Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities

Stochastic variational inequalities (SVI) model a large class of equilibrium problems subject to data uncertainty, and are closely related to stochastic optimization problems. The SVI solution is usually estimated by a solution to a sample average approximation (SAA) problem. This paper considers the normal map formulation of an SVI, and proposes a method to build asymptotically exact confidence regions and confidence intervals for the solution of the normal map formulation, based on the asymptotic distribution of SAA solutions. The confidence regions are single ellipsoids with high probability. We also discuss the computation of simultaneous and individual confidence intervals

Confidence Intervals for Stochastic Variational Inequalities

This dissertation examines the effects of uncertain data on a general class of optimization and equilibrium problems. The common framework used for modeling these problems is a stochastic variational inequality. Variational inequalities can be used to model conditions that characterize an equilibrium state, or describe necessary conditions for solutions to constrained optimization problems. For example, Cournot-Nash equilibrium problems and the Karush-Kuhn-Tucker conditions for nonlinear programming problems both fit in the framework of a variational inequality. Uncertain model data can be incorporated into a variational inequality through the use of an expectation function. A variational inequality defined in this manner is referred to as a stochastic variational inequality (SVI). For many problems of interest the SVI cannot be solved directly. This can be due to limited distributional information or an expectation function that lacks a closed form expression and is difficult to evaluate. When this is the case, the SVI must be replaced with a suitable approximation. A common approach is to solve a sample average approximation (SAA). The SAA problem is a variational inequality with the expectation function replaced by a function that depends on a sample of the uncertain data. A natural question is then how the solution of the SAA problem compares to the true solution of the SVI. To address this question, this dissertation examines the construction of simultaneous and individual confidence intervals for the true solution of an SVI.Doctor of Philosoph

Symmetric Confidence Regions and Confidence Intervals for Normal Map Formulations of Stochastic Variational Inequalities

Stochastic variational inequalities (SVI) model a large class of equilibrium problems subject to data uncertainty, and are closely related to stochastic optimization problems. The SVI solution is usually estimated by a solution to a sample average approximation (SAA) problem. This paper considers the normal map formulation of an SVI, and proposes a method to build asymptotically exact confidence regions and confidence intervals for the solution of the normal map formulation, based on the asymptotic distribution of SAA solutions. The confidence regions are single ellipsoids with high probability. We also discuss the computation of simultaneous and individual confidence intervals

Bayesian Quadrature for Multiple Related Integrals

Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to incomplete/finite information about the continuous mathematical problem being approximated. In this paper, we demonstrate that this paradigm can provide additional advantages, such as the possibility of transferring information between several numerical methods. This allows users to represent uncertainty in a more faithful manner and, as a by-product, provide increased numerical efficiency. We propose the first such numerical method by extending the well-known Bayesian quadrature algorithm to the case where we are interested in computing the integral of several related functions. We then prove convergence rates for the method in the well-specified and misspecified cases, and demonstrate its efficiency in the context of multi-fidelity models for complex engineering systems and a problem of global illumination in computer graphics.Comment: Proceedings of the 35th International Conference on Machine Learning (ICML), PMLR 80:5369-5378, 201

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

Data-driven linear decision rule approach for distributionally robust optimization of on-line signal control

We propose a two-stage, on-line signal control strategy for dynamic networks using a linear decision rule (LDR) approach and a distributionally robust optimization (DRO) technique. The first (off-line) stage formulates a LDR that maps real-time traffic data to optimal signal control policies. A DRO problem is solved to optimize the on-line performance of the LDR in the presence of uncertainties associated with the observed traffic states and ambiguity in their underlying distribution functions. We employ a data-driven calibration of the uncertainty set, which takes into account historical traffic data. The second (on-line) stage implements a very efficient linear decision rule whose performance is guaranteed by the off-line computation. We test the proposed signal control procedure in a simulation environment that is informed by actual traffic data obtained in Glasgow, and demonstrate its full potential in on-line operation and deployability on realistic networks, as well as its effectiveness in improving traffic