An Inexact Augmented Lagrangian Method for Second-order Cone Programming with Applications

In this paper, we adopt the augmented Lagrangian method (ALM) to solve convex quadratic second-order cone programming problems (SOCPs). Fruitful results on the efficiency of the ALM have been established in the literature. Recently, it has been shown in [Cui, Sun, and Toh, {\em Math. Program.}, 178 (2019), pp. 381--415] that if the quadratic growth condition holds at an optimal solution for the dual problem, then the KKT residual converges to zero R-superlinearly when the ALM is applied to the primal problem. Moreover, Cui, Ding, and Zhao [{\em SIAM J. Optim.}, 27 (2017), pp. 2332-2355] provided sufficient conditions for the quadratic growth condition to hold under the metric subregularity and bounded linear regularity conditions for solving composite matrix optimization problems involving spectral functions. Here, we adopt these recent ideas to analyze the convergence properties of the ALM when applied to SOCPs. To the best of our knowledge, no similar work has been done for SOCPs so far. In our paper, we first provide sufficient conditions to ensure the quadratic growth condition for SOCPs. With these elegant theoretical guarantees, we then design an SOCP solver and apply it to solve various classes of SOCPs, such as minimal enclosing ball problems, classical trust-region subproblems, square-root Lasso problems, and DIMACS Challenge problems. Numerical results show that the proposed ALM based solver is efficient and robust compared to the existing highly developed solvers, such as Mosek and SDPT3.Comment: 25 pages, 0 figur

Strong Variational Sufficiency for Nonlinear Semidefinite Programming and its Implications

Strong variational sufficiency is a newly proposed property, which turns out to be of great use in the convergence analysis of multiplier methods. However, what this property implies for non-polyhedral problems remains a puzzle. In this paper, we prove the equivalence between the strong variational sufficiency and the strong second order sufficient condition (SOSC) for nonlinear semidefinite programming (NLSDP), without requiring the uniqueness of multiplier or any other constraint qualifications. Based on this characterization, the local convergence property of the augmented Lagrangian method (ALM) for NLSDP can be established under strong SOSC in the absence of constraint qualifications. Moreover, under the strong SOSC, we can apply the semi-smooth Newton method to solve the ALM subproblems of NLSDP as the positive definiteness of the generalized Hessian of augmented Lagrangian function is satisfied.Comment: 23 page

MARS: A second-order reduction algorithm for high-dimensional sparse precision matrices estimation

Estimation of the precision matrix (or inverse covariance matrix) is of great importance in statistical data analysis. However, as the number of parameters scales quadratically with the dimension p, computation becomes very challenging when p is large. In this paper, we propose an adaptive sieving reduction algorithm to generate a solution path for the estimation of precision matrices under the $\ell_1$ penalized D-trace loss, with each subproblem being solved by a second-order algorithm. In each iteration of our algorithm, we are able to greatly reduce the number of variables in the problem based on the Karush-Kuhn-Tucker (KKT) conditions and the sparse structure of the estimated precision matrix in the previous iteration. As a result, our algorithm is capable of handling datasets with very high dimensions that may go beyond the capacity of the existing methods. Moreover, for the sub-problem in each iteration, other than solving the primal problem directly, we develop a semismooth Newton augmented Lagrangian algorithm with global linear convergence on the dual problem to improve the efficiency. Theoretical properties of our proposed algorithm have been established. In particular, we show that the convergence rate of our algorithm is asymptotically superlinear. The high efficiency and promising performance of our algorithm are illustrated via extensive simulation studies and real data applications, with comparison to several state-of-the-art solvers

Méthodes sans factorisation pour l’optimisation non linéaire

RÉSUMÉ : Cette thèse a pour objectif de formuler mathématiquement, d'analyser et d'implémenter deux méthodes sans factorisation pour l'optimisation non linéaire. Dans les problèmes de grande taille, la jacobienne des contraintes n'est souvent pas disponible sous forme de matrice; seules son action et celle de sa transposée sur un vecteur le sont. L'optimisation sans factorisation consiste alors à utiliser des opérateurs linéaires abstraits représentant la jacobienne ou le hessien. De ce fait, seules les actions > sont autorisées et l'algèbre linéaire directe doit être remplacée par des méthodes itératives. Outre ces restrictions, une grande difficulté lors de l'introduction de méthodes sans factorisation dans des algorithmes d'optimisation concerne le contrôle de l'inexactitude de la résolution des systèmes linéaires. Il faut en effet s'assurer que la direction calculée est suffisamment précise pour garantir la convergence de l'algorithme concerné. En premier lieu, nous décrivons l'implémentation sans factorisation d'une méthode de lagrangien augmenté pouvant utiliser des approximations quasi-Newton des dérivées secondes. Nous montrons aussi que notre approche parvient à résoudre des problèmes d'optimisation de structure avec des milliers de variables et contraintes alors que les méthodes avec factorisation échouent. Afin d'obtenir une méthode possédant une convergence plus rapide, nous présentons ensuite un algorithme qui utilise un lagrangien augmenté proximal comme fonction de mérite et qui, asymptotiquement, se transforme en une méthode de programmation quadratique séquentielle stabilisée. L'utilisation d'approximations BFGS à mémoire limitée du hessien du lagrangien conduit à l'obtention de systèmes linéaires symétriques quasi-définis. Ceux-ci sont interprétés comme étant les conditions d'optimalité d'un problème aux moindres carrés linéaire, qui est résolu de manière inexacte par une méthode de Krylov. L'inexactitude de cette résolution est contrôlée par un critère d'arrêt facile à mettre en œuvre. Des tests numériques démontrent l'efficacité et la robustesse de notre méthode, qui se compare très favorablement à IPOPT, en particulier pour les problèmes dégénérés pour lesquels la LICQ n'est pas respectée à la solution ou lors de la minimisation. Finalement, l'écosystème de développement d'algorithmes d'optimisation en Python, baptisé NLP.py, est exposé. Cet environnement s'adresse aussi bien aux chercheurs en optimisation qu'aux étudiants désireux de découvrir ou d'approfondir l'optimisation. NLP.py donne accès à un ensemble de blocs constituant les éléments les plus importants des méthodes d'optimisation continue. Grâce à ceux-ci, le chercheur est en mesure d'implémenter son algorithme en se concentrant sur la logique de celui-ci plutôt que sur les subtilités techniques de son implémentation.----------ABSTRACT : This thesis focuses on the mathematical formulation, analysis and implementation of two factorization-free methods for nonlinear constrained optimization. In large-scale optimization, the Jacobian of the constraints may not be available in matrix form; only its action and that of its transpose on a vector are. Factorization-free optimization employs abstract linear operators representing the Jacobian or Hessian matrices. Therefore, only operator-vector products are allowed and direct linear algebra is replaced by iterative methods. Besides these implementation restrictions, a difficulty inherent to methods without factorization in optimization algorithms is the control of the inaccuracy in linear system solves. Indeed, we have to guarantee that the direction calculated is sufficiently accurate to ensure convergence. We first describe a factorization-free implementation of a classical augmented Lagrangian method that may use quasi-Newton second derivatives approximations. This method is applied to problems with thousands of variables and constraints coming from aircraft structural design optimization, for which methods based on factorizations fail. Results show that it is a viable approach for these problems. In order to obtain a method with a faster convergence rate, we present an algorithm that uses a proximal augmented Lagrangian as merit function and that asymptotically turns in a stabilized sequential quadratic programming method. The use of limited-memory BFGS approximations of the Hessian of the Lagrangian combined with regularization of the constraints leads to symmetric quasi-definite linear systems. Because such systems may be interpreted as the KKT conditions of linear least-squares problems, they can be efficiently solved using an appropriate Krylov method. Inaccuracy of their solutions is controlled by a stopping criterion which is easy to implement. Numerical tests demonstrate the effectiveness and robustness of our method, which compares very favorably with IPOPT, especially for degenerate problems for which LICQ is not satisfied at the optimal solution or during the minimization process. Finally, an ecosystem for optimization algorithm development in Python, code-named NLP.py, is exposed. This environment is aimed at researchers in optimization and students eager to discover or strengthen their knowledge in optimization. NLP.py provides access to a set of building blocks constituting the most important elements of continuous optimization methods. With these blocks, users are able to implement their own algorithm focusing on the logic of the algorithm rather than on the technicalities of its implementation

Regularized interior point methods for convex programming

Interior point methods (IPMs) constitute one of the most important classes of optimization methods, due to their unparalleled robustness, as well as their generality. It is well known that a very large class of convex optimization problems can be solved by means of IPMs, in a polynomial number of iterations. As a result, IPMs are being used to solve problems arising in a plethora of fields, ranging from physics, engineering, and mathematics, to the social sciences, to name just a few. Nevertheless, there remain certain numerical issues that have not yet been addressed. More specifically, the main drawback of IPMs is that the linear algebra task involved is inherently ill-conditioned. At every iteration of the method, one has to solve a (possibly large-scale) linear system of equations (also known as the Newton system), the conditioning of which deteriorates as the IPM converges to an optimal solution. If these linear systems are of very large dimension, prohibiting the use of direct factorization, then iterative schemes may have to be employed. Such schemes are significantly affected by the inherent ill-conditioning within IPMs. One common approach for improving the aforementioned numerical issues, is to employ regularized IPM variants. Such methods tend to be more robust and numerically stable in practice. Over the last two decades, the theory behind regularization has been significantly advanced. In particular, it is well known that regularized IPM variants can be interpreted as hybrid approaches combining IPMs with the proximal point method. However, it remained unknown whether regularized IPMs retain the polynomial complexity of their non-regularized counterparts. Furthermore, the very important issue of tuning the regularization parameters appropriately, which is also crucial in augmented Lagrangian methods, was not addressed. In this thesis, we focus on addressing the previous open questions, as well as on creating robust implementations that solve various convex optimization problems. We discuss in detail the effect of regularization, and derive two different regularization strategies; one based on the proximal method of multipliers, and another one based on a Bregman proximal point method. The latter tends to be more efficient, while the former is more robust and has better convergence guarantees. In addition, we discuss the use of iterative linear algebra within the presented algorithms, by proposing some general purpose preconditioning strategies (used to accelerate the iterative schemes) that take advantage of the regularized nature of the systems being solved. In Chapter 2 we present a dynamic non-diagonal regularization for IPMs. The non-diagonal aspect of this regularization is implicit, since all the off-diagonal elements of the regularization matrices are cancelled out by those elements present in the Newton system, which do not contribute important information in the computation of the Newton direction. Such a regularization, which can be interpreted as the application of a Bregman proximal point method, has multiple goals. The obvious one is to improve the spectral properties of the Newton system solved at each IPM iteration. On the other hand, the regularization matrices introduce sparsity to the aforementioned linear system, allowing for more efficient factorizations. We propose a rule for tuning the regularization dynamically based on the properties of the problem, such that sufficiently large eigenvalues of the non-regularized system are perturbed insignificantly. This alleviates the need of finding specific regularization values through experimentation, which is the most common approach in the literature. We provide perturbation bounds for the eigenvalues of the non-regularized system matrix, and then discuss the spectral properties of the regularized matrix. Finally, we demonstrate the efficiency of the method applied to solve standard small- and medium-scale linear and convex quadratic programming test problems. In Chapter 3 we combine an IPM with the proximal method of multipliers (PMM). The resulting algorithm (IP-PMM) is interpreted as a primal-dual regularized IPM, suitable for solving linearly constrained convex quadratic programming problems. We apply few iterations of the interior point method to each sub-problem of the proximal method of multipliers. Once a satisfactory solution of the PMM sub-problem is found, we update the PMM parameters, form a new IPM neighbourhood, and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under standard assumptions. To our knowledge, this is the first polynomial complexity result for a primal-dual regularized IPM. The algorithm is guided by the use of a single penalty parameter; that of the logarithmic barrier. In other words, we show that IP-PMM inherits the polynomial complexity of IPMs, as well as the strong convexity of the PMM sub-problems. The updates of the penalty parameter are controlled by IPM, and hence are well-tuned, and do not depend on the problem solved. Furthermore, we study the behavior of the method when it is applied to an infeasible problem, and identify a necessary condition for infeasibility. The latter is used to construct an infeasibility detection mechanism. Subsequently, we provide a robust implementation of the presented algorithm and test it over a set of small to large scale linear and convex quadratic programming problems, demonstrating the benefits of using regularization in IPMs as well as the reliability of the approach. In Chapter 4 we extend IP-PMM to the case of linear semi-definite programming (SDP) problems. In particular, we prove polynomial complexity of the algorithm, under mild assumptions, and without requiring exact computations for the Newton directions. We furthermore provide a necessary condition for lack of strong duality, which can be used as a basis for constructing detection mechanisms for identifying pathological cases within IP-PMM. In Chapter 5 we present general-purpose preconditioners for regularized Newton systems arising within regularized interior point methods. We discuss positive definite preconditioners, suitable for iterative schemes like the conjugate gradient (CG), or the minimal residual (MINRES) method. We study the spectral properties of the preconditioned systems, and discuss the use of each presented approach, depending on the properties of the problem under consideration. All preconditioning strategies are numerically tested on various medium- to large-scale problems coming from standard test sets, as well as problems arising from partial differential equation (PDE) optimization. In Chapter 6 we apply specialized regularized IPM variants to problems arising from portfolio optimization, machine learning, image processing, and statistics. Such problems are usually solved by specialized first-order approaches. The efficiency of the proposed regularized IPM variants is confirmed by comparing them against problem-specific state--of--the--art first-order alternatives given in the literature. Finally, in Chapter 7 we present some conclusions as well as open questions, and possible future research directions