Fenchel-Rockafellar type duality for a non-convex non-differential optimization problem

AbstractA Fenchel-Rockafellar type duality theorem is obtained for a non-convex and non-differentiable maximization problem by embedding the original problem in a family of perturbed problems. The recent results of Ivan Singer are developed in this more general framework. A relationship is also established between the solutions and optimal values of the primal and dual problems using the theory of subdifferential calculus

Second-order subdifferential calculus with applications to tilt stability in optimization

The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of frst-order subdifferential mappings. We develop an extended second-order subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal secondorder chain rule for strongly and fully amenable compositions. The calculus results obtained in this way and computing the second-order subdifferentials for piecewise linear-quadratic functions and their major specifications are applied then to the study of tilt stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms

Rank properties of exposed positive maps

Let \cK and \cH be finite dimensional Hilbert spaces and let \fP denote the cone of all positive linear maps acting from \fB(\cK) into \fB(\cH). We show that each map of the form $\phi(X)=AXA^*$ or $\phi(X)=AX^TA^*$ is an exposed point of \fP. We also show that if a map $\phi$ is an exposed point of \fP then either $\phi$ is rank 1 non-increasing or \rank\phi(P)>1 for any one-dimensional projection P\in\fB(\cK).Comment: 6 pages, last section removed - it will be a part of another pape

Numerical Analysis of the Capacities for Two-Qubit Unitary Operations

We present numerical results on the capacities of two-qubit unitary operations for creating entanglement and increasing the Holevo information of an ensemble. In all cases tested, the maximum values calculated for the capacities based on the Holevo information are close to the capacities based on the entanglement. This indicates that the capacities based on the Holevo information, which are very difficult to calculate, may be estimated from the capacities based upon the entanglement, which are relatively straightforward to calculate.Comment: 9 pages, 10 figure

Minimum L1-distance projection onto the boundary of a convex set: Simple characterization

We show that the minimum distance projection in the L1-norm from an interior point onto the boundary of a convex set is achieved by a single, unidimensional projection. Application of this characterization when the convex set is a polyhedron leads to either an elementary minmax problem or a set of easily solved linear programs, depending upon whether the polyhedron is given as the intersection of a set of half spaces or as the convex hull of a set of extreme points. The outcome is an easier and more straightforward derivation of the special case results given in a recent paper by Briec.Comment: 5 page

Arbitrage and deflators in illiquid markets

This paper presents a stochastic model for discrete-time trading in financial markets where trading costs are given by convex cost functions and portfolios are constrained by convex sets. The model does not assume the existence of a cash account/numeraire. In addition to classical frictionless markets and markets with transaction costs or bid-ask spreads, our framework covers markets with nonlinear illiquidity effects for large instantaneous trades. In the presence of nonlinearities, the classical notion of arbitrage turns out to have two equally meaningful generalizations, a marginal and a scalable one. We study their relations to state price deflators by analyzing two auxiliary market models describing the local and global behavior of the cost functions and constraints

Higher-dimensional multifractal value sets for conformal infinite graph directed Markov systems

We give a description of the level sets in the higher dimensional multifractal formalism for infinite conformal graph directed Markov systems. If these systems possess a certain degree of regularity this description is complete in the sense that we identify all values with non-empty level sets and determine their Hausdorff dimension. This result is also partially new for the finite alphabet case.Comment: 20 pages, 1 figur

Favorable Classes of Lipschitz Continuous Functions in Subgradient Optimization

Clarke has given a robust definition of subgradients of arbitrary Lipschitz continuous functions f on R^n, but for purposes of minimization algorithms it seems essential that the subgradient multifunction partial f have additional properties, such as certain special kinds of semicontinuity, which are not automatic consequences of f being Lipschitz continuous. This paper explores properties of partial f that correspond to f being subdifferentially regular, another concept of Clarke's, and to f being a pointwise supremum of functions that are k times continuously differentiable

Augmented Lagrangians and Marginal Values in Parameterized Optimization Problems

When an optimization problem depends on parameters, the minimum value in the problem as a function of the parameters is typically far from being differentiable. Certain subderivatives nevertheless exist and can be intepreted as generalized marginal values. In this paper such subderivatives are studied in an abstract setting that allows for infinite dimensionality of the decision space. By means of the notion of proximal subgradients, a new general formula of subdifferentiation is established which provides an upper bound for the marginal values in question and a very broad criterion for local Lipschitz continuity of the optimal value function. Augmented Lagrangians are introduced and shown to lead to still sharper estimates in terms of special multiplier vectors. This approach opens a way to taking higher-order optimality conditions into account in such estimates

Lipschitzian Stability in Optimization: The Role of Nonsmooth Analysis

The motivations of nonsmooth analysis are discussed. Appiications are given to the sensitivity of optimal vaiues, the interpretation of Lagrange multipliers, and the stabiiity of constraint systems under perturbation