Convex Minimization Research Articles

Simple phase unwrapping method with continuous convex minimization : erratum (Optics Express, (2022), Vol. 30, Issue 18, p33395-33411)

Simple phase unwrapping method with continuous convex minimization : erratum (Optics Express, (2022), Vol. 30, Issue 18, p33395-33411)

An inertial projective forward-backward-forward algorithm for constrained convex minimization problems and application to cardiovascular disease prediction

An inertial projective forward-backward-forward algorithm for constrained convex minimization problems and application to cardiovascular disease prediction

Explicit A Posteriori Error Representation for Variational Problems and Application to TV-Minimization

AbstractIn this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise constant vector fields as well as a discrete integration-by-parts formula between the Crouzeix–Raviart and the Raviart–Thomas element, all convex duality relations are transferred to a discrete level, making the explicit a posteriori error representation –initially based on continuous arguments only– practicable from a numerical point of view. In addition, we provide a generalized Marini formula that determines a discrete primal solution in terms of a given discrete dual solution. We benchmark all these concepts via the Rudin–Osher–Fatemi model. This leads to an adaptive algorithm that yields a (quasi-optimal) linear convergence rate.

Hp-FEM for the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Mosolov problem: a priori and a posteriori error estimates

An hp-finite element discretization for the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Mosolov problem, a scalar variant of the Bingham flow problem but with the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Laplacian operator, is being analyzed. Its weak formulation is either a variational inequality of second kind or equivalently a non-smooth but convex minimization problem. For any α∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1,\\infty )$$\\end{document} we prove convergence, including guaranteed convergence rates in the mesh size h and polynomial degree p of the FE-solution of the corresponding discrete variational inequality. Moreover, we derive two families of reliable a posteriori error estimators which are applicable to any “approximation” of the exact solution and not only to the FE-solution and can therefore be coupled with an iterative solver. We prove that any quasi-minimizer of those families of a posteriori error estimators satisfies an efficiency estimate. All our results contain known results for the Mosolov problem by setting α=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =2$$\\end{document}. Numerical results underline our theoretical findings.

A quadratic optimization program for the inverse elastography problem

In this work we focus on the development of a numerical algorithm for the inverse elastography problem. The goal is to perform an efficient material parameter identification knowing the elastic displacement field induced by a mechanical load. We propose to define the inverse problem through a quadratic optimization program which uses the direct problem formulation to define the objective function. In this way, we end up with a convex minimization problem which attains its minimum at the solution of a linear system. The effectiveness of our method is illustrated through numeral examples.

Novel algorithms based on forward-backward splitting technique: effective methods for regression and classification

In this paper, we introduce two novel forward-backward splitting algorithms (FBSAs) for nonsmooth convex minimization. We provide a thorough convergence analysis, emphasizing the new algorithms and contrasting them with existing ones. Our findings are validated through a numerical example. The practical utility of these algorithms in real-world applications, including machine learning for tasks such as classification, regression, and image deblurring reveal that these algorithms consistently approach optimal solutions with fewer iterations, highlighting their efficiency in real-world scenarios.

Double Inertial Krasnosel'skii-Mann-Type Method for Approximating Fixed Point of Nonexpansive Mappings

In this paper, we investigate a new method motivated by current advancements in general inertial algorithms. Specifically, we incorporate double inertial extrapolation terms into an iterative sequence, derived from Krasnosel'skii-Mann techniques. The weak convergence theorem for fixed points of nonexpansive mappings in real Hilbert spaces is established. The theoretical developments are rigorously proven, extending existing methods in literature. We also utilize our convergence analysis to solve real-world problems, such as convex minimization problems and zero finding for sums of monotone operators.

Modified Inertial Krasnosel'skii-Mann Type Method for Solving Fixed PointProblems in Real Uniformly Convex Banach Spaces

We present an altered version of the inertial Krasnosel'skii-Mann algorithm and demonstrate convergence outcomes for mappings that are asymptotically nonexpansive within real, uniformly convex Banach spaces. To achieve our results, we skillfully construct the inequality in equation \eqref{6} and apply it accordingly. Our findings support and broadly generalize a number of significant findings from the literature. We demonstrate, as an application, the generation of maximal monotone operators' zeros via fixed point methods in Hilbert spaces. Additionally, we solve convex minimization issues using our fixed-point techniques.

Convex Predictor–Nonconvex Corrector Optimization Strategy with Application to Signal Decomposition

Many tasks in real life scenarios can be naturally formulated as nonconvex optimization problems. Unfortunately, to date, the iterative numerical methods to find even only the local minima of these nonconvex cost functions are extremely slow and strongly affected by the initialization chosen. We devise a predictor–corrector strategy that efficiently computes locally optimal solutions to these problems. An initialization-free convex minimization allows to predict a global good preliminary candidate, which is then corrected by solving a parameter-free nonconvex minimization. A simple algorithm, such as alternating direction method of multipliers works surprisingly well in producing good solutions. This strategy is applied to the challenging problem of decomposing a 1D signal into semantically distinct components mathematically identified by smooth, piecewise-constant, oscillatory structured and unstructured (noise) parts.

Discrete Weak Duality of Hybrid High-Order Methods for Convex Minimization Problems

Discrete Weak Duality of Hybrid High-Order Methods for Convex Minimization Problems

Solving split equality equilibrium and fixed point problems in Banach spaces

In this paper, we introduce a new algorithm for approximating a common solution of Split Equality Generalised Mixed Equilibrium Problem (SEGMEP) and Split Equality Fixed Point Problem (SEFPP) for two infinite families of closed uniformly Li-Lipschitz continuous and Ki-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings (i ∈ N and ϕ is the Lyapunov functional) in Banach spaces. Under standard and mild assumption of monotonicity and lower semicontinuity of the SEGMEP associated mappings, we establish the strong convergence of the scheme without imposing any compactness type conditions on either the operators or the spaces considered. We apply our result to approximate the solution of Split Equality Convex Minimization Problem (SECMP) and Split Equality Variational Inclusion Problem (SEVIP). A numerical example is presented to illustrate the performance and implementability of our method. Our results extend, generalize and complement several related works in the literature.

The Possibilistic Mean-Variance Model with Uncertain Possibility Distributions

The possibilistic mean–variance (MV) model is the counterpart of Markowitz’s MV model in the possibility theory. This study aims to examine the possibilistic MV model when the possibility distributions of stock returns are uncertain triangular fuzzy numbers. We define an uncertainty vector and use its ellipsoidal uncertainty set in a minimax optimization problem to model this uncertainty. We also show that this minimax optimization problem reduces to a strictly convex minimization problem. Thus, unlike the possibilistic MV model, we get diversified optimal portfolios uniquely with our approach. After laying down the theoretical points of our approach, we illustrate it with a real-world example in the literature by using a software package for convex optimization. To the best of our knowledge, this is the first paper that considers uncertain possibility distributions in the possibilistic MV model.

Perturbed Utility Stochastic Traffic Assignment

This paper develops a fast algorithm for computing the equilibrium assignment with the perturbed utility route choice (PURC) model. Without compromise, this allows the significant advantages of the PURC model to be used in large-scale applications. We formulate the PURC equilibrium assignment problem as a convex minimization problem and find a closed-form stochastic network loading expression that allows us to formulate the Lagrangian dual of the assignment problem as an unconstrained optimization problem. To solve this dual problem, we formulate a quasi-Newton accelerated gradient descent algorithm (qN-AGD*). Our numerical evidence shows that qN-AGD* clearly outperforms a conventional primal algorithm and a plain accelerated gradient descent algorithm. qN-AGD* is fast with a runtime that scales about linearly with the problem size, indicating that solving the perturbed utility assignment problem is feasible also with very large networks. Funding: This work has been financed by the European Union—NextGenerationEU.

Method for Delineating the Formula Limit of the Continental Shelf under the Maximum Area Principle Constraint

In current practices of determining continental shelf area, the measured sediment thickness data do not effectively reflect the distribution of sediments across the area due to its dispersed nature. This issue raises potential limitations in unknown optimal survey line layout strategies for maximizing the posterior determination area. This paper adopts the binary search algorithm, relies on existing sediment thickness grid data, and uses geodetic formulas to build an ellipsoidal surface grid distance calculation model. This model quickly screens potential areas for the 1% sediment thickness line candidate points set. By constraining the azimuth parameter values during the construction process of the ellipsoidal point feature buffer zones, efficiently select the candidate points set for the 1% sediment thickness line. Furthermore, by elucidating the essential meanings of points on the formula limit and combining the candidate points set of the foot of the continental slope (FOS)+60 n mile line, the polygon minimal convex hull construction technique and a baseline points optimization algorithm with a length threshold are used to efficiently select points on formula limit. Experimental results demonstrate that this method can effectively assist coastal states in optimizing the determination of continental shelf area to the fullest extent under the length threshold requirements of the United Nations Convention on the Law of the Sea. Experiments have proven that compared to the traditional intersection method, the method presented in this paper can help coastal countries delineate a larger continental shelf area. In typical application scenarios, the gain in area can reach 77,278,427 m2 accounting for 0.51% of the total area.

An optimal control problem for diffusion–precipitation model

We present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Posing controls on the porous part of the medium (distributed control), we set up a convex minimization problem. The main objective of this article is to characterize an arbitrary control to be an optimal control. We establish a relation between the optimal control and the corresponding adjoint state. At first, we analyse the microscopic description of the controlled system, then we homogenised the system by rigorous two-scale convergence method and periodic unfolding method.

An inertial‐like Tseng's extragradient method for solving pseudomonotone variational inequalities in reflexive Banach spaces

A survey of the existing results in the literature shows that several of the results on variational inequality problem were established under some stringent conditions and employed some form of linesearch technique even in the framework of Hilbert spaces. However, due to the loop nature of the linesearch technique, the implementation of such algorithms might be economically nonviable. In this article, we study the pseudomonotone variational inequality problem. We propose a new inertial‐like Tseng's extragradient method with non‐monotonic self‐adaptive step size that does not involve any linesearch technique for finding the solution of the problem in reflexive Banach spaces. Under some mild conditions, we prove a strong convergence result for the proposed method without the sequentially weakly continuity condition often assumed by authors to guarantee convergence when studying the pseudomonotone variational inequality problems. Moreover, we apply our result to study a constrained convex minimization problem. Finally, we present several theoretical and real applications to optimal control numerical experiments to illustrate and show the performance of our method in comparison with related methods in the literature.

Proximal point approach for solving convex minimization problems in positive curvature

In this research, we offer a new proximal point technique for approximating a common member of the set of all solutions of the convex minimization problems in the setting of CAT(1) spaces. We further show that, in some modest circumstances, the resulting process is convergent. Our results extend and enhance other related results in the literature.

Efficient Learning in Polyhedral Games via Best-Response Oracles

We study online learning and equilibrium computation in games with polyhedral decision sets, a property shared by normal-form games (NFGs) and extensive-form games (EFGs), when the learning agent is restricted to utilizing a best-response oracle. We show how to achieve constant regret in zero-sum games and O(T^0.25) regret in general-sum games while using only O(log t) best-response queries at a given iteration t, thus improving over the best prior result, which required O(T) queries per iteration. Moreover, our framework yields the first last-iterate convergence guarantees for self-play with best-response oracles in zero-sum games. This convergence occurs at a linear rate, though with a condition-number dependence. We go on to show a O(T^(-0.5)) best-iterate convergence rate without such a dependence. Our results build on linear-rate convergence results for variants of the Frank-Wolfe (FW) algorithm for strongly convex and smooth minimization problems over polyhedral domains. These FW results depend on a condition number of the polytope, known as facial distance. In order to enable application to settings such as EFGs, we show two broad new results: 1) the facial distance for polytopes in standard form is at least γ/k where γ is the minimum value of a nonzero coordinate of a vertex of the polytope and k≤n is the number of tight inequality constraints in the optimal face, and 2) the facial distance for polytopes of the form Ax=b, Cx≤d, x≥0 where x∈R^n, C≥0 is a nonzero integral matrix, and d≥0, is at least 1/(c√n), where c is the infinity norm of C. This yields the first such results for several problems such as sequence-form polytopes, flow polytopes, and matching polytopes.

Trust your source: quantifying source condition elements for variational regularisation methods

Abstract Source conditions are a key tool in regularisation theory that are needed to derive error estimates and convergence rates for ill-posed inverse problems. In this paper, we provide a recipe to practically compute source condition elements as the solution of convex minimisation problems that can be solved with first-order algorithms. We demonstrate the validity of our approach by testing it on two inverse problem case studies in machine learning and image processing: sparse coefficient estimation of a polynomial via LASSO regression and recovering an image from a subset of the coefficients of its discrete Fourier transform. We further demonstrate that the proposed approach can easily be modified to solve the machine learning task of identifying the optimal sampling pattern in the Fourier domain for a given image and variational regularisation method, which has applications in the context of sparsity promoting reconstruction from magnetic resonance imaging data.

Adaptive finite element methods for scalar double‐well problem

AbstractSome scalar double‐well problems eventually lead to a degenerate convex minimization problem with unique stress. We consider the adaptive conforming and nonconforming finite element methods for the scalar double‐well problem with the reliable a posteriori error analysis. A number of experiments confirm the effective decay rates of the methods.