Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization

Farshbaf-Shaker, M. Hassan; Henrion, René; Hömberg, Dietmar

doi:10.1007/s11228-017-0452-5

Cited by 42 publications

(42 citation statements)

References 29 publications

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“…Therefore we get that the probability constraint (1) can be expressed as − log ϕ(x) ≤ − log p establishing convexity of the constraint for all p ∈ [0, 1]. This result can be further generalized stating convexity of (1) provided that ξ admits a density disposing of generalized concavity properties and that g is jointly quasi-convex (see, e.g., [40, section 4.6], [4], [44] or [8,Theorem 4.39] for a modern version of such results as well as [14,Proposition 4]). In these results, the restrictive assumption is the joint quasiconvexity.…”

Section: Convexity Of Probability Constraintsmentioning

confidence: 90%

Eventual convexity of probability constraints with elliptical distributions

Ackooij¹,

Malick

2018

Math. Program.

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Probability constraints are often employed to intuitively define safety of given decisions in optimization problems. They simply express that a given system of inequalities depending on a decision vector and a random vector is satisfied with high enough probability. It is known that, even if this system is convex in the decision vector, the associated probability constraint is not convex in general. In this paper, we show that some degree of convexity is still preserved, for the large class of elliptical random vectors, encompassing for example Gaussian or Student random vectors. More precisely, our main result establishes that, under mild assumptions, eventual convexity holds, i.e. the probability constraint is convex when the safety level is large enough. We also provide tools to compute a concrete convexity certificate from nominal problem data. Our results are illustrated on several examples, including the situation of polyhedral systems with random technology matrices and arbitrary covariance structure.

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Section: Convexity Of Probability Constraintsmentioning

confidence: 90%

Eventual convexity of probability constraints with elliptical distributions

Ackooij¹,

Malick

2018

Math. Program.

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“…The main tool in these works to compute this probability is the so-called spheric-radial decomposition (see e.g. [21,22,6,9,8]). Theorem 1.1.…”

Section: Martin Gugat Rüdiger Schultz and Michael Schustermentioning

confidence: 99%

“…by using the definition of the function g (see (6)). With Σ k (b, β) defined in (21) the Lemma is proven.…”

Section: Convexity Of the Feasible Set In Linear Graphsmentioning

confidence: 99%

Convexity and starshapedness of feasible sets in stationary flow networks

Gugat¹,

Schultz²,

Schuster³

2020

Networks &Amp; Heterogeneous Media

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In this paper, we consider a stationary model for the flow through a network. The flow is determined by the values at the boundary nodes of the network. We call these values the loads of the network. In the applications, the feasible loads must satisfy some box constraints. We analyze the structure of the set of feasible loads. Our analysis is motivated by gas pipeline flows, where the box constraints are pressure bounds. We present sufficient conditions that imply that the feasible set is starshaped with respect to special points. Under stronger conditions, we prove the convexity of the set of feasible loads. All the results are given for passive networks with and without compressor stations. This analysis is motivated by the aim to use the spheric-radial decomposition for stochastic boundary data in this model. This paper can be used for simplifying the algorithmic use of the spheric-radial decomposition.

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“…Despite the fact that the state variables of most practical processes are constrained, almost all of the above studies have not considered state constraints with the exception of [23] and [52]. In [23] the authors studied continuity and convexity of chance (probabilistic) constraints in infinite dimensions (see also [62] for differentiability analysis). The work [52] studies a feed back control design for a state-constrained parabolic PDE system with distributed, but bounded uncertainties.…”

Section: Introductionmentioning

confidence: 99%

Chance constrained optimization of elliptic PDE systems with a smoothing convex approximation

et al. 2020

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In this paper, we consider chance constrained optimization of elliptic partial differential equation (CCPDE) systems with random parameters and constrained state variables. We demonstrate that, under standard assumptions, CCPDE is a convex optimization problem. Since chance constrained optimization problem are generally nonsmooth and difficult to solve directly, we propose a smoothing inner-outer approximation method to generate a sequence of smooth approximate problems for the CCPDE. Thus, the optimal solution of the convex CCPDE is approximable through optimal solutions of the inner-outer approximation problems. A numerical example demonstrates the viability of the proposed approach.

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Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization

Cited by 42 publications

References 29 publications

Eventual convexity of probability constraints with elliptical distributions

Eventual convexity of probability constraints with elliptical distributions

Convexity and starshapedness of feasible sets in stationary flow networks

Chance constrained optimization of elliptic PDE systems with a smoothing convex approximation

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