Approximate first and second order oracles and their use in optimization algorithms

Katya Scheinberg

Forecasting and decision-making in electricity markets

Pierre Pinson

Classical unconstrained continuous optimization algorithms, such as gradient descent and Newton methods rely on the ability to compute the gradient and the Hessian of the objective at the iterates. Many practical algorithms today relax the requirement of access to the exact derivatives and work with their approximations. We call these inexact computations - first- and second-order oracles. They appear in stochastic optimization, where the true gradient and Hessian are quantities that can only be exactly computed as an expectation over a distribution, but can be approximated by sample averages. Inexact oracles also appear in derivative free optimization where first and second order derivatives are approximated using function values. We will discuss a variety of derivative-free and stochastic optimization algorithms that rely on inexact oracles. We will try to extract common features and emphasize differences of these algorithms. We will specifically consider the connection between convergence properties of algorithms and properties of the oracles that they use.

Too often, forecasting and decision-making problems have been looked at independently. However, getting the most of probabilistic forecasting and decision-making under uncertainty requires an advanced understanding of both. The electricity market application is an ideal playground to look at them altogether. After a short introduction to the salient features of electricity markets, we will explore the state of the art in probabilistic forecasting. We will specifically discuss various types of probabilistic forecasts, univariate and multivariate cases, forecast quality and value, and how to assess them. We will also cover methods for online learning and forecast combination. In parallel, we will focus on various decision-making problems for electricity market participants, in both price-taker and price-maker setups. We will finally look at decision-making problems when one is not sure forecasts can be trusted.