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Neal Parikh and Stephen Boyd (2014), "Proximal Algorithms", Foundations and Trends® in Optimization: Vol. 1: No. 3, pp 127-239. http://dx.doi.org/10.1561/2400000003

© 2014 N. Parikh and S. Boyd

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**In this article:**

1. Introduction

2. Properties

3. Interpretations

4. Proximal Algorithms

5. Parallel and Distributed Algorithms

6. Evaluating Proximal Operators

7. Examples and Applications

8. Conclusions

References

This monograph is about a class of optimization algorithms called *proximal algorithms*. Much like Newton’s method is a
standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous
tool for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are
especially well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at
a higher level of abstraction than classical algorithms like Newton’s method: the base operation is evaluating the *proximal operator* of a
function, which itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting
a point onto a convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods. Here,
we discuss the many different interpretations of proximal operators and algorithms, describe their connections to many other topics in
optimization and applied mathematics, survey some popular algorithms, and provide a large number of examples of proximal operators that
commonly arise in practice.

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1. Introduction

2. Properties

3. Interpretations

4. Proximal Algorithms

5. Parallel and Distributed Algorithms

6. Evaluating Proximal Operators

7. Examples and Applications

8. Conclusions

References

*Proximal Algorithms* discusses proximal operators and proximal algorithms, and illustrates their applicability to
standard and distributed convex optimization in general and many applications of recent interest in particular. Much like Newton’s method is
a standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous tool
for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are especially
well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at a higher level
of abstraction than classical algorithms like Newton’s method: the base operation is evaluating the *proximal operator* of a function, which
itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting a point onto a
convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods.

*Proximal Algorithms* discusses different interpretations of proximal operators and algorithms, looks at their connections
to many other topics in optimization and applied mathematics, surveys some popular algorithms, and provides a large number of examples of
proximal operators that commonly arise in practice.