pyproximal.optimization.palm.PALM(H, proxf, proxg, x0, y0, gammaf=1.0, gammag=1.0, beta=0.5, niter=10, niterback=100, callback=None, show=False)[source]#

Proximal Alternating Linearized Minimization

Solves the following minimization problem using the Proximal Alternating Linearized Minimization (PALM) algorithm:

\[\mathbf{x}\mathbf{,y} = \argmin_{\mathbf{x}, \mathbf{y}} f(\mathbf{x}) + g(\mathbf{y}) + H(\mathbf{x}, \mathbf{y})\]

where \(f(\mathbf{x})\) and \(g(\mathbf{y})\) are any pair of convex functions that have known proximal operators, and \(H(\mathbf{x}, \mathbf{y})\) is a smooth function.


Bilinear function


Proximal operator of f function


Proximal operator of g function


Initial x vector


Initial y vector

gammaffloat, optional

Positive scalar weight for f function update. If None, use backtracking

gammagfloat, optional

Positive scalar weight for g function update. If None, use backtracking

betafloat, optional

Backtracking parameter (must be between 0 and 1)

niterint, optional

Number of iterations of iterative scheme

niterbackint, optional

Max number of iterations of backtracking

callbackcallable, optional

Function with signature (callback(x)) to call after each iteration where x and y are the current model vectors

showbool, optional

Display iterations log


Inverted x vector


Inverted y vector


PALM [1] can be expressed by the following recursion:

\[\begin{split}\mathbf{x}^{k+1} = \prox_{c_k f}(\mathbf{x}^{k} - \frac{1}{c_k}\nabla_x H(\mathbf{x}^{k}, \mathbf{y}^{k}))\\ \mathbf{y}^{k+1} = \prox_{d_k g}(\mathbf{y}^{k} - \frac{1}{d_k}\nabla_y H(\mathbf{x}^{k+1}, \mathbf{y}^{k}))\\\end{split}\]

Here \(c_k=\gamma_f L_x\) and \(d_k=\gamma_g L_y\), where \(L_x\) and \(L_y\) are the Lipschitz constant of \(\nabla_x H\) and \(\nabla_y H\), respectively. When such constants cannot be easily computed, a back-tracking algorithm can be instead employed to find suitable \(c_k\) and \(d_k\) parameters.


Bolte, J., Sabach, S., and Teboulle, M. “Proximal alternating linearized minimization for nonconvex and nonsmooth problems”, Mathematical Programming, vol. 146, pp. 459–494. 2014.

Examples using pyproximal.optimization.palm.PALM#

Low-Rank completion via Matrix factorization

Low-Rank completion via Matrix factorization