26 research outputs found
Nonmonotone globalization for Anderson acceleration via adaptive regularization
Anderson acceleration (AA) is a popular method for accelerating fixed-point iterations, but may suffer from instability and stagnation. We propose a globalization method for AA to improve stability and achieve unified global and local convergence. Unlike existing AA globalization approaches that rely on safeguarding operations and might hinder fast local convergence, we adopt a nonmonotone trust-region framework and introduce an adaptive quadratic regularization together with a tailored acceptance mechanism. We prove global convergence and show that our algorithm attains the same local convergence as AA under appropriate assumptions. The effectiveness of our method is demonstrated in several numerical experiments
Fragments d'Optimisation Différentiable - Théories et Algorithmes
MasterLecture Notes (in French) of optimization courses given at ENSTA (Paris, next Saclay), ENSAE (Paris) and at the universities Paris I, Paris VI and Paris Saclay (979 pages).Syllabus d’enseignements délivrés à l’ENSTA (Paris, puis Saclay), à l’ENSAE (Paris) et aux universités Paris I, Paris VI et Paris Saclay (979 pages)
A trust region-type normal map-based semismooth Newton method for nonsmooth nonconvex composite optimization
We propose a novel trust region method for solving a class of nonsmooth and
nonconvex composite-type optimization problems. The approach embeds inexact
semismooth Newton steps for finding zeros of a normal map-based stationarity
measure for the problem in a trust region framework. Based on a new merit
function and acceptance mechanism, global convergence and transition to fast
local q-superlinear convergence are established under standard conditions. In
addition, we verify that the proposed trust region globalization is compatible
with the Kurdyka-{\L}ojasiewicz (KL) inequality yielding finer convergence
results. We further derive new normal map-based representations of the
associated second-order optimality conditions that have direct connections to
the local assumptions required for fast convergence. Finally, we study the
behavior of our algorithm when the Hessian matrix of the smooth part of the
objective function is approximated by BFGS updates. We successfully link the KL
theory, properties of the BFGS approximations, and a Dennis-Mor{\'e}-type
condition to show superlinear convergence of the quasi-Newton version of our
method. Numerical experiments on sparse logistic regression and image
compression illustrate the efficiency of the proposed algorithm.Comment: 56 page
On a Nonsmooth Gauss–Newton Algorithms for Solving Nonlinear Complementarity Problems
In this paper, we propose a new version of the generalized damped Gauss–Newton method for solving nonlinear complementarity problems based on the transformation to the nonsmooth equation, which is equivalent to some unconstrained optimization problem. The B-differential plays the role of the derivative. We present two types of algorithms (usual and inexact), which have superlinear and global convergence for semismooth cases. These results can be applied to efficiently find all solutions of the nonlinear complementarity problems under some mild assumptions. The results of the numerical tests are attached as a complement of the theoretical considerations
Variational and Time-Distributed Methods for Real-time Model Predictive Control
This dissertation concerns the theoretical, algorithmic, and practical aspects of solving optimal control problems (OCPs) in real-time. The topic is motivated by Model Predictive Control (MPC), a powerful control technique for constrained, nonlinear systems that computes control actions by solving a parameterized OCP at each sampling instant. To successfully implement MPC, these parameterized OCPs need to be solved in real-time. This is a significant challenge for systems with fast dynamics and/or limited onboard computing power and is often the largest barrier to the deployment of MPC controllers. The contributions of this dissertation are as follows.
First, I present a system theoretic analysis of Time-distributed Optimization (TDO) in Model Predictive Control. When implemented using TDO, an MPC controller distributed optimization iterates over time by maintaining a running solution estimate for the optimal control problem and updating it at each sampling instant. The resulting controller can be viewed as a dynamic compensator which is placed in closed-loop with the plant. The resulting coupled plant-optimizer system is analyzed using input-to-state stability concepts and sufficient conditions for stability and constraint satisfaction are derived. When applied to time distributed sequential quadratic programming, the framework significantly extends the existing theoretical analysis for the real-time iteration scheme. Numerical simulations are presented that demonstrate the effectiveness of the scheme.
Second, I present the Proximally Stabilized Fischer-Burmeister (FBstab) algorithm for convex quadratic programming. FBstab is a novel algorithm that synergistically combines the proximal point algorithm with a primal-dual semismooth Newton-type method. FBstab is numerically robust, easy to warmstart, handles degenerate primal-dual solutions, detects infeasibility/unboundedness and requires only that the Hessian matrix be positive semidefinite. The chapter outlines the algorithm, provides convergence and convergence rate proofs, and reports some numerical results from model predictive control benchmarks and from the Maros-Meszaros test set. Overall, FBstab shown to be is competitive with state of the art methods and to be especially promising for model predictive control and other parameterized problems.
Finally, I present an experimental application of some of the approaches from the first two chapters: Emissions oriented supervisory model predictive control (SMPC) of a diesel engine. The control objective is to reduce engine-out cumulative NOx and total hydrocarbon (THC) emissions. This is accomplished using an MPC controller which minimizes deviation from optimal setpoints, subject to combustion quality constraints, by coordinating the fuel input and the EGR rate target provided to an inner-loop airpath controller. The SMPC controller is implemented using TDO and a variant of FBstab which allows us to achieve sub-millisecond controller execution times. We experimentally demonstrate 10-15% cumulative emissions reductions over the Worldwide Harmonized Light Vehicles Test Cycle (WLTC) drivecycle.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155167/1/dliaomcp_1.pd
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book
The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions.
This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
On affine scaling inexact dogleg methods for bound-constrained nonlinear systems
Within the framework of affine scaling trust-region methods for bound constrained problems, we discuss the use of a inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Focusing on bound-constrained systems of nonlinear equations, an inexact affine scaling method for large scale problems, employing the inexact dogleg procedure, is described. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given