The split feasibility problem (SFP), which provides a unified framework to model a wide range of inverse problems, has received much considerable attention in the literature.However, how to efficiently solve SFPs is still an interesting topic.In this paper, we introduce a block-wise formulation for algorithmic design.Specifically, we first introduce an auxiliary variable to formulate the original SFP as a constrained minimization problem with a block structure, which paves a new way to find solutions of SFPs.Then, we show that the employments of some classical gradient-type optimization algorithms produce very simple, yet quite efficient iterative schemes to find a solution of SFPs when the underlying block structure could be exploited.The parallel iterative schemes of the proposed blockwise algorithms are not only efficient to deal with the case that the projections onto the convex sets have explicit representations, but also are possibly valuable for solving large-scale SFPs without explicit projections onto the underlying sets.Some numerical results on synthetic examples support the idea of this paper.
We study a new class of variational inclusion problems in the framework of real Hilbert spaces.We propose two Tseng-type algorithms with inertial extrapolation for solving these problems and carry out the convergence analysis of these two methods.We also give an application of our results to solve convex bilevel optimization problems.
In this paper, we propose three self-adaptive relaxed CQ-algorithms with projections onto halfspaces for solving a split equality problem.The stepsize of the algorithms is dynamically calculated without any prior information regarding operator norms.Moreover, we prove the strong convergence to the minimum-norm solution of the split equality problem.Finally, we test the validity of our results by conducting some numerical experiments and consider signal recovery problems as applications.
In this paper, we propose a hybrid Halpern-based extragradient algorithm for finding a common solution of a pseudomonotone equilibrium problem and a fixed point problem of a nonexpansive mapping.We prove the strong convergence of the proposed algorithm under some mixed conditions.Some numerical examples are provided to illustrate the effectiveness of the proposed algorithm.
Entropy-regularized quadratic optimization problems are a special class of optimization problems with wide applications in various fields, such as transportation and machine learning.In this paper, we apply the augmented Lagrangian method to this problem with its subproblem solved by the block coordinate descent method.Under certain mild conditions, we analyze the global convergence of this algorithm.Numerical experiments demonstrate the effectiveness of this algorithm.
The paper proposes an iterative method for solving a variational inclusion with the sum of two operators in a Hilbert space.The method can be considered as a combination of the proximal contraction method, the regularization method, and the multi-step inertial technique.Theorem of strong convergence is established under mild conditions imposed on cost operators and control parameters.
We investigate the problem of minimizing Kullback-Leibler divergence between a linear model Ax and a positive vector b in different convex domains (positive orthant, n-dimensional box, probability simplex).Our focus is on the SMART method that employs efficient multiplicative updates.We explore the exponentiated gradient method, which can be viewed as a Bregman proximal gradient method and as a Riemannian gradient descent on the parameter manifold of a corresponding distribution of the exponential family.This dual interpretation enables us to establish connections and achieve accelerated SMART iterates while smoothly incorporating constraints.The performance of the proposed acceleration schemes is demonstrated by large-scale numerical examples.
In this paper, the notion of the ravine of real-valued functions is extended from the finitedimensional setting to an infinite-dimensional setting.Ravines of quadratic functions are studied in detail.The obtained results solve a problem raised by Professor Joachim Gwinner.In addition, it is proved that a weakly continuous real-valued convex function defined on a reflexive Banach space cannot have any ravine along the null subspace.
Generalized nonlinear programming is considered without any convexity assumption, capturing a variety of problems that include nonsmooth objectives, combinatorial structures, and set-membership nonlinear constraints. We extend the augmented Lagrangian framework to this broad problem class, preserving an implicit formulation and introducing slack variables merely as a formal device. This, however, gives rise to a generalized augmented Lagrangian function that lacks regularity, due to the marginalization with respect to slack variables. Based on parametric optimization, we develop a tailored stationarity concept to better qualify the iterates, generated as approximate solutions to a sequence of subproblems. Using this variational characterization and the lifted representation, a suitable multiplier update rule is derived, and then asymptotic properties and convergence guarantees are established for a safeguarded augmented Lagrangian scheme. An illustrative numerical example showcases the modelling versatility gained by dropping convexity assumptions and indicates the practical benefits of the advocated implicit approach.
Over the fast few years, the numerical success of the generalized alternating direction method of multipliers (GADMM) proposed by Eckstein & Bertsekas [Math. Prog., 1992] has inspired intensive attention in analyzing its theoretical convergence properties. In this paper, we devote to establishing the linear convergence rate of the semi-proximal GADMM (sPGADMM) for solving linearly constrained convex composite optimization problems. The semi-proximal terms contained in each subproblem possess the abilities of handling with multi-block problems efficiently. We initially present some important inequalities for the sequence generated by the sPGADMM, and then establish the local linear convergence rate under the assumption of calmness. As a by-product, the global convergence property is also discussed.
. In this paper, we establish optimality conditions for a nonsmooth multiobjective semi-infinite programming problem subject to switching constraints. In particular, we employ a surrogate problem and a suitable constraint qualification to state necessary M-stationary conditions in terms of Clarke sub-differentials. Moreover, we demonstrate that in different cases these M-stationary conditions becomes sufficient as well. Finally, we also present Weak and strong duality results of Wolfe an Mond-Weir types
Our purpose in this paper is to propose an iterative method involving a step-size selected in such a way that its implementation does not require the computation or an estimate of the spectral radius.Using our algorithm, we state and prove a strong convergence theorem of a common solution to a monotone inclusion problem and a fixed point problem of multi-valued Lipschitz hemicontractive-type mappings, whose image under a bounded linear operator is a fixed point of a demicontractive mapping.Our result generalizes some important and recent results in the literature.
This paper addresses the study of subdifferentials for set-valued mappings/multifunctions, which take values in ordered spaces.First we obtain the main calculus (sum and chain) rules for such subdifferentials.Then the developed subdifferential calculus is applied to establishing existence theorems for the so-called relative Pareto minimizers in general problems of set-valued optimization with constraints of various types.
In this paper, we investigate Lagrange dualities for a vector variational inequality problem.For a vector variational inequality problem with convex inclusion constraints, we develop an equivalent saddle point formulation via scalarization.For a vector variational inequality problem with linear constraints, we formulate a dual vector variational inequality via that of a linear multiobjective optimization problem and show that for a solution of a vector variational inequality problem, there is one corresponding solution for its dual.We give some examples to illustrate the results.
