arXiv subjects

Jean-Philippe Chancelier

Publications and source records attributed to Jean-Philippe Chancelier.

At least 37 records · Page 2Linked to original sources

arXiv2023

Decomposition Methods for Dynamically Monotone Two-Time-Scale Stochastic Optimization Problems

In energy management, it is common that strategic investment decisions (storage capacity, production units) are made at a slow time scale, whereas operational decisions (storage, production) are made at a fast time scale: for such problems, the total number of decision stages may be huge. In this paper, we consider multistage stochastic optimization problems with two time-scales, and we propose a time block decomposition scheme to address them numerically. More precisely, our approach relies on two assumptions. On the one hand, we suppose slow time scale stagewise independence of the noise process: the random variables that occur during a slow time scale interval are independent of those at another slow time scale interval. This makes it possible to use Dynamic Programming at the slow time scale. On the other hand, we suppose a dynamically monotone property for the problem under consideration, which makes it possible to obtain bounds. Then, we present two algorithmic methods to compute upper and lower bounds for slow time scale Bellman value functions. Both methods rely respectively on primal and dual decomposition of the Bellman equation applied at the slow time scale. We assess the methods tractability and validate their efficiency by solving a battery management problem where the fast time scale operational decisions have an impact on the storage current capacity, hence on the strategic decisions to renew the battery at the slow time scale.

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arXiv2023

Rank-Based Norms, Capra-Conjugacies and the Rank Function

We consider the space of matrices, with given number of rows and of columns, equipped with the classic trace scalar product. With any matrix (source) norm, we associate a coupling, called Capra, between the space of matrices and itself. Then, we compute the Capra conjugate and biconjugate of the rank function. They are expressed in function of a sequence of rank-based norms, more precisely generalized r-rank and dual r-rank matrix norms associated with the matrix source norm. We deduce a lower bound of the rank function given by a variational formula which involves the generalized r-rank norms. In the case of the Frobenius norm, we show that the rank function is equal to the variational formula.

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arXiv2022

Multistage Optimization of a Petroleum Production System with Material Balance Model

In this paper, we propose a mathematical formulation for the management of an oil production network as a multistage optimization problem. The reservoir is modeled as a controlled dynamical system by using material balance equations. We use a dynamic programming algorithm to solve the optimization problem. Two numerical applications illustrate our work: the first one consists in optimizing the production of a gas reservoir, whereas the second one tackles an oil reservoir with water injection.

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arXiv2022

Time Consistency for Multistage Stochastic Optimization Problems under Constraints in Expectation

We consider sequences-indexed by time (discrete stages)-of families of multistage stochastic optimization problems. At each time, the optimization problems in a family are parameterized by some quantities (initial states, constraint levels.. .). In this framework, we introduce an adapted notion of time consistent optimal solutions, that is, solutions that remain optimal after truncation of the past and that are optimal for any values of the parameters. We link this time consistency notion with the concept of state variable in Markov Decision Processes for a class of multistage stochastic optimization problems incorporating state constraints at the final time, either formulated in expectation or in probability. For such problems, when the primitive noise random process is stagewise independent and takes a finite number of values, we show that time consistent solutions can be obtained by considering a finite dimensional state variable. We illustrate our results on a simple dam management problem.

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arXiv2022

The Capra-subdifferential of the l0 pseudonorm

The l0 pseudonorm counts the nonzero coordinates of a vector. It is often used in optimization problems to enforce the sparsity of the solution. However, this function is nonconvex and noncontinuous, and optimization problems formulated with l0 in the objective function or in the constraints are hard to solve in general. Recently, a new family of coupling functions - called Capra (constant along primal rays) - has proved to induce relevant generalized Fenchel-Moreau conjugacies to handle the l0 pseudonorm. In particular, under a suitable choice of source norm on the Euclidean space used in the definition of the Capra coupling - the function l0 is Capra-subdifferentiable, hence is Capra-convex. In this article, we give explicit formulations for the Capra subdifferential of l0, when the source norm is a lp norm with p larger that 1. We illustrate our results with graphical visualizations of the Capra subdifferential of l0 for the Euclidean source norm.

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arXiv2022

Orthant-Strictly Monotonic Norms, Generalized Top-k and k-Support Norms and the L0 Pseudonorm

The so-called l0 pseudonorm on the Euclidean space Rd counts the number of nonzero components of a vector. We say that a sequence of norms is strictly increasingly graded (with respect to the l0 pseudonorm) if it is nondecreasing and that the sequence of norms of a vector~x becomes stationary exactly at the index l0(x). In this paper, with any (source) norm, we associate sequences of generalized top-k and k-support norms, and we also introduce the new class of orthant-strictly monotonic norms (that encompasses the lp norms, but for the extreme ones). Then, we show that an orthant-strictly monotonic source norm generates a sequence of generalized top-k norms which is strictly increasingly graded. With this, we provide a systematic way to generate sequences of norms with which the level sets of the l0 pseudonorm are expressed by means of the difference of two norms. Our results rely on the study of orthant-strictly monotonic norms.

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arXiv2022

Kuhn's Equivalence Theorem for Games in Product Form

We propose an alternative to the tree representation of extensive form games. Games in product form represent information with $σ$-fields over a product set, and do not require an explicit description of the play temporality, as opposed to extensive form games on trees. This representation encompasses games with a continuum of actions, randomness and players, as well as games for which the play order cannot be determined in advance. We adapt and prove Kuhn's theorem-regarding equivalence between mixed and behavioral strategies under perfect recall-for games in product form with continuous action sets.

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arXiv2022

The stochastic Auxiliary Problem Principle in Banach spaces: measurability and convergence

The stochastic Auxiliary Problem Principle (APP) algorithm is a general Stochastic Approximation (SA) scheme that turns the resolution of an original optimization problem into the iterative resolution of a sequence of auxiliary problems. This framework has been introduced to design decomposition-coordination schemes but also encompasses many well-known SA algorithms such as stochastic gradient descent or stochastic mirror descent. We study the stochastic APP in the case where the iterates lie in a Banach space and we consider an additive error on the computation of the subgradient of the objective. In order to derive convergence results or efficiency estimates for a SA scheme, the iterates must be random variables. This is why we prove the measurability of the iterates of the stochastic APP algorithm. Then, we extend convergence results from the Hilbert space case to the Banach space case. Finally, we derive efficiency estimates for the function values taken at the averaged sequence of iterates or at the last iterate, the latter being obtained by adapting the concept of modified Fej{é}r monotonicity to our framework.

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arXiv2022

Optimization of a domestic microgrid equipped with solar panel and battery: Model Predictive Control and Stochastic Dual Dynamic Programming approaches

In this study, a microgrid with storage (battery, hot water tank) and solar panel is considered. We benchmark two algorithms, MPC and SDDP, that yield online policies to manage the microgrid, and compare them with a rule based policy. Model Predictive Control (MPC) is a well-known algorithm which models the future uncertainties with a deterministic forecast. By contrast, Stochastic Dual Dynamic Programming (SDDP) models the future uncertainties as stagewise independent random variables with known probability distributions. We present a scheme, based on out-of-sample validation, to fairly compare the two online policies yielded by MPC and SDDP. Our numerical studies put to light that MPC and SDDP achieve significant gains compared to the rule based policy, and that SDDP overperforms MPC not only on average but on most of the out-of-sample assessment scenarios.

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arXiv2022

Daily Generation Scheduling : Decomposition Methods to Solve the Hydraulic Problems

Short-term hydro-generation management poses a non-convex or even non-continuous optimization problem. For this reason, the problem of systematically obtaining feasible and economically satisfying solutions has not yet been completely solved. Two decomposition methods, which, as far as we know, have not been applied in this field, are here proposed : $\bullet$ the first is based on a decomposition by prediction method and the coordination is a primal-dual relaxation algorithm, $\bullet$ handling the dynamic constraints by duality, the second achieves a price decomposition by an Augmented Lagrangian technique. Numerical tests show the efficiency of these algorithms. They will enable the process in use at Electricit{é} de France to be improved.

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arXiv2021

EMSx: a numerical benchmark for energy management systems

Inserting renewable energy in the electric grid in a decentralized manneris a key challenge of the energy transition. However, at local scale, both production and demand display erratic behavior, which makes it delicate to match them. It is the goal of Energy Management Systems (EMS) to achieve such balance at least cost. We present EMSx, a numerical benchmark for testing control algorithms for the management of electric microgrids equipped with a photovoltaic unit and an energy storage system. EMSx is made of three key components: the EMSx dataset, provided by Schneider Electric, contains a diverse pool of realistic microgrids with a rich collection of historical observations and forecasts; the EMSx mathematical framework is an explicit description of the assessment of electric microgrid control techniques and algorithms; the EMSx software EMSx.jl is a package, implemented in the Julia language, which enables to easily implement a microgrid controller and to test it. All components of the benchmark are publicly available, so that other researchers willing to test controllers on EMSx may reproduce experiments easily. Eventually, we showcase the results of standard microgrid control methods, including Model Predictive Control, Open Loop Feedback Control and Stochastic Dynamic Programming.

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arXiv2021

Causal Inference Theory with Information Dependency Models

Inferring the potential consequences of an unobserved event is a fundamental scientific question. To this end, Pearl's celebrated do-calculus provides a set of inference rules to derive an interventional probability from an observational one. In this framework, the primitive causal relations are encoded as functional dependencies in a Structural Causal Model (SCM), which are generally mapped into a Directed Acyclic Graph (DAG) in the absence of cycles. In this paper, by contrast, we capture causality without reference to graphs or functional dependencies, but with information fields and Witsenhausen's intrinsic model. The three rules of do-calculus reduce to a unique sufficient condition for conditional independence, the topological separation, which presents interesting theoretical and practical advantages over the d-separation. With this unique rule, we can deal with systems that cannot be represented with DAGs, for instance systems with cycles and/or 'spurious' edges. We treat an example that cannot be handled-to the extent of our knowledge-with the tools of the current literature. We also explain why, in the presence of cycles, the theory of causal inference might require different tools, depending on whether the random variables are discrete or continuous.

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arXiv2021

Capra-Convexity, Convex Factorization and Variational Formulations for the l0 Pseudonorm

The so-called l0 pseudonorm, or cardinality function, counts the number of nonzero components of a vector. In this paper, we analyze the l0 pseudonorm by means of so-called Capra (constant along primal rays) conjugacies, for which the underlying source norm and its dual norm are both orthant-strictly monotonic (a notion that we formally introduce and that encompasses the lp norms, but for the extreme ones). We obtain three main results. First, we show that the l0 pseudonorm is equal to its Capra-biconjugate, that is, is a Capra-convex function. Second, we deduce an unexpected consequence, that we call convex factorization: the l0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex lower semicontinuous function. Third, we establish a variational formulation for the l0 pseudonorm by means of generalized top-k dual~norms and k-support dual~norms (that we formally introduce).

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arXiv2021

Topological Conditional Separation

Pearl's d-separation is a foundational notion to study conditional independence between random variables. We define the topological conditional separation and we show that it is equivalent to the d-separation, extended beyond acyclic graphs, be they finite or infinite.

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arXiv2021

Minimization Interchange Theorem on Posets

Interchange theorems between minimization and integration are useful in optimization, especially in optimal control and in stochastic optimization. In this article, we establish a generalized minimization interchange theorem, where integration is replaced by a monotone mapping between posets (partially ordered sets). As an application, we recover, and slightly extend, classical results from the literature, and we tackle the case of the Choquet integral. Our result provides insight on the mechanisms behind existing interchange results.

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arXiv2021

Hidden Convexity in the l0 Pseudonorm

The so-called l0 pseudonorm counts the number of nonzero components of a vector of a Euclidian space. It is well-known that the l0 pseudonorm is not convex, as its Fenchel biconjugate is zero. In this paper, we introduce a suitable conjugacy, induced by a novel coupling, E-Capra, that has the property of being constant along primal rays like the l0 pseudonorm. The coupling E-Capra belongs to the class of one-sided linear couplings, that we introduce; we show that they induce conjugacies that share nice properties with the classic Fenchel conjugacy. For the E-Capra conjugacy, induced by the coupling E-Capra, we relate the E-Capra conjugate and biconjugate of the l0 pseudonorm, the characteristic functions of its level sets and the sequence of so-called top-k norms. In particular, we prove that the l0 pseudonorm is equal to its biconjugate: hence, the l0 pseudonorm is E-Capra-convex in the sense of generalized convexity. As a corollary, we show that there exists a proper convex lower semicontinuous function such that this function and the l0 pseudonorm coincide on the Euclidian unit sphere. This hidden convexity property is somewhat surprising as the l0 pseudonorm is a highly nonconvex function of combinatorial nature. We provide different expressions for this proper convex lower semicontinuous function, and we give explicit formulas in the two-dimensional case.

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arXiv2021

Mixed Spatial and Temporal Decompositions for Large Scale Multistage Stochastic Optimization Problems

We consider multistage stochastic optimization problems involving multiple units. Each unit is a (small) control system. Static constraints couple units at each stage. We present a mix of spatial and temporal decompositions to tackle such large scale problems. More precisely, we obtain theoretical bounds and policies by means of two methods, depending whether the coupling constraints are handled by prices or by resources. We study both centralized and decentralized information structures. We report the results of numerical experiments on the management of urban microgrids. It appears that decomposition methods are much faster and give better results than the standard Stochastic Dual Dynamic Programming method, both in terms of bounds and of policy performance.

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arXiv2021

Decentralized Multistage Optimization of Large-Scale Microgrids under Stochasticity

Microgrids are recognized as a relevant tool to absorb decentralized renewable energies in the energy mix. However, the sequential handling of multiple stochastic productions and demands, and of storage, make their management a delicate issue. We add another layer of complexity by considering microgrids where different buildings stand at the nodes of a network and are connected by the arcs; some buildings host local production and storage capabilities, and can exchange with others their energy surplus. We formulate the problem as a multistage stochastic optimization problem, corresponding to the minimization of the expected temporal sum of operational costs, while satisfying the energy demand at each node, for all time. The resulting mathematical problem has a large-scale nature, exhibiting both spatial and temporal couplings. However, the problem displays a network structure that makes it amenable to a mix of spatial decomposition-coordination with temporal decomposition methods. We conduct numerical simulations on microgrids of different sizes and topologies, with up to 48 nodes and 64 state variables. Decomposition methods are faster and provide more efficient policies than a state-of-the-art Stochastic Dual Dynamic Programming algorithm. Moreover, they scale almost linearly with the state dimension, making them a promising tool to address more complex microgrid optimal management problems.

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