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Measurements of light typically capture amplitude information, often overlooking crucial phase details. This oversight underscores the importance of phase retrieval (PR), essential in biomedical imaging, X-ray crystallography, and microscopy, for reconstructing complex signals from intensity-only measurements. Traditional methods frequently fall short, especially in noisy conditions or when restricted to single-shot measurements. To address these challenges, we introduce a novel model that combines non-convex regularization with physical constraints. The model adopts the smoothly clipped absolute deviation (SCAD) function as a sparsity regularization term for gradients, incorporating fundamental constraints on support and absorption to establish an inverse model. Using the alternating direction method of multipliers (ADMM), we break down the problem into manageable sub-problems, implementing SCAD shrinkage in the complex domain and applying Wirtinger gradient projection methods. A thorough convergence analysis validates the stability and robustness of the algorithm. Extensive simulations confirm significant improvements in reconstruction quality compared to existing methods, with evaluations demonstrating superior performance across various noise levels and parameter settings. [ABSTRACT FROM AUTHOR]

We develop a variable metric forward-reflected-Douglas-Rachford method for finding a zero point of a sum of one Lipschitzian monotone and two maximally monotone operators acting on a real Hilbert space. The weak convergence of the proposed algorithm is formally proved under standard conditions. We also consider the linear convergence of the proposed method in constant-case and obtain the desired result under the strong monotonicity and Lipschitz condition of maximally monotone operators. [ABSTRACT FROM AUTHOR]

To ensure the system stability of the H 2 -guaranteed cost optimal decentralized control (ODC) problem, we formulate an approximate semidefinite programming (SDP) problem that leverages the block diagonal structure of the decentralized controller’s gain matrix. To minimize data storage requirements and enhance computational efficiency, we employ the Kronecker product to vectorize the SDP problem into a conic programming (CP) problem. We then propose a proximal alternating direction method of multipliers (PADMM) to solve the dual of the resulting CP problem. By using the equivalence between the semi-proximal ADMM and the (partial) proximal point algorithm, we identify the non-expansive operator of PADMM, enabling the use of Halpern fixed-point iteration to accelerate the algorithm. Finally, we demonstrate that the sequence generated by the proposed accelerated PADMM exhibits a fast convergence rate for the Karush–Kuhn–Tucker residual. Numerical experiments confirm that the accelerated algorithm outperforms the well-known COSMO, MOSEK, and SCS solvers in efficiently solving large-scale CP problems, particularly those arising from H 2 -guaranteed cost ODC problems. [ABSTRACT FROM AUTHOR]

We present a gradient-based calibration algorithm to identify the system matrices of a linear port-Hamiltonian system from given input–output time data. Aiming for a direct structure-preserving approach, we employ techniques from optimal control with ordinary differential equations and define a constrained optimization problem. The input-to-state stability is discussed which is the key step towards the existence of optimal controls. Further, we derive the first-order optimality system taking into account the port-Hamiltonian structure. Indeed, the proposed method preserves the skew symmetry and positive (semi)-definiteness of the system matrices throughout the optimization iterations. Numerical results with perturbed and unperturbed synthetic data, as well as an example from the PHS benchmark collection [17] demonstrate the feasibility of the approach. [ABSTRACT FROM AUTHOR]

In this article, we study two methods for solving monotone inclusions in real Hilbert spaces involving the sum of a maximally monotone operator, a monotone-Lipschitzian operator, a cocoercive operator, and a normal cone to a vector subspace. Our algorithms split and exploits the intrinsic properties of each operator involved in the inclusion. We derive our methods by combining partial inverse techniques with the forward-half-reflected-backward algorithm and with the forward-shadow-Douglas–Rachford (FSDR) algorithm, respectively. Our methods inherit the advantages of those methods, requiring only one activation of the Lipschitzian operator, one activation of the cocoercive operator, two projections onto the closed vector subspace, and one calculation of the resolvent of the maximally monotone operator. Additionally, to allow larger step-sizes in one of the proposed methods, we revisit FSDR by extending its convergence for larger step-sizes. Furthermore, we provide methods for solving monotone inclusions involving a sum of maximally monotone operatores and for solving a system of primal-dual inclusions involving a mixture of sums, linear compositions, parallel sums, Lipschitzian operators, cocoercive operators, and normal cones. We apply our methods to constrained composite convex optimization problems as a specific example. Finally, in order to compare our methods with existing methods in the literature, we provide numerical experiments on constrained total variation least-squares optimization problems and computed tomography inverse problems. We obtain promising numerical results. [ABSTRACT FROM AUTHOR]

A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler–Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding primal problem is ill-posed and demonstrates an explosive 'Hadamard instability;' this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response. [ABSTRACT FROM AUTHOR]

This note discusses the computation of the distance function with respect to Finsler metrics. To this end, we show how the Finsler variants of the Eikonal equation can be solved by a primal-dual algorithm exploiting the variational structure. We also discuss the acceleration of the algorithm by preconditioning techniques, and illustrate the flexibility of the proposed method through a series of numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose two new unified splitting methods for monotone inclusion problems with three operators in real Hilbert spaces. These methods are based on the combination of Douglas-Rachford method and other methods, forward-backward-forward method and reflected-forward-backward method. The weak convergence of new algorithms under standard assumptions is established. We also give some numerical examples to demonstrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

We consider the proximal-gradient method for minimizing the sum of a smooth function and a convex non-smooth overlapping group- $ \ell _1 $ ℓ 1 regularizer, which is known to promote sparse solutions with respect to its groups. A feature that distinguishes our work from most in the literature is that the proximal operator for the overlapping group- $ \ell _1 $ ℓ 1 function does not admit a closed-form solution, which introduces challenges when proving convergence of the iterates and especially when proving that the iterates correctly identify the group-sparse structure of the optimal solution. To address these challenges, we present an implementable termination condition for the proximal-gradient algorithm, and a specialized primal-dual subproblem solver that is designed to ensure that a group-sparse structure identification property holds. In particular, we give an upper bound on the maximum number of iterations before the correct support (i.e. group-sparse structure) of an optimal solution is identified. [ABSTRACT FROM AUTHOR]

We introduce a generalized proximal point algorithm, and perform a detailed convergence analysis with the focus on the case of degenerate metric. The degeneracy leads to a well-defined resolvent form restricted to a reduced dimensional space. This approach unifies the algorithmic structures of Douglas-Rachford splitting and other related operator splitting schemes. Various aspects of these algorithms, in particular, the convergence of Douglas-Rachford splitting in terms of the solution itself and Chambolle-Pock algorithm under the limit setting, are investigated or revisited by the unified degenerate proximal point analysis. [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]

We address the numerical solution to multimarginal optimal transport (MMOT) with pairwise costs. MMOT, as a natural extension from the classical two-marginal optimal transport, has many important applications including image processing, density functional theory and machine learning, but lacks efficient and exact numerical methods. The popular entropy-regularized method may suffer numerical instability and blurring issues. Inspired by the back-and-forth method introduced by Jacobs and Léger, we investigate MMOT problems with pairwise costs. We show that such problems have a graphical representation and leverage this structure to develop a new computationally gradient ascent algorithm to solve the dual formulation of such MMOT problems. Our method produces accurate solutions which can be used for the regularization-free applications, including the computation of Wasserstein barycenters with high resolution imagery. [ABSTRACT FROM AUTHOR]

In the present paper, we examine a Crouzeix–Raviart approximation for non-linear partial differential equations having a (p, δ)-structure for some p ∈ (1, ∞) and δ ⩾ 0. We establish a priori error estimates, which are optimal for all p ∈ (1, ∞) and δ ⩾ 0, medius error estimates, i.e., best-approximation results, and a primal–dual a posteriori error estimate, which is both reliable and efficient. The theoretical findings are supported by numerical experiments. [ABSTRACT FROM AUTHOR]

Accounting for climate transition risks is one of the most important challenges in the transition to a low-carbon economy. Banks are encouraged to align their investment portfolios to CO2 trajectories fixed by international agreements, showing the necessity of a quantitative methodology to implement it. We propose a mathematical formulation for this problem and a multistage optimization criterion for a transition between the current bank portfolio and a target one. The optimization problem combines the Monge–Kantorovich formulation of optimal transport, for which the cost is defined according to the financial context, and a credit risk measure. We show that the problem is well-posed, and can be embedded into a saddle-point problem for which Primal–Dual algorithms can be used. We design a numerical scheme that is able to solve the problem in available time, with nice scalability properties according to the number of decision times; its numerical convergence is analysed. Last we test the model using real financial data, illustrating that the optimal portfolio alignment may differ from the naive interpolation between the initial portfolio and the target. [ABSTRACT FROM AUTHOR]

This work focuses on recovering images from various forms of corruption, for which a challenging non-smooth, non-convex optimization model is proposed. The model consists of a concave truncated data fidelity functional and a convex total-variational term. We introduce and study various novel equivalent mathematical models from the perspective of duality, leading to two new optimization frameworks in terms of two-stage and single-stage. The two-stage optimization approach follows a classical convex-concave optimization framework with an inner loop of convex optimization and an outer loop of concave optimization. In contrast, the single-stage optimization approach follows the proposed novel convex model, which boils down to the global optimization of all variables simultaneously. Moreover, the key step of both optimization frameworks can be formulated as linearly constrained convex optimization and efficiently solved by the augmented Lagrangian method. For this, two different implementations by projected-gradient and indefinite-linearization are proposed to build up their numerical solvers. The extensive experiments show that the proposed single-stage optimization algorithms, particularly with indefinite linearization implementation, outperform the multi-stage methods in numerical efficiency, stability, and recovery quality. [ABSTRACT FROM AUTHOR]

We introduce a numerical scheme that approximates solutions to linear PDE’s by minimizing a residual in the W - 1 , p ′ (Ω) norm with exponents p > 2 . The resulting problem is solved by regularized Kačanov iterations, allowing to compute the solution to the non-linear minimization problem even for large exponents p ≫ 2 . Such large exponents remedy instabilities of finite element methods for problems like convection-dominated diffusion. [ABSTRACT FROM AUTHOR]

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576]. [ABSTRACT FROM AUTHOR]

In this paper we discuss the use of implicit Runge–Kutta schemes for the time discretization of optimal control problems with evolution equations. The specialty of the considered discretizations is that the discretizations schemes for the state and adjoint state are chosen such that discretization and optimization commute. It is well known that for Runge–Kutta schemes with this property additional order conditions are necessary. We give sufficient conditions for which class of schemes these additional order condition are automatically fulfilled. The focus is especially on implicit Runge–Kutta schemes of Gauss, Radau IA, Radau IIA, Lobatto IIIA, Lobatto IIIB and Lobatto IIIC collocation type up to order six. Furthermore, we also use a SDIRK (singly diagonally implicit Runge–Kutta) method to demonstrate, that for general implicit Runge–Kutta methods the additional order conditions are not automatically fulfilled. Numerical examples illustrate the predicted convergence rates. [ABSTRACT FROM AUTHOR]

For unimodal random variable S with fixed moments E S i = m i (i = 1 , 2 , ... , n) and mode m, a formula on variance of any functions of S is provided along the line of Khintchine transform. Upper bounds are derived on the variance of European call options and Gap options. The techniques are based on symmetrization, domination by quadratic functions of two variables and change of measures. [ABSTRACT FROM AUTHOR]

This work aims to minimize a continuously differentiable convex function with Lipschitz continuous gradient under linear equality constraints. The proposed inertial algorithm results from the discretization of the second-order primal-dual dynamical system with asymptotically vanishing damping term addressed by Boţ and Nguyen (J. Differential Equations 303:369–406, 2021), and it is formulated in terms of the Augmented Lagrangian associated with the minimization problem. The general setting we consider for the inertial parameters covers the three classical rules by Nesterov, Chambolle–Dossal and Attouch–Cabot used in the literature to formulate fast gradient methods. For these rules, we obtain in the convex regime convergence rates of order O 1 / k 2 for the primal-dual gap, the feasibility measure, and the objective function value. In addition, we prove that the generated sequence of primal-dual iterates converges to a primal-dual solution in a general setting that covers the two latter rules. This is the first result which provides the convergence of the sequence of iterates generated by a fast algorithm for linearly constrained convex optimization problems without additional assumptions such as strong convexity. We also emphasize that all convergence results of this paper are compatible with the ones obtained in Boţ and Nguyen (J. Differential Equations 303:369–406, 2021) in the continuous setting. [ABSTRACT FROM AUTHOR]

We develop a novel stochastic primal–dual splitting method with Bregman distances for solving structured composite problems involving infimal convolutions in non-Euclidean spaces. The sublinear convergence in expectation of the primal–dual gap is proved under mild conditions on stepsize for the general case. The linear convergence rate is obtained under additional condition like the strong convexity relative to Bregman functions. [ABSTRACT FROM AUTHOR]

Entropy-regularized optimal transport and its multi-marginal generalization have attracted increasing attention in various applications, in particular due to efficient Sinkhorn-like algorithms for computing optimal transport plans. However, it is often desirable that the marginals of the optimal transport plan do not match the given measures exactly, which led to the introduction of the so-called unbalanced optimal transport. Since unbalanced methods were not examined for the multi-marginal setting so far, we address this topic in the present paper. More precisely, we introduce the unbalanced multi-marginal optimal transport problem and its dual and show that a unique optimal transport plan exists under mild assumptions. Furthermore, we generalize the Sinkhorn algorithm for regularized unbalanced optimal transport to the multi-marginal setting and prove its convergence. For cost functions decoupling according to a tree, the iterates can be computed efficiently. At the end, we discuss three applications of our framework, namely two barycenter problems and a transfer operator approach, where we establish a relation between the barycenter problem and the multi-marginal optimal transport with an appropriate tree-structured cost function. [ABSTRACT FROM AUTHOR]

The augmented Lagrangian method (ALM) is extended to a broader-than-ever setting of generalized nonlinear programming in convex and nonconvex optimization that is capable of handling many common manifestations of nonsmoothness. With the help of a recently developed sufficient condition for local optimality, it is shown to be derivable from the proximal point algorithm through a kind of local duality corresponding to an optimal solution and accompanying multiplier vector that furnish a local saddle point of the augmented Lagrangian. This approach leads to surprising insights into stepsize choices and new results on linear convergence that draw on recent advances in convergence properties of the proximal point algorithm. Local linear convergence is shown to be assured for a class of model functions that covers more territory than before. [ABSTRACT FROM AUTHOR]

The primal-dual method of Chambolle and Pock is a widely used algorithm to solve various optimization problems written as convex-concave saddle point problems. Each update step involves the application of both the forward linear operator and its adjoint. However, in practical applications like computerized tomography, it is often computationally favourable to replace the adjoint operator by a computationally more efficient approximation. This leads to an adjoint mismatch in the algorithm. In this paper, we analyze the convergence of Chambolle–Pock's primal-dual method under the presence of a mismatched adjoint in the strongly convex setting. We present an upper bound on the error of the primal solution and derive stepsizes and mild conditions under which convergence to a fixed point is still guaranteed. Furthermore we show linear convergence similar to the result of Chambolle–Pock's primal-dual method without the adjoint mismatch. Moreover, we illustrate our results both for an academic and a real-world inspired application. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of the relaxed primal-dual algorithm with critical preconditioners for solving composite monotone inclusions in real Hilbert spaces. We prove that this algorithm define Krasnosel'skiĭ-Mann (KM) iterations in the range of a particular monotone self-adjoint linear operator with non-trivial kernel. Our convergence result generalizes (Condat in J Optim Theory Appl 158: 460–479, 2013, Theorem 3.3) and follows from that of KM iterations defined in the range of linear operators, which is a real Hilbert subspace under suitable conditions. The Douglas–Rachford splitting (DRS) with a non-standard metric is written as a particular instance of the primal-dual algorithm with critical preconditioners and we recover classical results from this new perspective. We implement the algorithm in total variation reconstruction, verifying the advantages of using critical preconditioners and relaxation steps. [ABSTRACT FROM AUTHOR]

Thomas has also very important contributions to the finite element analysis of optimal control problems in particular to error estimates for Finite element discretizations with boundary control, [[14]]. Keywords: Finite Elements; Optimal Control; Error Estimates; Numerical Analysis; 35J05; 35J25; 35J75; 49M05; 49M25; 49M29; 65M10; 65M12; 65M15; 65M60; 65N12; 65N30 EN Finite Elements Optimal Control Error Estimates Numerical Analysis 35J05 35J25 35J75 49M05 49M25 49M29 65M10 65M12 65M15 65M60 65N12 65N30 813 815 3 10/06/23 20231001 NES 231001 Dedicated to Thomas Apel on the occasion of his 60th birthday This special issue of Computational Methods in Applied Mathematics is dedicated to Thomas Apel on the occasion of his 60th birthday in 2022. Thomas worked also together with Olaf Steinbach on Finite element analysis for optimal control problems, [[20]], and Gabriel Acosta on Raviart-Thomas interpolation on anisotropic tetrahedra, [[1]]. Appl. 52 (2012), no. 1, 3-28. 15 T. Apel, J. Pfefferer and M. Winkler, Local mesh refinement for the discretization of Neumann boundary control problems on polyhedra, Math. [Extracted from the article]

The paper presents primal–dual proximal splitting methods for convex optimization, in which generalized Bregman distances are used to define the primal and dual proximal update steps. The methods extend the primal and dual Condat–Vũ algorithms and the primal–dual three-operator (PD3O) algorithm. The Bregman extensions of the Condat–Vũ algorithms are derived from the Bregman proximal point method applied to a monotone inclusion problem. Based on this interpretation, a unified framework for the convergence analysis of the two methods is presented. We also introduce a line search procedure for stepsize selection in the Bregman dual Condat–Vũ algorithm applied to equality-constrained problems. Finally, we propose a Bregman extension of PD3O and analyze its convergence. [ABSTRACT FROM AUTHOR]

Many large-scale optimization problems can be expressed as composite optimization models. Accelerated first-order methods such as the fast iterative shrinkage–thresholding algorithm (FISTA) have proven effective for numerous large composite models. In this paper, we present a new variation of FISTA, to be called C-FISTA, which obtains global linear convergence for a broader class of composite models than many of the latest FISTA variants. We demonstrate the versatility and effectiveness of C-FISTA through multiple numerical experiments on group Lasso, group logistic regression and geometric programming models. Furthermore, we utilize Fenchel duality to show C-FISTA can solve the dual of a finite sum convex optimization model. [ABSTRACT FROM AUTHOR]

It is known [Mangasarian, A Newton method for linear programming, J. Optim. Theory Appl. 121 (2004), pp. 1–18] that every linear program can be solved exactly by minimizing an unconstrained quadratic penalty program. The penalty program is parameterized by a scalar t>0, and one is able to solve the original linear program in this manner when t is selected larger than a finite, but unknown t 0 > 0. In this paper, we show that every linear program can be solved using the solution to a parameter-free penalty program. We also characterize the solutions to the quadratic penalty programs using fixed points of certain nonexpansive maps. This leads to an iterative thresholding algorithm that converges to a desired limit point. We show in numerical experiments that this iterative method can outperform a variety of standard quadratic program solvers. Finally, we show that for every t ∈ R , the solution one obtains by solving a parameterized penalty program is guaranteed to lie in the feasible set of the original linear program. [ABSTRACT FROM AUTHOR]

In this paper, we propose a variable metric version of Tseng's algorithm (the forward-backward-forward algorithm: FBF) combined with extrapolation from the past that includes error terms for finding a zero of the sum of a maximally monotone operator and a monotone Lipschitzian operator in Hilbert spaces. The algorithm, which is a modified forward-reflected-backword method, is endowed with variable metrics and error terms. Primal-dual algorithms are also proposed for monotone inclusion problems involving compositions with linear operators. The primal-dual problem occurring in image deblurring demonstrates an application of our theoretical results. [ABSTRACT FROM AUTHOR]

We propose a new duality scheme based on a sequence of smooth minorants of the weighted- ℓ 1 penalty function, interpreted as a parametrized sequence of augmented Lagrangians, to solve non-convex constrained optimization problems. For the induced sequence of dual problems, we establish strong asymptotic duality properties. Namely, we show that (i) the sequence of dual problems is convex and (ii) the dual values monotonically increase to the optimal primal value. We use these properties to devise a subgradient based primal–dual method, and show that the generated primal sequence accumulates at a solution of the original problem. We illustrate the performance of the new method with three different types of test problems: A polynomial non-convex problem, large-scale instances of the celebrated kissing number problem, and the Markov–Dubins problem. Our numerical experiments demonstrate that, when compared with the traditional implementation of a well-known smooth solver, our new method (using the same well-known solver in its subproblem) can find better quality solutions, i.e. 'deeper' local minima, or solutions closer to the global minimum. Moreover, our method seems to be more time efficient, especially when the problem has a large number of constraints. [ABSTRACT FROM AUTHOR]

Recently, a golden ratio primal-dual algorithm (GRPDA) was proposed by Chang and Yang for solving structured convex optimization problems. It is a new adaptation of the classical Arrow-Hurwicz method by using a convex combination step, instead of the widely adopted extrapolation technique. The convex combination step is determined by a parameter ψ , which, to guarantee global convergence, is restricted to (1 , (1 + 5) / 2 ] . In this paper, by carrying out a refined analysis, we expand this region to (1 , 1 + 3) . Moreover, we establish ergodic sublinear convergence rate results based on function value residual and constraint violation of an equivalently reformulated constrained optimization problem, rather than the previously adopted so-called primal-dual gap function that could vanish at nonstationary points. For linear equality constrained and regularized least-squares problems, we further show that GRPDA and Chambolle-Pock’s primal-dual algorithm are equivalent provided that some parameters are chosen properly. Finally, experimental results on the LASSO and the basis pursuit problems are presented to demonstrate the performance of GRPDA with ψ being chosen in the expanded region. [ABSTRACT FROM AUTHOR]

In this paper we provide a splitting algorithm for solving coupled monotone inclusions in a real Hilbert space involving the sum of a normal cone to a vector subspace, a maximally monotone, a monotone-Lipschitzian, and a cocoercive operator. The proposed method takes advantage of the intrinsic properties of each operator and generalizes the method of partial inverses and the forward-backward-half forward splitting, among other methods. At each iteration, our algorithm needs two computations of the Lipschitzian operator while the cocoercive operator is activated only once. By using product space techniques, we derive a method for solving a composite monotone primal-dual inclusions including linear operators and we apply it to solve constrained composite convex optimization problems. Finally, we apply our algorithm to a constrained total variation least-squares problem and we compare its performance with efficient methods in the literature. [ABSTRACT FROM AUTHOR]

As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method based on the macroscopic partial differential equation, leveraging its formal structure as the gradient flow of the total variation energy, with respect to a weighted H - 1 norm. This gradient flow structure relates to several metric space gradient flows of recent interest, including 2-Wasserstein flows and their generalizations to nonlinear mobilities. We develop a novel semi-implicit time discretization of the gradient flow, inspired by the classical minimizing movements scheme (known as the JKO scheme in the 2-Wasserstein case). We then use a primal dual hybrid gradient (PDHG) method to compute each element of the semi-implicit scheme. In one dimension, we prove convergence of the PDHG method to the semi-implicit scheme, under general integrability assumptions on the mobility and its reciprocal. Finally, by taking finite difference approximations of our PDHG method, we arrive at a fully discrete numerical algorithm, with iterations that converge at a rate independent of the spatial discretization: in particular, the convergence properties do not deteriorate as we refine our spatial grid. We close with several numerical examples illustrating the properties of our method, including facet formation at local maxima, pinning at local minima, and convergence as the spatial and temporal discretizations are refined. [ABSTRACT FROM AUTHOR]

In this article, we provide a splitting method for solving monotone inclusions in a real Hilbert space involving four operators: a maximally monotone, a monotone-Lipschitzian, a cocoercive, and a monotone-continuous operator. The proposed method takes advantage of the intrinsic properties of each operator, generalizing the forward–backward–half-forward splitting and the Tseng's algorithm with line search. At each iteration, our algorithm defines the step size by using a line search in which the monotone-Lipschitzian and the cocoercive operators need only one activation. We also derive a method for solving nonlinearly constrained composite convex optimization problems in real Hilbert spaces. Finally, we implement our algorithm in a nonlinearly constrained least-square problem and we compare its performance with available methods in the literature. [ABSTRACT FROM AUTHOR]

This paper aims to study the mathematical properties of the l1−2 models that employ measurement matrices with correlated columns. We first show that the l1−2 model satisfies the grouping effect which ensures that coefficients corresponding to highly correlated columns in a measurement matrix have small differences. Then we provide the stability analysis based on the sparse approximation property. When the entries of the vectors have different signs, we show that the grouping effect also holds for the constraint l1+2 minimization model which is implicated by the linearized Bregman iteration. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel and effective approximation method for finding the value function for general utility maximization with closed convex control constraints and partial information. Using the separation principle and the weak duality relation, we transform the stochastic maximum principle of the fully observable dual control problem into an equivalent error minimization stochastic control problem and find the tight lower and upper bounds of the value function and its approximate value. Numerical examples show the goodness and usefulness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we provide the stability analysis for an evolutionary variational-hemivariational inequality in the reflexive Banach space, whose data including the constraint set are perturbed. First, by using its perturbed data and the duality mapping, the perturbed and regularized problems for the evolutionary variational-hemivariational inequality are constructed, respectively. Then, by proving the unique solvability for the evolutionary variational-hemivariational inequality and its perturbed and regularized problems, we obtain two sequences called approximating sequences of the solution to the evolutionary variational-hemivariational inequality, and prove their strong convergence to the unique solution to the evolutionary variational-hemivariational inequality under different mild conditions. [ABSTRACT FROM AUTHOR]

Multidomain mixed pairs of coupled stationary constrained boundary value problems, are formulated variationally in frameworks of reflexive Banach spaces as macro-hybrid systems, and analyzed. Interior synchronizing and interface coupling transmission constraints are modeled in terms of Lagrange multipliers as solutions of dual variational subpotential maximal monotone inclusions. Existence and uniqueness results are established via resolvent fixed-point characterizations of the corresponding primal and dual macro-hybridized problems. Multimedia mixed mechanical subsurface-surface Darcy/Stokes incompressible flow coupled pairs with intrinsic distributed control constraints exemplify the general macro-hybrid mixed systems. Coupling dual-primal and primal-dual interface macro-hybrid mixed variational conditions are lastly developed and applied to the Darcy/Stokes multimedia pair model. [ABSTRACT FROM AUTHOR]

This paper proposes two algorithms for solving stochastic control problems with deep learning, with a focus on the utility maximisation problem. The first algorithm solves Markovian problems via the Hamilton Jacobi Bellman (HJB) equation. We solve this highly nonlinear partial differential equation (PDE) with a second order backward stochastic differential equation (2BSDE) formulation. The convex structure of the problem allows us to describe a dual problem that can either verify the original primal approach or bypass some of the complexity. The second algorithm utilises the full power of the duality method to solve non-Markovian problems, which are often beyond the scope of stochastic control solvers in the existing literature. We solve an adjoint BSDE that satisfies the dual optimality conditions. We apply these algorithms to problems with power, log and non-HARA utilities in the Black-Scholes, the Heston stochastic volatility, and path dependent volatility models. Numerical experiments show highly accurate results with low computational cost, supporting our proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we develop an algorithm to efficiently solve risk-averse optimization problems posed in reflexive Banach space. Such problems often arise in many practical applications as, e.g., optimization problems constrained by partial differential equations with uncertain inputs. Unfortunately, for many popular risk models including the coherent risk measures, the resulting risk-averse objective function is nonsmooth. This lack of differentiability complicates the numerical approximation of the objective function as well as the numerical solution of the optimization problem. To address these challenges, we propose a primal–dual algorithm for solving large-scale nonsmooth risk-averse optimization problems. This algorithm is motivated by the classical method of multipliers and by epigraphical regularization of risk measures. As a result, the algorithm solves a sequence of smooth optimization problems using derivative-based methods. We prove convergence of the algorithm even when the subproblems are solved inexactly and conclude with numerical examples demonstrating the efficiency of our method. [ABSTRACT FROM AUTHOR]

Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs' underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations' strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method. [ABSTRACT FROM AUTHOR]

A two-stage continuous-discrete optimal partitioning-allocation problem is studied, and a method and an algorithm for its solving are proposed. This problem is a generalization of a classical transportation problem to the case when coordinates of the production points (collection, storage, processing) of homogeneous products are continuously allocated in the given domain and the production volumes at these points are unknown. These coordinates are found as a solution of the corresponding continuous optimal set-partitioning problem in a finite-dimensional Euclidean space with the placement (finding coordinates) of these subsets’ centers. Also, this problem generalizes discrete two-stage production-transportation problems to the case of continuously allocated consumers. The method and algorithm are illustrated by solving two model problems.

The purpose of this paper is to focus on continuous minimax problems with semi infinite constraints. We begin by proposing an algorithm for solving this kind of problems. The proposed algorithm combines the parametric approach and the Huard method of centers. That's we extend the Method of Centers of Roubi for solving minimax fractional programs to continuous minimax problems. On the other hand, calculating the exact optimal solution of the proposed parametric problems, requires an extraordinary computational work per iteration. To overcome this problem we use an iterative entropic regularization method which allows us to solve each auxiliary problem inexactly generating an approximate sequence of optimal values and then the continuous minimax problem is reduced into a sequence of finite minimax problems. Examples for illustration are given to test our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical approach for solving composite primal-dual monotone inclusions with a priori information. The underlying a priori information set is represented by the intersection of fixed point sets of a finite number of operators, and we propose an algorithm that activates the corresponding set by following a finite-valued random variable at each iteration. Our formulation is flexible and includes, for instance, deterministic and Bernoulli activations over cyclic schemes, and Kaczmarz-type random activations. The almost sure convergence of the algorithm is obtained by means of properties of stochastic Quasi-Fejér sequences. We also recover several primal-dual algorithms for monotone inclusions without a priori information and classical algorithms for solving convex feasibility problems and linear systems. In the context of convex optimization with inequality constraints, any selection of the constraints defines the a priori information set, in which case the operators involved are simply projections onto half spaces. By incorporating random projections onto a selection of the constraints to classical primal-dual schemes, we obtain faster algorithms as we illustrate by means of a numerical application to a stochastic arc capacity expansion problem in a transport network. [ABSTRACT FROM AUTHOR]