Solvers

MATLAB function	Method	Model	Assumptions	Available oracles
comd	CoMirror Descent		$f$ - Lipschitz, convex $g$ - vector-valued with convex components X -simplex/ball/box/spectahedron	function value of f subgradient of f function values of the components of g subgradients of the components of g
prox_subgradient	proximal subgradient	$\min f(x)+\lambda g(x)$	$f$ - Lipschitz, convex $g$ - proper closed convex proximable $\lambda$ - positive scalar	function value of f subgradient of f function value of g prox of a positive constant times g
prox_gradient	proximal gradient	$\min f(x)+\lambda g(x)$	$f$ - real-valued and differentiable^[*] $g$ - proper closed convex proximable $\lambda$ - positive scalar	function value of f gradient of f function value of g prox of a positive constant times g
fista	FISTA (Fast Iterative Shrinkage/Thresholding Algorithm) a.k.a. fast proximal gradient	$\min f(x)+\lambda g(x)$	$f$ - smooth^[*] $g$ - proper closed convex proximable $\lambda$ - positive scalar	function value of f gradient of f function value of g prox of a positive constant times g
sfista	sfista (smooth FISTA)	$\min_{x} f(x)+\lambda_G g(Ax)+\lambda_H h(x)$	$f$ - smooth and convex $g$ - real-valued, convex and proximable $h$ - proper closed convex and proximable A - linear transformation $\lambda_G,\lambda_H$ - positive scalars	function value of f gradient of f function value of g prox of a positive constant times g function value of h prox of a positive constant times h A - linear transformation A' - adjoint of A
fdpg	FDPG (Fast Dual Proximal Gradient)	$\min_x f(x)+\lambda g(Ax)$	$f$ - proper closed and strongly convex $g$ - proper closed convex proximable A - linear transformation	function value of f gradient of the conjugate of f function value of g prox of a positive constant times g A - linear transformation A' - adjoint transformation
adlpmm	ADLPMM (Alternating Directions Linearized Proximal Method of Mutipliers)	$\min_x f(x)+\lambda g(Ax)$	$f$ - proper closed convex and proximable $g$ - proper closed convex and proximable A - linear transformation $\lambda$ - positive scalar	function value of f prox of a positive constant times f function value of g prox of a positive constant times g A - linear transformation A' - adjoint of A
nested_fista	Nested FISTA	$\min_x \left \{ \varphi(f(x))+\lambda g(Ax) \right \}$	$\varphi$ - real-valued, Lipschitz, nondecreasing, convex and proximable $f$ - vector-valued and differentiable with convex components. $g$ - proper, closed, convex and proximable A - linear transformation $\lambda$ - positive scalar	function value of phi prox of a positive constant times phi function value of f gradient of f function value of g prox of a positive constant times g A - linear transformation A' - adjoint of A

[*] - in cases where f is nonconvex, the corresponding methods are only guaranteed to converge to a stationary point.

Subpages (6): comd fdpg fista prox_gradient prox_subgradient sfista