Infinite Dimensional Optimization and Control Theory

Fattorini, H. O.

doi:10.1017/cbo9780511574795

Cited by 351 publications

(255 citation statements)

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“…Monotone variational inequality theory, which was introduced in sixties, has emerged as an interesting and fascinating branch of applicable mathematics with a wide range of applications in finance, economics, optimization, engineering and medicine see, for example, [1], [8], [9]- [11], [17], [25], [26] and the references therein. This field is dynamic and is experiencing an explosive growth in both theory and applications; as a consequence, research techniques and problems are drawn from various fields.…”

Section: Introductionmentioning

confidence: 99%

Iterative common solutions of fixed point and variational inequality problems

Zhang¹,

Yuan²

2016

J. Nonlinear Sci. Appl.

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Section: Introductionmentioning

confidence: 99%

Iterative common solutions of fixed point and variational inequality problems

Zhang¹,

Yuan²

2016

J. Nonlinear Sci. Appl.

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“…Obviously, the E-SEM problem (19) is an infinite-dimensional optimization problem (see Fattorini (1999)), in which the solution (or strategy) is not a number or a vector, but rather a continuous quantity. By the second constraint (SIC), we immediately have:…”

Section: Problem Formulation Under Stochastic Informationmentioning

confidence: 99%

Combining Spot and Futures Markets: A Hybrid Market Approach to Dynamic Spectrum Access

Gao

Shou

Chen

et al. 2016

Operations Research

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Dynamic spectrum access is a new paradigm of secondary spectrum utilization and sharing. It allows unlicensed secondary users (SUs) to exploit opportunistically the under-utilized licensed spectrum. Market mechanism is a widely-used promising means to regulate the consuming behaviours of users and, hence, achieve the efficient allocation and consumption of limited resources. In this paper, we propose and study a hybrid secondary spectrum market consisting of both the futures market and the spot market, in which SUs (buyers) purchase under-utilized licensed spectrum from a spectrum regulator, either through predefined contracts via the futures market, or through spot transactions via the spot market. We focus on the optimal spectrum allocation among SUs in an exogenous hybrid market that maximizes the secondary spectrum utilization efficiency. The problem is challenging due to the stochasticity and asymmetry of network information. To solve this problem, we first derive an off-line optimal allocation policy that maximizes the ex-ante expected spectrum utilization efficiency based on the stochastic distribution of network information. We then propose an on-line VickreyCClarkeCGroves (VCG) auction that determines the real-time allocation and pricing of every spectrum based on the realized network information and the pre-derived off-line policy. We further show that with the spatial frequency reuse, the proposed VCG auction is NP-hard; hence, it is not suitable for on-line implementation, especially in a large-scale market. To this end, we propose a heuristics approach based on an on-line VCG-like mechanism with polynomial-time complexity, and further characterize the corresponding performance loss bound analytically. We finally provide extensive numerical results to evaluate the performance of the proposed solutions.

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“…It is well-known that under the assumption (2.2) the distributed optimal control problem (2.1a)-(2.1c) admits a unique solution (y, u) ∈ H 1 0 (Ω) × L 2 (Ω) (cf., e.g., [15,[21][22][23]) which is characterized by the existence of a co-state (adjoint state) p ∈ H 1 0 (Ω) and a Lagrange multiplier for the inequality constraint (adjoint control)…”

Section: The Distributed Elliptic Control Problemmentioning

confidence: 99%

Ana posteriorierror analysis of adaptive finite element methods for distributed elliptic control problems with control constraints

et al. 2007

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Abstract. We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method.

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Infinite Dimensional Optimization and Control Theory

Cited by 351 publications

References 0 publications

Iterative common solutions of fixed point and variational inequality problems

Iterative common solutions of fixed point and variational inequality problems

Combining Spot and Futures Markets: A Hybrid Market Approach to Dynamic Spectrum Access

Ana posteriorierror analysis of adaptive finite element methods for distributed elliptic control problems with control constraints

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