5,460 research outputs found
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
Quasiconvex Constrained Multicriteria Continuous Location Problems: Structure of Nondominated Solution Sets
In this paper, we consider constrained multicriteria continuous location problems in two-dimensional spaces. In the literature, the continuous multicriteria location problem in two-dimensional spaces has received special attention in the last years, although only particular instances of convex functions have been considered. Our approach only requires the functions to be strictly quasiconvex and inf-compact. We obtain a geometrical description that provides a unified approach to handle multicriteria location models in two-dimensional spaces which has been implemented in MATHEMATIC
Optimal designs for rational function regression
We consider optimal non-sequential designs for a large class of (linear and
nonlinear) regression models involving polynomials and rational functions with
heteroscedastic noise also given by a polynomial or rational weight function.
The proposed method treats D-, E-, A-, and -optimal designs in a
unified manner, and generates a polynomial whose zeros are the support points
of the optimal approximate design, generalizing a number of previously known
results of the same flavor. The method is based on a mathematical optimization
model that can incorporate various criteria of optimality and can be solved
efficiently by well established numerical optimization methods. In contrast to
previous optimization-based methods proposed for similar design problems, it
also has theoretical guarantee of its algorithmic efficiency; in fact, the
running times of all numerical examples considered in the paper are negligible.
The stability of the method is demonstrated in an example involving high degree
polynomials. After discussing linear models, applications for finding locally
optimal designs for nonlinear regression models involving rational functions
are presented, then extensions to robust regression designs, and trigonometric
regression are shown. As a corollary, an upper bound on the size of the support
set of the minimally-supported optimal designs is also found. The method is of
considerable practical importance, with the potential for instance to impact
design software development. Further study of the optimality conditions of the
main optimization model might also yield new theoretical insights.Comment: 25 pages. Previous version updated with more details in the theory
and additional example
On the Necessity of the Sufficient Conditions in Cone-Constrained Vector Optimization
The object of investigation in this paper are vector nonlinear programming
problems with cone constraints. We introduce the notion of a Fritz John
pseudoinvex cone-constrained vector problem. We prove that a problem with cone
constraints is Fritz John pseudoinvex if and only if every vector critical
point of Fritz John type is a weak global minimizer. Thus, we generalize
several results, where the Paretian case have been studied.
We also introduce a new Frechet differentiable pseudoconvex problem. We
derive that a problem with quasiconvex vector-valued data is pseudoconvex if
and only if every Fritz John vector critical point is a weakly efficient global
solution. Thus, we generalize a lot of previous optimality conditions,
concerning the scalar case and the multiobjective Paretian one.
Additionally, we prove that a quasiconvex vector-valued function is
pseudoconvex with respect to the same cone if and only if every vector critical
point of the function is a weak global minimizer, a result, which is a natural
extension of a known characterization of pseudoconvex scalar functions.Comment: 12 page
An Inequality Approach to Approximate Solutions of Set Optimization Problems in Real Linear Spaces
This paper explores new notions of approximate minimality in set optimization using a set approach. We propose characterizations of several approximate minimal elements of families of sets in real linear spaces by means of general functionals, which can be unified in an inequality approach. As particular cases, we investigate the use of the prominent Tammer–Weidner nonlinear scalarizing functionals, without assuming any topology, in our context. We also derive numerical methods to obtain approximate minimal elements of families of finitely many sets by means of our obtained results
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