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We study a mechanical system with a finite number of degrees of freedom, subjected to perfect time-dependent frictionless unilateral (possibly nonconvex) constraints with inelastic collisions on active constraints. The dynamic is described in the form of a second-order measure differential inclusion. Under some regularity assumptions on the data, we establish several properties of the set of admissible positions, which is not necessarily convex but assumed to be uniformly prox-regular. Our approach does not require any second-order information or boundedness of the Hessians of the constraints involved in the problem and are specific to moving sets represented by inequalities constraints. On that basis, we are able to discretize our problem by the time-stepping algorithm and construct a sequence of approximate solutions. It is shown that this sequence possesses a subsequence converging to a solution of the initial problem. This methodology is not only used to prove an existence result but could be also used to solve numerically the vibroimpact problem with time-dependent nonconvex constraints.

In a Hilbert space H, we study a dynamic inertial Newton method which aims to solve additively structured monotone equations involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator A=∇f+B, where ∇f is the gradient of a continuously differentiable convex function f and B is a nonpotential monotone and cocoercive operator. Besides a viscous friction term, the dynamic involves geometric damping terms which are controlled respectively by the Hessian of the potential f and by a Newton-type correction term attached to B. Based on a fixed point argument, we show the well-posedness of the Cauchy problem. Then we show the weak convergence as t→+∞ of the generated trajectories towards the zeros of ∇f+B. The convergence analysis is based on the appropriate setting of the viscous and geometric damping parameters. The introduction of these geometric dampings makes it possible to control and attenuate the known oscillations for the viscous damping of inertial methods. Rewriting the second-order evolution equation as a first-order dynamical system enables us to extend the convergence analysis to nonsmooth convex potentials. These results open the door to the design of new first-order accelerated algorithms in optimization taking into account the specific properties of potential and nonpotential terms. The proofs and techniques are original and differ from the classical ones due to the presence of the nonpotential term.

In this paper, we study numerical methods for solving eigenvalue complementarity problems involving the product of second-order cones (or Lorentz cones). We reformulate such problem to find the roots of a semismooth function. An extension of the Lattice Projection Method (LPM) to solve the second-order cone eigenvalue complementarity problem is proposed. The LPM is compared to the semismooth Newton methods, associated to the Fischer–Burmeister and the natural residual functions. The performance profiles highlight the efficiency of the LPM. A globalization of these methods, based on the smoothing and regularization approaches, are discussed.

We study several variants of a nonsmooth Newton-type algorithm for solving an eigenvalue problem of the form Such an eigenvalue problem arises in mechanics and in other areas of applied mathematics. The symbol K refers to a closed convex cone in the Euclidean space ℝ n and ( A , B ) is a pair of possibly asymmetric matrices of order n . Special attention is paid to the case in which K is the nonnegative orthant of ℝ n . The more general case of a possibly unpointed polyhedral convex cone is also discussed in detail.

This study investigates the workability of an emerging cement based on calcined clay, considered one of the sustainable binders for reducing the carbon footprint of construction materials. Despite existing experimental data, no comprehensive analysis has been conducted. In the present paper, a literature-derived dataset was analyzed using CPM-based packing density computation and interpretable statistical analyses (distribution statistics and Pearson correlation-based projections). The novelty of this study lies in integrating the domain-knowledge-informed hierarchical analysis to identify packing density as a primary, sustainable lever to enhance LC3 fluidity while limiting reliance on superplasticizers. PCE superplasticizers (0–2.5 wt.% in the dataset) improve fluidity across packing densities; noticeable gains are observed even for low dosages (≈0.5–1 wt.%) at packing 0.36–0.38. A paradigm shift is proposed through optimizing packing density by adjusting clay and limestone content in the mix. Prioritizing packing density, alongside conventional parameters, opens new avenues for sustainability by reducing reliance on organic fluidizers in low-carbon cements.

The main objective of this paper is to study the Lipschitz-like stability property, in particular the Aubin property, of the solution map for potentially nonmonotone variational systems with composite structure. Using the Mordukhovich coderivative criterion and second-order subdifferential analysis, we derive simple and geometrical characterizations of this property based on the data involved in the problem. Applications of these theoretical results are given in the context of market pricing strategies in economics. Furthermore, we explore Julia and Ipopt solver to compute the Lipschitz modulus of the solution map around a reference point, showing the practical implementation of our results.

We give new criteria for weak and strong invariant closed sets for differential inclusions in ℝn , and which are simultaneously governed by Lipschitz Cusco mapping and by maximal monotone operators. Correspondingly, we provide different characterizations for the associated strong Lyapunov functions and pairs. The resulting conditions only depend on the data of the system, while the invariant sets are assumed to be closed, and the Lyapunov pairs are assumed to be only lower semi-continuous.

In a Hilbert setting, our study focuses on the dynamical system introduced by Su-Boyd-Candès as a low resolution ODE of Nesterov’s accelerated gradient method (NAG). This inertial system, denoted by (AVD)α, is driven by the gradient of the function f to be minimized, and is damped with an asymptotic vanishing coefficient of the form α/t, with α≥3. Taking α large enough plays a crucial role in the asymptotic convergence properties of the trajectories. For a general convex function f, taking α>3 guarantees the asymptotic convergence rate of the values o(1/t2), as well as the convergence of the trajectories towards optimal solutions. For strongly convex f, the asymptotic rate of convergence is of order 1/t2α3, which increases with α. To analyze the effect of the parameter α in the convergence properties of (AVD)α, we show that a judicious time scaling of (AVD)α produces trajectories close to those of the continuous steepest descent method associated with f when α is sufficiently large. This limiting process involves a singular perturbation property, as we move from a second-order evolution equation to a first-order one. This transition enables us to understand the change in the rate of convergence from 1/t to 1/t2 between the steepest descent method and (NAG). Based on a complexity analysis over a finite time interval, new results are obtained regarding the optimal tuning of the parameter α and the involved constants Cα in the estimations. Numerical experiments have been conducted to illustrate and confirm the theoretical results.

In this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s (Moreau in Sém Anal Convexe Montpellier, 1971 ) with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset C ( t ) , supposed to have a bounded variation, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a first order sweeping process with velocity in the moving set C ( t ) . This class of problems subsumes as a particular case, the evolution variational inequalities [widely used in applied mathematics and unilateral mechanics (Duvaut and Lions in Inequalities in mechanics and physics. Springer, Berlin, 1976 ]. Assuming that the moving subset C ( t ) has a continuous variation for every t ∈ [ 0 , T ] with C ( 0 ) bounded, we show that the problem has at least a Lipschitz continuous solution. The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model (Krejčı in Eur J Appl Math 2:281–292, 1991 ), to the planning procedure in mathematical economy (Henry in J Math Anal Appl 41:179–186, 1973 and Cornet in J. Math. Anal. Appl. 96:130–147, 1983 ), and to nonregular electrical circuits containing nonsmooth electronic devices like diodes (Acary et al. Nonsmooth modeling and simulation for switched circuits. Lecture notes in electrical engineering. Springer, New York 2011 ). The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau’s seminal work.

We consider a class of time-dependent inclusions in Hilbert spaces for which we state and prove an existence and uniqueness result. The proof is based on arguments of variational inequalities, convex analysis and fixed point theory. Then, we use this result to prove the unique weak solvability of a new class of Moreau’s sweeping processes with constraints in velocity. Our results are useful in the study of mathematical models which describe the quasistatic evolution of deformable bodies in contact with an obstacle. To provide some examples, we consider three viscoelastic contact problems which lead to time-dependent inclusions and sweeping processes in which the unknowns are the displacement and the velocity fields, respectively. Then we apply our abstract results in order to prove the unique weak solvability of the corresponding contact problems.