Stochastic convex programming: basic duality

Rockafellar, R. T.; Wets, Roger J.-B.

doi:10.2140/pjm.1976.62.173

Cited by 88 publications

(43 citation statements)

References 7 publications

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“…Note that (Ψ) is a convex function of Ψ; thus max Ψ⪰0 (Ψ) is a stochastic convex optimization problem. Hence, employing the stochastic subgradient iterative method [35] with unknown channel CDF, we can get a stochastic subgradient iteration algorithm based on the channel realizations (h[ ] and g[ ]) at time slot for ∀ = 1, . .…”

Section: Stochastic Joint Time Slot and Power Allocation Algorithmmentioning

confidence: 99%

Fair Resource Allocation with QoS Guarantee in Secure Multiuser TDMA Networks

Bai

Wang

et al. 2018

Wireless Communications and Mobile Computing

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We investigate a secure multiuser time division multiple access (TDMA) system with statistical delay quality of service (QoS) guarantee in terms of secure effective capacity. An optimal resource allocation policy is proposed to minimize the -fair cost function of the average user power under the individual QoS constraint, which also balances the energy efficiency and fairness among the users. First, convex optimization problems associated with the resource allocation policy are formulated. Then, a subgradient iteration algorithm based on the Lagrangian duality theory and the dual decomposition theory is employed to approach the global optimal solutions. Furthermore, considering the practical channel conditions, we develop a stochastic subgradient iteration algorithm which is capable of dynamically learning the intended wireless channels and acquiring the global optimal solution. It is shown that the proposed optimal resource allocation policy depends on the delay QoS requirement and the channel conditions. The optimal policy can save more power and achieve the balance of the energy efficiency and the fairness compared with the other resource allocation policies.

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Section: Stochastic Joint Time Slot and Power Allocation Algorithmmentioning

confidence: 99%

Fair Resource Allocation with QoS Guarantee in Secure Multiuser TDMA Networks

Bai

Wang

et al. 2018

Wireless Communications and Mobile Computing

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“…ξ ∈ Ξ if and only if there is a subset A ∈ M such that μ(A) = 0 and the inequality g(x, ξ) ≤ 0 holds for every ξ ∈ Ξ \ A. This kind of formulation is relevant, for example, in stochastic programming (cf., [14,15,16]). Semi-infinite programming problems are studied in detail in [1,7,22,20].…”

Section: Theorem 62 [Criterion For Exact Penalty Representation] Assmentioning

confidence: 99%

Abstract Convexity and Augmented Lagrangians

Burachik¹,

Rubinov²

2007

SIAM J. Optim.

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The ultimate goal of this paper is to demonstrate that abstract convexity provides a natural language and a suitable framework for the examination of zero duality gap properties and exact multipliers of augmented Lagrangians. We study augmented Lagrangians in a very general setting and formulate the main definitions and facts describing the augmented Lagrangian theory in terms of abstract convexity tools. We illustrate our duality scheme with an application to stochastic semi-infinite optimization.

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“…The above can be formulated as a stochastic programming problem [13], [14]. We assume that there exists at least one interior feasible solution in the above problem.…”

Section: Opportunistic Power Scheduling With the Minimum Performmentioning

confidence: 99%

“…where s (n) is an index of the system state at iteration n, the algorithm in (11) converges to the optimal solution that solves problem (B) with a sequence of step sizes that satisfies conditions given in (13).…”

Section: Opportunistic Power Scheduling With the Minimum Performmentioning

confidence: 99%

“…If problem (A) is a stochastic convex programming problem, (i.e., U s,i (P s,i ), ∀i, ∀s is a concave function), there is no duality gap between it and its dual. Hence, we can obtain its optimal power scheduling by solving its dual [13]. However, since problem (A) may not be a stochastic convex programming problem, typically, we can guarantee neither optimality, nor feasibility (in terms of minimum performance constraints 4 ) of the solution that is obtained by solving the dual.…”

Section: Opportunistic Power Scheduling With the Minimum Performmentioning

confidence: 99%

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Opportunistic power scheduling for multi-server wireless systems with minimum performance constraints

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Abstract-In this paper, we present a power scheduling scheme, i.e., a joint time-slot and power allocation method for wireless cellular systems. We allow multiple transmissions in a timeslot that can interfere with each other. Hence, it is important to not only select the mobiles to be scheduled in a time-slot, but also important to allocate an appropriate power level for transmission to these scheduled mobiles in order to achieve high system performance and quality of service. We model the timevarying wireless channel as a stochastic process and formulate a stochastic programming problem that attempts to maximize the expected total system utility, with constraints on the minimum expected utility for each mobile. The power scheduling algorithm is obtained by using stochastic duality and implemented via stochastic subgradient techniques.

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Stochastic convex programming: basic duality

Cited by 88 publications

References 7 publications

Fair Resource Allocation with QoS Guarantee in Secure Multiuser TDMA Networks

Fair Resource Allocation with QoS Guarantee in Secure Multiuser TDMA Networks

Abstract Convexity and Augmented Lagrangians

Opportunistic power scheduling for multi-server wireless systems with minimum performance constraints

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