2010· Journal of Convex Analysis· Vol 17, pp. 13-33· 0 citations· 8 references
Abstract
Different methods are available to solve a constrained optimization problem where the objective function is convex and the constraint set is specified by a linear system of a finite number of linear inequalities. In particular, the problem can be formulated as an optimization problem with a unique constraint involving a polyhedral function. When the linear system has an arbitrary number of linear inequalities, the problem can also be transformed in such way that the constraint set is specified by a unique constraint involving a lower semi-continuous convex function. If this function is quasipolyhedral, it locally behaves like a polyhedral one, and this fact should allow to design an algorithm to resolve the optimization problem. In view of this new approach, this paper is devoted to study characterizations and properties of the class of quasipolyhedral functions, as well as their conjugate function and their subdifferential.
A new concept of upper semicontinuity for a vector-valued function and (weak and strong) duality statements under the assumption that the vector-valued objective function is D -quasiconvex and D -upper semicontinuous are proved.
J. Martínez-Legaz, Christiane Tammer· TOP - An Official Journal of...· 0 citations
We introduce Envelopt, a globally convergent iterative framework for a broad class of structured optimization problems where a smooth objective is augmented by a nonsmooth convex regularizer composed with a smooth mapping, and the variables are subject to general smooth constraints. All smooth functions may be nonconve...
For solving nonconvex equality-constrained optimization problems, a recent Gradient-Eigenstep Algorithm by Goyens et al.~is an iteration-efficient approach, based on minimizing Fletcher's augmented Lagrangian function, for finding an approximate second-order stationary point from an arbitrary starting point. In this pa...
F. Curtis, Ling-Jun Guo, Daniel P. Robinson· 0 citations
Recently, by using the derivatives of scalarized maps, associated with a vector optimization problem, new multiplier rules have been proven. The first objective of this paper is to show that those rules do not hold in infinite dimensional setting without imposing additional restrictions, even when the ordering cone has...
Akhtar A. Khan, M. Sama· Journal of Convex Analysis· 2 citations
We consider the quadratic fractional programming problem, which minimizes a ratio of two functions; a quadratic (not necessarily convex) function over an a affine function on an unbounded set. As is well-known, if the quadratic function is convex or quasiconvex, then the quadratic fractional function is pseudoconvex, a...
F. Lara· Journal of Convex Analysis· 8 citations· ⚡2
Recently, several particular problems in optimal design have been analyzed by using tools from non-convex, variational problems. As many of those have similarities, but also different features, we pretend to look at a full family of problems that includes most of those particular situations. Specifically, we examine an...
U. Prieto, Pablo Pedregal· Journal of Convex Analysis· 1 citation
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