Proximal operators

This "library" contains sample implementations of various proximal operators in Matlab. These implementations are intended to be pedagogical, not the most performant.

This code is associated with the paper Proximal Algorithms by Neal Parikh and Stephen Boyd.

Requirements

The C functions rely on the GNU Scientific Library (GSL). Some of these functions also contain OpenMP directives to parallelize some for loops, so compiling with OpenMP is optional, but some of the functions will be substantially faster if it is used.

The Matlab function prox_cvx.m requires CVX.

Examples

Evaluating the proximal operator of the l1 norm via CVX and the function here:

>> n = 100;
>> lambda = 1;
>> 
>> v = randn(n,1);
>> 
>> % CVX baseline
>> cvx_begin quiet
>>   variable x(n)
>>   minimize(norm(x,1) + (1/(2*lambda))*sum_square(x - v))
>> cvx_end
>> 
>> % Custom method
>> x2 = prox_l1(v, lambda);
>> 
>> % Comparison
>> norm(x - x2)
ans =
7.7871e-05

Evaluating the proximal operator of the nuclear norm:

>> m = 10;
>> n = 30;
>> lambda = 1;
>> 
>> V = randn(m,n);
>> 
>> % CVX baseline
>> cvx_begin quiet
>>   variable X(m,n)
>>   minimize(norm_nuc(X) + (1/(2*lambda))*square_pos(norm(X - V,'fro')))
>> cvx_end
>> 
>> % Custom method
>> X2 = prox_matrix(V, lambda, @prox_l1);
>> 
>> % Comparison
>> norm(X - X2)
ans =
1.9174e-05

This second example shows a case where one of the arguments is a function handle to another proximal operator.

The other Matlab functions work similarly; just use help in Matlab.

For a C example, see the file example.c in the C source directory.

Proximal operators

The Matlab functions include the following examples:

• Projection onto an affine set
• Projection onto a box
• Projection onto the consensus set (averaging)
• Projection onto the exponential cone
• Projection onto the nonnegative orthant
• Projection onto the second-order cone
• Projection onto the semidefinite cone
• Proximal operator of a generic function (via CVX)
• Proximal operator of the l1 norm
• Proximal operator of the max function
• Proximal operator of a quadratic function
• Proximal operator of a generic scalar function (vectorized)
• Proximal operator of an orthogonally invariant matrix function
• Precomposition of a proximal operator

Authors

• Neal Parikh
• Eric Chu
• Stephen Boyd

Other libraries

There are other libraries with implementations of proximal or projection operators that may be preferable or contain more examples:

• TFOCS (see prox/proj sections) by S. Becker, E. Candès, and M. Grant
• Python proximal operators by S. Vaiter

License

This code is released under a BSD license; see the "LICENSE" file.