Search results for "Mathematics - Optimization and Control"
showing 10 items of 52 documents
Regularity of sets under a reformulation in a product space of reduced dimension
2023
Different notions on regularity of sets and of collection of sets play an important role in the analysis of the convergence of projection algorithms in nonconvex scenarios. While some projection algorithms can be applied to feasibility problems defined by finitely many sets, some other require the use of a product space reformulation to construct equivalent problems with two sets. In this work we analyze how some regularity properties are preserved under a reformulation in a product space of reduced dimension. This allows us to establish local linear convergence of parallel projection methods which are constructed through this reformulation.
Monge Problem on infinite dimensional Hilbert space endowed with suitable Gaussian measure
2014
In this paper we solve the Monge problem on infinite dimensional Hilbert space endowed with a suitable Gaussian measure, that satisfies the Lebesgue differentiation theorem.
A globally convergent and locally quadratically convergent modified B-semismooth Newton method for $\ell_1$-penalized minimization
2015
We consider the efficient minimization of a nonlinear, strictly convex functional with $\ell_1$-penalty term. Such minimization problems appear in a wide range of applications like Tikhonov regularization of (non)linear inverse problems with sparsity constraints. In (2015 Inverse Problems (31) 025005), a globalized Bouligand-semismooth Newton method was presented for $\ell_1$-Tikhonov regularization of linear inverse problems. Nevertheless, a technical assumption on the accumulation point of the sequence of iterates was necessary to prove global convergence. Here, we generalize this method to general nonlinear problems and present a modified semismooth Newton method for which global converg…
On optimal control of free boundary problems of obstacle type
2018
A numerical study of an optimal control formulation for a shape optimization problem governed by an elliptic variational inequality is performed. The shape optimization problem is reformulated as a boundary control problem in a fixed domain. The discretized optimal control problem is a non-smooth and non-convex mathematical programing problem. The performance of the standard BFGS quasi-Newton method and the BFGS method with the inexact line search are tested.
An abstract inf-sup problem inspired by limit analysis in perfect plasticity and related applications
2020
This work is concerned with an abstract inf-sup problem generated by a bilinear Lagrangian and convex constraints. We study the conditions that guarantee no gap between the inf-sup and related sup-inf problems. The key assumption introduced in the paper generalizes the well-known Babuska-Brezzi condition. It is based on an inf-sup condition defined for convex cones in function spaces. We also apply a regularization method convenient for solving the inf-sup problem and derive a computable majorant of the critical (inf-sup) value, which can be used in a posteriori error analysis of numerical results. Results obtained for the abstract problem are applied to continuum mechanics. In particular, …
Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems
2019
In this paper, a new technique is shown for deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on upper bounds (majorants) for one of the cost functionals, both minorants and majorants lead to two-sided estimates of functional type for the optimal control problem. Both upper and lower bounds are derived for the second new cost functional subject to the same parabolic PDE-constraints, but where the target is a desired gradient. The time-periodic optimal control problems are discretized by the multiharmonic finite element method leading to lar…
Single-input perturbative control of a quantum symmetric rotor
2022
We consider the Schr\"odinger partial differential equation of a rotating symmetric rigid molecule (symmetric rotor) driven by a z-linearly polarized electric field, as prototype of degenerate infinite-dimensional bilinear control system. By introducing an abstract perturbative criterium, we classify its simultaneous approximate controllability; based on this insight, we numerically perform an orientational selective transfer of rotational population.
Dynamic Regret Analysis for Online Tracking of Time-varying Structural Equation Model Topologies
2020
Identifying dependencies among variables in a complex system is an important problem in network science. Structural equation models (SEM) have been used widely in many fields for topology inference, because they are tractable and incorporate exogenous influences in the model. Topology identification based on static SEM is useful in stationary environments; however, in many applications a time-varying underlying topology is sought. This paper presents an online algorithm to track sparse time-varying topologies in dynamic environments and most importantly, performs a detailed analysis on the performance guarantees. The tracking capability is characterized in terms of a bound on the dynamic re…
Non-cooperative Aerial Base Station Placement via Stochastic Optimization
2019
Autonomous unmanned aerial vehicles (UAVs) with on-board base station equipment can potentially provide connectivity in areas where the terrestrial infrastructure is overloaded, damaged, or absent. Use cases comprise emergency response, wildfire suppression, surveillance, and cellular communications in crowded events to name a few. A central problem to enable this technology is to place such aerial base stations (AirBSs) in locations that approximately optimize the relevant communication metrics. To alleviate the limitations of existing algorithms, which require intensive and reliable communications among AirBSs or between the AirBSs and a central controller, this paper leverages stochastic…
Strengthened splitting methods for computing resolvents
2021
In this work, we develop a systematic framework for computing the resolvent of the sum of two or more monotone operators which only activates each operator in the sum individually. The key tool in the development of this framework is the notion of the “strengthening” of a set-valued operator, which can be viewed as a type of regularisation that preserves computational tractability. After deriving a number of iterative schemes through this framework, we demonstrate their application to best approximation problems, image denoising and elliptic PDEs. FJAA and RC were partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund …