arXiv subjects

Michel de Lara

Publications and source records attributed to Michel de Lara.

47 records · Page 3Linked to original sources

arXiv2021

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.

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arXiv2021

Best Convex Lower Approximations of the l 0 Pseudonorm on Unit Balls

Whereas the norm of a vector measures amplitude (and is a 1-homogeneous function), sparsity is measured by the 0-homogeneous l0 pseudonorm, which counts the number of nonzero components. We propose a family of conjugacies suitable for the analysis of 0-homogeneous functions. These conjugacies are derived from couplings between vectors, given by their scalar product divided by a 1-homogeneous normalizing factor. With this, we characterize the best convex lower approximation of a 0-homogeneous function on the unit ''ball'' of a normalization function (i.e. a norm without the requirement of subadditivity). We do the same with the best convex and 1-homogeneous lower approximation. In particular, we provide expressions for the tightest convex lower approximation of the l0 pseudonorm on any unit ball, and we show that the tightest norm which minorizes the l0 pseudonorm on the unit ball of any lp-norm is the l1-norm. We also provide the tightest convex lower convex approximation of the l0 pseudonorm on the unit ball of any norm.

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arXiv2021

A Fresh Geometrical Look at the General S-Procedure

We revisit the S-procedure for general functions with "geometrical glasses". We thus delineate a necessary condition, and almost a sufficient condition, to have the S-procedure valid. Everything is expressed in terms of convexity of augmented sets (convex hulls, conical hulls) of images built from the data functions.

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arXiv2021

Conditional Infimum and Hidden Convexity in Optimization

Detecting hidden convexity is one of the tools to address nonconvex minimization problems. After giving a formal definition of hidden convexity, we introduce the notion of conditional infimum, as it will prove instrumental in detecting hidden convexity. We develop the theory of the conditional infimum, and we establish a tower property, relevant for minimization problems. Thus equipped, we provide a sufficient condition for hidden convexity in nonconvex minimization problems. We illustrate our result on nonconvex quadratic minimization problems. We conclude with perspectives for using the conditional infimum in relation to the so-called S-procedure, to couplings and conjugacies, and to lower bound convex programs.

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arXiv2020

Optimal Joint Allocation of Efforts in Inclusive Fitness by Related Individuals

Families are places of affection and cooperation, but also of conflict. In his famous paper Parent-Offspring Conflict, Robert L. Trivers builds upon W. D. Hamilton's concept of inclusive fitness to argue for genetic conflict in parent-offspring relationships, and to derive numerical predictions on the intensity of the conflict. We propose a mathematical model of game theory that depicts how each member of a family allocates her resource budget to maximize her inclusive fitness; this latter is made of the sum of personal fitness plus the sum of relatives fitnesses weighted by Wright's coefficients of relationship. We define an optimal allocation profile as a Nash equilibrium, and we characterize the solutions in function of resource budgets, coefficients of relationship and derivatives of personal fitnesses.

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arXiv2020

Constant Along Primal Rays Conjugacies and Generalized Convexity for Functions of the Support

The support of a vector in R d is the set of indices with nonzero entries. Functions of the support have the property to be 0-homogeneous and, because of that, the Fenchel conjugacy fails to provide relevant analysis. In this paper, we define the coupling Capra between R d and itself by dividing the classic Fenchel scalar product coupling by a given (source) norm on R d. Our main result is that, when both the source norm and its dual norm are orthant-strictly monotonic, any nondecreasing finite-valued function of the support mapping is Capra-convex, that is, is equal to its Capra-biconjugate (generalized convexity). We also establish that any such function is the composition of a proper convex lower semi continuous function on R d with the normalization mapping on the unit sphere (hidden convexity), and that, when normalized, it admits a variational formulation, which involves a family of generalized local-K-support dual norms.

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arXiv2020

Comparison Theorem for Viability Kernels via Conic Preorders

In natural resource management, decision-makers often aim at maintaining thestate of the system within a desirable set for all times.For instance, fisheries management procedures include keeping thespawning stock biomass over a critical threshold.Another example is given by the peak control of an epidemic outbreakthat encompasses maintaining thenumber of infected individuals below medical treatment capacities.In mathematical terms, one controls a dynamical system.Then, keeping the state of the system within a desirable set for all times is possible when the initial state belongs to the so-called viabilitykernel. We introduce the notion of conic quasimonotonicity reducibility.With this property, we provide a comparison theorem by inclusion between two viabilitykernels, corresponding to two control systems in the infinite horizon case. Wealso derive conditions for equality. We illustrate the method with a model for thebiocontrol of a vector-transmitted epidemic.

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arXiv2020

Kuhn's Equivalence Theorem for Games in Intrinsic Form

We state and prove Kuhn's equivalence theorem for a new representation of games, the intrinsic form. First, we introduce games in intrinsic form where information is represented by $σ$-fields over a product set. For this purpose, we adapt to games the intrinsic representation that Witsenhausen introduced in control theory. Those intrinsic games do not require an explicit description of the play temporality, as opposed to extensive form games on trees. Second, we prove, for this new and more general representation of games, that behavioral and mixed strategies are equivalent under perfect recall (Kuhn's theorem). As the intrinsic form replaces the tree structure with a product structure, the handling of information is easier. This makes the intrinsic form a new valuable tool for the analysis of games with information.

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arXiv2019

Upper and Lower Bounds for Large Scale Multistage Stochastic Optimization Problems: Application to Microgrid Management

We consider a microgrid where different prosumers exchange energy altogether by the edges of a given network. Each prosumer is located to a node of the network and encompasses energy consumption, energy production and storage capacities (battery, electrical hot water tank). The problem is coupled both in time and in space, so that a direct resolution of the problem for large microgrids is out of reach (curse of dimensionality). By affecting price or resources to each node in the network and resolving each nodal sub-problem independently by Dynamic Programming, we provide decomposition algorithms that allow to compute a set of decomposed local value functions in a parallel manner. By summing the local value functions together, we are able, on the one hand, to obtain upper and lower bounds for the optimal value of the problem, and, on the other hand, to design global admissible policies for the original system. Numerical experiments are conducted on microgrids of different size, derived from data given by the research and development centre Efficacity, dedicated to urban energy transition. These experiments show that the decomposition algorithms give better results than the standard SDDP method, both in terms of bounds and policy values. Moreover, the decomposition methods are much faster than the SDDP method in terms of computation time, thus allowing to tackle problem instances incorporating more than 60 state variables in a Dynamic Programming framework.

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arXiv2019

Payoffs-Beliefs Duality and the Value of Information

In decision problems under incomplete information, actions (identified to payoff vectors indexed by states of nature) and beliefs are naturally paired by bilinear duality. We exploit this duality to analyze the value of information, using concepts and tools from convex analysis. We define the value function as the support function of the set of available actions: the subdifferential at a belief is the set of optimal actions at this belief; the set of beliefs at which an action is optimal is the normal cone of the set of available actions at this point. Our main results are 1) a necessary and sufficient condition for positive value of information 2) global estimates of the value of information of any information structure from local properties of the value function and of the set of optimal actions taken at the prior belief only. We apply our results to the marginal value of information at the null, that is, when the agent is close to receiving no information at all, and we provide conditions under which the marginal value of information is infinite, null, or positive and finite.

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arXiv2019

Equivalence Between Time Consistency and Nested Formula

You are a financial analyst. At the beginning of every week, you are able to rank every pair of stochastic processes starting from that week up to the horizon. Suppose that two processes are equal at the beginning of the week. Your ranking procedure is time consistent if the ranking does not change between this week and the next one. In this paper, we propose a minimalist definition of Time Consistency (TC) between two (assessment) mappings. With very few assumptions, we are able to prove an equivalence between Time Consistency and a Nested Formula (NF) between the two mappings. Thus, in a sense, two assessments are consistent if and only if one is factored into the other. We review the literature and observe that the various definitions of TC (or of NF) are special cases of ours, as they always include additional assumptions. By stripping off these additional assumptions, we present an overview of the literature where the contribution of each author is enlightened.

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