arXiv subjects

Francois Glineur

Publications and source records attributed to Francois Glineur.

6 recordsLinked to original sources

arXiv2025

On the Convex Interpolation for Linear Operators

The worst-case performance of an optimization method on a problem class can be analyzed using a finite description of the problem class, known as interpolation conditions. In this work, we study interpolation conditions for linear operators given scalar products between discrete inputs and outputs. First, we show that if only convex constraints on the scalar products of inputs and outputs are allowed,it is only possible to characterize classes of linear operators or symmetric linear operator whose all singular values or eigenvalues belong to some subset of R. Then, we propose new interpolation conditions for linear operators with minimal and maximal singular values and linear operators whose eigenvalues or singular values belong to unions of subsets. Finally, we illustrate the new interpolation conditions through the analysis of the Gradient and Chambolle-Pock methods. It allows to obtain new numerical worst-case guarantees on these methods.

math.OC↗

arXiv2024

Proximal gradient methods with inexact oracle of degree q for composite optimization

We introduce the concept of inexact first-order oracle of degree q for a possibly nonconvex and nonsmooth function, which naturally appears in the context of approximate gradient, weak level of smoothness and other situations. Our definition is less conservative than those found in the existing literature, and it can be viewed as an interpolation between fully exact and the existing inexact first-order oracle definitions. We analyze the convergence behavior of a (fast) inexact proximal gradient method using such an oracle for solving (non)convex composite minimization problems. We derive complexity estimates and study the dependence between the accuracy of the oracle and the desired accuracy of the gradient or of the objective function. Our results show that better rates can be obtained both theoretically and in numerical simulations when q is large.

math.OC↗

arXiv2022

Real-time image-guided treatment of mobile tumors in proton therapy by a library of treatment plans: a simulation study

Purpose: To improve target coverage and reduce the dose in the surrounding organs-at-risks (OARs), we developed an image-guided treatment method based on a precomputed library of treatment plans controlled and delivered in real-time. Methods: A library of treatment plans is constructed by optimizing a plan for each breathing phase of a 4DCT. Treatments are delivered by simulation on a continuous sequence of synthetic CTs generated from real MRI sequences. During treatment, the plans for which the tumor is at a close distance to the current tumor position are selected to deliver their spots. The study is conducted on five liver cases. Results: We tested our approach under imperfect knowledge of the tumor positions with a 2 mm distance error. On average, compared to a 4D robustly optimized treatment plan, our approach led to a dose homogeneity increase of 5% (defined as 1-(D_5-D_95)/prescription) in the target and a mean liver dose decrease of 23. The treatment time was roughly increased by a factor of 2 but remained below 4 minutes on average. Conclusions: Our image-guided treatment framework outperforms state-of-the-art 4D-robust plans for all patients in this study on both target coverage and OARs sparing, with an acceptable increase in treatment time under the current accuracy of tumor tracking technology.

physics.med-ph↗

arXiv2020

Worst-case convergence analysis of inexact gradient and Newton methods through semidefinite programming performance estimation

We provide new tools for worst-case performance analysis of the gradient (or steepest descent) method of Cauchy for smooth strongly convex functions, and Newton's method for self-concordant functions, including the case of inexact search directions. The analysis uses semidefinite programming performance estimation, as pioneered by Drori and Teboulle [Mathematical Programming, 145(1-2):451-482, 2014], and extends recent performance estimation results for the method of Cauchy by the authors [Optimization Letters, 11(7), 1185-1199, 2017]. To illustrate the applicability of the tools, we demonstrate a novel complexity analysis of short step interior point methods using inexact search directions. As an example in this framework, we sketch how to give a rigorous worst-case complexity analysis of a recent interior point method by Abernethy and Hazan [PMLR, 48:2520-2528, 2016].

math.OC↗

arXiv2017

Complexity of first order inexact Lagrangian and penalty methods for conic convex programming

In this paper we present a complete iteration complexity analysis of inexact first order Lagrangian and penalty methods for solving cone constrained convex problems that have or may not have optimal Lagrange multipliers that close the duality gap. We first assume the existence of optimal Lagrange multipliers and study primal-dual first order methods based on inexact information and augmented Lagrangian smoothing or Nesterov type smoothing. For inexact (fast) gradient augmented Lagrangian methods we derive a total computational complexity of $\mathcal{O}\left(\frac{1}ε\right)$ projections onto a simple primal set in order to attain an $ε-$optimal solution of the conic convex problem. For the inexact fast gradient method combined with Nesterov type smoothing we derive computational complexity $\mathcal{O}\left(\frac{1}{ε^{3/2}}\right)$ projections onto the same set. Then, we assume that optimal Lagrange multipliers for the cone constrained convex problem might not exist, and analyze the fast gradient method for solving penalty reformulations of the problem. For the fast gradient method combined with penalty framework we also derive a total computational complexity of $\mathcal{O}\left(\frac{1}{ε^{3/2}}\right)$ projections onto a simple primal set to attain an $ε-$optimal solution for the original problem.

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arXiv2013

Weighted Sum Rate Maximization for Downlink OFDMA with Subcarrier-pair based Opportunistic DF Relaying

This paper addresses a weighted sum rate (WSR) maximization problem for downlink OFDMA aided by a decode-and-forward (DF) relay under a total power constraint. A novel subcarrier-pair based opportunistic DF relaying protocol is proposed. Specifically, user message bits are transmitted in two time slots. A subcarrier in the first slot can be paired with a subcarrier in the second slot for the DF relay-aided transmission to a user. In particular, the source and the relay can transmit simultaneously to implement beamforming at the subcarrier in the second slot. Each unpaired subcarrier in either the first or second slot is used for the source's direct transmission to a user. A benchmark protocol, same as the proposed one except that the transmit beamforming is not used for the relay-aided transmission, is also considered. For each protocol, a polynomial-complexity algorithm is developed to find at least an approximately optimum resource allocation (RA), by using continuous relaxation, the dual method, and Hungarian algorithm. Instrumental to the algorithm design is an elegant definition of optimization variables, motivated by the idea of regarding the unpaired subcarriers as virtual subcarrier pairs in the direct transmission mode. The effectiveness of the RA algorithm and the impact of relay position and total power on the protocols' performance are illustrated by numerical experiments. The proposed protocol always leads to a maximum WSR equal to or greater than that for the benchmark one, and the performance gain of using the proposed one is significant especially when the relay is in close proximity to the source and the total power is low. Theoretical analysis is presented to interpret these observations.

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