(α) t∈N converges to a (non-strict) global minimizer (P f 1 ,γ,ψ )
at the rate O(1/t) on the restricted duality gap.
3.3 Discussion
Algorithm 1 **and** 2 share some similarities, but exhibit also important differences. For instance, the **primal**-**dual** algo- rithm enjoys a convergence rate that is not known for the **primal** algorithm. Furthermore, the latter necessitates two operator inversions that can only be done efficiently for some Φ **and** H, while the former involves only application of these **linear** operators **and** their adjoints. Consequently, Algo- rithm 2 can virtually handle any **inverse** problem **with** a bounded **linear** H. In case where the inverses can be done efficiently, e.g. deconvolution **with** a tight frame, both algo- rithms have comparable computational burden. In general, if other regularizations/constraints are imposed on the solu- tion, in the form of additional proper lsc convex terms that would appear in (P f 1 ,γ,ψ ), both algorithms still apply by

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In this paper, we propose a framework for solving lin- ear **inverse** **problems** when the observations are corrupted by Poisson **noise**. In order to form the data fidelity term, we take the exact Poisson likelihood. As a prior, the images to restore are assumed to be positive **and** sparsely represented in a dic- tionary of atoms. The solution to the **inverse** problem is cast as the minimization of a non-smooth convex functional, for which we prove well-posedness of the optimization problem, characterize the corresponding minimizers, **and** solve them by means of **primal** **and** **primal**-**dual** proximal **splitting** al- gorithms originating from the realm of non-smooth convex optimization theory. Convergence of the algorithms is also shown. Experimental results **and** comparison to other algo- rithms on deconvolution are finally conducted.

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must be determined. We point out that for each time intervale [t 2n , t 2n+2 ] there are three of these
quantities. We must then write three **linear** independent equations that will allow us to obtain them **and** to couple both systems.
The coupling equations: Energy conservation The additional equations that we will add in order to couple the two systems in (19) will be chosen in such a way that the stability of the scheme will be ensured a priori. A simple way to do that is to impose a discrete version of the energy conservation property explained on the remarks 2.1 **and** 4.1. We introduce the discrete energy at the even time steps by

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This thesis is concerned **with** the analysis, implementation of interior-point methods . In particular, we focus on two type of **problems**: horizontal **linear** complementarity prob- lems (HLCPs) **and** Semidefinite **linear** complementarity **problems** (SDLCPs).
In chapter 1: we present the definitions **and** terms that will be used throughout the thesis. In chapter 2: we present a full-Newton feasible step interior-point algorithm for solving monotone horizontal **linear** complementarity **problems**. The idea of this algorithm is to follow the centers of the perturbed HLCP by using only full-Newton steps **with** the ad- vantage that no line search is required **and** restricts iterates in a small neighborhood of the central-path by introducing a suitable proximity measure during the solution process. Then we prove across a new appropriate choice of the defaults of the threshold of the parameter τ which defines the size of the neighborhood of the central-path **and** of the update barrier parameter θ that our algorithm is well-defined **and** the full-Newton step to the central-path is locally quadratically convergent. Moreover, we derive its complexity bound. which coincides **with** the best known iteration bound for such feasible IPMs. Fi- nally, we report some numerical results to show the ability of this approach.

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algorithm [ 37 , 21 , 13 , 36 , 14 , 15 ]. It has been used to solve a vast amount of state- of-the-art **problems**—to name a few examples in imaging: image denoising **with** the structure tensor [ 22 ], total generalized variation denoising [ 11 ], dynamic regularization [ 7 ], multi-modal medical imaging [ 27 ], multi-spectral medical imaging [ 43 ], computa- tion of non-**linear** eigenfunctions [ 26 ], regularization **with** directional total generalized variation [ 29 ]. Its popularity stems from two facts: First, it is very simple **and** there- fore easy to implement. Second, it involves only simple operations like matrix-vector multiplications **and** evaluations of proximal operators which are for many **problems** of interest simple **and** in closed-form or easy to compute iteratively, cf. e.g. [ 33 ]. However, for large **problems** that are encountered in many real world applications,

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Sparse regression consists in estimating unknown coefficients in a **linear** regression framework under the assumption that only a few regressors, also called variables or features, are predictive. In a word, most features are non-informative **and** their associated coefficients should be zero. The forward problem we consider in this work is the one proposed in [1] in the context of M/EEG. MEG **and** EEG are brain imaging methods that record the electromagnetic signals produced by active neurons using an array of sensors. The ambition is to use M/EEG to localize active brain regions non-invasively,

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1 Introduction
Variational methods have proven to be particularly useful to solve a number of ill-posed **inverse** imaging **problems**. They can be divided into two fundamentally different classes: Convex **and** non-convex **problems**. The advantage of convex **problems** over non-convex **problems** is that a global optimum can be computed, in general **with** a good precision **and** in a reasonable time, independent of the initialization. Hence, the quality of the solution solely depends on the accuracy of the model. On the other hand, non-convex **problems** have often the ability to model more precisely the process behind an image acquisition, but have the drawback that the quality of the solution is more sensitive to the initialization **and** the optimization algorithm.

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1 Introduction
There has been recently a growing interest in **primal**-**dual** approaches for finding a zero of a sum of monotone operators or minimizing a sum of proper lower-semicontinuous convex functions (see [ 36 ] **and** the references therein). When various **linear** operators are involved in the formulation of the problem under investigation, solving jointly its **primal** **and** **dual** forms allows the design of strategies where none of the **linear** operators needs to be inverted. Avoiding such inversions may offer a significant advantage in terms of computational complexity when dealing **with** large-scale **problems** (see e.g. [ 6 , 27 , 31 , 35 , 46 , 48 , 52 ]).

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Keywords: **inverse** **problems**, wavelets, redundant transforms, convex optimization, proximal operator, restoration
1. INTRODUCTION
Because of sensor imperfection **and** acquisition mode, observed data are often noisy **and** degraded by a **linear** operator. This operator **and** **noise** properties may actually depend on the considered application. For in- stance, in optical satellite imaging, the **linear** operator models a blur **and** the **noise** can be assumed Gaussian. In the biomedical area, the ill-conditioned **linear** operator in parallel Magnetic Resonance Imaging (pMRI) represents a sensitivity matrix **and** the **noise** is Gaussian, whereas in Positron Emission Tomography (PET), the acquisition process is modelled by a large-size projection matrix **and** the **noise** is Poisson distributed.

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These methods require only approximations of the functions used in the formulation of the optimization problem, which is of the utmost importance for solving online signal processing **problems**. The almost sure convergence of these algorithms has been established. The stochastic version of the **primal**- **dual** algorithm that we have investigated has been evaluated in an online image restoration problem in which the data are blurred by a stochastic point spread function **and** corrupted **with** **noise**.

1 Introduction
Duality theory occupies a central place in classical optimization [ 19 , 24 , 33 , 40 , 41 ]. Since the mid 1960s it has expanded in various directions, e.g., variational inequalities [ 2 , 17 , 21 , 23 , 26 , 34 ], minimax **and** saddle point **problems** [ 27 , 29 , 32 , 39 ], **and**, from a more global perspective, monotone inclusions [ 5 , 9 , 10 , 16 , 31 , 37 , 38 ]. In the present paper, we propose an algorithm for solving the following structured duality framework for monotone inclusions that encompasses the above cited works. In this formulation, we denote by B D the parallel sum of two set-valued operators B **and** D (see ( 2.5 )). This operation plays a central role in convex analysis **and** monotone operator theory. In particular, B D can be seen as a regularization of B by D, **and** is naturally connected to addition through duality since (B + D) −1 = B −1

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under constraints (2), (3), (4),(5), (6) **and** (8).
3 Neural network as a fast solver for **linear** quadratic programs
The basic idea for solving an optimization problem using a tailored neural network is to make sure that the neural network will converge asymptotically at a fast rate **and** that the equilibrium point of the neural network will correspond effectively to the solution of the original optimization problem. In 1986, Tank **and** Hopfield introduced a **linear** programming neural network solver realized **with** an analogic circuit which appeared to be well suited for applications requiring on-line solutions [5]. After this first successful attempt, many neural network models for solving **linear** **and** quadratic programming **problems** have been proposed in the literature. For a review see [6,7] **and** an application see [10].

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These Positivstellensatz-based relaxations provide inner approximations to the set {q | kM † qk ∗
A ≤ 1}, **and** therefore a
lower bound to the optimal value of the **dual** convex program (2). Consequently, they can be viewed as outer approxi- mations to the atomic norm ball C( A ). This interpretation leads to the following interesting conclusion: If a point ˜ x is an extreme point of the convex set C( A ) then the tangent cone at ˜ x **with** respect to C( A ) is smaller than the tangent cone at (an appropriately scaled) ˜ x **with** respect to the outer approximation. Thus we have a stronger requirement in terms of a larger number of measurements for exact recovery, which provides a trade-off between the complexity of the convex relaxation procedure **and** the amount of information required for exact recovery.

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The present paper aims at contributing to the solving of ultrasonic **inverse** **problems** by including attenuation **and** dispersion in the direct model. In particular, we propose to ac- count for physical attenuation profiles defined in the frequency domain such as power law attenuation models. Our objective is threefold. First, we improve the ultrasound model accuracy compared to the standard attenuation-free model. Second, in contrast **with** the methods described above [23–27], we yield a more constrained description of the data. Consequently, better performance of the inversion procedure is expected. In partic- ular, a more accurate model aims at improving echo detection for long propagation distances where the signal-to-**noise** ratio is low. Last, our framework yields a **linear** direct model which enables the use of many acknowledged inversion methods [8]. Related works [7, 28] proposed similar approaches but **with** an empirical description of attenuation within a time-domain signal model. The model that we propose is derived from the physics of wave propagation **and** is described in the frequency domain [3, 16].

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) [ 13 ] under strong convexity of either the **primal** or the **dual** problem. The iterates converge globally linearly if both the **primal** **and** **dual** **problems** are strongly convex [ 13 , 9 ], or locally linearly under certain regularity assumptions [ 35 ]. How- ever, in practice, local **linear** convergence of the sequences generated by Algorithm 1 has been observed for many **problems** **with** the absence of strong convexity (as conﬁrmed by our numerical experiments in Section 6 ). None of the existing theoretical analysis was able to explain this behaviour so far. Providing the theoretical underpinnings of this local behaviour is the main goal pursued in this paper. Our main ﬁndings can be summarized as follows.

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[10] S. G. Nash, “A survey of truncated-Newton methods,” Journal
of Computational **and** Applied Mathematics, vol. 124, pp. 45–
59, 2000.
[11] E. Chouzenoux, S. Moussaoui, **and** J. Idier, “Majorize- minimize linesearch for inversion methods involving barrier function optimization,” **Inverse** **Problems**, vol. 28, no. 6, 2012. [12] A. Conn, N. Gould, **and** L. Toint, “A **primal**-**dual** algo- rithm for minimizing a nonconvex function subject to bounds **and** nonlinear constraints,” Tech. Rep. RC 20639, York- town Heights, NY, 1996, citeseer.ist.psu.edu/

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I. I NTRODUCTION
The development of advanced decentralized collision avoidance algorithms for large scale systems is an issue of critical importance in future Air Traffic Control (ATC) architectures, where the increasing number of aircraft will render centralized approaches inefficient. In recent years the application of robotics collision avoidance potential field based methods to ATC has been explored [6],[15] as a promising alternative for such algorithms. A common prob- lem **with** potential field based path planning algorithms in multi-agent systems is the existence of local minima [7],[9]. The seminal work of Koditschek **and** Rimon [8] involved navigation of a single robot in an environment of spherical obstacles **with** guaranteed convergence. In previous work, the closed loop single robot navigation methodology of[8] was extended to multi-agent systems. In [10],[13] this method was extended to take into account the volume of each robot in a centralized multi-agent scheme, while a decentralized version has been presented in [4], [6]. Formation control for point agents using decentralized navigation functions was dealt **with** in [19], [3]. Decentralized navigation functions were also used for multiple UAV guidance in [2].

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bands from 383 nm to 2508 nm. A subset of P spectra is then randomly picked up to create synthetic
mixtures **with** abundances simulated from a Dirichlet distribution. Only realizations **with** maximum abundance value lower than a specified level A max are retained. Finally, a random Gaussian **noise** is
added to each resulting mixture spectrum, in order to get a signal to **noise** ratio (SNR) of 30 dB. The unmixing algorithms are implemented on Matlab 2012b **and** the calculations are performed using a HP Compaq Elite desktop having an Intel Core i7 3.4 GHz processor **and** 8 GB of RAM.

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Before going further, let us discuss how one quantifies convergence speed for saddle point **problems**. Several measures of optimality have been considered in the literature. The most natural one is feasibility error **and** optimality gap. It directly fits the definition of the optimization problem at stake. However, one cannot compute the optimality gap before the problem is solved. Hence, in algorithms, we usually use the Karush-Kuhn-Tucker (KKT) error instead. It is a computable quantity **and** if the Lagrangian’s gradient is metrically subregular [23], then a small KKT error implies that the current point is close to the set of saddle points. When the **primal** **and** **dual** domains are bounded, the duality gap is a very good way to measure optimality: it is often easily computable **and** it is an upper bound to the optimality gap. A generalization to unbounded domains has been proposed in [24]: the smoothed gap, based on the smoothing of nonsmooth functions [21], takes finite values for constrained **problems**, unlike the duality gap. Moreover, if the smoothness parameter is small **and** the smoothed gap is small, this means that optimality gap **and** feasibility error are both small. In the present paper, we shall reuse this concept not only for showing a convergence speed but also to define a new regularity assumption that we believe is better suited to the study of **primal**-**dual** algorithms.

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posedness, this term is of order n t−1 . Note that this term goes to zero when t is smaller
than 1, yet hampering the consistency rate. In other cases **and** in the severely ill-posed setting, this term becomes dominant in the upper bound.
Hence aggregation methods for **inverse** **problems** have the same kind of drawbacks than ℓ 1 penalization procedure since they cannot handle too badly ill-posed **inverse** **problems**.