New Directions in Stochastic Optimization

Recent work in stochastic programming draws on interaction with Algebraic and Combinatorial Models, Numerical Analysis of PDEs, Risk Analysis, Mathematical Equilibrium, Minimax, and Stochastic Games. For the first time ever, the workshop brought together scholars from these fields to exchange experience and identify promising topics for future research. Mathematics Subject Classification (2010): 90C15. Introduction by the Organisers The workshop New Directions in Stochastic Optimisation organized by Jesús De Loera (Davis), Darinka Dentcheva (Hoboken), Georg Ch. Pflug (Vienna) and Rüdiger Schultz (Essen) was well attended by 54 participants with broad geographic representation. By a surprising coincidence, the workshop took place precisely 50 years after the first Oberwolfach Workshop in Operations Research (Organizers: R. Henn (Karlsruhe), H. P. Künzi (Zurich) and H. Schubert (Kiel)). Topic: Stochastic programming offers mathematically rigorous optimisation models incorporating probabilistic information with uncertain data. Decisions are based exclusively on the information that is available at the moment of taking decisions (nonanticipativity). Depending on when missing information is unveiled and on how this interacts with decision making over time, different principal model setups arise, e.g., one-stage, two-stage, or multi-stage models. Selection and placement (in the objective or the constraints) of the statistical parameters according to which relevant random variables are to be evaluated is another important issue. 2304 Oberwolfach Report 38/2018 This allows to express perceptions such as reliability, risk neutrality, or risk aversion. Finally, the nature of the initial uncertain optimisation problem (linear or nonlinear, with or without integer variables, living in finite or infinite dimension) has crucial impact on the arising stochastic programming model. These aspects lead to a wide variety of stochastic optimization programs as well as to a wide variety of mathematical techniques for their analysis and algorithmic treatment. Course of the workshop: On the one hand, the workshop reflected the diversity of the involved areas. On the other hand, there was enough overlap among individual expertise to generate new ideas and obtain input from other directions. In particular, the 35 talks together with a brainstorming session on Thursday evening provided the basic ideas for stimulating discussions covering a broad spectrum of topics. We now will discuss some contributions on specific topics. Numerical Analysis of PDEs: The rapid development of PDE-constrained optimisation is accompanied by approaches to handle uncertainty in appropriate fashion. The approach via stochastic programming aims at finding best possible decisions under data uncertainty. Procedures for reaching optimality in terms of stochastic criteria may incorporate user attitudes such as being risk averse, risk neutral or risk seeking. A related contemporary approach to handling uncertainty, also discussed at the workshop, is uncertainty quantification that looks for “a computational framework for quantifying input and response uncertainties in a manner that facilitates predictions with quantified and reduced uncertainty”. Shape optimization under stochastic loading for elastic materials have been discussed in several talks. Risk neutral and risk averse objective functionals have been investigated and the concept of stochastic dominance constraints was explored. Thereby the expected excess and the excess probability are taken into account, first as objective functional involving the compliance cost of an elastic object under stochastic loading and then as a constraint when comparing a shape with a benchmark shape. Open problems concern the identification of subclasses allowing for duality and resulting algorithmic shortcuts; exploiting problem similarities for efficient repeated solution of PDE-constrained optimisation problems differing in the realisations of the random data; mathematical foundation and algorithm design of approximation via linearisation of full models, arriving at linear models, development of numerical PDE-solvers taking into account specifics of this problem class. PDE-constrained optimization under uncertainty has been addressed from further points of view.