quadratic program linear constraints.We demonstrate how strongly M-stationary solutions of this can be obtained by active set method without using enumeration techniques.We show that all limit points the sequence iterates generated our are least M-stationary.

Implicit variables of a mathematical program are which do not need to be optimized but used model feasibility conditions. They frequently appear in several different problem classes optimization theory comprising bilevel programming, evaluated multiobjective optimization, or nonlinear problems with slack variables. In order deal implicit variables, they often interpreted as explicit ones. Here, we first point out that this is light-headed approach induces artificial locally optimal...

In this paper, we readdress the classical topic of second-order sufficient optimality conditions for optimization problems with nonsmooth structure. Based on so-called second subderivative objective function and indicator associated feasible set, one easily obtains abstract form. order to exploit further structure problem, e.g., composite terms in or sets given as (images of) pre-images closed under smooth transformations, make these fully explicit, study calculus rules mild conditions. To...

Much is known about when a locally optimal solution depends in single-valued Lipschitz continuous way on the problem's parameters, including tilt perturbations. less known, however, that and uniquely determined multiplier vector associated with it exhibit dependence as primal-dual pair. In classical nonlinear programming, such advantageous behavior tied to combination of standard strong second-order sufficient condition (SSOC) for local optimality linear independent gradient (LIGC) active...

In this paper, we characterize Lipschitzian properties of different multiplier-free and multiplier-dependent perturbation mappings associated with the stationarity system a so-called generalized nonlinear program popularized by Rockafellar. Special emphasis is put on investigation isolated calmness property at around point. The latter decisive for locally fast convergence semismooth* Newton-type method Gfrerer Outrata. Our central result characterization point mapping via combination an...

As a starting point of our research, we show that, for fixed order $\gamma\geq 1$, each local minimizer rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary the classical sense (corresponding to stationarity $1$), satisfies conditions terms coderivative construction $\gamma$, or asymptotically stationary with respect critical direction as well $\gamma$ certain sense. By ruling out latter case constraint qualification not stronger than directional metric...

We propose an SQP algorithm for mathematical programs with vanishing constraints which solves at each iteration a quadratic program linear constraints. The is based on the newly developed concept of $\mathcal Q$-stationarity [5]. demonstrate how Q_M$-stationary solutions can be obtained. show that all limit points sequence iterates generated by basic method are least M-stationary and some extension we also guarantee stronger property Q_M$-stationarity points.

In this paper we consider a sufficiently broad class of nonlinear mathematical programs with disjunctive constraints, which, e.g., include complemetarity/vanishing constraints. We present an extension the concept ${\mathcal Q}$-stationarity as introduced in recent [2]. can be easily combined well-known notion M-stationarity to obtain stronger property so-called Q}_M$-stationarity. show how Q}_M$-stationarity (and thus also M-stationarity) efficiently verified for considered problem by...