# Monotone Operators and the Proximal Point Algorithm

@article{Rockafellar1976MonotoneOA, title={Monotone Operators and the Proximal Point Algorithm}, author={R. Tyrrell Rockafellar}, journal={Siam Journal on Control and Optimization}, year={1976}, volume={14}, pages={877-898} }

For the problem of minimizing a lower semicontinuous proper convex function f on a Hilbert space, the proximal point algorithm in exact form generates a sequence $\{ z^k \} $ by taking $z^{k + 1} $ to be the minimizes of $f(z) + ({1 / {2c_k }})\| {z - z^k } \|^2 $, where $c_k > 0$. This algorithm is of interest for several reasons, but especially because of its role in certain computational methods based on duality, such as the Hestenes-Powell method of multipliers in nonlinear programming. It…

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## References

SHOWING 1-10 OF 31 REFERENCES

### On the maximality of sums of nonlinear monotone operators

- Mathematics
- 1970

is called the effective domain of F, and F is said to be locally bounded at a point x e D(T) if there exists a neighborhood U of x such that the set (1.4) T(U) = (J{T(u)\ueU} is a bounded subset of…

### LEVEL SETS AND CONTINUITY OF CONJUGATE CONVEX FUNCTIONS

- Mathematics
- 1966

A finite-valued convex function on a nonempty convex set C in F can always be extended to a proper convex function on F by assigning it the value + 0o outside of C. Let F and G be real vector spaces…

### Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming

- Computer Science, MathematicsMath. Oper. Res.
- 1976

The theory of the proximal point algorithm for maximal monotone operators is applied to three algorithms for solving convex programs, one of which has not previously been formulated and is shown to have much the same convergence properties, but with some potential advantages.

### The multiplier method of Hestenes and Powell applied to convex programming

- Mathematics
- 1973

For nonlinear programming problems with equality constraints, Hestenes and Powell have independently proposed a dual method of solution in which squares of the constraint functions are added as…

### Multiplier and gradient methods

- Computer Science
- 1969

The main purpose of this paper is to suggest a method for finding the minimum of a functionf(x) subject to the constraintg(x)=0, which consists of replacingf byF=f+λg+1/2cg2, and computing the appropriate value of the Lagrange multiplier.

### Necessary and sufficient conditions for a penalty method to be exact

- MathematicsMath. Program.
- 1975

This paper identifies necessary and sufficient conditions for a penalty method to yield an optimal solution or a Lagrange multiplier of a convex programming problem by means of a single unconstrained…

### An example concerning fixed points

- Mathematics
- 1975

An example is given of a contractionT defined on a bounded closed convex subset of Hilbert space for which ((I+T)/2)n does not converge.

### Proximité et dualité dans un espace hilbertien

- Mathematics
- 1965

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