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

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

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

Ph.DDOCTOR OF PHILOSOPH

Polynomiality of Primal-Dual Algorithms for Semidefinite Linear Complementarity Problems Based on the Kojima-Shindoh-Hara Family of Directions

Kojima, Shindoh and Hara proposed a family of search directions for the semidefinite linear complementarity problem (SDLCP) and established polynomial convergence of a feasible shortstep path-following algorithm based on a particular direction of their family. The question of whether polynomiality could be established for any direction of their family thus remained an open problem. This paper answers this question in the affirmative by establishing the polynomiality of primal-dual interior-point algorithms for SDLCP based on any direction of the Kojima, Shindoh and Hara family of search directions. We show that the polynomial iterationcomplexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima et al. and Monteiro and Adler, and the predictor-corrector algorithm of Mizuno et al., carry over to the context of SDLCP. keywords: Semidefinite programming, interior-point methods, polynomial complexity, pathfollowing methods,..