Convergence of the Peaceman-Rachford Splitting Method for a Class of Nonconvex Programs

In this paper, we analyze the convergence of the Peaceman-Rachford splitting method (PRSM) for a type of nonconvex and nonsmooth optimization with linear constraints, whose objective function is the sum of a proper lower semicontinuous function and a strongly convex differential function. When a suitable penalty parameter is chosen and the iterative point sequence is bounded, we show the global convergence of the PRSM. Furthermore, under the assumption that the associated function satisfies the Kurdyka-Łojasiewicz property, we prove the strong convergence of the PRSM. We also provide sufficient conditions guaranteeing the boundedness of the generated sequence. Finally, we present some preliminary numerical results to show the effectiveness of the PRSM and also give a comparison with the Douglas-Rachford splitting method.     

A Proximal Point-Type Method for Multicriteria Optimization

In this paper, we present a proximal point algorithm for multicriteria optimization, by assuming an iterative process which uses a variable scalarization function. With respect to the convergence analysis, firstly we show that, for any sequence generated from our algorithm, each accumulation point is a Pareto critical point for the multiobjective function. A more significant novelty here is that our paper gets full convergence for quasi-convex functions. In the convex or pseudo-convex cases, we prove convergence to a weak Pareto optimal point. Another contribution is to consider a variant of our algorithm, obtaining the iterative step through an unconstrained subproblem. Then, we show that any sequence generated by this new algorithm attains a Pareto optimal point after a finite number of iterations under the assumption that the weak Pareto optimal set is weak sharp for the multiobjective problem.     

A smoothing Newton algorithm for weighted linear complementarity problem

In this paper, we present a new smoothing Newton method for solving monotone weighted linear complementarity problem (WCP). Our algorithm needs only to solve one linear system of equation and performs one line search per iteration. Any accumulation point of the iteration sequence generated by our algorithm is a solution of WCP. Under suitable conditions, our algorithm has local quadratic convergence rate. Numerical experiments show the feasibility and efficiency of the algorithm.     

A proximal algorithm with quasi distance. Application to habit's formation

We consider a proximal algorithm with quasi distance applied to nonconvex and nonsmooth functions involving analytic properties for a minimization problem. We show the behavioural importance of this proximal point model for the formation of habit in Decision and Making Sciences. The convergence of the sequence generated by our algorithm to a critical-limit point is guaranteed under standard conditions of coercivity and by using the Kurdyka–?ojasiewicz inequality. We present a definition of habit that contemplates that kind of convergence.     

On inertial proximal algorithm for split variational inclusion problems

ABSTRACT

In this paper, a hybrid proximal algorithm with inertial effect is introduced to solve a split variational inclusion problem in real Hilbert spaces. Under mild conditions on the parameters, we establish weak convergence results for the proposed algorithm. Unlike the earlier iterative methods, we do not impose any conditions on the sequence generated by the proposed algorithm. Also, we extend our results to find a common solution of a split variational inclusion problem and a fixed-point problem. Finally, some numerical examples are given to discuss the convergence and superiority of the proposed iterative methods.     

Approximation of Parametric Derivatives by the Empirical Interpolation Method

We introduce a general a priori convergence result for the approximation of parametric derivatives of parametrized functions. We consider the best approximations to parametric derivatives in a sequence of approximation spaces generated by a general approximation scheme, and we show that these approximations are convergent provided that the best approximation to the function itself is convergent. We also provide estimates for the convergence rates. We present numerical results with spaces generated by a particular approximation scheme—the Empirical Interpolation Method—to confirm the validity of the general theory.     

Dual Convergence for Penalty Algorithms in Convex Programming

Algorithms for convex programming, based on penalty methods, can be designed to have good primal convergence properties even without uniqueness of optimal solutions. Taking primal convergence for granted, in this paper we investigate the asymptotic behavior of an appropriate dual sequence obtained directly from primal iterates. First, under mild hypotheses, which include the standard Slater condition but neither strict complementarity nor second-order conditions, we show that this dual sequence is bounded and also, each cluster point belongs to the set of Karush–Kuhn–Tucker multipliers. Then we identify a general condition on the behavior of the generated primal objective values that ensures the full convergence of the dual sequence to a specific multiplier. This dual limit depends only on the particular penalty scheme used by the algorithm. Finally, we apply this approach to prove the first general dual convergence result of this kind for penalty-proximal algorithms in a nonlinear setting.     

Convergence of a Penalty Method for Mathematical Programming with Complementarity Constraints

We adapt the convergence analysis of the smoothing (Ref. 1) and regularization (Ref. 2) methods to a penalty framework for mathematical programs with complementarity constraints (MPCC); we show that the penalty framework shares convergence properties similar to those of these methods. Moreover, we give sufficient conditions for a sequence generated by the penalty framework to be attracted to a B-stationary point of the MPCC.     

MBFGS算法中迭代矩阵的收敛性

Li-Fukushima[3]提出了一种修正的BFGS方法MBFGS算法.本文研究MBFGS算法中迭代矩阵的收敛性.我们证明在一定条件下,MBFGS算法用于求解严格凸二次函数极小值时产生的迭代矩阵序列是收敛的.     

Weak sharp solutions for variational inequalities in Banach spaces

In this paper, we introduce the notion of a weak sharp set of solutions to a variational inequality problem (VIP) in a reflexive, strictly convex and smooth Banach space, and present its several equivalent conditions. We also prove, under some continuity and monotonicity assumptions, that if any sequence generated by an algorithm for solving (VIP) converges to a weak sharp solution, then we can obtain solutions for (VIP) by solving a finite number of convex optimization subproblems with linear objective. Moreover, in order to characterize finite convergence of an iterative algorithm, we introduce the notion of a weak subsharp set of solutions to a variational inequality problem (VIP), which is more general than that of weak sharp solutions in Hilbert spaces. We establish a sufficient and necessary condition for the finite convergence of an algorithm for solving (VIP) which satisfies that the sequence generated by which converges to a weak subsharp solution of (VIP), and show that the proximal point algorithm satisfies this condition. As a consequence, we prove that the proximal point algorithm possesses finite convergence whenever the sequence generated by which converges to a weak subsharp solution of (VIP).     

A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

We provide a local as well as a semilocal convergence analysis for two-point Newton-like methods in a Banach space setting under very general Lipschitz type conditions. Our equation contains a Fréchet differentiable operator F and another operator G whose differentiability is not assumed. Using more precise majorizing sequences than before we provide sufficient convergence conditions for Newton-like methods to a locally unique solution of equation F(x)+G(x)=0. In the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results. In the local case a larger radius of convergence is obtained. Several numerical examples are provided to show that our results compare favorably with earlier ones. As a special case we show that the famous Newton-Kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the Newton-Kantorovich theorem.     

Accelerated hybrid viscosity and steepest-descent method for proximal split feasibility problems

The purpose of this paper is to present an accelerated hybrid viscosity and steepest-descent method for solving proximal split feasibility problems (if solutions exist) in real Hilbert spaces. We obtain strong convergence of the sequence generated by our scheme under some suitable conditions on the parameters. In all our results in this work, our iterative schemes are proposed with a way of selecting the step sizes such that their implementation does not need any prior information about the bounded linear operator norm. Finally, we give numerical comparisons of our results with other established result to verify the efficiency and implementation of our new method.     

Regularized and inertial algorithms for common fixed points of nonlinear operators

This paper deals with a general fixed point iteration for computing a point in some nonempty closed and convex solution set included in the common fixed point set of a sequence of mappings on a real Hilbert space. The proposed method combines two strategies: viscosity approximations (regularization) and inertial type extrapolation. The first strategy is known to ensure the strong convergence of some successive approximation methods, while the second one is intended to speed up the convergence process. Under classical conditions on the operators and the parameters, we prove that the sequence of iterates generated by our scheme converges strongly to the element of minimal norm in the solution set. This algorithm works, for instance, for approximating common fixed points of infinite families of demicontractive mappings, including the classes of quasi-nonexpansive operators and strictly pseudocontractive ones.     

Tree wavelet approximations with applications

We construct a tree wavelet approximation by using a constructive greedy scheme (CGS). We define a function class which contains the functions whose piecewise polynomial approximations generated by the CGS have a prescribed global convergence rate and establish embedding properties of this class. We provide sufficient conditions on a tree index set and on bi-orthogonal wavelet bases which ensure optimal order of convergence for the wavelet approximations encoded on the tree index set using the bi-orthogonal wavelet bases. We then show that if we use the tree index set associated with the partition generated by the CGS to encode a wavelet approximation, it gives optimal order of convergence.     

Golden ratio algorithms with new stepsize rules for variational inequalities

In this paper, we introduce two golden ratio algorithms with new stepsize rules for solving pseudomonotone and Lipschitz variational inequalities in finite dimensional Hilbert spaces. The presented stepsize rules allow the resulting algorithms to work without the prior knowledge of the Lipschitz constant of operator. The first algorithm uses a sequence of stepsizes that is previously chosen, diminishing, and nonsummable, while the stepsizes in the second one are updated at each iteration and by a simple computation. A special point is that the sequence of stepsizes generated by the second algorithm is separated from zero. The convergence and the convergence rate of the proposed algorithms are established under some standard conditions. Also, we give several numerical results to show the behavior of the algorithms in comparison with other algorithms.     

On Weak and Strong Convergence of the Projected Gradient Method for Convex Optimization in Real Hilbert Spaces

This work focuses on convergence analysis of the projected gradient method for solving constrained convex minimization problems in Hilbert spaces. We show that the sequence of points generated by the method employing the Armijo line search converges weakly to a solution of the considered convex optimization problem. Weak convergence is established by assuming convexity and Gateaux differentiability of the objective function, whose Gateaux derivative is supposed to be uniformly continuous on bounded sets. Furthermore, we propose some modifications in the classical projected gradient method in order to obtain strong convergence. The new variant has the following desirable properties: the sequence of generated points is entirely contained in a ball with diameter equal to the distance between the initial point and the solution set, and the whole sequence converges strongly to the solution of the problem that lies closest to the initial iterate. Convergence analysis of both methods is presented without Lipschitz continuity assumption.     

Proximal point method for a special class of nonconvex multiobjective optimization functions

The proximal point method for a special class of nonconvex multiobjective functions is studied in this paper. We show that the method is well defined and that the accumulation points of any generated sequence, if any, are Pareto–Clarke critical points. Moreover, under additional assumptions, we show the full convergence of the generated sequence.     

A convergent decomposition method for box-constrained optimization problems

In this work we consider the problem of minimizing a continuously differentiable function over a feasible set defined by box constraints. We present a decomposition method based on the solution of a sequence of subproblems. In particular, we state conditions on the rule for selecting the subproblem variables sufficient to ensure the global convergence of the generated sequence without convexity assumptions. The conditions require to select suitable variables (related to the violation of the optimality conditions) to guarantee theoretical convergence properties, and leave the degree of freedom of selecting any other group of variables to accelerate the convergence.     

A proximal method with separable Bregman distances for quasiconvex minimization over the nonnegative orthant

We present an interior proximal method with Bregman distance, for solving the minimization problem with quasiconvex objective function under nonnegative constraints. The Bregman function is considered separable and zone coercive, and the zone is the interior of the positive orthant. Under the assumption that the solution set is nonempty and the objective function is continuously differentiable, we establish the well definedness of the sequence generated by our algorithm and obtain two important convergence results, and show in the main one that the sequence converges to a solution point of the problem when the regularization parameters go to zero.     

Directional Secant-Type Methods for Solving Equations

A semilocal convergence analysis for directional Secant-type methods in multidimensional space is provided. Using weaker hypotheses than the ones exploited by An and Bai, we provide a semilocal convergence analysis with the following advantages: weaker convergence conditions, larger convergence domain, finer error estimates on the distances involved, and more precise information on the location of the solution. A numerical example, where our results apply to solve an equation but not the ones of An and Bai, is also provided. In a second example, we show how to implement the method.