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The paper is devoted to the calmness from below/from above for the optimal value function of parametric optimization problems, where we are specifically interested in perturbed semi-infinite programs. A main intention is to revisit classical results and to derive refinements of them. In particular, we show in the context of semi-infinite optimization that calmness from below for j holds under quasiconvexity of the data functions and compactness of the solution set, which extends results on the lower semicontinuity of Illustrative examples are given, which demonstrate the significance of the imposed assumptions even in the case of linear and quadratic programs.

In this paper, we use a geometrical approach to sharpen a lower bound given in [5] for the Lipschitz modulus of the optimal value of (finite) linear programs under tilt perturbations of the objective function. The key geometrical idea comes from orthogonally projecting general balls on linear subspaces. Our new lower bound provides a computable expression for the exact modulus (as far as it only depends on the nominal data) in two important cases: when the feasible set has extreme points and when we deal with the Euclidean norm. In these two cases, we are able to compute or estimate the global Lipschitz modulus of the optimal value function in different perturbations frameworks.

We present conditions for Hölder calmness and upper Hölder continuity of optimal solution sets to perturbed optimization problems in finite dimensions. Studies on Hölder type stability were a popular subject in variational analysis already in the 1980s and 1990s, and have become a revived interest in the last decade. In this paper, we focus on conditions for Hölder calmness of the argmin mapping in the case of non-isolated minima. We recall known ideas and results in this context for general as well as special parametric programs, refine them and discuss particular settings, including nonlinear programs and convex semi-infinite optimization problems.

Recently, Cánovas et al. presented an interesting result: the argmin mapping of a linear semi-infinite program under canonical perturbations is calm if and only if some associated linear semi-infinite inequality system is calm. Using classical tools from parametric optimization, we show that the if-direction of this condition holds in a much more general framework of optimization models, while the opposite direction may fail in the general case. In applications to special classes of problems, we apply a more recent result on the intersection of calm multifunctions.

We give a framework for the globalization of a nonsmooth Newton method introduced by B. Kummer. We start with recalling Kummer's approach to convergence analysis of this method and state his results for local convergence. In a second part we give a globalized version of this method. In our approach we use a path-search idea to control the descent. After elaborating the single steps, we analyze and discuss the proof of global convergence resp. of local superlinear or quadratic convergence of the algorithm. We end with illustrating our ideas at some interesting examples.

In a recent article a globalization scheme for a local non-smooth Newton method from Kummer was proposed. This globalized method is particularly suited for non-linear optimization problems with only one time continuously differentiable objective and constraints. We show interesting applications from (generalized) semi-in nite optimization, which lead to such problems. We work out the necessary technical details for the application of the abstract globalized algorithm to non-linear optimization problems and show numerical results.

In Newton's method $0 \in f(x_k) + G(x_k) (x_{k+1} - x_k)$ for solving a nonsmooth equation $f(x) = 0$, the type of approximation of $f$ by some (generally multivalued) mapping $G$ determines not only the convergence behavior of the method, but also the difficulty and the concrete form of the auxiliary problems. With $G(x)=\partial f(x)$ (Clarke's Jacobian) -- like for locally convergent semismooth Newton methods -- and for various other generalized "derivatives", the inclusion is a canonical one, i.e., it describes the usual Newton step if $f$ is continuously differentiable near $x_k$. In our paper, we are interested in Kantorovich--type statements of convergence and study which meaningful hypotheses and auxiliary problems for particular pairs $(f,G)$ may replace those of the classical smooth case. In particular, we point out -- theoretically and by an example -- why the related hypotheses cannot be checked for canonical methods even if $f$ is piecewise linear.

A classical result due to Frank and Wolfe (1956) says that a quadratic function $f$ attains its supremum on a nonempty polyhedron $M$ if $f$ is bounded from above on $M$. In this note, we present a stringent proof of the extension of this result to cubic optimization (known from Andronov, Belousov and Shironin (1982)). Further, we discuss related results. In particular, we bring back to attention Kummer's (1977) generalization of the Frank-Wolfe theorem to the case that $f$ is quadratic, but $M$ is the Minkowski sum of a compact set and a polyhedral cone.

In a nonsmooth Newton method globalized with the aid of a path search was developed in an abstract framework. We refine the convergence analysis given there and adapt this algorithm to certain finite dimensional optimization problems with $C^{1,1}$ data. Such problems arise, for example, in semi-infinite programming under a reduction approach without strict complementarity and in generalized Nash equilibrium models. Using results from parametric optimization and variational analysis, we work out in detail the concrete Newton schemes and the construction of a path for these applications and discuss a series of numerical results for semi-infinite and generalized semi-infinite optimization problems.

We discuss conditions for the Aubin property of solutions to perturbed cone constrained programs, by using and refining results given in Klatte and Kummer (Nonsmooth equations in optimization. Kluwer, Dordrecht, 2002). In particular, we show that constraint nondegeneracy and hence uniqueness of the multiplier is necessary for the Aubin property of the critical point map. Moreover, we give conditions under which the critical point map has the Aubin property if and only if it is locally single-valued and Lipschitz

Our paper deals with the interrelation of optimization methods and Lipschitz stability of multifunctions in arbitrary Banach spaces. Roughly speaking, we show that linear convergence of several first order methods and Lipschitz stability mean the same. Particularly, we characterize calmness and the Aubin property by uniformly (with respect to certain starting points) linear convergence of descent methods and approximate projection methods. So we obtain, e.g., solution methods (for solving equations or variational problems) which require calmness only. The relations of these methods to several known basic algorithms are discussed, and errors in the subroutines as well as deformations of the given mappings are permitted. We also recall how such deformations are related to standard algorithms like barrier, penalty or regularization methods in optimization

The paper revisits characterizations of strong stability and strong regularity of KarushKuhn-Tucker solutions of nonlinear programs with twice differentiable data. We give a unified framework to handle both concepts simultaneously