Time Blocks Decomposition of Multistage Stochastic Optimization Problems

Multistage stochastic optimization problems are, by essence, complex because their solutions are indexed both by stages (time) and by uncertainties (scenarios). Their large scale nature makes decomposition methods appealing.The most common approaches are time decomposition --- and state-based resolution methods, like stochastic dynamic programming, in stochastic optimal control --- and scenario decomposition --- like progressive hedging in stochastic programming. We present a method to decompose multistage stochastic optimization problems by time blocks, which covers both stochastic programming and stochastic dynamic programming. Once established a dynamic programming equation with value functions defined on the history space (a history is a sequence of uncertainties and controls), we provide conditions to reduce the history using a compressed "state" variable. This reduction is done by time blocks, that is, at stages that are not necessarily all the original unit stages, and we prove areduced dynamic programming equation. Then, we apply the reduction method by time blocks to \emph{two time-scales} stochastic optimization problems and to a novel class of so-called \emph{decision-hazard-decision} problems, arising in many practical situations, like in stock management. The \emph{time blocks decomposition} scheme is as follows: we use dynamic programming at slow time scale where the slow time scale noises are supposed to be stagewise independent, and we produce slow time scale Bellman functions; then, we use stochastic programming at short time scale, within two consecutive slow time steps, with the final short time scale cost given by the slow time scale Bellman functions, and without assuming stagewise independence for the short time scale noises

Dynamic consistency for Stochastic Optimal Control problems

For a sequence of dynamic optimization problems, we aim at discussing a notion of consistency over time. This notion can be informally introduced as follows. At the very first time step $t_0$, the decision maker formulates an optimization problem that yields optimal decision rules for all the forthcoming time step $t_0, t_1, ..., T$; at the next time step $t_1$, he is able to formulate a new optimization problem starting at time $t_1$ that yields a new sequence of optimal decision rules. This process can be continued until final time $T$ is reached. A family of optimization problems formulated in this way is said to be time consistent if the optimal strategies obtained when solving the original problem remain optimal for all subsequent problems. The notion of time consistency, well-known in the field of Economics, has been recently introduced in the context of risk measures, notably by Artzner et al. (2007) and studied in the Stochastic Programming framework by Shapiro (2009) and for Markov Decision Processes (MDP) by Ruszczynski (2009). We here link this notion with the concept of "state variable" in MDP, and show that a significant class of dynamic optimization problems are dynamically consistent, provided that an adequate state variable is chosen

Variational Formulations for the l0 Pseudonorm and Application to Sparse Optimization

The so-called l0 pseudonorm on Rd counts the number of nonzero components of a vector. It is used in sparse optimization, either as criterion or in the constraints, to obtain solutions with few nonzero entries. However, the mathematical expression of the l0 pseudonorm, taking integer values, makes it difficult to handle in optimization problems on Rd. Moreover, the Fenchel conjugacy fails to provide relevant insight. In this paper, we analyze the l0 pseudonorm by means of a family of so-called Capra conjugacies. For this purpose, to each (source) norm on Rd, we attach a so-called Capra coupling between Rd and Rd, and sequences of generalized top-k dual norms and k-support dual norms. When we suppose that both the source norm and its dual norm are orthant-strictly monotonic, we obtain three main results. First, we show that the l0 pseudonorm is Capra biconjugate, that is, a Capra-convex function. Second, we deduce an unexpected consequence: the l0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex convex lower semicontinuous (lsc) function on Rd. Third, we establish variational formulations for the l0 pseudonorm and, with these novel expressions, we provide, on the one hand, a new family of lower and upper bounds for the l0 pseudonorm, as a ratio between two norms, and, on the other hand, reformulations for exact sparse optimization problems

Constant Along Primal Rays Conjugacies and the l0 Pseudonorm

The so-called l0 pseudonorm on Rd counts the number of nonzero components of a vector. It is used in sparse optimization, either as criterion or in the constraints, to obtain solutions with few nonzero entries. For such problems, the Fenchel conjugacy fails to provide relevant analysis: indeed, the Fenchel conjugate of the characteristic function of the level sets of the l0 pseudonorm is minus infinity, and the Fenchel biconjugate of the l0 pseudonorm is zero. In this paper, we display a class of conjugacies that are suitable for the l0 pseudonorm. For this purpose, we suppose given a (source) norm on Rd. With this norm, we define, on the one hand, a sequence of so-called coordinate-k norms and, on the other hand, a coupling between Rd and Rd , called Capra (constant along primal rays). Then, we provide formulas for the Capra-conjugate and biconjugate, and for the Capra subdifferentials, of functions of the l0 pseudonorm (hence, in particular, of the l0 pseudonorm itself and of the characteristic functions of its level sets), in terms of the coordinate-k norms. As an application, we provide a new family of lower bounds for the l0 pseudonorm, as a fraction between two norms, the denominator being any norm

La théorie de l’esprit dans l’école française de sociologie : éclairages analytiques

Philippe de Lara, maître de conférence à l’ENPC Au-delà d’un programme d’études positives, le foyer de l’entreprise de Durkheim est l’ambition d’élucider la nature de la société et des sociétés. L’objectif de la recherche est de dégager le système conceptuel de Durkheim et d’en éprouver la cohérence et la fécondité à partir de son aspect le plus paradoxal, la conscience collective, qu’il faut entendre comme l’idée de la société comme « être psychique d’une espèce nouvelle ». Mon hypothèse est..

La philosophie de l’esprit dans l’école française de sociologie : éclairages analytiques

Philippe de Lara, maître de conférences à l’ENPC Je cherche à dégager les conditions de sens du concept de conscience collective ou de sujet de la société. Il s’agit de prendre dans sa lettre ce que Durkheim comme ses adversaires nomment son « réalisme sociologique », à savoir la thèse de l’extériorité du social par rapport aux individus et de réévaluer la consistance et la portée de cette thèse, en dépit des difficultés qu’elle soulève. Le point le plus délicat et le plus controversé de ce r..

La philosophie de l’esprit dans l’école française de sociologie : éclairages analytiques

Philippe de Lara, maître de conférences à l’ENPC Je cherche à dégager les conditions de sens du concept de conscience collective ou de sujet de la société. Il s’agit de prendre dans sa lettre ce que Durkheim comme ses adversaires nomment son « réalisme sociologique », à savoir la thèse de l’extériorité du social par rapport aux individus et de réévaluer la consistance et la portée de cette thèse, en dépit des difficultés qu’elle soulève. Le point le plus délicat et le plus controversé de ce r..