Sujet de Th`ese
— Titre :R`egles de decision pour l’optimisation sous incertitudes
— Unit´e de recherche :IRMAR, UMR-6625
— Localisation :INSA de Rennes
— Th`eme :Optimisation stochastique, optimisation robuste
— Mots clefs : optimisation stochastique, optimisation robuste, r`egles de d´ecision, K-adaptabilit´e, m´ethodologies de d´ecomposition
— Les noms, pr´enoms et courriels des directeurs de th`ese Directeur Antonio Mucherino, antonio.mucherino@inria.fr Co-directeur Ay¸se Nur Arslan, aarslan@insa-rennes.fr Co-directeur J´er´emy Omer, jomer@insa-rennes.fr
— Profil recherch´e : Etudiant diplˆ´ om´e de master ou d’´ecole d’ing´enieur Connaissances requises :
Programmation lin´eaire en nombre entier Programmation C++ ou Python/Julia
Connaissances souhait´ees dans un ou plusieurs de ces domaines : M´ethodes de d´ecomposition pour la programmation lin´eaire Programmation lin´eaire en nombres entiers
Programmation stochastique Optimisation robuste
Objectif de la th` ese
In the majority of mathematical programming applications, the input data is assumed to be known with certainty. However, real world problems almost always include some uncertain parameters. For instance, in power generation, energy demand is highly uncertain at the time of the generation. Moreover, for hydro/wind plants, the amount of future water/wind inflows is uncertain.
In production planning problems, demand is the major source of uncertainty, while in transportation applications travel time uncertainty plays an important role in addition to demand uncertainty. With the current rise of data analytics and the increasing availability of historical data, it is possible to model uncer- tainty and in turn exploit the available data to model optimization problems taking uncertainty into account. Such optimization models provide very useful information to decision-makers, but their solutions are quite challenging from a computational and theoretical perspective.
This thesis focuses on optimization under uncertainty using the paradigms of stochastic programming and robust optimization. In stochastic programming, the probability distribution governing uncertainty is assumed to be known, and the objective function involves a statistical risk measure such as the expected value or the conditional value-at-risk. Whereas, in robust optimization, distri- butions are replaced by uncertainty sets, and the objective function involves the worst-case realization within this set. Stochastic programming is more sui- table when the decision-making process is repeated regularly as in production planning, whereas robust optimization proves useful when the decision-making
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process involves high risks as in disaster management. The common characte- ristic between these two frameworks is the difficulty in solving realistically-sized instances. This difficulty increases when the possibility of taking certain de- cisions after the realization of uncertainty is introduced. This is referred to as multistage/two-stage stochastic programming or adjustable robust optimi- zation. The resulting optimization problems become extremely difficult when (some) decisions are additionally required to be integer, which is called integer recourse. To mitigate this computational difficulty while still providing good solutions, approximate solution methodologies are proposed.
One class of approximate solution methods for the problems described above restrict the set of recourse solutions to more or less simple functions of un- certain parameters. These are referred to, in general, as decision rules. The subclass of decision rules where continuous recourse decisions are expressed as affine functions of uncertain parameters, are further referred to aslinear deci- sion rules (LDRs) [2]. In [7], LDRs are applied both in the primal and dual spaces to evaluate the optimality gap resulting from using LDRs. This latter study is important in the sense that it provides methodological tools for pro- viding a measure of the quality of solutions obtained through decision rules.
Recently, two new classes of decision rules, two-stage decision rules and Lagran- gian dual decision rules, were introduced for multistage stochastic programming with (mixed-)integer recourse and shown to be numerically promising ([3], [5]).
These decision rules exploit the problem structure to reduce multi-stage sto- chastic programs to two-stage stochastic programs, and are designed to provide a measure of the quality of solutions obtained.
While the linear decision rules described above may provide good solutions in the absence of integer variables, they are not adapted to handle integer re- course. To mitigate this problem, decision rules known as finite adaptability or K-adaptability have been introduced, [6], [8]. The premise of these rules is to choose K recourse policies to be implemented before the realization of un- certainty and to implement the best among them after the realization of uncer- tainty. Aside from providing approximations for robust adjustable optimization, these problems have an applied appeal in that they mimic the decision-making process where policy makers may prefer to choose among various implementable policies. Although some progress have been made in the numerical solution of these problems ([4],[1]), current applicability is still contingent upon our ability to efficiently solve them.
The objective of this thesis is twofold : studying the interactions between decision rules in the stochastic programming and the robust optimization li- teratures and facilitating the adaptation of various advances in each field to the other, and improving our current ability to solve optimization under un- certainty problems with integer recourse (potentially under the aforementioned decision rule restrictions). In order to achieve both of these objectives we will use a combination of mathematical programming, combinatorial optimization and statistics techniques. Our theoretical developments will be strongly sup- ported by practical applications and numerical implementation. To this end, special attention will be given to the emerging field of information discovery in optimization under uncertainty where the decision-makers may undertake some cost to uncover part of the exogenous uncertainty.
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R´ ef´ erences
[1] A. N. Arslan, M. Poss, and M. Silva. Min-max-min robust combinatorial optimization with few recourse solutions. working paper or preprint, Oct.
2020.
[2] A. Ben-Tal, A. Goryashko, E. Guslitzer, and A. Nemirovski. Adjustable robust solutions of uncertain linear programs. Mathematical programming, 99(2) :351–376, 2004.
[3] M. Bodur and J. R. Luedtke. Two-stage linear decision rules for multi-stage stochastic programming. Mathematical Programming, pages 1–34, 2018.
[4] A. Chassein, M. Goerigk, J. Kurtz, and M. Poss. Faster algorithms for min- max-min robustness for combinatorial problems with budgeted uncertainty.
European Journal of Operational Research, 279(2) :308–319, 2019.
[5] M. Daryalal, M. Bodur, and J. R. Luedtke. Lagrangian dual decision rules for multistage stochastic mixed integer programming, 2020.
[6] G. A. Hanasusanto, D. Kuhn, and W. Wiesemann. K-adaptability in two- stage robust binary programming. Operations Research, 63(4) :877–891, 2015.
[7] D. Kuhn, W. Wiesemann, and A. Georghiou. Primal and dual linear decision rules in stochastic and robust optimization. Mathematical Programming, 130(1) :177–209, 2011.
[8] A. Subramanyam, C. E. Gounaris, and W. Wiesemann. K-adaptability in two-stage mixed-integer robust optimization. Mathematical Programming Computation, pages 1–32, 2019.
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