98 research outputs found
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
Stat Optim Inf Comput
In this paper, an improved Interior-Point Method (IPM) for solving symmetric optimization problems is presented. Symmetric optimization (SO) problems are linear optimization problems over symmetric cones. In particular, the method can be efficiently applied to an important instance of SO, a Controlled Tabular Adjustment (CTA) problem which is a method used for Statistical Disclosure Limitation (SDL) of tabular data. The presented method is a full Nesterov-Todd step infeasible IPM for SO. The algorithm converges to |-approximate solution from any starting point whether feasible or infeasible. Each iteration consists of the feasibility step and several centering steps, however, the iterates are obtained in the wider neighborhood of the central path in comparison to the similar algorithms of this type which is the main improvement of the method. However, the currently best known iteration bound known for infeasible short-step methods is still achieved.CC999999/ImCDC/Intramural CDC HHSUnited States/2022-01-01T00:00:00Z34141814PMC820532010747vault:3716
Interior-point algorithms for convex optimization based on primal-dual metrics
We propose and analyse primal-dual interior-point algorithms for convex
optimization problems in conic form. The families of algorithms we analyse are
so-called short-step algorithms and they match the current best iteration
complexity bounds for primal-dual symmetric interior-point algorithm of
Nesterov and Todd, for symmetric cone programming problems with given
self-scaled barriers. Our results apply to any self-concordant barrier for any
convex cone. We also prove that certain specializations of our algorithms to
hyperbolic cone programming problems (which lie strictly between symmetric cone
programming and general convex optimization problems in terms of generality)
can take advantage of the favourable special structure of hyperbolic barriers.
We make new connections to Riemannian geometry, integrals over operator spaces,
Gaussian quadrature, and strengthen the connection of our algorithms to
quasi-Newton updates and hence first-order methods in general.Comment: 36 page
Numerical optimization for frictional contact problems
International audienc
Numerical optimization for rolling frictional contact problems
International audienceThe scope of this report is to study the optimal solution of a model for the problem of unilateralcontact between solid bodies with rolling friction, by means of an interior-point algorithm. Themodel is based on the second-order cone programming (also SOCP) but with a conical constraintgiven by the rolling friction cone that is not a standard cone in convex optimization as the Lorentzcone, for instance. The difficulties are that the KKT system related to this model is not square(the number of variables and equations are not equal) and the rolling friction cone is not symmet-ric. That means it is necessary to find out the way that transforms the KKT system into a squareversion, and as well as symmetrizes the conical constraint in order to be able to apply the solvingtechniques for the standard SOCP. Some prerequisites will be firstly known in annex, then one ofthe best possibility is proposed to solve the model in this report, along with difficulties and theirsolutions. In the last section, more than 200 experiments are introduced to apply this proposedapproach and then present the results and observations of exceptional phenomena
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