Primal-Dual Algorithms for Semidefinit Optimization Problems based on generalized trigonometric barrier function

Recently, M. Bouafoa, et al. (Journal of optimization Theory and Applications, August, 2016), investigated a new kernel function which differs from the self-regular kernel functions. The kernel function has a trigonometric Barrier Term. In this paper we generalize the analysis presented in the above paper for Semidefinit Optimization Problems (SDO). It is shown that the interior-point methods based on this function for large-update methods, the iteration bound is improved significantly. For small-update interior point methods the iteration bound is the best currently known bound for primal-dual interior point methods. The analysis for SDO deviates significantly from the analysis for linear optimization. Several new tools and techniques are derived in this paper.publishedVersio

A second order cone formulation of continuous CTA model

The final publication is available at link.springer.comIn this paper we consider a minimum distance Controlled Tabular Adjustment (CTA) model for statistical disclosure limitation (control) of tabular data. The goal of the CTA model is to find the closest safe table to some original tabular data set that contains sensitive information. The measure of closeness is usually measured using l1 or l2 norm; with each measure having its advantages and disadvantages. Recently, in [4] a regularization of the l1 -CTA using Pseudo-Huber func- tion was introduced in an attempt to combine positive characteristics of both l1 -CTA and l2 -CTA. All three models can be solved using appro- priate versions of Interior-Point Methods (IPM). It is known that IPM in general works better on well structured problems such as conic op- timization problems, thus, reformulation of these CTA models as conic optimization problem may be advantageous. We present reformulation of Pseudo-Huber-CTA, and l1 -CTA as Second-Order Cone (SOC) op- timization problems and test the validity of the approach on the small example of two-dimensional tabular data set.Peer ReviewedPostprint (author's final draft

OSQP: An Operator Splitting Solver for Quadratic Programs

We present a general-purpose solver for convex quadratic programs based on the alternating direction method of multipliers, employing a novel operator splitting technique that requires the solution of a quasi-definite linear system with the same coefficient matrix at almost every iteration. Our algorithm is very robust, placing no requirements on the problem data such as positive definiteness of the objective function or linear independence of the constraint functions. It can be configured to be division-free once an initial matrix factorization is carried out, making it suitable for real-time applications in embedded systems. In addition, our technique is the first operator splitting method for quadratic programs able to reliably detect primal and dual infeasible problems from the algorithm iterates. The method also supports factorization caching and warm starting, making it particularly efficient when solving parametrized problems arising in finance, control, and machine learning. Our open-source C implementation OSQP has a small footprint, is library-free, and has been extensively tested on many problem instances from a wide variety of application areas. It is typically ten times faster than competing interior-point methods, and sometimes much more when factorization caching or warm start is used. OSQP has already shown a large impact with tens of thousands of users both in academia and in large corporations

Interior-point methods for P∗(κ)-linear complementarity problem based on generalized trigonometric barrier function

Recently, M.~Bouafoa, et al. investigated a new kernel function which differs from the self-regular kernel functions. The kernel function has a trigonometric Barrier Term. In this paper we generalize the analysis presented in the above paper for $P_{*}(\kappa)$ Linear Complementarity Problems (LCPs). It is shown that the iteration bound for primal-dual large-update and small-update interior-point methods based on this function is as good as the currently best known iteration bounds for these type methods. The analysis for LCPs deviates significantly from the analysis for linear optimization. Several new tools and techniques are derived in this paper.publishedVersio

Conic Optimization: Optimal Partition, Parametric, and Stability Analysis

A linear conic optimization problem consists of the minimization of a linear objective function over the intersection of an affine space and a closed convex cone. In recent years, linear conic optimization has received significant attention, partly due to the fact that we can take advantage of linear conic optimization to reformulate and approximate intractable optimization problems. Steady advances in computational optimization have enabled us to approximately solve a wide variety of linear conic optimization problems in polynomial time. Nevertheless, preprocessing methods, rounding procedures and sensitivity analysis tools are still the missing parts of conic optimization solvers. Given the output of a conic optimization solver, we need methodologies to generate approximate complementary solutions or to speed up the convergence to an exact optimal solution. A preprocessing method reduces the size of a problem by finding the minimal face of the cone which contains the set of feasible solutions. However, such a preprocessing method assumes the knowledge of an exact solution. More importantly, we need robust sensitivity and post-optimal analysis tools for an optimal solution of a linear conic optimization problem. Motivated by the vital importance of linear conic optimization, we take active steps to fill this gap.This thesis is concerned with several aspects of a linear conic optimization problem, from algorithm through solution identification, to parametric analysis, which have not been fully addressed in the literature. We specifically focus on three special classes of linear conic optimization problems, namely semidefinite and second-order conic optimization, and their common generalization, symmetric conic optimization. We propose a polynomial time algorithm for symmetric conic optimization problems. We show how to approximate/identify the optimal partition of semidefinite optimization and second-order conic optimization, a concept which has its origin in linear optimization. Further, we use the optimal partition information to either generate an approximate optimal solution or to speed up the convergence of a solution identification process to the unique optimal solution of the problem. Finally, we study the parametric analysis of semidefinite and second-order conic optimization problems. We investigate the behavior of the optimal partition and the optimal set mapping under perturbation of the objective function vector