arXiv subjects

Mounir Haddou

Publications and source records attributed to Mounir Haddou.

14 recordsLinked to original sources

arXiv2026

Optimal Control Problem with Mixed Control and State Constraints for Cancer Chemotherapy and Treatment Optimization

The success of chemotherapy depends on the effectiveness of the drug delivery strategy and its ability to destroy cancer cells while minimizing damage to healthy tissues. The main objective of this work is to minimize the density of invasive tumour cells by controlling the chemotherapeutic agents. For this, we address an optimal control problem with mixed control and state constraints. The concentration of chemotherapeutic drugs is represented as a control variable. We use a nonlinear reaction-diffusion equation to describe the effect of drugs on the progression of invasive tumours. We start with the mathematical analysis of this initial boundary value problem. Then, we formulate the optimal control problem, explain the role of different constraints, and derive first-order necessary conditions of optimality. Finally, in order to demonstrate the efficiency of the proposed strategy, numerical simulations in the case of the eradication of malignant lung tumours, are presented and analysed.

math.OC↗

arXiv2026

Approximation of an optimal control problem on a network with a perturbed problem in the whole space

A classical optimal control problem posed in the whole space R^2 is perturbed by a singular term of magnitude $ε$^{-1} aimed at driving the trajectories to a prescribed network $Γ$. We are interested in the link between the limit problem, as $ε$ $\rightarrow$ 0, and some optimal control problems on networks studied in the literature. We prove that the sequence of trajectories admits a subsequential limit evolving on $Γ$. Moreover, in the case of the Eikonal equation, we show that the sequence of value functions associated with the perturbed optimal control problems converges to a limit which, in particular, coincides with the value function of the expected optimal control problem set on the network $Γ$.

math.AP↗

arXiv2026

Solving Nonlinear Absolute Value Equations

In this work, we show that several problems naturally represented as Nonlinear Absolute Value Equations (NAVE) can be reformulated as Nonlinear Complementarity Problems (NCP) and efficiently solved using smoothing regularization techniques under mild assumptions. As far as we know, this is the first numerical approach that directly deals with NAVE. We also identify a technical assumption commonly utilized in smoothing techniques and prove its equivalence to a classical __ojasiewicz inequality at infinity, validating its non-restrictive nature. Furthermore, we extend established error estimates for NCP solvers to derive error bounds for NAVE problems under weaker assumptions. We illustrate the effectiveness of our approach through applications including asymmetric ridge optimization and nonlinear ordinary differential equations.

math.OC↗

arXiv2023

Analysis and Comparison of Two-Level KFAC Methods for Training Deep Neural Networks

As a second-order method, the Natural Gradient Descent (NGD) has the ability to accelerate training of neural networks. However, due to the prohibitive computational and memory costs of computing and inverting the Fisher Information Matrix (FIM), efficient approximations are necessary to make NGD scalable to Deep Neural Networks (DNNs). Many such approximations have been attempted. The most sophisticated of these is KFAC, which approximates the FIM as a block-diagonal matrix, where each block corresponds to a layer of the neural network. By doing so, KFAC ignores the interactions between different layers. In this work, we investigate the interest of restoring some low-frequency interactions between the layers by means of two-level methods. Inspired from domain decomposition, several two-level corrections to KFAC using different coarse spaces are proposed and assessed. The obtained results show that incorporating the layer interactions in this fashion does not really improve the performance of KFAC. This suggests that it is safe to discard the off-diagonal blocks of the FIM, since the block-diagonal approach is sufficiently robust, accurate and economical in computation time.

cs.LG↗

arXiv2022

Efficient Approximations of the Fisher Matrix in Neural Networks using Kronecker Product Singular Value Decomposition

Several studies have shown the ability of natural gradient descent to minimize the objective function more efficiently than ordinary gradient descent based methods. However, the bottleneck of this approach for training deep neural networks lies in the prohibitive cost of solving a large dense linear system corresponding to the Fisher Information Matrix (FIM) at each iteration. This has motivated various approximations of either the exact FIM or the empirical one. The most sophisticated of these is KFAC, which involves a Kronecker-factored block diagonal approximation of the FIM. With only a slight additional cost, a few improvements of KFAC from the standpoint of accuracy are proposed. The common feature of the four novel methods is that they rely on a direct minimization problem, the solution of which can be computed via the Kronecker product singular value decomposition technique. Experimental results on the three standard deep auto-encoder benchmarks showed that they provide more accurate approximations to the FIM. Furthermore, they outperform KFAC and state-of-the-art first-order methods in terms of optimization speed.

cs.NE↗

arXiv2022

A new smoothing method for nonlinear complementarity problems involving P0 function

In this paper, we propose a smoothing method to solve nonlinear complementarity problems involving P 0-functions. We propose a nonparametric algorithm to solve the nonlinear corresponding system of equations and prove some global and local convergence results. We also present several numerical experiments and applications that show the efficiency of our approach. Our main contribution relies in the fact that the regularization parameter r is considered as a variable and we do not need any complicated strategy to update it.

math.OC↗

arXiv2021

A convex function satisfying the Lojasiewicz inequality but failing the gradient conjecture both at zero and infinity

We construct an example of a smooth convex function on the plane with a strict minimum at zero, which is real analytic except at zero, for which Thom's gradient conjecture fails both at zero and infinity. More precisely, the gradient orbits of the function spiral around zero and at infinity. Besides, the function satisfies the Lojasiewicz gradient inequality at zero.

math.DS↗

arXiv2021

A New Approach for Solving the Linear Complementarity Problem using Smoothing Functions

Based on smoothing techniques, we propose two new methods to solve linear complementarity problems (LCP) called TLCP and Soft-Max. The idea of these two new methods takes inspiration from interior-point methods in optimization. The technique that we propose avoids any parameter management while ensuring good theoretical convergence results. In our approach we do not need any complicated strategy to update the smoothing parameter r since we will consider it as a new variable. Our methods are validated by extensive numerical tests, in which we compare our methods to several other classical methods.

math.OC↗

arXiv2020

A new Relaxation Method for Optimal Control of Semilinear Elliptic Variational Inequalities Obstacle Problems

In this paper, we investigate optimal control problems governed by semilinear elliptic variational inequalities involving constraints on the state, and more precisely the obstacle problem. Since we adopt a numerical point of view, we first relax the feasible domain of the problem, then using both mathematical programming methods and penalization methods we get optimality conditions with smooth Lagrange multipliers. Some numerical experiments using IPOPT algorithm are presented to verify the efficiency of our approach.

math.OC↗

arXiv2018

Explicit formulas for $C^{1,1}$ Glaeser-Whitney extensions of 1-fields in Hilbert spaces

We give a simple alternative proof for the $C^{1,1}$--convex extension problem which has been introduced and studied by D. Azagra and C. Mudarra [2]. As an application, we obtain an easy constructive proof for the Glaeser-Whitney problem of $C^{1,1}$ extensions on a Hilbert space. In both cases we provide explicit formulae for the extensions. For the Gleaser-Whitney problem the obtained extension is almost minimal, that is, minimal up to a factor $\frac{1+\sqrt{3}}{2}$ in the sense of Le Gruyer [15].

math.FA↗

arXiv2010

Smoothing Methods for Nonlinear Complementarity Problems

In this paper, we present a new smoothing approach to solve general nonlinear complementarity problems. Under the $P_0$ condition on the original problems, we prove some existence and convergence results . We also present an error estimate under a new and general monotonicity condition. The numerical tests confirm the efficiency of our proposed methods.

math.OC↗

arXiv2008

Optimization of operational aircraft parameters Reducing Noise Emission

The objective of this paper is to develop a model and a minimization method to provide flight path optimums reducing aircraft noise in the vicinity of airports. Optimization algorithm has solved a complex optimal control problem, and generates flight paths minimizing aircraft noise levels. Operational and safety constraints have been considered and their limits satisfied. Results are here presented and discussed.

math.OC↗

arXiv2007

A new class of smoothing methods for mathematical programs with equilibrium constraints

A class of smoothing methods is proposed for solving mathematical programs with equimibrium constraints. We introduce new and very simple regularizations of the complementarity constraints. Some estimate distance to optimal solution and expansions of the optimal value function are presented. Limited numerical experiments using SNOPT algorithm are presented to verify the efficiency of our approach.

math.OC↗

arXiv2006

A regularization method for ill-posed bilevel optimization problems

We present a regularization method to approach a solution of the pessimistic formulation of ill -posed bilevel problems . This allows to overcome the difficulty arising from the non uniqueness of the lower level problems solutions and responses. We prove existence of approximated solutions, give convergence result using Hoffman-like assumptions. We end with objective value error estimates.

math.OC↗