Stochastic Optimization in the widest sense is concerned with optimization problems influenced by random parameters in the objective or constraints. The solution of such problems aims in general at finding cost optimal decisions which at the same time are robust against the effect of randomness. A typical such problem class is defined by so-called probabilistic (or chance) constraints. Here, the decisions guarantee that a given random inequality system (e.g., satisfaction of the random demand of a certain good) is fulfilled at a specified minimum probability. A typical application is the control of a water reservoir under random inflows and bounds for the water level in the reservoir (see picture left). In the context of gas network optimization under random loads, the maximization of free capacities in the nodes under load coverage with given probability plays an important role (see picture right). The mathematical challenge of these constraints consists in the absence of an explicit formula for the occuring probability functions which can be approximated only with a limited precision. This complicates in particular the derivation of important structural properties like convexity or differentiability. A major research topic at WIAS is therefore the derivation of gradient formulae for probabilistic constraints and the development of algorithmic solution approaches on their basis.

In another typical problem class, the objective is given as an expectation of a function depending on random parameters. The goal is to develop an algorithm which with high probability gives a good approximation to the minimum of this objective. The main mathematical challenge is to obtain non-asymptotic convergence rates for the proposed algorithm. Such algorithms, developed for stochastic optimization problems, turn out to be efficient for solving complex deterministic problems. The idea behind this approach is usually called "randomization". A deterministic objective function is represented as an expectation of a simple random function. Then, a stochastic optimization algorithm with much cheaper iteration is used to solve the deterministic problem with high probability.

Publications

  Monographs

  • H. Heitsch, R. Henrion, H. Leövey, R. Mirkov, A. Möller, W. Römisch, I. Wegner-Specht, Chapter 13: Empirical Observations and Statistical Analysis of Gas Demand Data, in: Evaluating Gas Network Capacities, Th. Koch, B. Hiller, M.E. Pfetsch, L. Schewe, eds., MOS-SIAM Series on Optimization, SIAM, Philadelphia, 2015, pp. 273--290, (Chapter Published).

  • B. Hiller, Ch. Hayn, H. Heitsch, R. Henrion, H. Leövey, A. Möller, W. Römisch, Chapter 14: Methods for Verifying Booked Capacities, in: Evaluating Gas Network Capacities, Th. Koch, B. Hiller, M.E. Pfetsch, L. Schewe, eds., MOS-SIAM Series on Optimization, SIAM, Philadelphia, 2015, pp. 291--315, (Chapter Published).

  • P. Deuflhard, M. Grötschel, D. Hömberg, U. Horst, J. Kramer, V. Mehrmann, K. Polthier, F. Schmidt, Ch. Schütte, M. Skutella, J. Sprekels, eds., MATHEON -- Mathematics for Key Technologies, 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, 453 pages, (Collection Published).

  Articles in Refereed Journals

  • A.L. Diniz, R. Henrion, On probabilistic constraints with multivariate truncated Gaussian and lognormal distributions, Energy Systems, 8 (2017) pp. 149--167, DOI 10.1007/s12667-015-0180-6 .

  • W. VAN Ackooij, R. Henrion, (Sub-) Gradient formulae for probability functions of random inequality systems under Gaussian distribution, SIAM/ASA Journal on Uncertainty Quantification, 5 (2017) pp. 63--87, DOI 10.1137/16M1061308 .
    Abstract
    We consider probability functions of parameter-dependent random inequality systems under Gaussian distribution. As a main result, we provide an upper estimate for the Clarke subdifferential of such probability functions without imposing compactness conditions. A constraint qualification ensuring continuous differentiability is formulated. Explicit formulae are derived from the general result in case of linear random inequality systems. In the case of a constant coefficient matrix an upper estimate for even the smaller Mordukhovich subdifferential is proven.

  • H. Heitsch, H. Leövey, W. Römisch, Are quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?, Computational Optimization and Applications. An International Journal, 65 (2016) pp. 567--603.
    Abstract
    Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The two-stage integrands are piecewise linear, but neither smooth nor lie in the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial derivatives. Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence not depending on the dimension if the effective superposition dimension is at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol' point sets accompanied with PCA for dimension reduction.

  • R. Hildebrand, Spectrahedral cones generated by rank 1 matrices, Journal of Global Optimization. An International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering, 64 (2016) pp. 349--397.

  • C. Gotzes, H. Heitsch, R. Henrion, R. Schultz, On the quantification of nomination feasibility in stationary gas networks with random load, Mathematical Methods of Operations Research, 84 (2016) pp. 427--457.
    Abstract
    The paper considers the computation of the probability of feasible load constellations in a stationary gas network with uncertain demand. More precisely, a network with a single entry and several exits with uncertain loads is studied. Feasibility of a load constellation is understood in the sense of an existing flow meeting these loads along with given pressure bounds in the pipes. In a first step, feasibility of deterministic exit loads is characterized algebraically and these general conditions are specified to networks involving at most one cycle. This prerequisite is essential for determining probabilities in a stochastic setting when exit loads are assumed to follow some (joint) Gaussian distribution when modeling uncertain customer demand. The key of our approach is the application of the spheric-radial decomposition of Gaussian random vectors coupled with Quasi Monte-Carlo sampling. This approach requires an efficient algorithmic treatment of the mentioned algebraic relations moreover depending on a scalar parameter. Numerical results are illustrated for different network examples and demonstrate a clear superiority in terms of precision over simple generic Monte-Carlo sampling. They lead to fairly accurate probability values even for moderate sample size.

  • A.V. Gasnikov, P. Dvurechensky, Y.E. Nesterov, Stochastic gradient methods with inexact oracle, Proceedings of Moscow Institute of Physics and Technology, 8:1 (2016) pp. 41--91.

  • A. Gasnikov, P. Dvurechensky, I. Usmanova, On accelerated randomized methods, Proceedings of Moscow Institute of Physics and Technology, 8:2 (2016) pp. 67--100.

  • A. Gasnikov, P. Dvurechensky, Stochastic intermediate gradient method for convex optimization problems, Doklady Mathematics. Maik Nauka/Interperiodica Publishing, Moscow. English. Translation of the Mathematics Section of: Doklady Akademii Nauk. (Formerly: Russian Academy of Sciences. Doklady. Mathematics)., 93 (2016) pp. 148--151.

  • P. Dvurechensky, A. Gasnikov, Stochastic intermediate gradient method for convex problems with inexact stochastic oracle, Journal of Optimization Theory and Applications, 171 (2016) pp. 121--145.

  • A. Gasnikov, E. Gasnikova, P. Dvurechensky, E. Ershov, A. Lagunovskaia, Searching for the stochastic equilibria in the transport models of equilibrium flow distribution (in Russian), Proceedings of Moscow Institute of Physics and Technology, 7 (2015) pp. 114--128.

  • A. Gasnikov, P. Dvurechensky, D. Kamzolov, Y. Nesterov, V. Spokoiny, P. Stetsyuk, A. Suvorikova, A. Chernov, Searching for equilibriums in multistage transport models (in Russian), Proceedings of Moscow Institute of Physics and Technology, 7 (2015) pp. 143--155.

  • I. Bremer, R. Henrion, A. Möller, Probabilistic constraints via SQP solver: Application to a renewable energy management problem, Computational Management Science, 12 (2015) pp. 435--459.
    Abstract
    The aim of this paper is to illustrate the efficient solution of nonlinear optimization problems with joint probabilistic constraints by means of an SQP method. Here, the random vector is assumed to obey some multivariate Gaussian distribution. The numerical solution approach is applied to a renewable energy management problem. We consider a coupled system of hydro and wind power production used in order to satisfy some local demand of energy and to sell/buy excessive or missing energy on a day-ahead and intraday market, respectively. A short term planning horizon of 2 days is considered and only wind power is assumed to be random. In the first part of the paper, we develop an appropriate optimization problem involving a probabilistic constraint reflecting demand satisfaction. Major attention will be payed to formulate this probabilistic constraint not directly in terms of random wind energy produced but rather in terms of random wind speed, in order to benefit from a large data base for identifying an appropriate distribution of the random parameter. The second part presents some details on integrating Genz' code for Gaussian probabilities of rectangles into the environment of the SQP solver SNOPT. The procedure is validated by means of a simplified optimization problem which by its convex structure allows to estimate the gap between the numerical and theoretical optimal values, respectively. In the last part, numerical results are presented and discussed for the original (nonconvex) optimization problem.

  • TH. Arnold, R. Henrion, A. Möller, S. Vigerske, A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints, Pacific Journal of Optimization. An International Journal, 10 (2014) pp. 5--20.
    Abstract
    We illustrate the solution of a mixed-integer stochastic nonlinear optimization problem in an application of power management. In this application, a coupled system consisting of a hydro power station and a wind farm is considered. The objective is to satisfy the local energy demand and sell any surplus energy on a spot market for a short time horizon. Generation of wind energy is assumed to be random, so that demand satisfaction is modeled by a joint probabilistic constraint taking into account the multivariate distribution. The turbine is forced to either operate between given positive limits or to be shut down. This introduces additional binary decisions. The numerical solution procedure is presented and results are illustrated.

  • K. Emich, R. Henrion, W. Römisch, Conditioning of linear-quadratic two-stage stochastic optimization problems, Mathematical Programming. A Publication of the Mathematical Programming Society, 148 (2014) pp. 201--221.
    Abstract
    In this paper a condition number for linear-quadratic two-stage stochastic optimization problems is introduced as the Lipschitz modulus of the multifunction assigning to a (discrete) probability distribution the solution set of the problem. Being the outer norm of the Mordukhovich coderivative of this multifunction, the condition number can be estimated from above explicitly in terms of the problem data by applying appropriate calculus rules. Here, a chain rule for the extended partial second-order subdifferential recently proved by Mordukhovich and Rockafellar plays a crucial role. The obtained results are illustrated for the example of two-stage stochastic optimization problems with simple recourse.

  • W. VAN Ackooij, R. Zorgati, R. Henrion, A. Möller, Joint chance constrained programming for hydro reservoir management, Optimization and Engineering. International Multidisciplinary Journal to Promote Optimization Theory & Applications in Engineering Sciences, 15 (2014) pp. 509--531.

  • W. VAN Ackooij, R. Henrion, Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distributions, SIAM Journal on Optimization, 24 (2014) pp. 1864--1889.
    Abstract
    Probabilistic constraints represent a major model of stochastic optimization. A possible approach for solving probabilistically constrained optimization problems consists in applying nonlinear programming methods. In order to do so, one has to provide sufficiently precise approximations for values and gradients of probability functions. For linear probabilistic constraints under Gaussian distribution this can be successfully done by analytically reducing these values and gradients to values of Gaussian distribution functions and computing the latter, for instance, by Genz' code. For nonlinear models one may fall back on the spherical-radial decomposition of Gaussian random vectors and apply, for instance, Deák's sampling scheme for the uniform distribution on the sphere in order to compute values of corresponding probability functions. The present paper demonstrates how the same sampling scheme can be used in order to simultaneously compute gradients of these probability functions. More precisely, we prove a formula representing these gradients in the Gaussian case as a certain integral over the sphere again. Later, the result is extended to alternative distributions with an emphasis on the multivariate Student (or T-) distribution.

  Contributions to Collected Editions

  • TH. Arnold, R. Henrion, M. Grötschel, W. Römisch ET AL., B4 -- A Jack of all trades? Solving stochastic mixed-integer nonlinear constraint programs, in: MATHEON -- Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 135--146.

  Preprints, Reports, Technical Reports

  • P. Dupuis, V. Laschos, K. Ramanan, Exit time risk-sensitive stochastic control problems related to systems of cooperative agents, Preprint no. 2407, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2407 .
    Abstract, PDF (443 kByte)
    We study sequences, parametrized by the number of agents, of exit time stochastic control problems with risk-sensitive costs structures generate by unbounded costs. We identify a fully characterizing assumption, under which, each of them corresponds to a risk-neutral stochastic control problem with additive cost, and also to a risk-neutral stochastic control problem on the simplex, where the specific information about the state of each agent can be discarded. We finally prove that, under some additional assumptions, the sequence of value functions converges to the value function of a deterministic control problem.

  • T. González Grandón, H. Heitsch, R. Henrion, A joint model of probabilistic/robust constraints for gas transport management in stationary networks, Preprint no. 2401, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2401 .
    Abstract, PDF (316 kByte)
    We present a novel mathematical algorithm to assist gas network operators in managing uncertainty, while increasing reliability of transmission and supply. As a result, we solve an optimization problem with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise in the presence of uncertain parameters having partially stochastic and partially non-stochastic character. The application that drives this new approach is a stationary network with uncertain demand (which are stochastic due to the possibility of fitting statistical distributions based on historical measurements) and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of attainable measurements). We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is allowed (shaping a 'maximal' uncertainty set) around nominal roughness coefficients, such that random demands in a stationary gas network can be satisfied at given high probability level for no matter which realization of true roughness coefficients within the uncertainty set. One ends up with a constraint, which is probabilistic with respect to the load of gas and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions. The numerical solution of a corresponding optimization problem is illustrated. The results might assist the network operator with the implementation of cost-intensive roughness measurements.

  • M. Eigel, J. Neumann, R. Schneider, S. Wolf, Stochastic topology optimisation with hierarchical tensor reconstruction, Preprint no. 2362, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2362 .
    Abstract, PDF (8552 kByte)
    A novel approach for risk-averse structural topology optimization under uncertainties is presented which takes into account random material properties and random forces. For the distribution of material, a phase field approach is employed which allows for arbitrary topological changes during optimization. The state equation is assumed to be a high-dimensional PDE parametrized in a (finite) set of random variables. For the examined case, linearized elasticity with a parametric elasticity tensor is used. Instead of an optimization with respect to the expectation of the involved random fields, for practical purposes it is important to design structures which are also robust in case of events that are not the most frequent. As a common risk-aware measure, the Conditional Value at Risk (CVaR) is used in the cost functional during the minimization procedure. Since the treatment of such high-dimensional problems is a numerically challenging task, a representation in the modern hierarchical tensor train format is proposed. In order to obtain this highly efficient representation of the solution of the random state equation, a tensor completion algorithm is employed which only required the pointwise evaluation of solution realizations. The new method is illustrated with numerical examples and compared with a classical Monte Carlo sampling approach.

  • R. Hildebrand, J.G.M. Schoenmakers, J. Zhang, F. Dickmann, Regression based duality approach to optimal control with application to hydro electricity storage, Preprint no. 2330, WIAS, Berlin, 2016, DOI 10.5072/WIAS.PREPRINT.2330 .
    Abstract, PDF (341 kByte)
    In this paper we consider the problem of optimal control of stochastic processes. We employ the dual martingale method brought forward in [Brown, Smith, and Sun, 2010]. The martingale constituting the solution of the dual problem is determined by linear regression within a Monte-Carlo approach. We apply the solution algorithm to a model of a hydro electricity storage and production system coupled with a model of the electricity wholesale market.

  • V. Guigues, R. Henrion, Joint dynamic probabilistic constraints with projected linear decision rules, Preprint no. 2271, WIAS, Berlin, 2016.
    Abstract, PDF (413 kByte)
    We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (infinite dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of linear decision rules and a generalized linear model for the underlying stochastic process with noises that are Gaussian or truncated Gaussian, we show that the value and gradient of the objective and constraint functions of the approximating problems can be computed analytically.

  • R. Hildebrand, Copositive matrices with circulant zero pattern, Preprint no. 2241, WIAS, Berlin, 2016.
    Abstract, PDF (335 kByte)
    Let n be an integer not smaller than 5 and let u1,...,un be nonnegative real n-vectors such that the indices of their positive elements form the sets 1,2,...,n-2,2,3,...,n-1,...,n,1,...,n-3, respectively. Here each index set is obtained from the previous one by a circular shift. The set of copositive forms which vanish on the vectors u1,...,un is a face of the copositive cone. We give an explicit semi-definite description of this face and of its subface consisting of positive semi-definite matrices, and study their properties. If the vectors u1,...,un and their positive multiples exhaust the zero set of an exceptional copositive form belonging to this face, then we call this form regular, otherwise degenerate. We show that degenerate forms are always extremal, and regular forms can be extremal only if n is odd. We construct explicit examples of extremal degenerate forms for any order n, and examples of extremal regular forms for any odd order n. The set of all degenerate forms, i.e., defined by different collections u1,...,un of zeros, is a submanifold of codimension 2n, the set of all regular forms a submanifold of codimension n.

  Talks, Poster

  • H. Heitsch, A probabilistic approach to optimization problems in gas transport networks, SESO 2017 International Thematic Week ``Smart Energy and Stochastic Optimization'', May 30 - June 1, 2017, ENSTA ParisTech and École des Ponts ParisTech, Paris, France, June 1, 2017.

  • R. Henrion, Contraintes en probabilité: Formules du gradient et applications, Workshop ``MAS-MODE 2017", Institut Henri Poincaré, Paris, France, January 9, 2017.

  • R. Henrion, On M-stationnary condition for a simple electricity spot market model, Workshop ``Variational Analysis and Applications for Modelling of Energy Exchange'', May 4 - 5, 2017, Université Perpignan, France, May 4, 2017.

  • R. Henrion, Subdifferential characterization of Gaussian probability functions, SESO 2017 International Thematic Week ``Smart Energy and Stochastic Optimization'', May 30 - June 1, 2017, ENSTA ParisTech and École des Ponts ParisTech, Paris, France, June 1, 2017.

  • R. Henrion, Subdifferential estimates for Gaussian probability functions, HCM Workshop: Nonsmooth Optimization and its Applications, May 15 - 19, 2017, Hausdorff Center for Mathematics, Bonn, May 17, 2017.

  • H. Heitsch, Nonlinear probabilistic constraints in gas transportation problems, WIAS-PGMO Workshop on Nonsmooth and Stochastic Optimization with Applications to Energy Management, May 10 - 12, 2016, WIAS Berlin, Australia, May 11, 2016.

  • H. Heitsch, Optimization in gas transport networks using nonlinear probabilistic constraints, XIV International Conference on Stochastic Programming (ICSP 2016), Thematic Session: Probabilistic Constraints: Applications and Theory, June 25 - July 1, 2016, Búzios, Brazil, June 28, 2016.

  • R. Hildebrand, Canonical barriers on convex cones, Oberseminar Geometrische Analysis, Johann Wolfgang Goethe-Universität Frankfurt am Main, Fachbereich Mathematik, April 26, 2016.

  • J. Neumann, The phase field approach for topology optimization under uncertainties, ZIB Computational Medicine and Numerical Mathematics Seminar, Konrad-Zuse-Zentrum für Informationstechnik Berlin, August 25, 2016.

  • I. Bremer, Dealing with probabilistic constraints under multivariate normal distribution in a standard SQP solver by using Genz' method, WIAS-PGMO Workshop on Nonsmooth and Stochastic Optimization with Applications to Energy Management, May 10 - 12, 2016, WIAS Berlin, May 11, 2016.

  • R. Henrion, (Sub-)Gradient formulae for Gaussian probability functions, XIV International Conference on Stochastic Programming (ICSP 2016), Thematic Session: Probabilistic Constraints: Applications and Theory, June 25 - July 1, 2016, Búzios, Brazil, June 28, 2016.

  • R. Henrion, Aspects of nondifferentiability for probability functions, 7th International Seminar on Optimization and Variational Analysis, June 1 - 3, 2016, Universidad de Alicante, Spain, June 2, 2016.

  • R. Henrion, Aspects of nonsmoothness for Gaussian probability functions, PGMO Days 2016 -- Gaspard Monge Program for Optimization and Operations Research, November 8 - 9, 2016, Electricité de France, Palaiseau, France, November 9, 2016.

  • R. Henrion, Formules du gradient pour des fonctions probabilistes Gaussiennes, Workshop on Offshore Wind Generation, September 9, 2016, Electricité de France R&D, Paris, France, September 9, 2016.

  • R. Henrion, Initiation aux problèmes d'optimisation sous contraintes en probabilité, Workshop ``Optimisation en Milieu Aléatoire'', November 8, 2016, Institut des Sciences Informatiques et de leurs Interactions, GdR 720 ISIS (Information, Signal, Image et ViSion), Paristech Télécom, Paris, France, November 8, 2016.

  • R. Henrion, Optimisation sous contraintes en probabilité, Séminaire du Groupe de Travail ``Modèles Stochastiques en Finance'', Ecole Nationale Supérieure des Techniques Avancées (ENSTA) ParisTech, Palaiseau, France, November 28, 2016.

  • R. Henrion, Robust-stochastic optimization problems in stationary gas networks, Conference ``Mathematics of Gas Transport'', October 6 - 7, 2016, Zuse Institut Berlin, October 6, 2016.

  • H. Heitsch, Optimization of booked capacity in gas transport networks using nonlinear probabilistic constraints, 2nd International Symposium on Mathematical Programming (ISMP 2015), Cluster ``Optimization in Energy Systems'', July 13 - 17, 2015, Pittsburgh, USA, July 17, 2015.

  • R. Hildebrand, Geometry of barriers for 3-dimensional cones, Optimization and Applications in Control and Data Science, May 13 - 15, 2015, Moscow Institute of Physics and Technology, PreMoLab, Moscow, Russian Federation, May 15, 2015.

  • R. Hildebrand, Rank 1 generated spectrahedral cones, Frontiers of High Dimensional Statistics, Optimization, and Econometrics, February 26 - 27, 2015, Higher School of Economics, Moscow, Russian Federation, February 26, 2015.

  • R. Hildebrand , Optimizing strategies in energy and storage markets, Matheon Center Days, Technische Universität Berlin, April 16, 2015.

  • P. Dvurechensky, Semi-supervised pagerank model learning with gradient-free optimization methods, Traditional Youth School ``Control, Information and Optimization'', June 14 - 20, 2015, Moscow, Russian Federation, June 17, 2015.

  • P. Dvurechensky, Stochastic intermediate gradient method: Convex and strongly convex cases, Information Technologies and Systems 2015, September 6 - 11, 2015, Russian Academy of Sciences, Institute for Information Transmission Problems, Sochi, Russian Federation, September 9, 2015.

  • R. Henrion, (Sub-) Gradient formulae for probability functions with Gaussian distribution, PGMO Days 2015 -- Gaspard Monge Program for Optimization and Operations Research, October 27 - 28, 2015, ENSTA ParisTech, Palaiseau, France, October 28, 2015.

  • R. Henrion, (Sub-)Gradient formulae for probability functions with applications to power management, Universidad de Chile, Centro de Modelamiento Matemático, Santiago de Chile, Chile, November 25, 2015.

  • R. Henrion, Conditioning of linear-quadratic two-stage stochastic optimization problems, Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic, March 26, 2015.

  • R. Henrion, On some relations between probability functions and variational analysis, International Workshop ``Variational Analysis and Applications'', August 28 - September 5, 2015, Erice, Italy, August 31, 2015.

  • R. Henrion, Application of chance constraints in a coupled model of hydro-wind energy production, Charles University in Prague, Faculty of Mathematics and Physics, Czech Republic, March 6, 2014.

  • R. Henrion, Application of probabilistic constraints to problems of energy management under uncertainty, Eidgenössische Technische Hochschule Zürich, Power Systems Laboratory, Switzerland, September 30, 2014.

  • R. Henrion, Conditioning of linear-quadratic two-stage stochastic optimization problems, 5th Conference on Optimization Theory and its Applications (ALEL 2014), June 5 - 7, 2014, Universidad de Sevilla, Spain, June 6, 2014.

  • R. Henrion, Gradient formulae in probabilistic programming, Conference on Optimization & Practices in Industry (PGMO-COPI'14), October 28 - 31, 2014, Ecole Polytechnique, Paris, France, October 29, 2014.

  • R. Henrion, Gradient formulae in probabilistic programming, Université Paul Sabatier, Laboratoire d'analyse et d'architecture des systèmes, Toulouse, France, December 8, 2014.

  • R. Henrion, Nonlinear programming for solving chance constrained optimization problems: Application to renewable energies, Winter School on Stochastic Programming with Applications in Energy, Finance and Insurance, March 23 - 28, 2014, Bad Hofgastein, Austria, March 25, 2014.

  • R. Henrion, Probabilistic constraints in hydro reservoir management, XIII Symposium of Specialists in Electric Operational and Expansion Planning (SEPOPE), May 18 - 21, 2014, Foz do Iguassu, Brazil, May 19, 2014.

  • R. Henrion, Probabilistic constraints via nonlinear programming: Application to energy management problems, Euro Mini Conference on Stochastic Programming and Energy Applications (EuroCSP2014), September 24 - 26, 2014, Institut Henri Poincaré, Paris, France, September 25, 2014.

  • R. Henrion, Probabilistic constraints: A structure-oriented introduction, Optimization and Applications Seminar, Eidgenössische Technische Hochschule Zürich, Switzerland, September 29, 2014.

  • R. Henrion, Problèmes d'optimisation sous contraintes en probabilité: une initiation, December 9 - 10, 2014, Université Paul Sabatier, Institut de Mathématiques de Toulouse, France.