On the Polyak convexity principle and its application to variational analysis

According to a result due to B.T. Polyak, a mapping between Hilbert spaces, which is $C^{1,1}$ around a regular point, carries a ball centered at that point to a convex set, provided that the radius of the ball is small enough. The present paper considers the extension of such result to mappings defined on a certain subclass of uniformly convex Banach spaces. This enables one to extend to such setting a variational principle for constrained optimization problems, already observed in finite dimension, that establishes a convex behaviour for proper localizations of them. Further variational consequences are explored.Comment: 13 page

New Constraint Qualifications for Optimization Problems in Banach Spaces based on Asymptotic KKT Conditions

Optimization theory in Banach spaces suffers from the lack of available constraint qualifications. Despite the fact that there exist only a very few constraint qualifications, they are, in addition, often violated even in simple applications. This is very much in contrast to finite-dimensional nonlinear programs, where a large number of constraint qualifications is known. Since these constraint qualifications are usually defined using the set of active inequality constraints, it is difficult to extend them to the infinite-dimensional setting. One exception is a recently introduced sequential constraint qualification based on asymptotic KKT conditions. This paper shows that this so-called asymptotic KKT regularity allows suitable extensions to the Banach space setting in order to obtain new constraint qualifications. The relation of these new constraint qualifications to existing ones is discussed in detail. Their usefulness is also shown by several examples as well as an algorithmic application to the class of augmented Lagrangian methods

Convexity of the images of small balls through perturbed convex multifunctions

In the present paper, the following convexity principle is proved: any closed convex multifunction, which is metrically regular in a certain uniform sense near a given point, carries small balls centered at that point to convex sets, even if it is perturbed by adding C^{1,1} smooth mappings with controlled Lipschizian behaviour. This result, which is valid for mappings defined on a subclass of uniformly convex Banach spaces, can be regarded as a set-valued generalization of the Polyak convexity principle. The latter, indeed, can be derived as a special case of the former. Such an extension of that principle enables one to build large classes of nonconvex multifunctions preserving the convexity of small balls. Some applications of this phenomenon to the theory of set-valued optimization are proposed and discussed

Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods

This paper deals with quasi-variational inequality problems (QVIs) in a generic Banach space setting. We provide a theoretical framework for the analysis of such problems which is based on two key properties: the pseudomonotonicity (in the sense of Brezis) of the variational operator and a Mosco-type continuity of the feasible set mapping. We show that these assumptions can be used to establish the existence of solutions and their computability via suitable approximation techniques. In addition, we provide a practical and easily verifiable sufficient condition for the Mosco-type continuity property in terms of suitable constraint qualifications. Based on the theoretical framework, we construct an algorithm of augmented Lagrangian type which reduces the QVI to a sequence of standard variational inequalities. A full convergence analysis is provided which includes the existence of solutions of the subproblems as well as the attainment of feasibility and optimality. Applications and numerical results are included to demonstrate the practical viability of the method.Comment: 27 pages, 3 figure

On a class of convex sets with convex images and its application to nonconvex optimization

In the present paper, conditions under which the images of uniformly convex sets through $C^{1,1}$ regular mappings between Banach spaces remain convex are established. These conditions are expressed by a certain quantitative relation betweeen the modulus of convexity of a given set and the global regularity behaviour of the mapping on it. Such a result enables one to extend to a wide subclass of convex sets the Polyak's convexity principle, which was originally concerned with images of small balls around points of Hilbert spaces. In particular, the crucial phenomenon of the preservation of convexity under regular $C^{1,1}$ transformations is shown to include the class of $r$-convex sets, where the value of $r$ depends on the regularity behaviour of the involved transformation. Two consequences related to nonconvex optimization are discussed: the first one is a sufficient condition for the global solution existence for infinite-dimensional constrained extremum problems; the second one provides a zero-order Lagrangian type characterization of optimality in nonlinear mathematical programming.Comment: This paper has been withdrawn by the author due to errors found in a proo

An extension of the Polyak convexity principle with application to nonconvex optimization

The main problem considered in the present paper is to single out classes of convex sets, whose convexity property is preserved under nonlinear smooth transformations. Extending an approach due to B.T. Polyak, the present study focusses on the class of uniformly convex subsets of Banach spaces. As a main result, a quantitative condition linking the modulus of convexity of such kind of set, the regularity behaviour around a point of a nonlinear mapping and the Lipschitz continuity of its derivative is established, which ensures the images of uniformly convex sets to remain uniformly convex. Applications of the resulting convexity principle to the existence of solutions, their characterization and to the Lagrangian duality theory in constrained nonconvex optimization are then discussed

Subdifferential Stability Analysis for Convex Optimization Problems via Multiplier Sets

This paper discusses differential stability of convex programming problems in Hausdorff locally convex topological vector spaces. Among other things, we obtain formulas for computing or estimating the subdifferential and the singular subdifferential of the optimal value function via suitable multiplier sets

Pontryagin Maximum Principle for Control Systems on Infinite Dimensional Manifolds

We discuss a mathematical framework for analysis of optimal control problems on infinite-dimensional manifolds. Such problems arise in study of optimization for partial differential equations with some symmetry. It is shown that some nonsmooth analysis methods and Lagrangian charts techniques can be used for study of global variations of optimal trajectories of such control systems and derivation of maximum principle for them

On Error Bounds and Multiplier Methods for Variational Problems in Banach Spaces

This paper deals with a general form of variational problems in Banach spaces which encompasses variational inequalities as well as minimization problems. We prove a characterization of local error bounds for the distance to the (primal-dual) solution set and give a sufficient condition for such an error bound to hold. In the second part of the paper, we consider an algorithm of augmented Lagrangian type for the solution of such variational problems. We give some global convergence properties of the method and then use the error bound theory to provide estimates for the rate of convergence and to deduce boundedness of the sequence of penalty parameters. Finally, numerical results for optimal control, Nash equilibrium problems, and elliptic parameter estimation problems are presented.Comment: 27 page

Strong Duality of Linear Optimisation Problems over Measure Spaces

In this work we present two particular cases of the general duality result for linear optimisation problems over signed measures with infinitely many constraints in the form of integrals of functions with respect to the decision variables (the measure in question) for which strong duality holds. In the first case the optimisation problems are over measures with $L^p$ density functions with $1 < p < \infty$. In the second case we consider a semi-infinite optimisation problem where finitely many constraints are given in form of bounds on integrals. The latter case has a particular importance in practice where the model can be applied in robust risk management and model-free option pricing