6,056 research outputs found
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 or is an
exposed point of \fP. We also show that if a map is an exposed point
of \fP then either 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
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