Dual gradient method for linearly constrained, strongly convex, separable mathematical programming problems

Cheng, Yongling

doi:10.1007/bf00939216

Cited by 15 publications

(20 citation statements)

References 6 publications

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“…Section IV will elaborate on how condition (14) can be checked in the OPF context. Under current modeling assumptions, it follows that the duality gap is zero, and the dual function q({λ i }) is concave, differentiable, and it has a continuous first derivative [37]. Consider then utilizing a gradient method to solve the dual problem, which amounts to iteratively performing [37]:…”

Section: Assumption 3: Vectorsmentioning

confidence: 99%

“…Thus, (18) coincides with standard dual gradient method in (16), and the convergence results in [35,Prop. 8.2.6], [37] carry over to this ideal setup. In this work, convergence of the system outputs {y i (t)} i∈ND to the solution of (P1) is assessed in (3) is solved in a distributed fashion by using steps (16); once the problem is solved (i.e., iterates in (16) have converged to the optimal primal and dual values), the optimal reference signals {u opt i } i∈N D are dispatched to the PV-inverters.…”

Section: B Controller Synthesismentioning

confidence: 99%

See 1 more Smart Citation

Photovoltaic Inverter Controllers Seeking AC Optimal Power Flow Solutions

Dall’Anese

Dhople

Giannakis

2016

IEEE Trans. Power Syst.

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Abstract-This paper considers future distribution networks featuring inverter-interfaced photovoltaic (PV) systems, and addresses the synthesis of feedback controllers that seek real-and reactive-power inverter setpoints corresponding to AC optimal power flow (OPF) solutions. The objective is to bridge the temporal gap between long-term system optimization and realtime inverter control, and enable seamless PV-owner participation without compromising system efficiency and stability. The design of the controllers is grounded on a dual ǫ-subgradient method, and semidefinite programming relaxations are advocated to bypass the non-convexity of AC OPF formulations. Global convergence of inverter output powers is analytically established for diminishing stepsize rules for cases where: i) computational limits dictate asynchronous updates of the controller signals, and ii) inverter reference inputs may be updated at a faster rate than the power-output settling time.Index Terms-Distribution systems, photovoltaic inverter control, distributed optimization and control; optimal power flow.

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Section: Assumption 3: Vectorsmentioning

confidence: 99%

Section: B Controller Synthesismentioning

confidence: 99%

Photovoltaic Inverter Controllers Seeking AC Optimal Power Flow Solutions

Dall’Anese

Dhople

Giannakis

2016

IEEE Trans. Power Syst.

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“…In this section, we provide some basic relations for the algorithm (5)- (7) that are fundamental to establishing the almost sure convergence of the sequences produced by the algorithm. The proofs of all the results can be found in [27].…”

Section: Basic Properties Of the Algorithmmentioning

confidence: 99%

“…Our next lemma provides a relation for the iterates θ k i related to the learning scheme of the algorithm. Lemma 3: Let Assumptions 2 and 5 hold, and let the iterates θ k i be generated by the algorithm (5)- (7). Then, almost surely, we have for all k ≥ 0,…”

Section: A Iterate Relationsmentioning

confidence: 99%

Distributed stochastic optimization under imperfect information

Kannan

Nedić

Shanbhag

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Abstract-We consider a stochastic convex optimization problem that requires minimizing a sum of misspecified agentspecific expectation-valued convex functions over the intersection of a collection of agent-specific convex sets. This misspecification is manifested in a parametric sense and may be resolved through solving a distinct stochastic convex learning problem. Our interest lies in the development of distributed algorithms in which every agent makes decisions based on the knowledge of its objective and feasibility set while learning the decisions of other agents by communicating with its local neighbors over a time-varying connectivity graph. While a significant body of research currently exists in the context of such problems, we believe that the misspecified generalization of this problem is both important and has seen little study, if at all. Accordingly, our focus lies on the simultaneous resolution of both problems through a joint set of schemes that combine three distinct steps: (i) An alignment step in which every agent updates its current belief by averaging over the beliefs of its neighbors; (ii) A projected (stochastic) gradient step in which every agent further updates this averaged estimate; and (iii) A learning step in which agents update their belief of the misspecified parameter by utilizing a stochastic gradient step. Under an assumption of mere convexity on agent objectives and strong convexity of the learning problems, we show that the sequences generated by this collection of update rules converge almost surely to the solution of the correctly specified stochastic convex optimization problem and the stochastic learning problem, respectively.

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“…Other examples of problems expressed as a Transportation Problem with convex costs have been discussed in [1] (pp.547-551: Area Transfers in Communication Networks, Matrix Balancing, the Stick Percolation Problem). Three other examples of problems that are transformable to convex network flow problems are listed by Cheng in [11]. These are water distribution, electrical network analysis and equilibrium export-import trade problems.…”

Section: Introductionmentioning

confidence: 99%

The Nonlinear Generalized Transportation Problem with convex costs

Anholcer

2015

Croat. Oper. Res. Rev.

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Abstract. We investigate the Nonlinear Generalized Transportation Problem (NGTP), where the transportation costs and the costs that depend on the amount of good delivered to the destination points are strictly convex functions. The amounts of goods change during the transportation process. This model may correspond, for example, with a congested network where the time costs are involved. First, we present the NGTIP model, then provide a method of solving the problems of this type and then we prove convergence of the method. The numerical experiments prove the effectiveness of the presented algorithm.

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Dual gradient method for linearly constrained, strongly convex, separable mathematical programming problems

Cited by 15 publications

References 6 publications

Photovoltaic Inverter Controllers Seeking AC Optimal Power Flow Solutions

Photovoltaic Inverter Controllers Seeking AC Optimal Power Flow Solutions

Distributed stochastic optimization under imperfect information

The Nonlinear Generalized Transportation Problem with convex costs

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