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The variational inequality problem in finite-dimensional Euclidean space is addressed in this paper, and two inexact variants of the extragradient method are proposed to solve it. Instead of computing exact projections on the constraint set, as in previous versions extragradient method, the proposed methods compute feasible inexact projections on the constraint set using a relative error criterion. The first version of the proposed method provided is a counterpart to the classic form of the extragradient method with constant steps. In order to establish its convergence we need to assume that the operator is pseudo-monotone and Lipschitz continuous, as in the standard approach. For the second version, instead of a fixed step size, the method presented finds a suitable step size in each iteration by performing a line search. Like the classical extragradient method, the proposed method does just two projections into the feasible set in each iteration. A full convergence analysis is provided, with no Lipschitz continuity assumption of the operator defining the variational inequality problem.

In the present paper, we formulate two versions of Frank--Wolfe algorithm or conditional gradient method to solve the DC optimization problem with an adaptive step size. The DC objective function consists of two components; the first is thought to be differentiable with a continuous Lipschitz gradient, while the second is only thought to be convex. The second version is based on the first and employs finite differences to approximate the gradient of the first component of the objective function. In contrast to past formulations that used the curvature/Lipschitz-type constant of the objective function, the step size computed does not require any constant associated with the components. For the first version, we established that the algorithm is well-defined of the algorithm and that every limit point of the generated sequence is a stationary point of the problem. We also introduce the class of weak-star-convex functions and show that, despite the fact that these functions are non-convex in general, the rate of convergence of the first version of the algorithm to minimize these functions is ${\cal O}(1/k)$. The finite difference used to approximate the gradient in the second version of the Frank-Wolfe algorithm is computed with the step-size adaptively updated using two previous iterations. Unlike previous applications of finite difference in the Frank-Wolfe algorithm, which provided approximate gradients with absolute error, the one used here provides us with a relative error, simplifying the algorithm analysis. In this case, we show that all limit points of the generated sequence for the second version of the Frank-Wolfe algorithm are stationary points for the problem under consideration, and we establish that the rate of convergence for the duality gap is ${\cal O}(1/\sqrt{k})$.

Intrinsic core generalises the finite-dimensional notion of the relative interior to arbitrary (real) vector spaces. Our main goal is to provide a self-contained overview of the key results pertaining to the intrinsic core and to elucidate the relations between intrinsic core and facial structure of convex sets in this general context. We gather several equivalent definitions of the intrinsic core, cover much of the folklore, review relevant recent results and present examples illustrating some of the phenomena specific to the facial structure of infinite-dimensional sets.

In this paper, we propose a new algorithm combining the Douglas-Rachford (DR) algorithm and the Frank-Wolfe algorithm, also known as the conditional gradient (CondG) method, for solving the classic convex feasibility problem. Within the algorithm, which will be named {\it Approximate Douglas-Rachford (ApDR) algorithm}, the CondG method is used as a subroutine to compute feasible inexact projections on the sets under consideration, and the ApDR iteration is defined based on the DR iteration. The ApDR algorithm generates two sequences, the main sequence, based on the DR iteration, and its corresponding shadow sequence. When the intersection of the feasible sets is nonempty, the main sequence converges to a fixed point of the usual DR operator, and the shadow sequence converges to the solution set. We provide some numerical experiments to illustrate the behaviour of the sequences produced by the proposed algorithm.

In this paper we develop an optimisation based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalised rational approximation. In the second case the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimisation problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood.In this paper we demonstrate that in the case of multivariate generalised rational approximation the corresponding optimisation problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimisation. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalised rational approximation with the same number of decision variables., Comment: 14 pages, 5 figures

The motivation of this paper is the development of an optimisation method for solving optimisation problems appearing in Chebyshev rational and generalised rational approximation problems, where the approximations are constructed as ratios of linear forms (linear combinations of basis functions). The coefficients of the linear forms are subject to optimisation and the basis functions are continuous function. It is known that the objective functions in generalised rational approximation problems are quasi-convex. In this paper we also prove a stronger result, the objective functions are pseudo-convex in the sense of Penot and Quang. Then we develop numerical methods, that are efficient for a wide range of pseudo-convex functions and test them on generalised rational approximation problems., Comment: 17 pages, 5 figures

The classical convex feasibility problem in a finite dimensional Euclidean space is studied in the present paper. We are interested in two cases. First, we assume to know how to compute an exact project onto one of the sets involved and the other set is compact such that the conditional gradient (CondG) method can be used for computing efficiently an inexact projection on it. Second, we assume that both sets involved are compact such that the CondG method can be used for computing efficiently inexact projections on them. We combine alternating projection method with CondG method to design a new method, which can be seen as an inexact feasible version of alternate projection method. The proposed method generates two different sequences belonging to each involved set, which converge to a point in the intersection of them whenever it is not empty. If the intersection is empty, then the sequences converge to points in the respective sets whose distance is equal to the distance between the sets in consideration., Comment: 20 pages, 6 figures, 2 tables, 4 algorithms

We focus on the convergence analysis of averaged relaxations of cutters, specifically for variants that---depending upon how parameters are chosen---resemble \emph{alternating projections}, the \emph{Douglas--Rachford method}, \emph{relaxed reflect-reflect}, or the \emph{Peaceman--Rachford} method. Such methods are frequently used to solve convex feasibility problems. The standard convergence analyses of projection algorithms are based on the \emph{firm nonexpansivity} property of the relevant operators. However if the projections onto the constraint sets are replaced by cutters (projections onto separating hyperplanes), the firm nonexpansivity is lost. We provide a proof of convergence for a family of related averaged relaxed cutter methods under reasonable assumptions, relying on a simple geometric argument. This allows us to clarify fine details related to the allowable choice of the relaxation parameters, highlighting the distinction between the exact (firmly nonexpansive) and approximate (strongly quasi-nonexpansive) settings. We provide illustrative examples and discuss practical implementations of the method.

We introduce and study the convergence properties of a projection-type algorithm for solving the variational inequality problem for point-to-set operators. No monotoni\-city assumption is used in our analysis. The operator defining the problem is only assumed to be continuous in the point-to-set sense, i.e., inner- and outer-semicontinuous. Additionally, we assume non-emptiness of the so-called dual solution set. We prove that the whole sequence of iterates converges to a solution of the variational inequality. Moreover, we provide numerical experiments illustrating the behavior of our iterates. Through several examples, we provide a comparison with a recent similar algorithm.

The goal of this paper is to present two algorithms for solving systems of inclusion problems, with all component of the systems being a sum of two maximal monotone operators. The algorithms are variants of the forward-backward splitting method and one being a hybrid with the alternating projection method. They consist of approximating the solution sets involved in the problem by separating halfspaces which are a well-studied strategy. The schemes contain two part, the first one is an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. The second part is the projection step, being this the main difference between the algorithms. While the first algorithm computes the projection onto the intersection of the separating halfspaces, the second choose one component of the system and project onto the separating halfspace of this case. In the iterative process, the forward-backward operator is computed once per inclusion problem, representing a relevant computational saving if compared with similar algorithms in the literature. The convergence analysis of the proposed methods is given assuming monotonicity in all operators, without Lipschitz continuity assumption. We also present some numerical experiments., Comment: 16 pages. arXiv admin note: text overlap with arXiv:1503.03142, arXiv:1604.01866

In this paper, we propose an algorithm combining the forward-backward splitting method and the alternative projection method for solving the system of splitting inclusion problem. We want to find a point in the interception of a finite number of sets that we don't know, the solution of each component of the system. The algorithm consists of approximate the sets involved in the problem by separates halfspaces which are a known strategy. By finding these halfspaces in each iteration we use only one inclusion problem of the system. The iterations consist of two parts, the first contains an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. In the iterative process, the operator forward-backward is computed only one time for each inclusion problem, this represents a great computational saving because the computational cost of this operator is nothing cheap. The second part consists of special projection step, projecting in the separating halfspace. The convergence analysis of the proposed scheme is given assuming monotonicity all operators, without any Lipschitz continuity assumption., Comment: 10 pages, 0 figures. arXiv admin note: substantial text overlap with arXiv:1503.03142

In this paper, we propose variants of forward-backward splitting method for solving the system of splitting inclusion problem. We propose a conceptual algorithm containing three variants, each having a different projection steps. The algorithm consists in two parts, the first and main contains an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. In the iterative process the operator forward-backward is computed only one time for each inclusion problem, this represent a great computational saving if we compare with Tseng's algorithm, because the computational cost of this operator is very high. The second part of the scheme consists in special projection steps. The convergence analysis of the proposed scheme is given assuming monotonicity on both operators, without assuming Lipschitz continuity on the forward operators.

In this paper, we generalize the classical extragradient algorithm for solving variational inequality problems by utilizing nonzero normal vectors of the feasible set. In particular, conceptual algorithms are proposed with two different linesearchs. We then establish convergence results for these algorithms under mild assumptions. Our study suggests that nonzero normal vectors may significantly improve convergence if chosen appropriately.

We introduce a relaxed-projection splitting algorithm for solving variational inequalities in Hilbert spaces for the sum of nonsmooth maximal monotone operators, where the feasible set is defined by a nonlinear and nonsmooth continuous convex function inequality. In our scheme, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each iteration of the proposed method consists of simple subgradient-like steps, which does not demand the solution of a nontrivial subproblem, using only individual operators, which exploits the structure of the problem. Assuming monotonicity of the individual operators and the existence of solutions, we prove that the generated sequence converges weakly to a solution., Comment: 18 pages