Abstract : Optimization problems involving stochastic models or randomized
algorithms are at the core of various applications areas such as
machine learning, finance, energy production, signal processing,
telecommunications, and medical imaging. Modern applications
involve complex models and large dimensions. They require
sophisticated analysis and algorithmic tools to obtain efficiently
reliable solutions. This minisymposium will feature several
advances illustrating the fertile interface between stochastic
analysis and optimization through talks presented by junior and
senior researchers. It will cover theoretical advances, as well as
practical applications in finance, optimal control, inverse
problems, optimal transportation, and petroleum production.
[00650] Random activations in primal-dual splittings for monotone inclusions with a priori information
Format : Talk at Waseda University
Author(s) :
Luis Briceño-Arias (Universidad Técnica Federico Santa María)
Abstract : In this paper, we propose a numerical approach for solving composite primal-dual
monotone inclusions with a priori information. The underlying a priori information
set is represented by the intersection of fixed point sets of a finite number of operators, and we propose an algorithm that activates the corresponding set by following a finite-valued random variable at each iteration. Our formulation is flexible and includes, for instance, deterministic and Bernoulli activations over cyclic schemes, and Kaczmarz-type random activations. The almost sure convergence of the algorithm is obtained by means of properties of stochastic Quasi-Fejér sequences. We also recover several primal-dual algorithms for monotone inclusions without a priori information and classical algorithms for solving convex feasibility problems and linear systems. In the context of convex optimization with inequality constraints, any selection of the constraints defines the a priori information set, in which case the operators involved are simply projections onto half spaces. By incorporating random projections onto a selection of the constraints to classical primal-dual schemes, we obtain faster algorithms as we illustrate by means of a numerical application to a stochastic arc capacity expansion problem in a transport network.
[01376] New results on Carathéodory integral functions, applications to stochastic optimization
Format : Talk at Waseda University
Author(s) :
Minh Nhut Bui (Universität Graz)
Patrick L. Combettes (North Carolina State University)
Abstract : We present new results on Carath\'eodory integral functions and discuss applications to stochastic optimization.
[03790] Adaptive partition-based method for 2-stage stochastic linear programming
Format : Talk at Waseda University
Author(s) :
Eduardo Moreno (Universidad Adolfo Ibañez)
Ivana Ljubic (ESSEC Business School)
Abstract : The Generalized Adaptive partition-based method for solving 2-stage stochastic linear problems is based on iteratively and automatically aggregating the uncertainty space into scenarios and disaggregating them based on the dual subproblem variables. It can be seen as a Benders-like method, where optimality cuts are adaptively aggregated based on the dual solutions. Computational experiments show significant improvements compared to classical methods, including the possibility of solving problems in continuous probability spaces using discrete optimization.
[03978] Multistage optimization of a partially observed petroleum production system
Format : Talk at Waseda University
Author(s) :
Jean-Philippe Chancelier (CERMICS -- Ecole des Ponts ParisTech)
Pierre Carpentier (UMA, ENSTA PARIS)
Michel De Lara (CERMICS - Ecole des Ponts ParisTech)
Cyrille Vessaire (CERMICS -- Ecole des Ponts ParisTech)
Alejandro Rodriguez-Martinez (Total Energies, SE Pau)
Abstract : An oil production network is composed of one or more reservoirs (geological formations containing oil) connected through a network of wells and pipes. At the beginning of the reservoir exploitation, we have only partial knowledge of the content of the reservoir, namely a probability distribution of the initial state of the reservoir.
We propose a formulation of the management of an oil production network where the reservoir is a partially observed controlled dynamical system. This approach leads to a well-known class of problems: Partially Observed Markovian Decision Process (POMDP). However, general POMDPs are often untractable due to the curse of dimensionality. In the case of the proposed formulation, we consider a subclass of POMDPs: deterministic-POMDPs (deterministic transition and observation). We highlight and exploit structure in the deterministic-POMDP formulation to push back the curse of dimensionality. Then, we are able to use Dynamic Programming to find the optimal production planning.
Finally, we present numerical applications.
Ari-Pekka Perkkiö (Ludwig-Maximilian University of Munich)
Abstract : We study dynamic programming, duality and optimality conditions in general convex stochastic optimization problems introduced by Rockafellar and Wets in the 70s. We give a general formulation of the dynamic programming recursion and derive an explicit dual problem in terms of two dual variables, one of which is the shadow price of information while the other one gives the marginal cost of a perturbation much like in classical Lagrangian duality. Existence of primal solutions and the absence of duality gap are obtained without compactness or boundedness assumptions. In the context of financial mathematics, the relaxed assumptions are satisfied under the well-known no-arbitrage condition and the reasonable asymptotic elasticity condition of the utility function. We extend classical portfolio optimization duality theory to problems of optimal semi-static hedging. Besides financial mathematics, we obtain several new results in stochastic programming and stochastic optimal control.
