# Convex programming in Hilbert space

@article{Goldstein1964ConvexPI, title={Convex programming in Hilbert space}, author={Allen A. Goldstein}, journal={Bulletin of the American Mathematical Society}, year={1964}, volume={70}, pages={709-710} }

This note gives a construction for minimizing certain twice-differentiable functions on a closed convex subset C, of a Hubert Space, H. The algorithm assumes one can constructively "project" points onto convex sets. A related algorithm may be found in Cheney-Goldstein [ l ] , where a constructive fixed-point theorem is employed to construct points inducing a minimum distance between two convex sets. In certain instances when such projections are not too difficult to construct, say on spheres…

## 539 Citations

### Outer approximation methods for solving variational inequalities in Hilbert space

- Mathematics
- 2017

Abstract In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set C. We…

### Methods for Variational Inequality Problem Over the Intersection of Fixed Point Sets of Quasi-Nonexpansive Operators

- Mathematics
- 2013

Many convex optimization problems in a Hilbert space ℋ can be written as the following variational inequality problem VIP (ℱ, C): Find such that for all z ∈ C, where C ⊂ ℋ is closed convex and ℱ: ℋ →…

### Minimizing the Moreau Envelope of Nonsmooth Convex Functions over the Fixed Point Set of Certain Quasi-Nonexpansive Mappings

- MathematicsFixed-Point Algorithms for Inverse Problems in Science and Engineering
- 2011

A nontrivial integration of the ideas of the hybrid steepest descent method and the Moreau–Yosida regularization is proposed, yielding a useful approach to the challenging problem of nonsmooth convex optimization over Fix(T).

### Error Bound and Reduced-Gradient Projection Algorithms for Convex Minimization over a Polyhedral Set

- Computer Science, MathematicsSIAM J. Optim.
- 1993

A class of reduced-gradient projection algorithms for solving the case where the simpler polyhedral set is a box is proposed and this bound is used to show that algorithms in this class attain a linear rate of convergence.

### Convergence of the Gradient Projection Method for Generalized Convex Minimization

- MathematicsComput. Optim. Appl.
- 2000

This paper develops convergence theory of the gradient projection method by Calamai and Moré which, for minimizing a continuously differentiable optimization problem min{f(x) : x ε Ω} where Ω is a nonempty closed convex set, generates a sequence xk+1 = P( xk − αk ∇ f(xk) where the stepsize αk > 0 is chosen suitably.

### Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces

- MathematicsAppl. Math. Comput.
- 2015

### PROPERTIES OF A CLASS OF APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS

- Mathematics
- 2014

In this paper we present an application of a class of quasi-nonexpansive operators to iterative methods for solving the following variational inequality problem VIP(F,C): Find ū ∈ C such that 〈F ū,…

### Convex Optimization in Normed Spaces

- Mathematics
- 2015

Functional and convex analysis are closely intertwined. In this chapter we recall the basic concepts and results from functional analysis and calculus that will be needed throughout this book. A…

### A projection and contraction method for a class of linear complementarity problems and its application in convex quadratic programming

- Mathematics
- 1992

In this paper we propose a new iterative method for solving a class of linear complementarity problems:u ≥ 0,Mu + q ≥ 0, uT(Mu + q)=0, where M is a givenl ×l positive semidefinite matrix (not…

### Outer Approximation Methods for Solving Variational Inequalities Defined over the Solution Set of a Split Convex Feasibility Problem

- MathematicsNumerical Functional Analysis and Optimization
- 2020

Abstract We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set S. We assume that is the nonempty solution set of…

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