On the convergence time of the drift-plus-penalty algorithm for strongly convex programs

Yu, Hao; Neely, Michael J.

doi:10.1109/cdc.2015.7402619

Cited by 13 publications

(31 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Section 2, we provide, for several classes of problems, conditions ensuring that the dual function of an optimization problem is strongly concave and give formulas for computing the corresponding constant of strong concavity when possible. It turns out that our results improve Theorem 10 in [19] (the only reference we are aware of on the strong concavity of the dual function) which proves the strong concavity of the dual function under stronger assumptions. The tools developped in Sections 2 and 3 allow us to build the inexact black boxes necessary for the Inexact Stochastic Mirror Descent (ISMD) algorithm and its convergence analysis presented in Section 4.…”

Section: Introductionsupporting

confidence: 75%

See 1 more Smart Citation

Inexact stochastic mirror descent for two-stage nonlinear stochastic programs

Guigues

2020

Math. Program.

View full text Add to dashboard Cite

We introduce an inexact variant of Stochastic Mirror Descent (SMD), called Inexact Stochastic Mirror Descent (ISMD), to solve nonlinear two-stage stochastic programs where the second stage problem has linear and nonlinear coupling constraints and a nonlinear objective function which depends on both first and second stage decisions. Given a candidate first stage solution and a realization of the second stage random vector, each iteration of ISMD combines a stochastic subgradient descent using a prox-mapping with the computation of approximate (instead of exact for SMD) primal and dual second stage solutions. We provide two convergence analysis of ISMD, under two sets of assumptions. We show in particular that under some assumptions, ISMD has the same convergence rate as SMD. The first convergence analysis is based on the formulas for inexact cuts of value functions of convex optimization problems shown recently in [6]. The second convergence analysis relies on new formulas that we derive for inexact cuts of value functions of convex optimization problems assuming that the dual function of the second stage problem for all fixed first stage solution and realization of the second stage random vector, is strongly concave. We show that this assumption of strong concavity is satisfied for some classes of problems and present the results of numerical experiments on two simple two-stage problems which show that solving approximately the second stage problem for the first iterations of ISMD can help us obtain a good approximate first stage solution quicker than with SMD.

show abstract

Section: Introductionsupporting

confidence: 75%

“…The local strong concavity of the dual function of (2.13) was shown recently in Theorem 10 in [19] assuming (A3), assuming instead of (A1) that f is strongly convex and second-order continuously differentiable (which is stronger than (A1)), and assuming instead of (A2) that g i , i = 1, . .…”

Section: We Assume Thatmentioning

confidence: 78%

Inexact stochastic mirror descent for two-stage nonlinear stochastic programs

Guigues

2020

Math. Program.

View full text Add to dashboard Cite

show abstract

“…From a broader perspective, the work in this paper is also related to the backpressure algorithm, first proposed in the context of stochastic network optimization [30]. As shown in [31], the dual subgradient algorithm when applied to deterministic resource allocation problems, may also be viewed as the so-called drift-plus-penalty algorithm. The analysis in [31] however does not translate to convergence rate results for the SDSD algorithm.…”

Section: A Related Workmentioning

confidence: 99%

“…As shown in [31], the dual subgradient algorithm when applied to deterministic resource allocation problems, may also be viewed as the so-called drift-plus-penalty algorithm. The analysis in [31] however does not translate to convergence rate results for the SDSD algorithm.…”

Section: A Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Network Resource Allocation via Stochastic Subgradient Descent: Convergence Rate

Bedi

Rajawat

2018

IEEE Trans. Commun.

View full text Add to dashboard Cite

This paper considers a general stochastic resource allocation problem that arises widely in wireless networks, cognitive radio, networks, smart-grid communications, and cross-layer design. The problem formulation involves expectations with respect to a collection of random variables with unknown distributions, representing exogenous quantities such as channel gain, user density, or spectrum occupancy.We consider the constant step-size stochastic dual subgradient descent (SDSD) method that has been widely used for online resource allocation in networks. The problem is solved in dual domain which results in a primal resource allocation subproblem at each time instant. The goal here is to characterize the non-asymptotic behavior of such stochastic resource allocations in an almost sure sense. It is well known that with a step size of ǫ, SDSD converges to an O(ǫ)-sized neighborhood of the optimum. In practice however, there exists a trade-off between the rate of convergence and the choice of ǫ. This paper establishes a convergence rate result for the SDSD algorithm that precisely characterizes this trade-off. Towards this end, a novel stochastic bound on the gap between the objective function and the optimum is developed. The asymptotic behavior of the stochastic term is characterized in an almost sure sense, thereby generalizing the existing results for the stochastic subgradient methods. For the stochastic resource allocation problem at hand, the result explicates the rate with which the allocated resources become near-optimal. As an application, the power and user-allocation problem in device-to-device networks is formulated and solved using the SDSD algorithm. Further intuition on the rate results is obtained from the verification of the regularity conditions and accompanying simulation results.

show abstract