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

Nonsmooth and derivative-free optimization based hybrid methods and applications

"In this thesis, we develop hybrid methods for solving global and in particular, nonsmooth optimization problems. Hybrid methods are becoming more popular in global optimization since they allow to apply powerful smooth optimization techniques to solve global optimization problems. Such methods are able to efficiently solve global optimization problems with large number of variables. To date global search algorithms have been mainly applied to improve global search properties of the local search methods (including smooth optimization algorithms). In this thesis we apply rather different strategy to design hybrid methods. We use local search algorithms to improve the efficiency of global search methods. The thesis consists of two parts. In the first part we describe hybrid algorithms and in the second part we consider their various applications." -- taken from Abstract.Operational Research and Cybernetic

Inference, Computation, and Games

In this thesis, we use statistical inference and competitive games to design algorithms for computational mathematics. In the first part, comprising chapters two through six, we use ideas from Gaussian process statistics to obtain fast solvers for differential and integral equations. We begin by observing the equivalence of conditional (near-)independence of Gaussian processes and the (near-)sparsity of the Cholesky factors of its precision and covariance matrices. This implies the existence of a large class of dense matrices with almost sparse Cholesky factors, thereby greatly increasing the scope of application of sparse Cholesky factorization. Using an elimination ordering and sparsity pattern motivated by the screening effect in spatial statistics, we can compute approximate Cholesky factors of the covariance matrices of Gaussian processes admitting a screening effect in near-linear computational complexity. These include many popular smoothness priors such as the Matérn class of covariance functions. In the special case of Green's matrices of elliptic boundary value problems (with possibly unknown elliptic operators of arbitrarily high order, with possibly rough coefficients), we can use tools from numerical homogenization to prove the exponential accuracy of our method. This result improves the state-of-the-art for solving general elliptic integral equations and provides the first proof of an exponential screening effect. We also derive a fast solver for elliptic partial differential equations, with accuracy-vs-complexity guarantees that improve upon the state-of-the-art. Furthermore, the resulting solver is performant in practice, frequently beating established algebraic multigrid libraries such as AMGCL and Trilinos on a series of challenging problems in two and three dimensions. Finally, for any given covariance matrix, we obtain a closed-form expression for its optimal (in terms of Kullback-Leibler divergence) approximate inverse-Cholesky factorization subject to a sparsity constraint, recovering Vecchia approximation and factorized sparse approximate inverses. Our method is highly robust, embarrassingly parallel, and further improves our asymptotic results on the solution of elliptic integral equations. We also provide a way to apply our techniques to sums of independent Gaussian processes, resolving a major limitation of existing methods based on the screening effect. As a result, we obtain fast algorithms for large-scale Gaussian process regression problems with possibly noisy measurements. In the second part of this thesis, comprising chapters seven through nine, we study continuous optimization through the lens of competitive games. In particular, we consider competitive optimization, where multiple agents attempt to minimize conflicting objectives. In the single-agent case, the updates of gradient descent are minimizers of quadratically regularized linearizations of the loss function. We propose to generalize this idea by using the Nash equilibria of quadratically regularized linearizations of the competitive game as updates (linearize the game). We provide fundamental reasons why the natural notion of linearization for competitive optimization problems is given by the multilinear (as opposed to linear) approximation of the agents' loss functions. The resulting algorithm, which we call competitive gradient descent, thus provides a natural generalization of gradient descent to competitive optimization. By using ideas from information geometry, we extend CGD to competitive mirror descent (CMD) that can be applied to a vast range of constrained competitive optimization problems. CGD and CMD resolve the cycling problem of simultaneous gradient descent and show promising results on problems arising in constrained optimization, robust control theory, and generative adversarial networks. Finally, we point out the GAN-dilemma that refutes the common interpretation of GANs as approximate minimizers of a divergence obtained in the limit of a fully trained discriminator. Instead, we argue that GAN performance relies on the implicit competitive regularization (ICR) due to the simultaneous optimization of generator and discriminator and support this hypothesis with results on low-dimensional model problems and GANs on CIFAR10.</p

User-Oriented Methodology and Techniques of Decision Analysis and Support

This volume contains 26 papers selected from Workshop presentations. The book is divided into two sections; the first is devoted to the methodology of decision analysis and support and related theoretical developments, and the second reports on the development of tools -- algorithms, software packages -- for decision support as well as on their applications. Several major contributions on constructing user interfaces, on organizing intelligent DSS, on modifying theory and tools in response to user needs -- are included in this volume

Energy Management Systems and Potential Applications of Quantum Computing in the Energy Sector

The combined use of technologies plays a key role in the energy transition towards a green and sustainable economy, driven by the European Green Deal initiatives and the Paris Agreement to achieve climate neutrality in the European Union (EU) by 2050. Indeed, all viable solutions with no barriers to innovation should be considered if a fair, cost-effective, competitive, and green transition is to be ensured.Energy hubs enable the synergy of different forms of energy by exploiting their specific vir-tues. However, their management in an integrated context must be entrusted to automated manage-ment systems capable of making real-time decisions.This PhD thesis aims to assess the main potential applications of quantum computing to the energy sector in the current development scenario of quantum technologies, as well as provide the elements for modelling an energy hub and managing uncertainties.The thesis is organized as follows. Chapter 1 provides an introduction to energy manage-ment systems. The concept of an energy hub and its mathematical modelling are introduced in chap-ter 2. Chapter 3 introduces the fundamentals of energy supply. Chapter 4 examines potential use cases for quantum computing in the energy sector. Chapter 5 addresses the modelling of uncertain parameters. Chapter 6 concludes the thesis with a case study of two urban districts modelled as mul-ticarrier energy hubs connected by a multicarrier energy infrastructure providing electricity, gas and hydrogen. The conclusions are drawn in chapter 7. The appendices with additional insights enrich the thesis, which is full of comments and bibliographical references