An inexact interior-point algorithm for conic convex optimization problems

In this dissertation we study an algorithm for convex optimization problems in conic form. (Without loss of generality, any convex problem can be written in conic form.) Our algorithm belongs to the class of interior-point methods (IPMs), which have been associated with many recent theoretical and algorithmic advances in mathematical optimization. In an IPM one solves a family of slowly-varying optimization problems that converge in some sense to the original optimization problem. Each problem in the family depends on a so-called barrier function that is associated with the problem data. Typically IPMs require evaluation of the gradient and Hessian of a suitable (``self-concordant'') barrier function. In some cases such evaluation is expensive; in other cases formulas in closed form for a suitable barrier function and its derivatives are unknown. We show that even if the gradient and Hessian of a suitable barrier function are computed inexactly, the resulting IPM can possess the desirable properties of polynomial iteration complexity and global convergence to the optimal solution set. In practice the best IPMs are primal-dual methods, in which a convex problem is solved together with its dual, which is another convex problem. One downside of existing primal-dual methods is their need for evaluation of a suitable barrier function, or its derivatives, for the dual problem. Such evaluation can be even more difficult than that required for the barrier function associated with the original problem. Our primal-dual IPM does not suffer from this drawback---it does not require exact evaluation, or even estimates, of a suitable barrier function for the dual problem. Given any convex optimization problem, Nesterov and Nemirovski showed that there exists a suitable barrier function, which they called the universal barrier function. Since this function and its derivatives may not be available in closed form, we explain how a Monte Carlo method can be used to estimate the derivatives. We make probabilistic statements regarding the errors in these estimates, and give an upper bound on the minimum Monte Carlo sample size required to ensure that with high probability, our primal-dual IPM possesses polynomial iteration complexity and global convergence

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

Speed limits and locality in many-body quantum dynamics

We review the mathematical speed limits on quantum information processing in many-body systems. After the proof of the Lieb-Robinson Theorem in 1972, the past two decades have seen substantial developments in its application to other questions, such as the simulatability of quantum systems on classical or quantum computers, the generation of entanglement, and even the properties of ground states of gapped systems. Moreover, Lieb-Robinson bounds have been extended in non-trivial ways, to demonstrate speed limits in systems with power-law interactions or interacting bosons, and even to prove notions of locality that arise in cartoon models for quantum gravity with all-to-all interactions. We overview the progress which has occurred, highlight the most promising results and techniques, and discuss some central outstanding questions which remain open. To help bring newcomers to the field up to speed, we provide self-contained proofs of the field's most essential results.Comment: review article. 93 pages, 10 figures, 1 table. v2: minor change

Asymptotic safety in QFT: from quantum gravity to graphene

In this work we investigate properties of scale invariant theories. This kind of theories describe a variety of phenomena, and two particular examples are discussed. On the one hand, more than 100 years after the discovery of General Relativity by Einstein, we still don't know how to unify gravity and quantum mechanics. One possibility is that on very small scales, gravity could be scale invariant, allowing for a finite ultraviolet completion. On the other hand, we will study the phase diagram of graphene and related materials. Scale invariant points in phase diagrams are related to second order phase transitions, and near these universal behaviour is found. To investigate these systems, nonperturbative renormalisation group methods are used. In order to achieve trustworthy results, also technical progress, both analytical and numerical, had to be made. On the analytical side, the Mathematica package xAct is used to derive the equations underlying the scale invariance of the theories. To solve these numerically, pseudo-spectral methods are systematically introduced in the present context for the first time. The results thus obtained support the ultraviolet completion of gravity by a scale invariant point. The dependence on gauge fixing and parametrisation is investigated, and found to be reasonably small. The 2-loop counterterm, being the hallmark of the perturbative nonrenormalisability of gravity, is shown to be irrelevant at the scale invariant point. Finally, the split-Ward identities are partially solved by resolving correlation functions. Regarding graphene and similar materials, different levels of approximation show a very good convergence of results for critical exponents and anomalous dimensions at the phase transition studied. The combined power of both analytical and numerical methods excels particularly - without either, the calculations wouldn't be possible

Numerical Solution of Optimal Control Problems with Explicit and Implicit Switches

This dissertation deals with the efficient numerical solution of switched optimal control problems whose dynamics may coincidentally be affected by both explicit and implicit switches. A framework is being developed for this purpose, in which both problem classes are uniformly converted into a mixed–integer optimal control problem with combinatorial constraints. Recent research results relate this problem class to a continuous optimal control problem with vanishing constraints, which in turn represents a considerable subclass of an optimal control problem with equilibrium constraints. In this thesis, this connection forms the foundation for a numerical treatment. We employ numerical algorithms that are based on a direct collocation approach and require, in particular, a highly accurate determination of the switching structure of the original problem. Due to the fact that the switching structure is a priori unknown in general, our approach aims to identify it successively. During this process, a sequence of nonlinear programs, which are derived by applying discretization schemes to optimal control problems, is solved approximatively. After each iteration, the discretization grid is updated according to the currently estimated switching structure. Besides a precise determination of the switching structure, it is of central importance to estimate the global error that occurs when optimal control problems are solved numerically. Again, we focus on certain direct collocation discretization schemes and analyze error contributions of individual discretization intervals. For this purpose, we exploit a relationship between discrete adjoints and the Lagrange multipliers associated with those nonlinear programs that arise from the collocation transcription process. This relationship can be derived with the help of a functional analytic framework and by interrelating collocation methods and Petrov–Galerkin finite element methods. In analogy to the dual-weighted residual methodology for Galerkin methods, which is well–known in the partial differential equation community, we then derive goal–oriented global error estimators. Based on those error estimators, we present mesh refinement strategies that allow for an equilibration and an efficient reduction of the global error. In doing so we note that the grid adaption processes with respect to both switching structure detection and global error reduction get along with each other. This allows us to distill an iterative solution framework. Usually, individual state and control components have the same polynomial degree if they originate from a collocation discretization scheme. Due to the special role which some control components have in the proposed solution framework it is desirable to allow varying polynomial degrees. This results in implementation problems, which can be solved by means of clever structure exploitation techniques and a suitable permutation of variables and equations. The resulting algorithm was developed in parallel to this work and implemented in a software package. The presented methods are implemented and evaluated on the basis of several benchmark problems. Furthermore, their applicability and efficiency is demonstrated. With regard to a future embedding of the described methods in an online optimal control context and the associated real-time requirements, an extension of the well–known multi–level iteration schemes is proposed. This approach is based on the trapezoidal rule and, compared to a full evaluation of the involved Jacobians, it significantly reduces the computational costs in case of sparse data matrices