Function list

Main TFOCS programs
tfocs	Minimize a convex problem using a first-order algorithm.
tfocs_SCD	Smoothed conic dual form of TFOCS, for problems with non-trivial linear operators.
continuation	Meta-wrapper to run TFOCS_SCD in continuation mode.
Miscellaneous functions
tfocs_version	Version information.
tfocs_where	Returns the location of the TFOCS system.
Operator calculus
linop_adjoint	Computes the adjoint operator of a TFOCS linear operator
linop_compose	Composes two TFOCS linear operators
linop_scale	Scaling linear operator.
prox_dualize	Define a proximity function by its dual
prox_scale	Scaling a proximity/projection function.
tfunc_scale	Scaling a function.
tfunc_sum	Sum of functions.
tfocs_normsq	Squared norm.
linop_normest	Estimates the operator norm.
Linear operators
linop_matrix	Linear operator, assembled from a matrix.
linop_dot	Linear operator formed from a dot product.
linop_fft	Fast Fourier transform linear operator.
linop_TV	2D Total-Variation (TV) linear operator.
linop_TV3D	3D Total-Variation (TV) linear operator.
linop_handles	Linear operator from user-supplied function handles.
linop_spot	Linear operator, assembled from a SPOT operator.
linop_reshape	Linear operator to perform reshaping of matrices.
linop_subsample	Subsampling linear operator.
linop_vec	Matrix to vector reshape operator
Projection operators (proximity operators for indicator functions)
proj_0	Projection onto the set {0} (for equality constraints)
proj_box	Projection onto box constraints.
proj_l1	Projection onto the scaled 1-norm ball $\{x\,\|\,\\|x\\|_1\leq q\}$ .
proj_l2	Projection onto the scaled 2-norm ball $\{x\,\|\,\\|x\\|_2\leq q\}$ .
proj_linf	Projection onto the scaled $\infty$ -norm ball $\{x\,\|\,\\|x\\|_\infty\leq q\}$ .
proj_linfl2	Projection of each row of a matrix onto the scaled $2$ -norm ball $\{x\,\|\,\\|x\\|_2\leq q\}$ .
proj_max	Projection onto the scaled set of vectors $\{x\,\|\,\max_i x_i\leq q\}$ .
proj_nuclear	Projection onto the set of matrices with bounded nuclear norm: $\{X\,\|\,\sum_i \sigma_i(X) \leq q\}$
proj_psd	Projection onto the positive semidefinite (PSD) cone: $\{X\,\|\,X=X^T,\,\lambda_{\text{min}}(X)\geq 0\}$
proj_psdUTrace	Projection onto the PSD cone with fixed trace: $\{X\,\|\,X=X^T,\,\lambda_{\text{min}}(X)\geq 0,\,\mathop{\textrm{Tr}}(X)=q\}$
proj_Rn	“Projection” onto the entire space.
proj_Rplus	Projection onto the nonnegative orthant $\{x\,\|\,\min_i x_i\geq 0\}$ .
proj_simplex	Projection onto the simplex $\{x\,\|\,\min_i x_i\geq 0\,\sum_i x_i=1\}$ .
proj_conic	Projection onto the second-order (aka Lorentz) cone.
proj_singleAffine	Projection onto a single affine equality or in-equality constraint.
proj_boxAffine	Projection onto a single affine equality along with box constraints.
proj_affine	Projection onto general affine equations, e.g., solutions of linear equations.
proj_l2group	Projection of each group of coordinates onto 2-norm balls.
proj_spectral	Projection onto the set of matrices with bounded spectral norm: $\{X\,\|\,\sigma_{\text{max}}(X) \leq q\}$
proj_maxEig	Projection onto the set of symmetric matrices with bounded maximum eigenvalue: $\{X\,\|\,X=X^T,\,\lambda_{\text{max}}(X)\leq q\}$ .
Proximity operators of general convex functions
prox_0	The zero proximity function:
prox_boxDual	Dual function of box indicator function { l <= x <= u }
prox_hinge	Hinge-loss function.
prox_hingeDual	Dual function of the Hinge-loss function.
prox_l1	$\ell_1$ norm.
prox_Sl1	Sorted/ordered $\ell_1$ norm; see the ordered l1 page for info.
prox_l1l2	$\ell_1$ - $\ell_2$ block norm: the sum of the $\ell_2$ norms of rows.
prox_l1linf	$\ell_1$ - $\ell_\infty$ block norm: the max of the $\ell_2$ norms of rows.
prox_l1pos	$\ell_1$ norm, restricted to $x \geq 0$ .
prox_l2	$\ell_2$ norm.
prox_linf	$\ell_\infty$ norm.
prox_max	Maximum function.
prox_nuclear	Nuclear norm.
prox_spectral	Spectral norm, i.e. max singular value.
prox_maxEig	Maximum eigenvalue of a symmetric matrix.
prox_trace	Nuclear norm, for positive semidefinite matrices. Equivalent to trace.
Smooth functions
smooth_constant	Constant function generation.
smooth_entropy	The entropy function $-\sum_i x_i\log x_i$
smooth_handles	Smooth function from separate $f$ / $g$ handles.
smooth_huber	Huber function generation.
smooth_linear	Linear function generation.
smooth_logdet	The $-\log\det(X)$ function.
smooth_logLLogistic	Log-likelihood function of a logistic: $\sum_i y_i \mu_i – log( 1+\exp(\mu_i) )$
smooth_logLPoisson	Log-likelihood of a Poisson: $\sum_i -\lambda_i + x_i \cdot \log( \lambda_i)$
smooth_logsumexp	The function $\log(\sum_i \exp(x_i))$
smooth_quad	Quadratic function generation.
Testing functions
test_nonsmooth	Runs diagnostic tests to ensure a non-smooth function conforms to TFOCS conventions
test_proxPair	Runs diagnostics on a pair of functions to check if they are Legendre conjugates.
test_smooth	Runs diagnostic checks on a TFOCS smooth function object.
linop_test	Performs an adjoint test on a linear operator.
Premade solvers for specific problems (vector variables)
solver_L1RLS	l1-regularized least squares problem, sometimes called the LASSO. $\min_x .5\\|Ax-b\\|_2^2 + \lambda \\|x\\|_1$
solver_LASSO	Minimize residual subject to $\ell_1$ -norm constraints. $\min_x .5\\|Ax-b\\|_2^2$ subject to $\\|x\\|_1 \le \tau$
solver_SLOPE	Sorted L-One Penalized Estimation; like LASSO but with an ordered $\ell_1$ -norm; see documentation.
solver_sBP	Basis pursuit (l1-norm with equality constraints). Uses smoothing. $\min_x \\|x\\|_1$ subject to $Ax=b$
solver_sBPDN	Basis pursuit de-noising. BP with relaxed constraints. Uses smoothing. $\min_x \\|x\\|_1$ subject to $\\|Ax-b\\|_2 \le \epsilon$
solver_sBPDN_W	Weighted BPDN problem. Uses smoothing. $\min_x \\|Wx\\|_1$ subject to $\\|Ax-b\\|_2 \le \epsilon$
solver_sBPDN_WW	BPDN with two separate (weighted) $\ell_1$ -norm terms. Uses smoothing. $\min_x \\|W^{(1)}x\\|_1+\\|W^{(2)}x\\|_1$ subject to $\\|Ax-b\\|_2 \le \epsilon$
solver_sDantzig	Dantzig selector problem. Uses smoothing. $\min_x \\|x\\|_1$ subject to $\\|A^T(Ax-b)\\|_\infty \le \delta$
solver_sDantzig_W	Weighted Dantzig selector problem. Uses smoothing. $\min_x \\|Wx\\|_1$ subject to $\\|A^T(Ax-b)\\|_\infty \le \delta$
solver_sLP	Generic linear programming in standard form. Uses smoothing. $\min_x c^Tx$ subject to $x \ge 0$ and $Ax=b$
solver_sLP_box	Generic linear programming with box constraints. Uses smoothing. $\min_x c^Tx$ subject to $\ell \le x \le u$ and $Ax=b$
Premade solvers for specific problems (matrix variables)
	Here, $\mathcal{A}$ is a generic linear operator and $\mathcal{A}_\Omega$ is the sub-sampling operator that samples entries in $\Omega$ . The nuclear/trace norm is written $\\|\cdot\\|_{\text{tr}}$
solver_psdComp	Matrix completion for PSD matrices. $\min_X .5\\| \mathcal{A}_\Omega(X) – b\\|_2^2$ subject to $X \succeq 0$
solver_psdCompConstrainedTrace	Matrix completion with constrained trace, for PSD matrices. $\min_X .5\\| \mathcal{A}_\Omega(X) – b\\|_2^2$ subject to $X \succeq 0$ and \( \textrm{trace}(X)=\tau)
solver_TraceLS	Unconstrained form of trace-regularized least-squares problem. $\min_X .5\\| \mathcal{A}(X) – b\\|_2^2 + \lambda \textrm{trace}(X)$ subject to $X \succeq 0$
solver_sNuclearBP	Nuclear norm basis pursuit problem (i.e. matrix completion). Uses smoothing. $\min_X \\|X\\|_{\text{tr}}$ subject to $\mathcal{A}_\Omega(X) = b$
solver_sNuclearBPDN	Nuclear norm basis pursuit problem with relaxed constraints. Uses smoothing. $\min_X \\|X\\|_{\text{tr}}$ subject to $\\|\mathcal{A}_\Omega(X) – b\\|_2 \le \epsilon$
solver_sSDP	Generic semi-definite programs (SDP). Uses smoothing. $\min_X \text{tr}(C^TX)$ subject to $X \succeq 0$ and $\mathcal{A}(X) = b$
solver_sLMI	Generic linear matrix inequality problems (LMI is the dual of a SDP). Uses smoothing. $\min_y b^Ty$ subject to $A_0 + \sum_i A_i y_i \succeq 0$
Algorithm variants
tfocs_AT	Auslender and Teboulle’s accelerated method. Reference: Auslender, A., Teboulle, M.: Interior gradient and proximal methods for convex and conic optimization. SIAM J. Optim. 16(3), 697–725 (2006)
tfocs_GRA	Gradient descent.
tfocs_LLM	Lan, Lu and Monteiro’s accelerated method. Reference: Lan, G., Lu, Z., Monteiro, R.D.C.: Primal-dual first-order methods with O(1/eps) iteration-complexity for cone programming. Math. Program. Ser. A (2009)
tfocs_N07	Nesterov’s 2007 accelerated method. References: Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course. Applied Optimization, vol. 87. Kluwer, Boston (2004); Nesterov,Y.: Gradient methods for minimizing composite objective function. Technical report (later published in Math. Prog. Series B), CORE 2007/76, Université Catholique de Louvain, Louvain-la-Neuve, Belgium (2007)
tfocs_N83	Nesterov’s 1983 accelerated method; also by Beck and Teboulle 2005 (FISTA). References: Nesterov, Y.: A method for unconstrained convex minimization problem with the rate of convergence O(1/k2). Doklady AN USSR; Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Program. Ser. A 103, 127– 152 (2005); Beck, A., Teboulle, M.:A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)
tfocs_TS	Tseng’s modification of Nesterov’s 2007 method. Reference: Tseng, P.: On accelerated proximal gradient methods for convex-concave optimization. (2008).