Algorithm Engineering in Robust Optimization

Robust optimization is a young and emerging field of research having received a considerable increase of interest over the last decade. In this paper, we argue that the the algorithm engineering methodology fits very well to the field of robust optimization and yields a rewarding new perspective on both the current state of research and open research directions. To this end we go through the algorithm engineering cycle of design and analysis of concepts, development and implementation of algorithms, and theoretical and experimental evaluation. We show that many ideas of algorithm engineering have already been applied in publications on robust optimization. Most work on robust optimization is devoted to analysis of the concepts and the development of algorithms, some papers deal with the evaluation of a particular concept in case studies, and work on comparison of concepts just starts. What is still a drawback in many papers on robustness is the missing link to include the results of the experiments again in the design

Robust optimization of a bi‑objective tactical resource allocation problem with uncertain qualification costs

In the presence of uncertainties in the parameters of a mathematical model, optimal solutions using nominal or expected parameter values can be misleading. In practice, robust solutions to an optimization problem are desired. Although robustness is a key research topic within single-objective optimization, little attention is received within multi-objective optimization, i.e. robust multi-objective optimization. This work builds on recent work within robust multi-objective optimization and presents a new robust efficiency concept for bi-objective optimization problems with one uncertain objective. Our proposed concept and algorithmic contribution are tested on a real-world\ua0multi-item capacitated resource planning\ua0problem, appearing at a large aerospace company manufacturing high precision engine parts. Our algorithm finds all the robust efficient solutions required by the decision-makers in significantly less time than the approach of Kuhn et al. (Eur J Oper Res 252(2):418–431, 2016) on 28 of the 30 industrial instances

Probabilistic Optimization Techniques in Smart Power System

Uncertainties are the most significant challenges in the smart power system, necessitating the use of precise techniques to deal with them properly. Such problems could be effectively solved using a probabilistic optimization strategy. It is further divided into stochastic, robust, distributionally robust, and chance-constrained optimizations. The topics of probabilistic optimization in smart power systems are covered in this review paper. In order to account for uncertainty in optimization processes, stochastic optimization is essential. Robust optimization is the most advanced approach to optimize a system under uncertainty, in which a deterministic, set-based uncertainty model is used instead of a stochastic one. The computational complexity of stochastic programming and the conservativeness of robust optimization are both reduced by distributionally robust optimization.Chance constrained algorithms help in solving the constraints optimization problems, where finite probability get violated. This review paper discusses microgrid and home energy management, demand-side management, unit commitment, microgrid integration, and economic dispatch as examples of applications of these techniques in smart power systems. Probabilistic mathematical models of different scenarios, for which deterministic approaches have been used in the literature, are also presented. Future research directions in a variety of smart power system domains are also presented.publishedVersio

Advancing stability analysis of mean-risk stochastic programs: Bilevel and two-stage models

Measuring and managing risk has become crucial in modern decision making under stochastic uncertainty. In two-stage stochastic programming, mean-risk models are essentially defined by a parametric recourse problem and a quantification of risk. The thesis addresses sufficient conditions for weak continuity of the resulting objective functions with respect to perturbations of the underlying probability measure. The approach is based on so called psi-weak topologies that are finer than the topology of weak convergence and allows to unify and extend known results for a comprehensive class of risk measures and recourse problems. In particular, stability of mean-risk models with mixed-integer quadratic and general mixed-integer convex recourse problems is derived for any law-invariant, convex and nondecreasing quantification of risk. From a conceptual point of view, two-stage stochastic programs and bilevel problems under stochastic uncertainty are closely related. Assuming that only the follower can observe the realization of the randomness, the optimistic and pessimistic setting give rise to two-stage problems where only optimal solutions of the lower level are feasible for the recourse problem. So far, stability in stochastic bilevel programming has only been examined for a specific model based on a quantile criterion. The novel approach allows to identify sufficient conditions for stability of stochastic bilevel problems with quadratic lower level and is applicable for a comprehensive class of risk measures.Die Bewertung und das Management von Risken sind ein wesentlicher Aspekt von Entscheidungsproblemen unter stochastischer Unsicherheit. Zielfunktionsbasierte risikoaverse Modelle der zweistufigen stochastischen Optimierung lassen sich im Wesentlichen durch ihr parametrisches Zweitstufenproblem und das betrachtete Risikomaß charakterisieren. Die Arbeit befasst sich mit hinreichenden Bedingungen für Stetigkeit der resultierenden Zielfunktion unter Störungen des zu Grunde liegenden Wahrscheinlichkeitsmaßes bezüglich der Topologie schwacher Konvergenz. Der Ansatz basiert auf so genannten psi-schwachen Topologien, die feiner als die Topologie schwacher Konvergenz sind. Für eine umfassende Klasse von Risikomaßen und Zweitstufenproblemen werden so bestehende Resultate vereinheitlicht und erweitert. Insbesondere lassen sich für jedes verteilungsinvariante, konvexe und nichtfallende Risikomaß Stabilitätsaussagen für Aufgaben mit quadratischem oder konvexem gemischt-ganzzahligen Zweitstufenproblem treffen. Aus konzeptioneller Sicht sind zweistufige stochastische Programme und Bilevel Probleme unter stochastischer Unsicherheit eng miteinander verbunden. Unter der Annnahme, dass nur der Entscheider auf der unteren Ebene die Realisierung des Zufalls beobachten kann, führen sowohl der optimistische als auch der pessimistische Ansatz auf ein zweistufiges stochastisches Programm. Bei diesem sind nur die Optimallösungen der unteren Ebene zulässig für das Zweitstufenproblem. Bisher ist die Stabilität solcher Aufgaben nur für Modelle mit einem speziellen Quantilkriterium untersucht worden. Der neue Ansatz erlaubt es, hinreichende Bedingungen für die Stabilität von stochastischen Bilevel Problemen mit quadratischem Nachfolgerproblem zu identifizieren und ist auf eine reichhaltige Klasse von Risikomaßen anwendbar

Hybrid optimisation and formation of index tracking portfolio in TSE

Asset allocation and portfolio optimisation are some of the most important steps in an investors decision making process. In order to manage uncertainty and maximise returns, it is assumed that active investment is a zero-sum game. It is possible however, that market inefficiencies could provide the necessary opportunities for investors to beat the market. In this study we examined a core-satellite approach to gain higher returns than that of the market. The core component of the portfolio consists of an index-tracking portfolio which has been formulated using a meta-heuristic genetic algorithm, allowing for the efficient search of the solution space for an optimal (or near-optimal) solution. The satellite component is made up of publicly traded active managed funds and the weights of each component are optimised using mathematical modelling (quadratics) to maximise the returns of the resultant portfolio.In order to address uncertainty within the model variables, robustness is introduced into the objective function of the model in the form of risk tolerance (Degree of uncertainty). The introduction of robustness as a variable allows us to assess the resultant model in worst-case circumstances and determine suitable levels of risk tolerance. Further attempts at implementing additional robustness within the model using an artificial neural network in an LSTM configuration were inconclusive, suggesting that LSTM networks were unable to make informative predictions on the future returns of the index because market efficiencies render historical data irrelevant and market movement is akin to a random walk. A framework is offered for the formation and optimisation of a hybrid multi-stage core-satellite portfolio which manages risk through the implementation of robustness and passive investment, whilst attempting to beat the market in terms of returns. Using daily returns data from the Tehran Stock Exchange for a four-year period, it is shown that the resultant core-satellite portfolio is able to beat the market considerably after training.Results indicate that the tracking ability of the portfolio is affected by the number of its constituents, that there is a specific time frame of 70 days after which the resultant portfolio needs to be re assessed and readjusted and that the implementation of robustness as a degree of uncertainty variable within the objective function increases the correlation coefficient and reduces tracking error.Keywords: Index Funds, Index Tracking, Passive Portfolio Management, Robust Optimisation, Core Satellite Investment, Quadratic Optimisation, Genetic Algorithms, LSTM, Heuristic Neural Networks, Efficient Market Hypothesis, Modern Portfolio Theory, Portfolio optimisatio