An Indirect Method of Nonconvex Variational Problems in Asplund Spaces: The Case for Saturated Measure Spaces

Sagara, Nobusumi

doi:10.1137/140965570

Cited by 7 publications

(4 citation statements)

References 34 publications

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“…The proof of Corollary 4.1 is based on the "direct method" of the calculus of variations via the relaxation technique. For the existence result without the relaxation technique, based on the "indirect method" exploiting the duality theory in Asplund spaces in the nonsmooth setting, see [39]. As investigated thoroughly in [35], Asplund spaces are suitable places for exploring subdifferential calculus fully.…”

Section: The Minimization Principlementioning

confidence: 99%

Relaxation and Purification for Nonconvex Variational Problems in Dual Banach Spaces: The Minimization Principle in Saturated Measure Spaces

Sagara¹

2017

SIAM J. Control Optim.

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We formulate bang-bang, purification, and minimization principles in dual Banach spaces with Gelfand integrals and provide a complete characterization of the saturation property of finite measure spaces. We also present an application of the relaxation technique to large economies with infinite-dimensional commodity spaces, where the space of agents is modeled as a finite measure space. We propose a "relaxation" of large economies, which is regarded as a reasonable convexification of original economies. Under the saturation hypothesis, the relaxation and purification techniques enable us to prove the existence of Pareto optimal allocations without convexity assumptions.

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Section: The Minimization Principlementioning

confidence: 99%

Relaxation and Purification for Nonconvex Variational Problems in Dual Banach Spaces: The Minimization Principle in Saturated Measure Spaces

Sagara¹

2017

SIAM J. Control Optim.

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“…As this example shows, in the general infinite-dimensional setting, the systematic convexity of the value function would require an infinite-dimensional version of the convexity Lyapunov theorem. This in turn would necessitate, as expected, a strong nonatomic-like assumption on the underlying measure space, namely, the super-atomless or, equivalently, the saturation property of the given measure [20]: for each A \in \scrE with positive measure, the space L 1 (A, \mu ) given with respect to the relative \sigma -algebra \scrE A := \{ E \cap A : E \in \scrE \} is nonseparable [23]. In other words, the measure space (T, E, \mu ) is saturated if and only if ( [13]; see, also, [14,19,23,24] for other equivalences) \int T M (t)d\mu is convex for every multifunction M : T \rightri Z with measurable graph.…”

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confidence: 93%

Convex Representatives of the Value Function and Aumann Integrals in Normed Spaces

Flores-Bazán¹,

Hantoute²

2022

SIAM J. Optim.

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Convex representatives are proposed for the value function of an infinite-dimensional constrained nonconvex variational problem. All the involved variables in this problem take their values in (possibly of infinite dimension, not necessarily separable or complete) normed spaces, while the associated measure can be any \sigma -finite, nonnegative, and nonatomic complete measure. This in particular shows that the closure hull of the (possibly nonconvex) value function is always convex, as long as the sense of the integral within the cone-valued functional constraint is given and the type of the closure is appropriately determined. Correspondingly, similar convexity properties for the Aumann integral in general normed spaces of infinite dimension are established. Applications are given in a fairly general positively homogeneous framework.

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“…The state constraints are formulated in the Bochner and Gelfand integral settings with control systems in separable metrizable spaces. The problem under consideration is a general reduced form of the isometric problem studied in [24,32,33], which is an infinite-dimensional analogue of [6] followed by the forementioned works. We provide a characterization of optimality via the subgradient like inequality for the integrand and the state constraint function, which describes the maximum principle for the Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

Optimality Conditions for Nonconvex Variational Problems with Integral Constraints in Banach Spaces

Sagara

2019

Preprint

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This paper exemplifies that saturation is an indispensable structure on measure spaces to obtain the existence and characterization of solutions to nonconvex variational problems with integral constraints in Banach spaces and their dual spaces. We provide a characterization of optimality via the maximum principle for the Hamiltonian and an existence result without the purification of relaxed controls, in which the Lyapunov convexity theorem in infinite dimensions under the saturation hypothesis on the underlying measure space plays a crucial role. We also demonstrate that the existence of solutions for certain class of primitives is necessary and sufficient for the measure space to be saturated.

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An Indirect Method of Nonconvex Variational Problems in Asplund Spaces: The Case for Saturated Measure Spaces

Cited by 7 publications

References 34 publications

Relaxation and Purification for Nonconvex Variational Problems in Dual Banach Spaces: The Minimization Principle in Saturated Measure Spaces

Relaxation and Purification for Nonconvex Variational Problems in Dual Banach Spaces: The Minimization Principle in Saturated Measure Spaces

Convex Representatives of the Value Function and Aumann Integrals in Normed Spaces

Optimality Conditions for Nonconvex Variational Problems with Integral Constraints in Banach Spaces

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