843 research outputs found
Stable strong Fenchel and Lagrange duality for evenly convex optimization problems
By means of a conjugation scheme based on generalized convex conjugation theory instead of Fenchel conjugation, we build an alternative dual problem, using the perturbational approach, for a general optimization one defined on a separated locally convex topological space. Conditions guaranteeing strong duality for primal problems which are perturbed by continuous linear functionals and their respective dual problems, which is named stable strong duality, are established. In these conditions, the fact that the perturbation function is evenly convex will play a fundamental role. Stable strong duality will also be studied in particular for Fenchel and Lagrange primal–dual problems, obtaining a characterization for Fenchel case.This research was partially supported by MINECO of Spain and FEDER of EU, [grant numberMIM2014-59179-C2-1-P]; Conselleria de la Educacion de la Generalitat Valenciana, Spain, Pre-doc Program Vali+d, DOCV 6791/07.06.2012, [grant number ACIF-2013-156]
New duality results for evenly convex optimization problems
We present new results on optimization problems where the involved functions are evenly convex. By means of a generalized conjugation scheme and the perturbation theory introduced by Rockafellar, we propose an alternative dual problem for a general optimization one defined on a separated locally convex topological space. Sufficient conditions for converse and total duality involving the even convexity of the perturbation function and c-subdifferentials are given. Formulae for the c-subdifferential and biconjugate of the objective function of a general optimization problem are provided, too. We also characterize the total duality by means of the saddle-point theory for a notion of Lagrangian adapted to the considered framework.Research partially supported by MINECO of Spain and ERDF of EU, Grant MTM2014-59179-C2-1-P, Austrian Science Fund (FWF), Project M-2045, and German Research Foundation (DFG), Project GR3367/4-1
E′-Convex Sets and Functions: Properties and Characterizations
The main properties of evenly convex sets and functions have been deeply studied by different authors, and a duality theory for evenly convex optimization problems has been well developed as well. In this theory, the notion of e′-convexity appears as a necessary requirement for obtaining important results in strong and stable strong duality. This fact has motivated the authors to study possible properties of this kind of convexity in sets and functions, which is closely connected to even convexity
Unified and Distributed QoS-Driven Cell Association Algorithms in Heterogeneous Networks
This paper addresses the cell association problem in the downlink of a
multi-tier heterogeneous network (HetNet), where base stations (BSs) have
finite number of resource blocks (RBs) available to distribute among their
associated users. Two problems are defined and treated in this paper: sum
utility of long term rate maximization with long term rate quality of service
(QoS) constraints, and global outage probability minimization with outage QoS
constraints. The first problem is well-suited for low mobility environments,
while the second problem provides a framework to deal with environments with
fast fading. The defined optimization problems in this paper are solved in two
phases: cell association phase followed by the optional RB distribution phase.
We show that the cell association phase of both problems have the same
structure. Based on this similarity, we propose a unified distributed algorithm
with low levels of message passing to for the cell association phase. This
distributed algorithm is derived by relaxing the association constraints and
using Lagrange dual decomposition method. In the RB distribution phase, the
remaining RBs after the cell association phase are distributed among the users.
Simulation results show the superiority of our distributed cell association
scheme compared to schemes that are based on maximum signal to interference
plus noise ratio (SINR)
Rotation Averaging and Strong Duality
In this paper we explore the role of duality principles within the problem of
rotation averaging, a fundamental task in a wide range of computer vision
applications. In its conventional form, rotation averaging is stated as a
minimization over multiple rotation constraints. As these constraints are
non-convex, this problem is generally considered challenging to solve globally.
We show how to circumvent this difficulty through the use of Lagrangian
duality. While such an approach is well-known it is normally not guaranteed to
provide a tight relaxation. Based on spectral graph theory, we analytically
prove that in many cases there is no duality gap unless the noise levels are
severe. This allows us to obtain certifiably global solutions to a class of
important non-convex problems in polynomial time.
We also propose an efficient, scalable algorithm that out-performs general
purpose numerical solvers and is able to handle the large problem instances
commonly occurring in structure from motion settings. The potential of this
proposed method is demonstrated on a number of different problems, consisting
of both synthetic and real-world data
Set-Valued Evenly Convex Functions: Characterizations and C-Conjugacy
In this work we deal with set-valued functions with values in the power set of a separated locally convex space where a nontrivial pointed convex cone induces a partial order relation. A set-valued function is evenly convex if its epigraph is an evenly convex set, i.e., it is the intersection of an arbitrary family of open half-spaces. In this paper we characterize evenly convex set-valued functions as the pointwise supremum of its set-valued e-affine minorants. Moreover, a suitable conjugation pattern will be developed for these functions, as well as the counterpart of the biconjugation Fenchel-Moreau theorem.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Research partially supported by MINECO of Spain and ERDF of EU, Grant MTM2014-59179-C2-1-P
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