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In recent years, the notion of neural ODEs has connected deep learning with the field of ODEs and optimal control. In this setting, neural networks are defined as the mapping induced by the corresponding time-discretization scheme of a given ODE. The learning task consists in finding the ODE parameters as the optimal values of a sampled loss minimization problem. In the limit of infinite time steps, and data samples, we obtain a notion of continuous formulation of the problem. The practical implementation involves two discretization errors: a sampling error and a time-discretization error. In this work, we develop a general optimal control framework to analyze the interplay between the above two errors. We prove that to approximate the solution of the fully continuous problem at a certain accuracy, we not only need a minimal number of training samples, but also need to solve the control problem on the sampled loss function with some minimal accuracy. The theoretical analysis allows us to develop rigorous adaptive schemes in time and sampling, and gives rise to a notion of adaptive neural ODEs. The performance of the approach is illustrated in several numerical examples. [ABSTRACT FROM AUTHOR]

This work is devoted to show the efficiency of a new numerical approach in solving geometrical shape optimization problems constrained to partial differential equations, on a family of convex domains. More precisely, we are interested to an improved numerical optimization process based on the new shape derivative formula, using the Minkowski deformation of convex domains, recently established in Boulkhemair and Chakib (J. Convex Anal. 21( n ∘ 1 ), 67–87 2014), Boulkhemair (SIAM J. Control Optim. 55( n ∘ 1 ), 156–171 2017). This last formula allows to express the shape derivative by means of the support function, in contrast to the classical one expressed in term of vector fields Henrot and Pierre 2005, Delfour and Zolésio 2011, Sokolowski and Zolesio 1992. This avoids some of the disadvantages related to the classical shape derivative approach, when one use the finite elements discretization for approximating the auxiliary boundary value problems in shape optimization processes Allaire 2007. So, we investigate here the performance of the proposed shape optimization approach through the numerical resolution of some shape optimization problems constrained to boundary value problems governed by Laplace or Stokes operator. Notably, we carry out a comparative numerical study between its resulting numerical optimization process and the classical one. Finally, we give some numerical results showing the efficiency of the proposed approach and its ability in producing good quality solutions and in providing better accuracy for the optimal solution in less CPU time compared to the classical approach. [ABSTRACT FROM AUTHOR]

Two-dimensional transport codes for the simulation of tokamak plasma are reduced version of full 3D fluid models where plasma turbulence has been smoothed out by averaging. One of the main issues nowadays in such reduced models is the accurate modelling of transverse transport fluxes resulting from the averaging of stresses due to fluctuations. Transverse fluxes are assumed driven by local gradients, and characterised by ad hoc diffusion coefficients (turbulent eddy viscosity), adjusted by hand in order to match numerical solutions with experimental measurements. However, these coefficients vary substantially depending on the machine used, type of experiment and even the location inside the device, reducing drastically the predictive capabilities of these codes for a new configuration. To mitigate this issue, we recently proposed an innovative path for fusion plasma simulations by adding two supplementary transport equations to the mean-flow system for turbulence characteristic variables (here the turbulent kinetic energy k and its dissipation rate ϵ ) to estimate the turbulent eddy viscosity. The remaining free parameters are more driven by the underlying transport physics and hence vary much less between machines and between locations in the plasma. In this paper, as a proof of concept, we explore, on the basis of digital twin experiments, the efficiency of the assimilation of data to fix these free parameters involved in the transverse turbulent transport models in the set of averaged equations in 2D. [ABSTRACT FROM AUTHOR]

We explore the potential for using a nonsmooth loss function based on the max-norm in the training of an artificial neural network without hidden layers. We hypothesise that this may lead to superior classification results in some special cases where the training data are either very small or the class size is disproportional. Our numerical experiments performed on a simple artificial neural network with no hidden layer appear to confirm our hypothesis. [ABSTRACT FROM AUTHOR]

We aim to develop efficient and robust algorithms for nonsmooth nonlinear systems arising from complementarity problems. The semismooth Newton algorithm is popular due to its reliability and efficiency. However, it struggles with issues with imbalanced nonlinearities of the problems, leading to degraded convergence rates or failure despite help from the globalization techniques like linesearch or trust region. We introduce a right nonlinearly preconditioned semismooth Newton algorithm to address this difficulty. The critical success ingredient is that before each global Newton update, a nonlinear preconditioning step implicitly removes the so-called ‘bad components’ causing trouble via nonlinear subspace correction, inspired by Gaussian elimination but adapted nonlinearly to balance system nonlinearities. Additionally, our method integrates with a domain decomposition framework, enhancing parallelism. Numerical results on two classes of problems demonstrate significantly improved convergence over standard semismooth Newton methods. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we focus on the solution of a class of monotone inclusion problems in reflexive Banach spaces. To reflect the geometry of the space and the operator, a more general proximal point iterative algorithm with Bregman divergence is proposed and some strong convergence results for the proposed scheme under standard assumptions are obtained. Meanwhile, the convex optimization problems and the critical point problems are studied in the applications, and the recovery of the sparse signal is simulated in the numerical experiments. [ABSTRACT FROM AUTHOR]

Recently, Gowda and Sossa [Math. Program., 2019, 177: 149–171] studied the existence of solutions to weakly homogeneous variational inequalities. In particular, their main result, based on a degree-theoretic condition and a constraint on the corresponding cone complementarity problem, covers a majority of existence results on the subcategory problems of weakly homogeneous variational inequalities. In this paper, what we achieve is a new copositivity-type existence result for the weakly homogeneous variational inequality. The conditions we used are easier to check than the degree-theoretic condition and our result crosses each other with the main result established by Gowda and Sossa and the main result given by Ma, Zheng and Huang [SIAM J. Optim., 2020, 30(1): 132–148], respectively. Besides, we show the distinctiveness of our existence result by comparing it with the well-known coercivity result obtained for variational inequalities and a norm-coercivity result obtained for complementarity problems, respectively. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we study an inverse problem for a linear third-order pseudoparabolic equation. The investigated inverse problem consists of determining a space-dependent coefficient of the right-hand side of a pseudoparabolic equation. As an additional information a final overdetermination condition is considered. Under the suitable conditions on the data of the problem, the unique solvability of the considered inverse problem is established. The stability of solutions is also proved. The established results are also true for inverse problems for parabolic equations, which could be obtained as a regularization of the studied pseudoparabolic equation. In addition, the pseudoparabolic problem is discretized using the cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for three benchmark test examples are presented and discussed. Moreover, the von Neumann stability analysis is also discussed. [ABSTRACT FROM AUTHOR]

Using quasi-Newton methods in stochastic optimization is not a trivial task given the difficulty of extracting curvature information from the noisy gradients. Moreover, pre-conditioning noisy gradient observations tend to amplify the noise. We propose a Bayesian approach to obtain a Hessian matrix approximation for stochastic optimization that minimizes the secant equations residue while retaining the extreme eigenvalues between a specified range. Thus, the proposed approach assists stochastic gradient descent to converge to local minima without augmenting gradient noise. We propose maximizing the log posterior using the Newton-CG method. Numerical results on a stochastic quadratic function and an $ \ell _2 $ ℓ2-regularized logistic regression problem are presented. In all the cases tested, our approach improves the convergence of stochastic gradient descent, compensating for the overhead of solving the log posterior maximization. In particular, pre-conditioning the stochastic gradient with the inverse of our Hessian approximation becomes more advantageous the larger the condition number of the problem is. [ABSTRACT FROM AUTHOR]

We propose and analyze a model-based derivative-free (DFO) algorithm for solving bound-constrained optimization problems where the objective function is the composition of a smooth function and a vector of black-box functions. We assume that the black-box functions are smooth and the evaluation of them is the computational bottleneck of the algorithm. The distinguishing feature of our algorithm is the use of approximate function values at interpolation points which can be obtained by an application-specific surrogate model that is cheap to evaluate. As an example, we consider the situation in which a sequence of related optimization problems is solved and present a regression-based approximation scheme that uses function values that were evaluated when solving prior problem instances. In addition, we propose and analyze a new algorithm for obtaining interpolation points that handles unrelaxable bound constraints. Our numerical results show that our algorithm outperforms a state-of-the-art DFO algorithm for solving a least-squares problem from a chemical engineering application when a history of black-box function evaluations is available. [ABSTRACT FROM AUTHOR]

In this paper, a Euler wavelets method for solving the variational problems is presented. The operational matrices of integration and product of Euler wavelets are calculated. Then, by using Euler wavelets and the operational matrices, the variational problems are reduced into the system of algebraic equations. Furthermore, the convergence analysis and error bound of the Euler wavelets method are given. Some examples are included to demonstrate the applicability and validity of the schemes. [ABSTRACT FROM AUTHOR]

This paper is concerned with the growth-driven shape-programming problem, which involves determining a growth tensor that can produce a deformation on a hyperelastic body reaching a given target shape. We consider the two cases of globally compatible growth, where the growth tensor is a deformation gradient over the undeformed domain, and the incompatible one, which discards such hypothesis. We formulate the problem within the framework of optimal control theory in hyperelasticity. The Hausdorff distance is used to quantify dissimilarities between shapes; the complexity of the actuation is incorporated in the cost functional as well. Boundary conditions and external loads are allowed in the state law, thus extending previous works where the stress-free hypothesis turns out to be essential. A rigorous mathematical analysis is then carried out to prove the well-posedness of the problem. The numerical approximation is performed using gradient-based optimisation algorithms. Our main goal in this part is to show the possibility to apply inverse techniques for the numerical approximation of this problem, which allows us to address more generic situations than those covered by analytical approaches. Several numerical experiments for beam-like and shell-type geometries illustrate the performance of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

Due to the importance of vibration effects on the functional accuracy of mechanical systems, this research aims to develop a precise model of a nonlinearly vibrating single-link mobile flexible manipulator. The manipulator consists of an elastic arm, a rotary motor, and a rigid carrier, and undergoes general in-plane rigid body motion along with elastic transverse deformation. To accurately model the elastic behavior, Timoshenko's beam theory is used to describe the flexible arm, which accounts for rotary inertia and shear deformation effects. By applying Newton's second law, the nonlinear governing equations of motion for the manipulator are derived as a coupled system of ordinary differential equations (ODEs) and partial differential equations (PDEs). Then, the assumed mode method (AMM) is used to solve this nonlinear system of governing equations with appropriate shape functions. The assumed modes can be obtained after solving the characteristic equation of a Timoshenko beam with clamped boundary conditions at one end and an attached mass/inertia at the other. In addition, the effect of the transverse vibration of the inextensible arm on its axial behavior is investigated. Despite the axial rigidity, the effect makes the rigid body dynamics invalid for the axial behavior of the arm. Finally, numerical simulations are conducted to evaluate the performance of the developed model, and the results are compared with those obtained by the finite element approach. The comparison confirms the validity of the proposed dynamic model for the system. According to the mentioned features, this model can be reliable for investigating the system's vibrational behavior and implementing vibration control algorithms. [ABSTRACT FROM AUTHOR]

We extend our study of the swarm-based gradient descent method for non-convex optimization, (Lu et al., Swarm-based gradient descent method for non-convex optimization, 2022, arXiv:2211.17157), to allow random descent directions. We recall that the swarm-based approach consists of a swarm of agents, each identified with a position, x , and mass, m . The key is the transfer of mass from high ground to low(-est) ground. The mass of an agent dictates its step size: lighter agents take larger steps. In this paper, the essential new feature is the choice of direction: rather than restricting the swarm to march in the steepest gradient descent, we let agents proceed in randomly chosen directions centered around — but otherwise different from — the gradient direction. The random search secures the descent property while at the same time, enabling greater exploration of ambient space. Convergence analysis and benchmark optimizations demonstrate the effectiveness of the swarm-based random descent method as a multi-dimensional global optimizer. [ABSTRACT FROM AUTHOR]

This paper studies the interplay between the concepts of error bounds and the Kurdyka–Łojasiewicz (KL) inequality on Hadamard manifolds. To this end, we extend some properties and existence results of a solution for differential inclusions on Hadamard manifolds. As a second contribution, we show how the KL inequality can be used to obtain the convergence of the gradient method for solving convex feasibility problems on Hadamard manifolds. The convergence results of the alternating projection method are also established for cyclic and random projections on Hadamard manifolds and, more generally, CAT(0) spaces. [ABSTRACT FROM AUTHOR]

In the framework of a real Hilbert space, we address the problem of finding the zeros of the sum of a maximally monotone operator A and a cocoercive operator B. We study the asymptotic behaviour of the trajectories generated by a second order equation with vanishing damping, attached to this problem, and governed by a time-dependent forward–backward-type operator. This is a splitting system, as it only requires forward evaluations of B and backward evaluations of A. A proper tuning of the system parameters ensures the weak convergence of the trajectories to the set of zeros of A + B , as well as fast convergence of the velocities towards zero. A particular case of our system allows to derive fast convergence rates for the problem of minimizing the sum of a proper, convex and lower semicontinuous function and a smooth and convex function with Lipschitz continuous gradient. We illustrate the theoretical outcomes by numerical experiments. [ABSTRACT FROM AUTHOR]

In this work, we are interested in the famous FISTA algorithm. We show that FISTA is an automatic geometrically optimized algorithm for functions satisfying a quadratic growth assumption. This explains why FISTA works better than the standard Forward-Backward algorithm (FB) in such a case, although FISTA is known to have a polynomial asymptotic convergence rate while FB is exponential. We provide a simple rule to tune the α parameter within the FISTA algorithm to reach an ε -solution with an optimal number of iterations. These new results highlight the efficiency of FISTA algorithms, and they rely on new non asymptotic bounds for FISTA. [ABSTRACT FROM AUTHOR]

This paper proposes an entropy-weighted local intensity clustering-based model for segmenting intensity inhomogeneous images caused by the bias field. The variational model minimizes an energy functional consisting of a regularization term of the total length of object boundaries and a data-fitting term partitioning the image. Specifically, the total length is approximated by the convolution of the heat kernel with the characteristic functions of the segmented regions of interest. The data-fitting term is derived from the multiplicative bias field resulting in a local intensity clustering property and further weighted by the local entropy. One of the most advantageous features of the proposed model is that it can simultaneously segment the image and estimate the bias field for intensity inhomogeneity correction. Moreover, a simple and efficient iterative convolution-thresholding scheme can be applied to realize the model, exhibiting the energy-decaying property. Finally, numerical simulations are carried out to validate the superior performance of the approach. [ABSTRACT FROM AUTHOR]

In this paper, we present a new nonsmooth Newton-type algorithm for solving the generalized complementarity problem based on its reformulation as a system of nonlinear equations using a one-parametric family of complementarity functions. We demonstrate, under suitable hypotheses, that this algorithm converges locally and q-quadratically. In addition, we show numerical experiments that allow us to see the good performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to discuss hp spectral element method for optimal control problem governed by a nonlinear elliptic equation with L 2 -norm constraint for control variable. We then set up its weak formulation and hp spectral element approximation scheme. A priori error estimates of hp spectral element approximation based on some suitable projection operators are proved carefully. Using some properties of projection operators, a posteriori error estimates for both the state and the control approximation under some reasonable assumptions are established rigorously. Such estimates are useful tools, which can be used to construct reliable adaptive spectral element methods for optimal control problems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a forward-backward-forward algorithm with inertial extrapolation to solve the problem of finding a common solution to the variational inequality problem (CSVIP) in a Hadamard manifold. Using a self-adaptive step size, we obtain convergence results under some standard conditions. Numerical examples are given to illustrate the theoretical analysis. Our result is a generalization and extension of previously announced results in this direction in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear scalarization function for a variable domination structure in an arbitrary linear space without assuming any particular topology. Conditions are provided under which the nonlinear scalarization function possesses several useful properties such as finiteness, properness, positive homogeneity, subadditivity, (strict) monotonicity, convexity or continuity. These properties are employed to characterize approximate efficiency in linear spaces. [ABSTRACT FROM AUTHOR]

This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, our experiments showed that the relaxation by rank-one convexification delivering an approximation to the quasiconvex envelope prevents mesh dependence of the solutions of finite element discretizations. By the combination, modification and parallelization of the underlying convexification algorithms, the novel approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step-size control prevent stability issues related to local minima in the energy landscape and the computation of derivatives. Numerical techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations. An interpretation in terms of microstructural damage evolution is given, based on the rank-one lamination process. [ABSTRACT FROM AUTHOR]

In this work, we present a fast, globally convergent, iterative algorithm for computing the asymptotically stable states of nonlinear large-scale systems of quadratic autonomous ordinary differential equations (ODE) modeling, e.g., the dynamic of complex chemical reaction networks. Toward this aim, we reformulate the problem as a box-constrained optimization problem where the roots of a set of nonlinear equations need to be determined. Then, we propose to use a projected Newton's approach combined with a gradient descent algorithm so that every limit point of the sequence generated by the overall algorithm is a stationary point. More importantly, we suggest replacing the standard orthogonal projector with a novel operator that ensures the final solution to satisfy the box constraints while lowering the probability that the intermediate points reached at each iteration belong to the boundary of the box where the Jacobian of the objective function may be singular. The effectiveness of the proposed approach is shown in a practical scenario concerning a chemical reaction network modeling the signaling network of colorectal cancer cells. Specifically, in this scenario the proposed algorithm is proved to be faster and more accurate than a classical dynamical approach where the asymptotically stable states are computed as the limit points of the flux of the Cauchy problem associated with the ODE system. [ABSTRACT FROM AUTHOR]

In this paper, by using the inexact scalarization proximal methods, we solve quasiconvex multiobjective optimization problems on Hadamard manifolds. Under some assumptions on the multifunction of problem and vector fields, our methods are proved to be convergent to a Pareto critical point of the problem. For the convex case, the sequences generated by the methods converge to a weak Pareto solution. [ABSTRACT FROM AUTHOR]

The problem of variational assimilation of observational data for the sea thermodynamics model is considered with the aim to reconstruct heat fluxes on the sea surface. The local sensitivity of functionals of the solution to the observational data and input data for heat fluxes is studied in the considered problem and the results of numerical experiments are presented for the Black Sea dynamics model. [ABSTRACT FROM AUTHOR]

Geometric numerical integration has recently been exploited to design symplectic accelerated optimization algorithms by simulating the Bregman Lagrangian and Hamiltonian systems from the variational framework introduced by Wibisono et al. In this paper, we discuss practical considerations which can significantly boost the computational performance of these optimization algorithms and considerably simplify the tuning process. In particular, we investigate how momentum restarting schemes ameliorate computational efficiency and robustness by reducing the undesirable effect of oscillations and ease the tuning process by making time-adaptivity superfluous. We also discuss how temporal looping helps avoiding instability issues caused by numerical precision, without harming the computational efficiency of the algorithms. Finally, we compare the efficiency and robustness of different geometric integration techniques and study the effects of the different parameters in the algorithms to inform and simplify tuning in practice. From this paper emerge symplectic accelerated optimization algorithms whose computational efficiency, stability and robustness have been improved, and which are now much simpler to use and tune for practical applications. [ABSTRACT FROM AUTHOR]

In this study, we concentrate on solving the problem of non-Lipschitz absolute value equations (NAVE). A new Bezier curve based smoothing technique is introduced and a new Levenberg–Marquardt type algorithm is developed depending on the smoothing technique. The numerical performance of the algorithm is analysed by considering some well-known and randomly generated test problems. Finally, the comparison with other methods is illustrated to demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We consider the question whether starting from a smooth initial condition 3D inviscid Euler flows on a periodic domain may develop singularities in a finite time. Our point of departure is the well-known result by Kato (J Funct Anal 9(3):296–305, 1972), which asserts the local existence of classical solutions to the Euler system in the Sobolev space H m for m > 5 / 2 . Thus, the potential formation of a singularity must be accompanied by an unbounded growth of the H m norm of the velocity field as the singularity time is approached. We perform a systematic search for “extreme” Euler flows that may realize such a scenario by formulating and solving a PDE-constrained optimization problem where the H 3 norm of the solution at a certain fixed time T > 0 is maximized with respect to the initial data subject to suitable normalization constraints. This problem is solved using a state-of-the-art Riemannian conjugate gradient method where the gradient is obtained from solutions of an adjoint system. Computations performed with increasing numerical resolutions demonstrate that, as asserted by the theorem of Kato (1972), when the optimization time window [0, T] is sufficiently short, the H 3 norm remains bounded in the extreme flows found by solving the optimization problem, which indicates that the Euler system is well-posed on this “short” time interval. On the other hand, when the window [0, T] is long, possibly longer than the time of the local existence asserted by Kato’s theorem, then the H 3 norm of the extreme flows diverges upon resolution refinement, which indicates a possible singularity formulation on this “long” time interval. The extreme flow obtained on the long time window has the form of two colliding vortex rings and is characterized by certain symmetries. In particular, the region of the flow in which a singularity might occur is nearly axisymmetric. [ABSTRACT FROM AUTHOR]

One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms. [ABSTRACT FROM AUTHOR]

First-order optimization algorithms can be considered as a discretization of ordinary differential equations (ODEs) (Su et al. in Adv Neural Inf Process Syst 27, 2014). In this perspective, studying the properties of the corresponding trajectories may lead to convergence results which can be transfered to the numerical scheme. In this paper we analyse the following ODE introduced by Attouch et al. (J Differ Equ 261(10):5734–5783, 2016): ∀ t ⩾ t 0 , x ¨ (t) + α t x ˙ (t) + β H F (x (t)) x ˙ (t) + ∇ F (x (t)) = 0 , where α > 0 , β > 0 and H F denotes the Hessian of F. This ODE can be derived to build numerical schemes which do not require F to be twice differentiable as shown in Attouch et al. (Math Program 1–43, 2020) and Attouch et al. (Optimization 72:1–40, 2021). We provide strong convergence results on the error F (x (t)) - F ∗ and integrability properties on ‖ ∇ F (x (t)) ‖ under some geometry assumptions on F such as quadratic growth around the set of minimizers. In particular, we show that the decay rate of the error for a strongly convex function is O (t - α - ε) for any ε > 0 . These results are briefly illustrated at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we propose a sparse equity portfolio optimization model that aims at minimizing transaction cost by avoiding small investments while promoting diversification to help mitigate the volatility in the portfolio. The former is achieved by including the ℓ 0 -norm regularization of the asset weights to promote sparsity. Subjected to a minimum expected return, the proposed model turns out to be an objective function consisting of discontinuous and nonconvex terms. The complexity of the model calls for proximal method, which allows us to handle the objective terms separately via the corresponding proximal operators. We develop an efficient algorithm to find the optimal portfolio and prove its global convergence. The efficiency of the algorithm is demonstrated using real stock data and the model is promising in portfolio selection in terms of generating higher expected return while maintaining good level of sparsity, and thus minimizing transaction cost. [ABSTRACT FROM AUTHOR]

The multistage stochastic variational inequality is reformulated into a variational inequality with separable structure through introducing a new variable. The prediction–correction ADMM which was originally proposed in He et al. (J Comput Math 24:693–710, 2006) for solving deterministic variational inequalities in finite-dimensional spaces is adapted to solve the multistage stochastic variational inequality. Weak convergence of the sequence generated by that algorithm is proved under the conditions of monotonicity and Lipschitz continuity. When the sample space is a finite set, the corresponding multistage stochastic variational inequality is actually defined on a finite-dimensional Hilbert space and the strong convergence of the algorithm naturally holds true. Some numerical examples are given to show the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we use the calculus of variations to derive a sensitivity analysis for ordinary differential equations with events. This sweeping gradient method (SGM) requires a forward sweep to evaluate the original model and a backwards sweep of the adjoint to compute the sensitivity. The method is applied to canonical optimal control problems with numerical examples, including the sampled linear quadratic regulator and the optimal time-switching and state-switching for minimum-time transfer of the double integrator. We show that the application of the SGM for these examples matches the gradient determined analytically. Numerical examples are produced using gradient-based optimization algorithms. The emphasis of this work is on modeling considerations for the effective application of this method. [ABSTRACT FROM AUTHOR]

We present a general version of bundle trust region method for minimizing convex functions. The trust region is constructed by generic p -norm with p ∈ [ 1 , + ∞ ] . In each iteration the algorithm solves a subproblem with a constraint involving p -norm. We show the convergence of the generic bundle trust region algorithm. In implementation, the infinity norm is chosen so that a linear programming subproblem is solved in each iteration. Preliminary numerical experiments show that our algorithm performs comparably with the traditional bundle trust region method and has advantages in solving large-scale problems. [ABSTRACT FROM AUTHOR]

Discrete inverse problems correspond to solving a system of equations in a stable way with respect to noise in the data. A typical approach to select a meaningful solution is to introduce a regularizer. While for most applications the regularizer is convex, in many cases it is neither smooth nor strongly convex. In this paper, we propose and study two new iterative regularization methods, based on a primal-dual algorithm, to regularize inverse problems efficiently. Our analysis, in the noise free case, provides convergence rates for the Lagrangian and the feasibility gap. In the noisy case, it provides stability bounds and early stopping rules with theoretical guarantees. The main novelty of our work is the exploitation of some a priori knowledge about the solution set: we show that the linear equations determined by the data can be used more than once along the iterations. We discuss various approaches to reuse linear equations that are at the same time consistent with our assumptions and flexible in the implementation. Finally, we illustrate our theoretical findings with numerical simulations for robust sparse recovery and image reconstruction. We confirm the efficiency of the proposed regularization approaches, comparing the results with state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Let H be a real Hilbert space, and f : H → R be a convex twice differentiable function whose solution set argmin f is nonempty. We investigate the long time behavior of the trajectories of the vanishing damped dynamical system with Tikhonov regularizing term and Hessian-driven damping x ¨ (t) + α x ˙ (t) + δ ∇ 2 f (x (t)) x ˙ (t) + β (t) ∇ f (x (t)) + c x (t) = 0 where α , c , δ are three positive constants, and the time scale parameter β is a positive nondecreasing function such that lim t → + ∞ β (t) = + ∞ . Under some assumptions on the parameter β , we will show rapid convergence of values, strong convergence toward the minimum norm element of argmin f , and rapid convergence of the gradients toward zero. Note that the Hessian-driven damping significantly reduces the oscillatory aspects, and the time scale parameter β improves the rate of convergences mentioned above. As particular cases of β , we set β (t) = t p ln q (t) , for (p , q) ∈ (R +) 2 ∖ { (0 , 0) } , and β (t) = e γ t p , for p ∈ ] 0 , 1 [ and γ > 0 . The manuscript concludes with two numerical examples and comments on their performance. [ABSTRACT FROM AUTHOR]

In this paper, we present a general oligopolistic market equilibrium model in which each firm produces several commodities in a time interval. To this aim, we introduce tensor variational inequalities in Hilbert spaces which are a powerful tool to analyse the model. Indeed we characterize the equilibrium condition by means of a suitable time-dependent tensor variational inequality. In addition, we prove some existence and regularity results and a numerical scheme to compute the solution. Finally we provide a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we propose three kinds of inertial approximation methods based on the proximal gradient algorithm to accelerate the convergence of the algorithm for solving convex bilevel optimization problems. Under reasonable parameters, we prove that our algorithms converge strongly to some solution of the problem, which is the unique solution of a variational inequality problem. Firstly, two alternated inertial methods are presented. Secondly, a multi-step inertial method is proposed to accelerate the convergence of the algorithm. Thirdly, an alternated multi-step inertial method is further introduced. In addition, we also consider that the inertial parameters can be chosen as positive or negative parameters, and we get better results. Our numerical results illustrate the performances of our algorithms and present some comparisons with related algorithms. Our results improve and extend the corresponding results reported by some authors recently. [ABSTRACT FROM AUTHOR]

In this work we are interested in stochastic particle methods for multi-objective optimization. The problem is formulated via scalarization using parametrized, single-objective sub-problems which are solved simultaneously. To this end a consensus based multi-objective optimization method on the search space combined with an additional heuristic strategy to adapt parameters during the computations is proposed. The adaptive strategy aims to distribute the particles uniformly over the image space, in particular over the Pareto front, by using energy-based measures to quantify the diversity of the system. The resulting gradient-free metaheuristic algorithm is mathematically analyzed using a mean-field approximation of the algorithm iteration and convergence guarantees towards Pareto optimal points are rigorously proven. In addition, we analyze the dynamics when the Pareto front corresponds to the unit simplex, and show that the adaptive mechanism reduces to a gradient flow in this case. Several numerical experiments show the validity of the proposed stochastic particle dynamics, investigate the role of the algorithm parameters and validate the theoretical findings. [ABSTRACT FROM AUTHOR]

For bounded, convex sets Ω ⊂ R d , the sharp Poincaré constant C (Ω) , which appears in | | f - f ¯ Ω | | L ∞ (Ω) ≤ C (Ω) | | ∇ f | | L ∞ (Ω) , is given by C (Ω) = max ∂ Ω ζ for a specific convex function ζ [Bevan et al. in Proc Am Math Soc 151:1071–1085, 2023 (Theorem 1.1)]. We study C (·) as a function on convex sets, in particular on polyhedra, and find that while a geometric characterization of C (Ω) for triangles is possible, for other polyhedra the problem of ordering ζ (V i) , where V i are the vertices of Ω , can be formidable. In these cases, we develop estimates of C (Ω) from above and below in terms of more tractable quantities. We find, for example, that a good proxy for C(Q) when Q is a planar polygon with vertices V i and centroid γ (Q) is the quantity D (Q) = max i | V i - γ (Q) | , with an error of up to ∼ 8 % . A numerical study suggests that a similar statement holds for k-gons, this time with a maximal error across all k-gons of ∼ 13 % . We explore the question of whether there is, for each Ω , at least one point M capable of ordering the ζ (V i) according to the ordering of the | V i - M | . For triangles, M always exists; for quadrilaterals, M seems always to exist; for 5-gons and beyond, they seem not to. [ABSTRACT FROM AUTHOR]

In a Hilbert framework, for general convex differentiable optimization, we consider accelerated gradient dynamics combining Tikhonov regularization with Hessian-driven damping. The temporal discretization of these dynamics leads to a new class of first-order optimization algorithms with favorable properties. The Tikhonov regularization parameter is assumed to tend to zero as time tends to infinity, which preserves equilibria. The presence of the Tikhonov regularization term induces a strong property of convexity which vanishes asymptotically. To take advantage of the fast convergence rates attached to the heavy ball method in the strongly convex case, we consider inertial dynamics where the viscous damping coefficient is proportional to the square root of the Tikhonov regularization parameter, and hence converges to zero. The geometric damping, controlled by the Hessian of the function to be minimized, induces attenuation of the oscillations. Under an appropriate setting of the parameters, based on Lyapunov's analysis, we show that the trajectories provide at the same time several remarkable properties: fast convergence of values, fast convergence of gradients towards zero, and strong convergence to the minimum norm minimizer. We show that the corresponding proximal algorithms share the same properties as continuous dynamics. The numerical illustrations confirm the results obtained. This study extends a previous paper by the authors regarding similar problems without the presence of Hessian driven damping. [ABSTRACT FROM AUTHOR]

In this work, we approach the minimization of a continuously differentiable convex function under linear equality constraints by a second-order dynamical system with an asymptotically vanishing damping term. The system under consideration is a time rescaled version of another system previously found in the literature. We show fast convergence of the primal-dual gap, the feasibility measure, and the objective function value along the generated trajectories. These convergence rates now depend on the rescaling parameter, and thus can be improved by choosing said parameter appropriately. When the objective function has a Lipschitz continuous gradient, we show that the primal-dual trajectory asymptotically converges weakly to a primal-dual optimal solution to the underlying minimization problem. We also exhibit improved rates of convergence of the gradient along the primal trajectories and of the adjoint of the corresponding linear operator along the dual trajectories. We illustrate the theoretical outcomes and also carry out a comparison with other classes of dynamical systems through numerical experiments. [ABSTRACT FROM AUTHOR]

We adopt the integral definition of the fractional Laplace operator and analyze solution techniques for fractional, semilinear, and elliptic optimal control problems posed on Lipschitz polytopes. We consider two strategies of discretization: a semidiscrete scheme where the admissible control set is not discretized and a fully discrete scheme where such a set is discretized with piecewise constant functions. As an instrumental step, we derive error estimates for finite element discretizations of fractional semilinear elliptic partial differential equations (PDEs) on quasi-uniform and graded meshes. With these estimates at hand, we derive error bounds for the semidiscrete scheme and improve the ones that are available in the literature for the fully discrete scheme. [ABSTRACT FROM AUTHOR]

This paper studies the convergence of three temporal semi-discretizations for a backward semilinear stochastic evolution equation. For general terminal value and general coefficient with Lipschitz continuity, the convergence of the first two temporal semi-discretizations is established, and an explicit convergence rate is derived for the third temporal semi-discretization. The third temporal semi-discretization is applied to a general stochastic linear quadratic control problem, and the convergence of a temporally semi-discrete approximation to the optimal control is established. [ABSTRACT FROM AUTHOR]

In this paper we provide a rigorous convergence analysis for the renowned particle swarm optimization method by using tools from stochastic calculus and the analysis of partial differential equations. Based on a continuous-time formulation of the particle dynamics as a system of stochastic differential equations, we establish convergence to a global minimizer of a possibly nonconvex and nonsmooth objective function in two steps. First, we prove consensus formation of an associated mean-field dynamics by analyzing the time-evolution of the variance of the particle distribution, which acts as Lyapunov function of the dynamics. We then show that this consensus is close to a global minimizer by employing the asymptotic Laplace principle and a tractability condition on the energy landscape of the objective function. These results allow for the usage of memory mechanisms, and hold for a rich class of objectives provided certain conditions of well-preparation of the hyperparameters and the initial datum. In a second step, at least for the case without memory effects, we provide a quantitative result about the mean-field approximation of particle swarm optimization, which specifies the convergence of the interacting particle system to the associated mean-field limit. Combining these two results allows for global convergence guarantees of the numerical particle swarm optimization method with provable polynomial complexity. To demonstrate the applicability of the method we propose an efficient and parallelizable implementation, which is tested in particular on a competitive and well-understood high-dimensional benchmark problem in machine learning. [ABSTRACT FROM AUTHOR]

In this article, we introduce a forward–backward splitting method with a new step size rule for finding a singularity point of an inclusion problem which is defined by means of a sum of a single-valued vector field and a multi-valued vector field on a Hadamard manifold. Using a Mann, viscosity and an inertial extrapolation method, we establish a convergence result without prior knowledge of the Lipschitz constant of the underlying operator. We present some applications of our result to variational inequality problem. Finally, we present some numerical examples to demonstrate the numerical behavior of our proposed method. The result discuss in this article extends and complements many related results in the literature. [ABSTRACT FROM AUTHOR]

We study the question as to when the closed convex hull of the graph of a K-convex map equals its K-epigraph. In particular, we shed light onto the smallest cone K such that a given map has convex and closed K-epigraph, respectively. We apply our findings to several examples in matrix space as well as to convex composite functions. [ABSTRACT FROM AUTHOR]

The exterior Bernoulli problem — a prototype stationary free boundary problem — is rephrased into a shape optimization setting using an energy-gap type cost functional that is subject to two auxiliary problems: a pure Dirichlet problem and a mixed Dirichlet-Robin boundary value problem. It is demonstrated here that depending on what method is used, the shape gradient of the cost functional may appear in a different form. The dissimilarity in structure comes from the way the adjoint variable was utilized in the computation — then resulting to a different adjoint problem. The shape derivative is first obtained via Delfour-Zolésio's minimax formulation, and then by using the weak form of the Eulerian derivative of the states coupled with the adjoint method. The latter approach is accomplished by first showing the existence of the derivatives. A fast iterative scheme based on finite element method is then formulated to numerically solve the proposed shape optimization formulation. The feasibility of the method — highlighting its efficiency and practicality — is illustrated through numerical examples in two and three dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze an inexact version of the symmetric proximal alternating direction method of multipliers (ADMM) for solving linearly constrained optimization problems. Basically, the method allows its first subproblem to be solved inexactly in such way that a relative approximate criterion is satisfied. In terms of the iteration number k, we establish global O (1 / k) pointwise and O (1 / k) ergodic convergence rates of the method for a domain of the acceleration parameters, which is consistent with the largest known one in the exact case. Since the symmetric proximal ADMM can be seen as a class of ADMM variants, the new algorithm as well as its convergence rates generalize, in particular, many others in the literature. Numerical experiments illustrating the practical advantages of the method are reported. To the best of our knowledge, this work is the first one to study an inexact version of the symmetric proximal ADMM. [ABSTRACT FROM AUTHOR]