Convergence analysis of an Inexact Infeasible Interior Point method for Semidefinite Programming

In this paper we present an extension to SDP of the well known infeasible Interior Point method for linear programming of Kojima,Megiddo and Mizuno (A primal-dual infeasible-interior-point algorithm for Linear Programming, Math. Progr., 1993). The extension developed here allows the use of inexact search directions; i.e., the linear systems defining the search directions can be solved with an accuracy that increases as the solution is approached. A convergence analysis is carried out and the global convergence of the method is prove

A polynomial-time interior-point method for conic optimization, with inexact barrier evaluations

We consider a primal-dual short-step interior-point method for conic convex optimization problems for which exact evaluation of the gradient and Hessian of the primal and dual barrier functions is either impossible or prohibitively expensive. As our main contribution, we show that if approximate gradients and Hessians of the primal barrier function can be computed, and the relative errors in such quantities are not too large, then the method has polynomial worst-case iteration complexity. (In particular, polynomial iteration complexity ensues when the gradient and Hessian are evaluated exactly.) In addition, the algorithm requires no evaluation---or even approximate evaluation---of quantities related to the barrier function for the dual cone, even for problems in which the underlying cone is not self-dual

A polynomial-time inexact interior-point method for convex quadratic symmetric cone programming

Abstract. In this paper, we design an inexact primal-dual infeasible path-following algorithm for convex quadratic programming over symmetric cones. Our algorithm and its polynomial iteration complexity analysis give a unified treatment for a number of previous algorithms and their complexity analysis. In particular, our algorithm and analysis includes the one designed for linear semidefinite programming in "Math. Prog. 99 (2004), pp. 261-282". Under a mild condition on the inexactness of the search direction at each interior-point iteration, we show that the algorithm can find an ϵ-approximate solution in O(n 2 log(1/ϵ)) iterations, where n is the rank of the underlying Euclidean Jordan algebra

An interior point-proximal method of multipliers for linear positive semi-definite programming

In this paper we generalize the Interior Point-Proximal Method of Multipliers (IP-PMM) presented in Pougkakiotis and Gondzio (Comput Optim Appl 78:307–351, 2021. https://doi.org/10.1007/s10589-020-00240-9) for the solution of linear positive Semi-Definite Programming (SDP) problems, allowing inexactness in the solution of the associated Newton systems. In particular, we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM) and interpret the algorithm (IP-PMM) as a primal-dual regularized IPM, suitable for solving SDP problems. We apply some iterations of an IPM to each sub-problem of the PMM until a satisfactory solution is found. We then update the PMM parameters, form a new IPM neighbourhood, and repeat this process. Given this framework, 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.</p

A distributed primal-dual interior-point method for loosely coupled problems using ADMM

In this paper we propose an efficient distributed algorithm for solving loosely coupled convex optimization problems. The algorithm is based on a primal-dual interior-point method in which we use the alternating direction method of multipliers (ADMM) to compute the primal-dual directions at each iteration of the method. This enables us to join the exceptional convergence properties of primal-dual interior-point methods with the remarkable parallelizability of ADMM. The resulting algorithm has superior computational properties with respect to ADMM directly applied to our problem. The amount of computations that needs to be conducted by each computing agent is far less. In particular, the updates for all variables can be expressed in closed form, irrespective of the type of optimization problem. The most expensive computational burden of the algorithm occur in the updates of the primal variables and can be precomputed in each iteration of the interior-point method. We verify and compare our method to ADMM in numerical experiments.Comment: extended version, 50 pages, 9 figure

Inexact Interior-Point Methods for Large Scale Linear and Convex Quadratic Semidefinite Programming

Ph.DDOCTOR OF PHILOSOPH

Quantum Interior Point Methods for Semidefinite Optimization

We present two quantum interior point methods for semidefinite optimization problems, building on recent advances in quantum linear system algorithms. The first scheme, more similar to a classical solution algorithm, computes an inexact search direction and is not guaranteed to explore only feasible points; the second scheme uses a nullspace representation of the Newton linear system to ensure feasibility even with inexact search directions. The second is a novel scheme that might seem impractical in the classical world, but it is well-suited for a hybrid quantum-classical setting. We show that both schemes converge to an optimal solution of the semidefinite optimization problem under standard assumptions. By comparing the theoretical performance of classical and quantum interior point methods with respect to various input parameters, we show that our second scheme obtains a speedup over classical algorithms in terms of the dimension of the problem $n$, but has worse dependence on other numerical parameters

On inexact Newton directions in interior point methods for linear optimization

In each iteration of the interior point method (IPM) at least one linear system has to be solved. The main computational effort of IPMs consists in the computation of these linear systems. Solving the corresponding linear systems with a direct method becomes very expensive for large scale problems. In this thesis, we have been concerned with using an iterative method for solving the reduced KKT systems arising in IPMs for linear programming. The augmented system form of this linear system has a number of advantages, notably a higher degree of sparsity than the normal equations form. We design a block triangular preconditioner for this system which is constructed by using a nonsingular basis matrix identified from an estimate of the optimal partition in the linear program. We use the preconditioned conjugate gradients (PCG) method to solve the augmented system. Although the augmented system is indefinite, short recurrence iterative methods such as PCG can be applied to indefinite system in certain situations. This approach has been implemented within the HOPDM interior point solver. The KKT system is solved approximately. Therefore, it becomes necessary to study the convergence of IPM for this inexact case. We present the convergence analysis of the inexact infeasible path-following algorithm, prove the global convergence of this method and provide complexity analysis

Conception pas à pas d'un solveur par points intérieurs en optimisation conique auto-duale, avec applications

MasterThese notes present a project in numerical optimization dealing with the implementation of an interior-point method for solving a self-dual conic optimization (SDCO) problem. The cone is the Cartesian product of cones of positive semidefinite matrices of various dimensions (imposing to matrices to be positive semidefinite) and of a positive orthant. Therefore, the solved problem encompasses semidefinite and linear optimization.The project was given in a course entitled 'Advanced Continuous Optimization II' at the University Paris-Saclay, in 2016-2020. The solver is designed step by step during a series of 5 sessions of 4 hours each. Each session corresponds to a chapter of these notes (or a part of it). The correctness of the SDCO solver is verified during each session on small academic problems, having diverse properties. During the last session, the developed piece of software is used to minimize a univariate polynomial on an interval and to solve a few small size rank relaxations of QCQO (quadratically constrained quadratic optimization) problems, modeling various instances of the OPF (optimal power flow) problem. The student has to master not ony the implementation of the interior-point solver, but is also asked to understand the underliying theory by solving exercises consisting in proving some properties of the implemented algorithms.The goal of the project is not to design an SDCO solver that would beat the best existing solver but to help the students to understand and demystify what there is inside such a piece of software. As a side outcome, this course also shows that a rather performent SDCO solver can be realized in a relatively short time.Ces notes présentent un projet d'optimisation numérique dans lequel on implémente une méthode de points intérieurs pour résoudre un problème d'optimisation conique auto-duale (OCAD). Le cône est le produit cartésien de cônes de matrices semi-définies positives de dimensions variables et d'un orthant positif. Dès lors, le problème contient l'optimisation semi-défiinie et l'optimisation linéaire.Ce projet a été proposé dans un cours intitulé 'Advanced Continuous Optimization II' à l'université Paris-Saclay, en 2016-1020. Le solveur est conçu pas à pas durant une suite de 5 leçons de 4 heures chacune. Chaque session fait l'objet d'un chapitre de ces notes. La bonne marche du solveur OCAD est vérifiée à chaque session sur de petits problèmes académiques, ayant diverses propriétés. Durant la dernière session, le code développé est utilisé pour minimiser un polynôme d'une variable sur un intervalle et pour résoudre la relaxation de rang de la formulation QCQP (quadratically constrained quadratic programming) de quelques problèmes d'optimisation de flux d'énergie (OPF) dans de petits réseaux de distribution d'électricité. L'étudiant doit maîtriser non seulement l'implémentation du solveur de points-intérieurs, mais aussi la théorie sous-jacente de manière à pouvoir résoudre des exercices qui consistent à démontrer des propriétés des algorithmes implémentés.Le but de ce cours n'est pas de concevoir un solveur OCAD qui surpasserait le meilleur solveur existant, mais d'aider l'étudiant à comprendre et à démythifier ce que contient un tel solveur. Une conséquence secondaire de cet exercice est de montrer qu'un code OCAD assez performant peut être réalisé en très peu de temps