## Software

### LMAOPT: Low-rank Matrix OPTimization

• LMAOPT-v1.0 is a MATLAB code colection for solving three special cases of the following low-rank matrix optimization problem: $$\Phi^{\star} := \min_{U\in\mathbf{R}^{m\times r}, V\in\mathbb{R}^{n\times r}}\Big\{ \Phi(U,V) := \phi(\mathcal{A}(UV^T) - B) \Big\},$$ where $\phi$ is a proper, closed and convex function from $\mathbb{R}^l\to\mathbb{R}\cup\{+\infty\}$, $\mathcal{A}$ is a linear operator from $\mathbb{R}^{m\times n}$ to $\mathbb{R}^l$, and $B\in\mathbb{R}^l$ is a given observed vector. Here, we are more interested in the case $r \ll \min\{m, n\}$.
Currently, we provide the code to solve three special cases of the above problem:
• Quadratic loss: $\phi(\cdot) = \frac{1}{2}\Vert\cdot\Vert_2^2$;
• Quadratic loss and symmetric case: $\phi(\cdot) = \frac{1}{2}\Vert\cdot\Vert_2^2$ and $U = V$; and
• Nonsmooth objective loss with tractably proximal operators: For instance, $\phi(\cdot) = \Vert\cdot\Vert_1$.

LMAOPT is implemented by Quoc Tran-Dinh at the Department of Statistics and Operations Research (STAT&OR), University of North Carolina at Chapel Hill (UNC). This is a joint work with Zheqi Zhang at STAT&OR, UNC.

• REFERENCES

The theory and algorithms implemented in LMAOPT can be found in the following manuscript:
1. Q. Tran-Dinh, and Z. Zhang. Extended Gauss-Newton and Gauss-Newton ADMM algorithms for low-rank matrix optimization. STAT&OR Tech. Report UNC-REPORT-2016.a, (2016).
Available at: http://arxiv.org/abs/1606.03358
Abstract: We develop a generic Gauss-Newton (GN) framework for solving a class of nonconvex optimization problems involving low-rank matrix variables. As opposed to standard Gauss-Newton method, our framework allows one to handle general smooth convex cost function via its surrogate. The main complexity-per-iteration consists of the inverse of two rank-size matrices and at most six small matrix multiplications to compute a closed form Gauss-Newton direction, and a backtracking linesearch. We show, under mild conditions, that the proposed algorithm globally and locally converges to a stationary point of the original nonconvex problem. We also show empirically that the Gauss-Newton algorithm achieves much higher accurate solutions compared to the well studied alternating direction method (ADM). Then, we specify our Gauss-Newton framework to handle the symmetric case and prove its convergence, where ADM is not applicable without lifting variables. Next, we incorporate our Gauss-Newton scheme into the alternating direction method of multipliers (ADMM) to design a GN-ADMM algorithm for solving the low-rank optimization problem. We prove that, under mild conditions and a proper choice of the penalty parameter, our GN-ADMM globally converges to a stationary point of the original problem. Finally, we apply our algorithms to solve several problems in practice such as low-rank approximation, matrix completion, robust low-rank matrix recovery, and matrix recovery in quantum tomography. The numerical experiments provide encouraging results to motivate the use of nonconvex optimization.
• ### DECOPT: DEcomposition OPTimization

• DECOPT-v1.0 is a MATLAB software package for solving constrained convex optimization problems of the form: $$\begin{array}{ll} \displaystyle\min_{\mathbf{x}\in\mathbf{R}^p, \mathbf{r}\in\mathbb{R}^n} & f(\mathbf{x}) + g(\mathbf{r}) \\ \text{s.t.} & \mathbf{A}\mathbf{x} - \mathbf{r} = \mathbf{b}, ~\mathbf{l}\leq \mathbf{x}\leq \mathbf{u}, \end{array}$$ where $f$ and $g$ are two convex functions, $\mathbf{A}\in\mathbb{R}^{n\times p}$, $\mathbf{b}\in\mathbb{R}^n$, $\mathbf{l}, \mathbf{u}\in\mathbb{R}^p$ are the lower and upper bound of $\mathbf{x}$. Here, we assume that $f$ and $g$ are proximally tractable, i.e., the proximal operators $\mathrm{prox}_{f}$ and $\mathrm{prox}_g$ are efficient to compute (see below).
DECOPT aims at solving constrained convex optimization problem for any convex functions $f$ and $g$, where their proximal operator is provided. The following special cases have been customized in DECOPT:
• Basis pursuit: $$\min_{\mathbf{x}}\Big\{ \Vert\mathbf{x}\Vert_1 : \mathbf{A}(\mathbf{x}) = \mathbf{b}, ~\mathbf{l} \leq \mathbf{x}\leq\mathbf{u} \Big\}.$$
• $\ell_1/\ell_2$-unconstrained LASSO problem: $$\min_{\mathbf{x}}\Big\{ \frac{1}{2}\Vert\mathbf{A}(\mathbf{x}) - \mathbf{b}\Vert_2^2 + \lambda\Vert\mathbf{x}\Vert_1\Big\}.$$
• $\ell_1/\ell_1$-convex problem: $$\min_{\mathbf{x}}\Big\{ \Vert\mathbf{A}(\mathbf{x}) - \mathbf{b}\Vert_1 + \lambda\Vert\mathbf{x}\Vert_1 \Big\}.$$
• Square-root $\ell_1/\ell_2$ LASSO problem: $$\min_{\mathbf{x}}\Big\{ \Vert\mathbf{A}(\mathbf{x}) - \mathbf{b}\Vert_2 + \lambda\Vert\mathbf{x}\Vert_1 \Big\}.$$
• $\ell_1/\ell_2$ - the basis denosing (BPDN) problem: $$\min_{\mathbf{x}}\Big\{ \Vert\mathbf{x}\Vert_1 : \Vert\mathbf{A}(\mathbf{x}) - \mathbf{b}\Vert_2 \leq \delta\Big\}.$$
• $\ell_2/\ell_1$ - the $\ell_1$ - constrained LASSO problem: $$\min_{\mathbf{x}}\Big\{ \frac{1}{2}\Vert\mathbf{A}(\mathbf{x}) - \mathbf{b}\Vert_2^2 : \Vert\mathbf{x}\Vert_1 \leq \delta \Big\}.$$
Here, $\lambda > 0$ is a penalty parameter, $\delta > 0$ is a given noise level parameter, $\mathbf{b}$ is the observed measurement vector, and $\mathbf{A}$ is a linear operator.

DECOPT is developed by Quoc Tran-Dinh at the Laboratory for Information and Inference Systems (LIONS), EPFL, Lausanne, Switzerland. This is a joint work with V. Cevher at EPFL/LIONS.

• REFERENCES

The theory and algorithms implemented in DECOPT can be found in the following papers:

1. Q. Tran-Dinh, and V. Cevher. Constrained convex minimization via model-based excessive gap. Proceedings of the annual conference on Neural Information Processing Systems Foundation (NIPS), Montreal, Canada, (2014).
Available at: http://papers.nips.cc/paper/5574-constrained-convex-minimization-via-model-based-excessive-gap.
Abstract: Abstract We introduce a model-based excessive gap technique to analyze first-order primal- dual methods for constrained convex minimization. As a result, we construct first-order primal-dual methods with optimal convergence rates on the primal objec-tive residual and the primal feasibility gap of their iterates separately. Through a dual smoothing and prox- center selection strategy, our framework subsumes the augmented Lagrangian, alternating direction, and dual fast-gradient methods as special cases, where our rates apply.

2. Q. Tran Dinh, and V. Cevher. A Primal-Dual Algorithmic Framework for Constrained Convex Minimization. LIONS Tech. Report EPFL-REPORT-199844, (2014).
Available at: http://arxiv.org/abs/1406.5403
Abstract: We present a primal-dual algorithmic framework to obtain approximate solutions to a prototypical constrained convex optimization problem, and rigorously characterize how common structural assumptions affect the numerical efficiency. Our main analysis technique provides a fresh perspective on Nesterov's excessive gap technique in a structured fashion and unifies it with smoothing and primal-dual methods. For instance, through the choices of a dual smoothing strategy and a center point, our framework subsumes decomposition algorithms, augmented Lagrangian as well as the alternating direction method-of-multipliers methods as its special cases, and provides optimal convergence rates on the primal objective residual as well as the primal feasibility gap of the iterates for all.

### SCOPT: Self-Concordant OPTimization

• SCOPT-v1.0 - A MATLAB package for Composite Self-concordant (like) Minimization
SCOPT is a MATLAB implementation of the proximal-gradient, proximal quasi-Newton, proximal Newton, and path-following interior-point algorithms for solving composite convex minimization problems involving self-concordant and self-concordant-like cost functions: $$\displaystyle\min_{\mathbf{x}\in\mathbb{R}^n}\left\{ F(\mathbf{x}) := f(\mathbf{x}) + g(\mathbf{x}) \right\},$$ where $f$ is a self-concordant or self-concordant-like function, and $g$ is a possibly nonsmooth and convex function.

• Briefly, the function $f$ is called self-concordant (with the parameter $M_f >0$) if $\vert\varphi{'''}(t)\vert \leq M_f\varphi{''}(t)^{3/2}$, where $\varphi(t):= f(\mathbf{x} + t\mathbf{u})$ for $\mathbf{x}\in\mathrm{dom}(f)$, $\mathbf{u}\in\mathbb{R}^m$ and $\mathbf{x} + t\mathbf{u}\in\mathrm{dom}(f)$.
Examples: $f(\mathbf{X}) := -\log(\det(\mathbf{X}))$ (for a symmetric and positive definite matrix $\mathbf{X}$), $f(\mathbf{x},t) := -\log(t^2 - \Vert\mathbf{x}\Vert_2^2)$, and $f(\mathbf{x}) := -\sum_{i=1}^n\log(b_i - \mathbf{a}_i^T\mathbf{x})$ are all self-concordant.
• The function $g$ is usually assumed that its proximal operator: $$\mathrm{prox}_{g}(\mathbf{x}) := \mathrm{arg}\!\displaystyle\min_{\mathbf{z}\in\mathbb{R}^n}\left\{g(\mathbf{z}) + \frac{1}{2}\Vert\mathbf{z} - \mathbf{x}\Vert_2^2\right\},$$ is "efficient" to compute (i.e., in a closed form or by polynomial-time algorithms). If the proximal operator of $g$ can be computed efficently, then we say that $g$ is tractably proximal.
SCOPT is developed by Quoc Tran-Dinh at the Laboratory for Information and Inference Systems (LIONS), EPFL, Lausanne, Switzerland. This is a joint work with A. Kyrillidis and V. Cevher at EPFL/LIONS.

SCOPT consists of several algorithms customized for solving the following convex optimization problems (but not limitted to those):
• Sparse Inverse Covariance Estimation: An instance of this problem class can be expressed as follows: $$\min_{\mathbf{X}\succ 0}\left\{ -\log(\det(\mathbf{X})) + \mathrm{trace}(\boldsymbol{\Sigma}\mathbf{X}) + \rho\Vert\mathrm{vec}(\mathbf{X})\Vert_1 \right\},$$ where $\rho > 0$ is a penalty parameter.
Reference: J. Friedman, T. Hastie, and R. Tibshirani. Sparse Inverse Covariance Estimation with the graphical LASSO. Biostatistics, vol. 9, no. 3, pp. 432--441, 2007.
• Sparse Poisson Intensity Reconstruction: As an example, this problem has the following form: $$\min_{\mathbf{x}\geq 0}\left\{ -\sum_{i=1}^m\mathbf{a}_i^T\mathbf{x} - \sum_{i=1}^my_i\log(\mathbf{a}_i^T\mathbf{x}) + \rho\Vert\mathbf{x}\Vert_{\mathrm{TV}}\right\},$$ where $y_i\in\mathbb{N}$, $\rho > 0$ is a penalty parameter, and $\Vert\cdot\Vert_{\mathrm{TV}}$ is the TV (total-variation) norm.
Reference: Z. T. Harmany, R. F. Marcia, and R. M. Willett. This is SPIRAL-TAP: Sparse Poisson Intensity Reconstruction Algorithms - Theory and Practice. IEEE Trans. Image Processing, vol. 21, no. 3, pp. 1084--1096, 2012.
• Heteroscedastic LASSO: This problem is formulated as follows: $$\min_{\sigma > 0, \boldsymbol{\beta}\in\mathbb{R}^p}\left\{ -\log(\sigma) + \frac{1}{2n}\Vert\mathbf{X}\boldsymbol{\beta} - \sigma\mathbf{y}\Vert_2^2 + \rho\Vert\boldsymbol{\beta}\Vert_1\right\},$$ where $\rho > 0$ is a penalty parameter.
Reference: N. Stadler, P. Buhlmann, and S. V. de Geer. L1-Penalization for mixture regression models. TEST, vol. 19, no. 2, pp. 209--256, 2010.
• Sparse logistic and multimonomial logistic regression: This problem can be written in the following mathematical form: $$\min_{\mathbf{X}\in\mathbb{R}^{m\times p}}\left\{ \frac{1}{N}\sum_{i=1}^N\left[ \log\left(1 + \sum_{j=1}^m\mathrm{exp}(\mathbf{W}_{(j)}^T\mathbf{X}_{(i)})\right) -\sum_{i=1}^my_i^{(j)}\mathbf{W}_{(j)^T\mathbf{X}_{(i)}}\right] + \rho\Vert\mathrm{vec}(\mathbf{X})\Vert_1 \right\},$$ where $\mathbf{W}_{(j)}$ and $y_i^{(j)}$ are input data, and $\rho > 0$ is a penalty parameter.
Reference: B. Krishnapuram, M. Figueredo, L. Carin, and A. Hertemink. Sparse multinomial logistic regression: Fast algorithms and generalization bounds. IEEE Trans. Pattern Analysis and Machine Intelligence (PAMI), vol. 27, pp. 957--968, 2005.
• In addition to the above mentioned problems, SCOPT also contains the codes which can be customized to solve the following constrained convex optimization problem: $$g^{\star} := \min_{\mathbf{x}\in\Omega}g(\mathbf{x}),$$ where $g$ is a convex function with a tractable proximity operator, and $\Omega$ is a nonempty, closed and convex set in $\mathbb{R}^p$ endowed with a self-concordant barrier $f$ (i.e., $f$ is self-concordant with $M_f = 2$ and $\vert\varphi'(t)\vert \leq \sqrt{\nu}\varphi''(t)^{1/2}$ for given $\nu > 0$ and $\varphi(t) := f(\mathbf{x} + t\mathbf{u})$).
Example: The clustering problem in unsupervised learning can be reformulated as the above constrained convex problem using the $\max$-norm: $$\begin{array}{ll} \min_{\mathbf{K},\mathbf{R}, \mathbf{L}}& \Vert\mathrm{vec}(\mathbf{K} - \mathbf{A})\Vert_1 \\ \textrm{s.t.} & \begin{bmatrix}\mathbf{R} & \mathbf{K}\\ \mathbf{K}^T & \mathbf{L}\end{bmatrix}\succeq 0, ~\mathbf{R}_{ii} \leq 1, \mathbf{L}_{ii}\leq 1, ~i=1,\dots, p. \end{array}$$ Reference: Jalali, A., and Srebro, N. Clustering using max-norm constrained optimization. In Proc. of International Conference on Machine Learning (ICML2012) (2012), pp. 1–17.

• #### REFERENCES

The theory and algorithms implemented in SCOPT can be found in the following papers:

1. Q. Tran-Dinh, A. Kyrillidis, and V. Cevher (2013). A proximal Newton framework for composite minimization: Graph learning without Cholesky decomposition and matrix inversions. Proceedings of the 30th International Conference on Machine Learning (ICML), JMLR W&CP 28(2): 271--279, 2013.
Available at: https://arxiv.org/abs/1301.1459.
Abstract: We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function, endowed with an easily computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption on the smooth part. An important highlight of our work is a new set of analytic step-size selection and correction procedures based on the structure of the problem. We describe concrete algorithmic instances of our framework for several interesting applications and demonstrate them numerically on both synthetic and real data.

2. Q. Tran-Dinh, A. Kyrillidis and V. Cevher. Composite Self-concordant Minimization, Journal of Machine Learning Research, 2014 (to appear).
Available at: arxiv.org/abs/1308.2867
Abstract: We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function, endowed with an easily computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption on the smooth part. An important highlight of our work is a new set of analytic step-size selection and correction procedures based on the structure of the problem. We describe concrete algorithmic instances of our framework for several interesting applications and demonstrate them numerically on both synthetic and real data.

3. Q. Tran-Dinh, A. Kyrillidis and V. Cevher. An inexact proximal path-following algorithm for constrained convex minimization, SIAM J. Optimization, vol. 24, no. 4, pp. 1718--1745, 2014.
Available at: http://arxiv.org/abs/ 1311.1756.
Abstract: Many scientific and engineering applications feature nonsmooth convex minimization problems over convex sets. In this paper, we address an important instance of this broad class where we assume that the nonsmooth objective is equipped with a tractable proximity operator and that the convex constraint set affords a self-concordant barrier. We provide a new joint treatment of proximal and self-concordant barrier concepts and illustrate that such problems can be efficiently solved, without the need of lifting the problem dimensions, as in disciplined convex optimization approach. We propose an inexact path-following algorithmic framework and theoretically characterize the worst-case analytical complexity of this framework when the proximal subproblems are solved inexactly. To show the merits of our framework, we apply its instances to both synthetic and real-world applications, where it shows advantages over standard interior point methods. As a by-product, we describe how our framework can obtain points on the Pareto frontier of regularized problems with self-concordant objectives in a tuning free fashion.

### BMISolver: Optimization involving Bilinear Matrix Inequalities

• BMIsolver-v1.1 - a MATLAB software package for BMI optimization
BMIsolver is a MATLAB package for solving optimization problems with BMI (bilinear matrix inequality) constraints arising in feedback controller design (spectral abscissa, H2, H-infinity and mixed H2/Hinf optimization problems). It is implemented based on the methods proposed in [1] and [2] which is a joint work with S. Gummusoy and W. Michiels. The code is developed by Quoc Tran Dinh at Department of Electrical Engineering (ESAT/SCD) and Optimization in Engineering Center (OPTEC), KU Leuven, Belgium (under the supervision of Prof. M. Diehl). The current version requires YALMIP as a modeling language. We recommend SeDuMi as an SDP solver.

References
1. Q. Tran Dinh, S. Gumussoy, W. Michiels and M. Diehl. Combining Convex-Concave Decompositions and Linearization Approaches for solving BMIs, with Application to Static Output Feedback. Tech. Report, July, 2011, [pdf].
2. Q. Tran Dinh, W. Michiels and M. Diehl. An inner convex approximation algorithm for BMI optimization and applications in control. Tech. Report, December, 2011, [pdf].

## Software and codes that I use for my research

I have been using many open source as well as commercial software packages for my research. I find that they are useful and valuable. Here are some of them.

• IPOPT - Sparse Nonlinear Optimization (open-source).

• CasADI - A minimalistic computer algebra system with automatic differentiation (open-source), Joel Andersson (for C++ and Python).

• qpOASES - Parametric Quadratic Programming for MPC (open-source), Hans Joachim Ferreau.

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