Alois Pichler

Stochastic multi-objective optimization: a survey on non-scalarizing methods

Currently, stochastic optimization on the one hand and multi-objective optimization on the other hand are rich and well-established special fields of Operations Research. Much less developed, however, is their intersection: the analysis of decision problems involving multiple objectives and stochastically represented uncertainty simultaneously. This is amazing, since in economic and managerial applications, the features of multiple decision criteria and uncertainty are very frequently co-occurring. Part of the existing quantitative approaches to deal with problems of this class apply scalarization techniques in order to reduce a given stochastic multi-objective problem to a stochastic single-objective one. The present article gives an overview over a second strand of the recent literature, namely methods that preserve the multi-objective nature of the problem during the computational analysis. We survey publications assuming a risk-neutral decision maker, but also articles addressing the situation where the decision maker is risk-averse. In the second case, modern risk measures play a prominent role, and generalizations of stochastic orders from the univariate to the multivariate case have recently turned out as a promising methodological tool. Modeling questions as well as issues of computational solution are discussed.

A Distance For Multistage Stochastic Optimization Models

We describe multistage stochastic programs in a purely in-distribution setting, i.e., without any reference to a concrete probability space. The concept is based on the notion of nested distributions, which encompass in one mathematical object the scenario values as well as the information structure under which decisions have to be made. The nested distance between these distributions is introduced and turns out to be a generalization of the Wasserstein distance for stochastic two-stage problems. We give characterizations of this distance and show its usefulness in examples. The main result states that the difference of the optimal values of two multistage stochastic programs, which are Lipschitz and differ only in the nested distribution of the stochastic parameters, can be bounded by the nested distance of these distributions. This theorem generalizes the well-known Kantorovich-Rubinstein theorem, which is applicable only in two-stage situations, to multistage. Moreover, a dual characterization for the nested distance is established. The setup is applicable both for general stochastic processes and for finite scenario trees. In particular, the nested distance between general processes and scenario trees is well defined and becomes the important tool for judging the quality of the scenario tree generation. Minimizing—at least heuristically—this distance is what good scenario tree generation is all about.

Multistage stochastic optimization

Introduction.- The Nested Distance.- Risk and Utility Functionals.- From Data to Models.- Time Consistency.- Approximations and Bounds.- The Problem of Ambiguity in Stochastic Optimization.- Examples.

Approximations for probability distributions and stochastic optimization problems

In this chapter, an overview of the scenario generation problem is given. After an introduction, the basic problem of measuring the distance between two single-period probability models is described in Section 15.2. Section 15.3 deals with finding good single-period scenarios based on the results of the first section. The distance concepts are extended to the multi-period situation in Section 15.4. Finally, Section 15.5 deals with the construction and reduction of scenario trees.

Time-inconsistent multistage stochastic programs: Martingale bounds

Multistage stochastic programs show time-inconsistency in general, if the objective is neither the expectation nor the maximum functional.

Premiums and reserves, adjusted by distortions

The net premium principle is considered to be the most genuine and fair premium principle in actuarial applications. However, actuarial due diligence requires additional caution in pricing of insurance contracts to avoid, for example, at least bankruptcy of the insurer. This paper addresses the distorted premium principle from various angles. Distorted premiums are typically computed by underweighting or ignoring low, but overweighting high losses. Dual characterizations, which are elaborated in a first part of the paper, support this interpretation. The main contribution consists in an opposite point of view—an alternative characterization—which leaves the probability measure unchanged, but modifies (increases) the outcomes instead in a consistent way. It turns out that this new point of view is natural in actuarial practice,as it can be used for premium calculations, but equally well to determine the reserve process in subsequent years in a time consistent way.

Time-consistent decisions and temporal decomposition of coherent risk functionals

In management and planning it is commonplace for additional information to become available gradually over time. It is well known that most risk measures (risk functionals) are time in consistent in the following sense: it may happen that at a given time period, some loss distribution appears to be less risky than another one, but looking at the conditional distribution at a later time, the opposite relation holds almost surely.The extended conditional risk functionals introduced in this paper enable a temporal decomposition of the initial risk functional that can be used to ensure consistency between past and future preferences. The central result is a decomposition theorem, which allows recomposing the initial coherent risk functional by compounding the conditional risk functionals without losing information or preferences. It follows from our results that the revelation of partial information in time must change the decision maker’s preferences—for consistency reasons—among the remaining courses of action. Further, in many situations, the extended conditional risk functional allows ranking of different policies, even based on incomplete information.In addition, we use counterexamples to show that without change-of-measures, the only time-consistent risk functionals are the expectation and the essential supremum.

Evaluations of Risk Measures for Different Probability Measures

Stochastic optimization problems, which arise in different areas, such as simple asset allocation problems or problems in insurance, often involve coherent risk measures. In these real-world problems the considered risk measure is frequently built on empirical distributions. It is therefore of interest to understand the potential deviation, which occurs when evaluating a risk measure for a perturbed, or slightly changed, probability measure. This paper addresses the potential deviation. It turns out that the Wasserstein distance, a well-known distance for probability measures, provides a valuable notion of distance in the present context: Many risk measures allow a precise quantification in terms of the Wasserstein distance, and important risk measures are continuous with respect to this distance. For specific random variables, which often occur in concrete, real-world problems, it is moreover demonstrated that the derived constants, describing the continuity properties, cannot be improved. The associated...

Tree approximation for discrete time stochastic processes: a process distance approach

Approximating stochastic processes by scenario trees is important in decision analysis. In this paper we focus on improving the approximation quality of trees by smaller, tractable trees. In particular we propose and analyze an iterative algorithm to construct improved approximations: given a stochastic process in discrete time and starting with an arbitrary, approximating tree, the algorithm improves both, the probabilities on the tree and the related path-values of the smaller tree, leading to significantly improved approximations of the initial stochastic process. The quality of the approximation is measured by the process distance (nested distance), which was introduced recently. For the important case of quadratic process distances the algorithm finds locally best approximating trees in finitely many iterations by generalizing multistage k-means clustering.

Stochastic short-term hydropower planning with inflow scenario trees

This paper presents an optimization approach to solve the short-term hydropower unit commitment and loading problem with uncertain inflows. A scenario tree is built based on a forecasted fan of inflows, which is developed using the weather forecast and the historical weather realizations. The tree-building approach seeks to minimize the nested distance between the stochastic process of historical inflow data and the multistage stochastic process represented in the scenario tree. A two-phase multistage stochastic model is used to solve the problem. The proposed approach is tested on a 31 day rolling-horizon with daily forecasted inflows for three power plants situated in the province of Quebec, Canada, that belong to the company Rio Tinto.