Convex Minimization Problem Research Articles

Strong convergence results for variational inclusions, systems of variational inequalities and fixed point problems using composite viscosity implicit methods

Let the VI indicate a variational inclusion, the CFPP denote a common fixed point problem of countably many nonexpansive mappings, and the SVI represent a system of variational inequalities. We introduce a composite viscosity implicit method for solving the VI and CFPP with the SVI constraint in the framework of uniformly convex and q-uniformly smooth Banach space where . Moreover, we prove the strong convergence of the sequences generated by the proposed implicit method to a solution of a certain hierarchical variational inequality (HVI). In addition, our results are also applied for solving the fixed point problem (FPP) of nonexpansive mapping, variational inequality problem, convex minimization problem and split feasibility problem in Hilbert spaces.

Fast Convergence of Dynamical ADMM via Time Scaling of Damped Inertial Dynamics

In this paper, we propose in a Hilbertian setting a second-order time-continuous dynamic system with fast convergence guarantees to solve structured convex minimization problems with an affine constraint. The system is associated with the augmented Lagrangian formulation of the minimization problem. The corresponding dynamics brings into play three general time-varying parameters, each with specific properties, and which are, respectively, associated with viscous damping, extrapolation and temporal scaling. By appropriately adjusting these parameters, we develop a Lyapunov analysis which provides fast convergence properties of the values and of the feasibility gap. These results will naturally pave the way for developing corresponding accelerated ADMM algorithms, obtained by temporal discretization.

Extremal Functions for Morrey’s Inequality

We give a qualitative description of extremals for Morrey’s inequality. Our theory is based on exploiting the invariances of this inequality, studying the equation satisfied by extremals, and the observation that extremals are optimal for a related convex minimization problem.

On solving variational inequality for SM-iteration

The fundamental motivation behind this paper is to solve variational inequality for SM-iterative method under some gentle conditions. We focus on calculating the common results of variational inequality and of SM-iteration. As application part, convex minimization problem is solved under modified SM-algorithm. Numerical example is supplied to validate our main result. Our result holds comparison between the SM- and S-algorithms.

APPROXIMATION OF ZEROS OF SUM OF MONOTONE MAPPINGS WITH APPLICATIONS TO VARIATIONAL INEQUALITY AND IMAGE RESTORATION PROBLEMS

In this paper, an inertial Halpern-type forward backward iterative algorithm for approximating solution of a monotone inclusion problem whose solution is also a fixed point of some nonlinear mapping is introduced and studied. Strong convergence theorem is established in a real Hilbert space. Furthermore, our theorem is applied to variational inequality problems, convex minimization problems and image restoration problems. Finally, numerical illustrations are presented to support the main theorem and its applications.

Modified Tseng’s Method with Inertial Viscosity Type for Solving Inclusion Problems and Its Application to Image Restoration Problems

In this paper, we study a monotone inclusion problem in the framework of Hilbert spaces. (1) We introduce a new modified Tseng’s method that combines inertial and viscosity techniques. Our aim is to obtain an algorithm with better performance that can be applied to a broader class of mappings. (2) We prove a strong convergence theorem to approximate a solution to the monotone inclusion problem under some mild conditions. (3) We present a modified version of the proposed iterative scheme for solving convex minimization problems. (4) We present numerical examples that satisfy the image restoration problem and illustrate our proposed algorithm’s computational performance.

Nonconforming discretizations of convex minimization problems and precise relations to mixed methods

This article discusses nonconforming finite element methods for convex minimization problems and systematically derives dual mixed formulations. Duality relations lead to simple error estimates that avoid an explicit treatment of nonconformity errors. A reconstruction formula provides the discrete solution of the dual problem via a simple postprocessing procedure which implies a strong duality relation and is of interest in a posteriori error estimation. The framework applies to differentiable and nonsmooth problems, examples include p-Laplace, total-variation regularized, and obstacle problems. Numerical experiments illustrate advantages of nonconforming over standard conforming methods.

A modified proximal point algorithm involving nearly asymptotically quasi-nonexpansive mappings

In this paper, we propose a modified proximal point algorithm based on the Thakur iteration process to approximate the common element of the set of solutions of convex minimization problems and the fixed points of two nearly asymptotically quasi-nonexpansive mappings in the framework of operatorname{CAT}(0) spaces. We also prove the Δ-convergence of the proposed algorithm. We also provide an application and numerical result based on our proposed algorithm as well as the computational result by comparing our modified iteration with previously known Sahu’s modified iteration.

On Lagrangian L2-norm pinball twin bounded support vector machine via unconstrained convex minimization

With the introduction of the regularization term in the formulation of the well-known twin support vector machine (TWSVM) for classification , twin bounded support vector machine (TBSVM) method was proposed recently as an improved version by implementing the structural risk minimization principle. However, TBSVM employs hinge loss function and it is sensitive to noise and unstable to re-sampling. Since the pinball loss function related to quantile distance enjoys noise insensitivity property , a novel TBSVM method with squared pinball loss function for classification is proposed. The noise insensitivity and scatter minimization properties are discussed. Our formulation is further simplified as a pair of unconstrained strongly convex minimization problems in the dual space free of matrix inversion terms and having only m variables where m is the number of training examples. As opposed to TWSVM and TBSVM wherein approximate kernel generated surfaces are constructed, kernel trick is applied directly in our formulation and thereby elegant formulation as in the classical support vector machine (SVM) is achieved. Numerical experiments performed on a synthetic and thirteen benchmark datasets with noise where better or comparable generalization performance with faster learning speed by the proposed method confirms its suitability and applicability to problems of interest.

A Projected Forward-Backward Algorithm for Constrained Minimization with Applications to Image Inpainting

In this research, we study the convex minimization problem in the form of the sum of two proper, lower-semicontinuous, and convex functions. We introduce a new projected forward-backward algorithm using linesearch and inertial techniques. We then establish a weak convergence theorem under mild conditions. It is known that image processing such as inpainting problems can be modeled as the constrained minimization problem of the sum of convex functions. In this connection, we aim to apply the suggested method for solving image inpainting. We also give some comparisons to other methods in the literature. It is shown that the proposed algorithm outperforms others in terms of iterations. Finally, we give an analysis on parameters that are assumed in our hypothesis.

New Convergence Theorems for Maximal Monotone Operators in Banach Spaces

The purpose of this paper is to introduce a new hybrid iterative scheme for resolvents of maximal monotone operators in Banach spaces by using the notion of generalized fprojection. Next, we apply this result to the convex minimization and variational inequality problems in Banach spaces. The results presented in this paper improve and extend important recent results in the literature.

An accelerated viscosity forward-backward splitting algorithm with the linesearch process for convex minimization problems

In this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting algorithm is one of the popular optimization methods for approximating a minimizer of the function; however, the stepsize of this algorithm depends on the Lipschitz constant of the gradient of the function, which is not an easy work to find in general practice. By using a new modification of the linesearches of Cruz and Nghia [Optim. Methods Softw. 31:1209–1238, 2016] and Kankam et al. [Math. Methods Appl. Sci. 42:1352–1362, 2019] and an inertial technique, we introduce an accelerated viscosity-type algorithm without any Lipschitz continuity assumption on the gradient. A strong convergence result of the proposed algorithm is established under some control conditions. As applications, we apply our algorithm to solving image and signal recovery problems. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature.

Inertial generalized proximal Peaceman–Rachford splitting method for separable convex programming

The Peaceman–Rachford splitting method (PRSM) is a preferred method for solving the two-block separable convex minimization problems with linear constraints at present. In this paper, we propose an inertial generalized proximal PRSM (abbreviated as IGPRSM) to improve computing efficiency, which unify the ideas of inertial proximal point and linearization technique. Both subproblems are linearized by positive semi-definite proximal matrices, and we explain why the matrix cannot be indefinite. The global convergence and the worst-case asymptotic iteration complexity are derived theoretically via the variational inequality framework. Numerical experiments on LASSO, total variation (TV) based denoising models and image decomposition problems are presented to show the effectiveness of the introduced method even compared with the state-of-the-art methods.

An algorithm for approximating a common solution of some nonlinear problems in Banach spaces with an application

In this paper, we construct a new Halpern-type subgradient extragradient iterative algorithm. The sequence generated by this algorithm converges strongly to a common solution of a variational inequality, an equilibrium problem, and a J-fixed point of a continuous J-pseudo-contractive map in a uniformly smooth and two-uniformly convex real Banach space. Also, the theorem is applied to approximate a common solution of a variational inequality, an equilibrium problem, and a convex minimization problem. Moreover, a numerical example is given to illustrate the implementability of our algorithm. Finally, the theorem proved complements, improves, and unifies some related recent results in the literature.

A Common Solution of Equilibrium, Constrained Convex Minimization, and Fixed Point Problems

In this paper, we propose a new iterative scheme with the help of the gradient- projection algorithm (GPA) for finding a common solution of an equilibrium problem, a constrained convex minimization problem, and a fixed point problem. Then, we prove some strong convergence theorems which improve and extend some recent results. Moreover, we give a numerical result to show the validity of our main theorem.

Regularized based implicit Lagrangian twin extreme learning machine in primal for pattern classification

In this paper, we suggest a novel approach termed as regularized based implicit Lagrangian twin extreme learning machine in primal as a pair of unconstrained convex minimization problem (RILTELM) where regularization term is added to follow the structural risk minimization principle. Here, we consider 2-norm of the slack vector of variables to make the problem strongly convex which results in a unique solution. Since it has non-smooth plus functions in their objective function, so we find an approximate solution by replacing the non-smooth plus function with smooth approximation function because to find an approximation solution in primal space is always superior to its dual. Due to non-smooth plus function, we solve the problem by either smooth approximation approach or generalized derivative approach. In addition, a functional iterative scheme is also suggested to find the optimal solution. Hence, no external optimization toolbox is required unlike in twin extreme learning machine (TELM) and twin support vector machine (TWSVM). The numerical experiments are demonstrated on artificial and real-world datasets and compared with TWSVM, ELM, TELM and LSTELM to establish the efficacy and applicability of proposed RILTELM.

A New Gradient Projection Algorithm for Convex Minimization Problem and its Application to Split Feasibility Problem

In this paper, we study convergence analysis of a new gradient projection algorithm for solving convex minimization problems in Hilbert spaces. We observe that the proposed gradient projection algorithm weakly converges to a minimum of convex function f which is defined from a closed and convex subset of a Hilbert space to $\mathbb {R}$ . Also, we give a nontrivial example to illustrate our result in an infinite dimensional Hilbert space. We apply our result to solve the split feasibility problem.

An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings

In this paper, we study a classical monotone and Lipschitz continuous variational inequality and fixed point problems defined on a level set of a convex function in the framework of Hilbert spaces. First, we introduce a new iterative scheme which combines the inertial subgradient extragradient method with viscosity technique and with self-adaptive stepsize. Unlike in many existing subgradient extragradient techniques in literature, the two projections of our proposed algorithm are made onto some half-spaces. Furthermore, we prove a strong convergence theorem for approximating a common solution of the variational inequality and fixed point of an infinite family of nonexpansive mappings under some mild conditions. The main advantages of our method are: the self-adaptive stepsize which avoids the need to know a priori the Lipschitz constant of the associated monotone operator, the two projections made onto some half-spaces, the strong convergence and the inertial technique employed which accelerates convergence rate of the algorithm. Second, we apply our theorem to solve generalised mixed equilibrium problem, zero point problems and convex minimization problem. Finally, we present some numerical examples to demonstrate the efficiency of our algorithm in comparison with other existing methods in literature. Our results improve and extend several existing works in the current literature in this direction.

Convex duality for principal frequencies

<abstract><p>We consider the sharp Sobolev-Poincaré constant for the embedding of $ W^{1, 2}_0(\Omega) $ into $ L^q(\Omega) $. We show that such a constant exhibits an unexpected dual variational formulation, in the range $ 1 < q < 2 $. Namely, this can be written as a convex minimization problem, under a divergence–type constraint. This is particularly useful in order to prove lower bounds. The result generalizes what happens for the torsional rigidity (corresponding to $ q = 1 $) and extends up to the case of the first eigenvalue of the Dirichlet-Laplacian (i.e., to $ q = 2 $).</p></abstract>

An accelerated forward-backward algorithm with a new linesearch for convex minimization problems and its applications

We study and investigate a convex minimization problem of the sum of two convex functions in the setting of a Hilbert space. By assuming the Lipschitz continuity of the gradient of the function, many optimization methods have been invented, where the stepsizes of those algorithms depend on the Lipschitz constant. However, finding such a Lipschitz constant is not an easy task in general practice. In this work, by using a new modification of the linesearches of Cruz and Nghia [ 7 ] and Kankam et al. [ 14 ] and an inertial technique, we introduce an accelerated algorithm without any Lipschitz continuity assumption on the gradient. Subsequently, a weak convergence result of the proposed method is established. As applications, we apply and analyze our method for solving an image restoration problem and a regression problem. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature.