Application of a Smoothing Technique to Decomposition in Convex Optimization

Necoara, Ion; Suykens, Johan A. K.

doi:10.1109/tac.2008.2007159

Cited by 154 publications

(229 citation statements)

References 18 publications

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“…An alternative approach developed by Necoara and Suykens (2008) is to smooth the dual problem through the addition of strongly concave (prox) functions to the objective in (A.2) (and, if necessary, the objective J in the inner sup problems). In our calculations, we follow Necoara and Suykens (2008) by adding terms c v r 2 to the objective in (A.2) and allowing c v → 0 with successive iterations. We use the optimizer SNOPT to solve these (smoothed) optimizations.…”

Section: ≤Dmentioning

confidence: 99%

The Dual Approach to Recursive Optimization: Theory and Examples

Pavoni¹,

Sleet²,

Messner³

2018

Econometrica

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APPENDIX A: NUMERICAL IMPLEMENTATION THIS SECTION DESCRIBES how to implement the recursive dual approach numerically. Under the conditions of Theorem 2, the dual Bellman operator is a contraction and, consequently, it is natural to calculate D * via value iteration. Numerical approximation of candidate dual value functions is facilitated by their sub-linearity and the simplicity of their domain. The dual Bellman involves an (outer) minimization over a set of multipliers; these multipliers are passed to (and "coordinate") a family of simple (inner) maximizations over current actions and states. Additive separability in the objective may be exploited to decompose the inner maximizations into a family of simpler maximizations that in parametric settings often have analytical solutions.Dual Value Function Approximation. Numerical implementation of a value function iteration algorithm requires approximations to candidate value functions. Our implementation exploits the sub-linearity of dual value functions and uses a piecewise linear approximation (on the spherical domain C). Piecewise linear approximations to value functions defined on spheres were first applied in economics by Judd, Yeltekin, and Conklin (2003). We apply their approximation procedure to our setting. 31 Recall that under the conditions of Theorem 2, the domain for the dual Bellman operator may be identified with an interval of functions D : S × Y → R, each of which is sub-linear in its second argument. As noted, these functions are fully determined on S × C (or a subset thereof). Moreover, their sub-linearity implies that, for all y ∈ C, Judd, Yeltekin, and Conklin (2003) used this approach to approximate the support function of a payoff set in a repeated game; we use it to approximate the recursive dual value function. In other aspects, our (recursive dual) formulation is different from that of Judd, Yeltekin, and Conklin (2003). Alternative approaches to approximation on spherical domains are described in Sloan and Womersley (2000). 32 For this and other properties of sub-linear functions used below, see Florenzano and Van (2001).

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Section: ≤Dmentioning

confidence: 99%

The Dual Approach to Recursive Optimization: Theory and Examples

Pavoni¹,

Sleet²,

Messner³

2018

Econometrica

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“…Also, it is shown in [32] that through distributed Newton method the convergence speed of the algorithm can be significantly improved by achieving super-linear convergence rates. In [98] a smoothing technique is used to develop a new decomposition method which significantly improves the efficiency of conventional dual decomposition methods for distributed optimization. Figure 22b shows the same system, where MG C4 (shown red) is faulted and completely lost.…”

Section: An Mg-based Power System Architecture With Resiliency and Sementioning

confidence: 99%

A Survey on Smart Agent-Based Microgrids for Resilient/Self-Healing Grids

2017

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This paper presents an overview of our body of work on the application of smart control techniques for the control and management of microgrids (MGs). The main focus here is on the application of distributed multi-agent system (MAS) theory in multi-objective (MO) power management of MGs to find the Pareto-front of the MO power management problem. In addition, the paper presents the application of Nash bargaining solution (NBS) and the MAS theory to directly obtain the NBS on the Pareto-front. The paper also discusses the progress reported on the above issues from the literature. We also present a MG-based power system architecture for enhancing the resilience and self-healing of the system.

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“…Furthermore, they also suffer from slow convergence speeds due, in part, to the manually adjusted step-sizes. However, it is important to remark that in the last decade a significant progress has been made in dualdecomposition-based solutions using smoothing [19] or path-following [20] strategies, improving the number of iterations of the classical dual decomposition by an order of magnitude. Notwithstanding, these methods tackle problems with linear constraints and are not designed under a decentralized implementation perspective.…”

Section: Primal-dual Techniquesmentioning

confidence: 99%

Coupled-decompositions: exploiting primal–dual interactions in convex optimization problems

Morell

Vicario

2013

EURASIP J. Adv. Signal Process.

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Decomposition techniques implement the so-called "divide and conquer" in convex optimization problems, being primal and dual decompositions the two classical approaches. Although both solutions achieve the goal of splitting the original program into several smaller problems (called the subproblems), these techniques exhibit in general slow speed of convergence. This is a limiting factor in practice and in order to circumvent this drawback, we develop in this article the coupled-decompositions method. As a result, the number of iterations can be reduced by more than one order of magnitude. Furthermore, the new technique is self-adjustable, i.e., it does not depend on user-defined parameters, as opposite to what happens with classical strategies. Given that in signal processing applied to communications and networking we usually deal with a variety of problems that exhibit certain coupling structures, our method is useful to design decentralized as well as centralized optimization schemes with advantages over the existing techniques in the literature. In this article, we expose there different resource allocation problems where the proposed method is successfully applied.

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Application of a Smoothing Technique to Decomposition in Convex Optimization

Cited by 154 publications

References 18 publications

The Dual Approach to Recursive Optimization: Theory and Examples

The Dual Approach to Recursive Optimization: Theory and Examples

A Survey on Smart Agent-Based Microgrids for Resilient/Self-Healing Grids

Coupled-decompositions: exploiting primal–dual interactions in convex optimization problems

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