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## Time-average optimization with nonconvex decision set and its convergence,” in CDC, (2014)

Venue: | Proceedings IEEE, |

Citations: | 1 - 0 self |

### Citations

7431 | Convex Optimization
- BOYD, VANDENBERGHE
- 2004
(Show Context)
Citation Context ...allows a convergence time analysis of a drift-plus-penalty algorithm that solves problem (1). Further, this paper shows that faster convergence can be achieved by starting time averages after a suitable transient period. Another area of literature focuses on convergence time of first-order algorithms to an -optimal solution to a convex problem, including problem (2). For unconstrained optimization, the optimal first-order method has O(1/ √ ) convergence time [6], [7], while gradient (without strong convexity of objective function) and subgradient methods take O(1/) and O(1/2) respectively [8], [3], [4]. Two O(1/) first-order methods for constrained optimization are developed in [9], [10], but the results rely on special convex formulations. A second-order method for constrained optimization [11] has a fast convergence rate but rely on special a convex formulation. All of these results rely on convexity assumption that do not hold in formulation (1). This paper develops an algorithm for the formulation (1) and analyzes its convergence time. The algorithm is shown to have O(1/2) convergence time for general problems. However, inspired by results in [12], under a uniqueness assumpt... |

539 |
Introductory lectures on convex optimization: a basic course
- Nesterov
- 2003
(Show Context)
Citation Context ...aper removes the stochastic characteristic and focuses on the connection between the technique and a general convex optimization. This allows a convergence time analysis of a drift-plus-penalty algorithm that solves problem (1). Further, this paper shows that faster convergence can be achieved by starting time averages after a suitable transient period. Another area of literature focuses on convergence time of first-order algorithms to an -optimal solution to a convex problem, including problem (2). For unconstrained optimization, the optimal first-order method has O(1/ √ ) convergence time [6], [7], while gradient (without strong convexity of objective function) and subgradient methods take O(1/) and O(1/2) respectively [8], [3], [4]. Two O(1/) first-order methods for constrained optimization are developed in [9], [10], but the results rely on special convex formulations. A second-order method for constrained optimization [11] has a fast convergence rate but rely on special a convex formulation. All of these results rely on convexity assumption that do not hold in formulation (1). This paper develops an algorithm for the formulation (1) and analyzes its convergence time. The alg... |

357 | Dynamic power allocation and routing for time-varying wireless networks
- Neely, Modiano, et al.
- 2005
(Show Context)
Citation Context ...-penalty algorithm solving (5). Algorithm 1 generates sequence {x(t), y(t)}∞t=0, which is an O()-optimal solution to the auxiliary problem (5) by setting V = 1/ [5]. For an O()-optimal solution to the timeaverage problem (1), decision x(t) made by Algorithm 1 is implemented every iteration t, which coincides with Theorem 1. F. Relation to dual subgradient algorithm It is interesting to note that the drift-plus-penalty algorithm is identical to a classic dual subgradient method [13] with a fixed stepsize 1/V , with the exception that it takes a time average of x(t) values. This was noted in [14], [12] for related problems. To see this for the problem of this paper, consider the following convex program, called the embedded formulation of the time-average problem (5): Minimize f(y) (9) Subject to gj(y) ≤ 0 j ∈ {1, . . . , J} xi = yi i ∈ {1, . . . , I} x ∈ X , y ∈ Y. This problem is convex. It is not difficult to show that the above problem has an optimal value f (opt) that is the same as that of problems (1)–(2), (5). Now consider the dual of embedded formulation (9). Let vectors w and z be dual variables of the first and second constraints in problem (9), where the feasible set of (w... |

320 |
Layering as optimization decomposition: A mathematical theory of network architectures,”
- Chiang, Low, et al.
- 2007
(Show Context)
Citation Context ...rithm, related to a classical dual subgradient algorithm, converges to an -optimal solution within O(1/2) time steps. However, when the problem has a unique vector of Lagrange multipliers, the algorithm is shown to have a transient phase and a steady state phase. By restarting the time averages after the transient phase, the total convergence time is improved to O(1/) under a locally-polyhedron assumption, and to O(1/1.5) under a locally-smooth assumption. I. INTRODUCTION Convex optimization is often used to optimally control communication networks and distributed multi-agent systems (see [1] and references therein). This framework utilizes both convexity properties of an objective function and a feasible decision set. However, various systems have inherent discrete (and hence nonconvex) decision sets. For example, a packet switch system makes a binary (0/1) decision about connecting a given link. Further, a wireless system might constrain transmission rates to a finite set corresponding to a fixed set of coding options. This discreteness restrains the application of convex optimization. Let I and J be positive integers. This paper considers time-average optimization where decisio... |

312 |
Convex Analysis and Optimization. Athena Scientific
- Bertsekas, Nedić, et al.
- 2003
(Show Context)
Citation Context ...[V f(y) +W (t) >g(y)− Z(t)>y] W (t+ 1) = [W (t) + g(y(t))]+ Z(t+ 1) = Z(t) + x(t)− y(t) end Algorithm 1: Drift-plus-penalty algorithm solving (5). Algorithm 1 generates sequence {x(t), y(t)}∞t=0, which is an O()-optimal solution to the auxiliary problem (5) by setting V = 1/ [5]. For an O()-optimal solution to the timeaverage problem (1), decision x(t) made by Algorithm 1 is implemented every iteration t, which coincides with Theorem 1. F. Relation to dual subgradient algorithm It is interesting to note that the drift-plus-penalty algorithm is identical to a classic dual subgradient method [13] with a fixed stepsize 1/V , with the exception that it takes a time average of x(t) values. This was noted in [14], [12] for related problems. To see this for the problem of this paper, consider the following convex program, called the embedded formulation of the time-average problem (5): Minimize f(y) (9) Subject to gj(y) ≤ 0 j ∈ {1, . . . , J} xi = yi i ∈ {1, . . . , I} x ∈ X , y ∈ Y. This problem is convex. It is not difficult to show that the above problem has an optimal value f (opt) that is the same as that of problems (1)–(2), (5). Now consider the dual of embedded formulation (9). Let... |

201 | Fair Resource Allocation in Wireless Networks using Queue-length-based Scheduling and Congestion Control.
- Eryilmaz, Srikant
- 2005
(Show Context)
Citation Context ...ers of dual problem (11). The transient phase is defined as the period before a generated dual variables arrives at that set. With this idea, we analyze convergence time in two cases of dual function (10) satisfying locally-polyhedron and locally-smooth properties under the following mild assumption. Assumption 1: The dual formulation (11) has a unique Lagrange multiplier denoted by λ∗,(w∗, z∗). This assumption is assumed throughout Section IV, and replaces the Slater assumption (which is no longer needed). Note that this is a mild assumption when practical systems are considered, e.g., [12], [15]. In addition, Section V shows that the convergences times derived in this section still hold without this uniqueness assumption. We first provide a general result that will be used later. Locally polyhedron Locally smooth Fig. 1. Illustration of locally-polyhedron and locally-smooth functions Lemma 5: Let {λ(t)}∞t=0 be a sequence generated by Algorithm 2. The following relation holds: ‖λ(t+ 1)− λ∗‖2 ≤ ‖λ(t)− λ∗‖2 + 2 V [d(λ(t))− d(λ∗)] + 2C3 V 2 , t ∈ {0, 1, 2, . . .}. (23) Proof: Recall that λ(t) = (w(t), z(t)), h(t) = (g(y(t)), x(t) − y(t)). From the non-expansive property, we have that ‖λ(... |

180 |
On accelerated proximal gradient methods for convex-concave optimization.
- Tseng
- 2008
(Show Context)
Citation Context ...removes the stochastic characteristic and focuses on the connection between the technique and a general convex optimization. This allows a convergence time analysis of a drift-plus-penalty algorithm that solves problem (1). Further, this paper shows that faster convergence can be achieved by starting time averages after a suitable transient period. Another area of literature focuses on convergence time of first-order algorithms to an -optimal solution to a convex problem, including problem (2). For unconstrained optimization, the optimal first-order method has O(1/ √ ) convergence time [6], [7], while gradient (without strong convexity of objective function) and subgradient methods take O(1/) and O(1/2) respectively [8], [3], [4]. Two O(1/) first-order methods for constrained optimization are developed in [9], [10], but the results rely on special convex formulations. A second-order method for constrained optimization [11] has a fast convergence rate but rely on special a convex formulation. All of these results rely on convexity assumption that do not hold in formulation (1). This paper develops an algorithm for the formulation (1) and analyzes its convergence time. The algorith... |

143 | Primal-dual subgradient methods for convex problems
- Nesterov
- 2005
(Show Context)
Citation Context ...s Angeles, CA 90089-2565, supittay@usc.edu, mjneely@usc.edu L. Huang is with Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China, 100084, longbohuang@tsinghua.edu.cn Formulation (1) has an optimal solution which can be converted (by averaging) to the following: Minimize f(x) (2) Subject to gj(x) ≤ 0 j ∈ {1, . . . , J} x ∈ X . However, an optimal solution to formulation (2) may not be in the nonconvex decision set X . Nevertheless, problems (1) and (2) have the same optimal value. Although there have been several techniques utilizing timeaveraged solutions [2], [3], [4], those works are limited to convex formulations. This paper is inspired by the Lyapunov optimization technique [5] which solves stochastic and time-averaged optimization problems. This paper removes the stochastic characteristic and focuses on the connection between the technique and a general convex optimization. This allows a convergence time analysis of a drift-plus-penalty algorithm that solves problem (1). Further, this paper shows that faster convergence can be achieved by starting time averages after a suitable transient period. Another area of literature focuses on convergen... |

79 | Approximate primal solutions and rate analysis for dual subgradient methods,”
- Nedic, Ozdaglar
- 2009
(Show Context)
Citation Context ...eles, CA 90089-2565, supittay@usc.edu, mjneely@usc.edu L. Huang is with Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China, 100084, longbohuang@tsinghua.edu.cn Formulation (1) has an optimal solution which can be converted (by averaging) to the following: Minimize f(x) (2) Subject to gj(x) ≤ 0 j ∈ {1, . . . , J} x ∈ X . However, an optimal solution to formulation (2) may not be in the nonconvex decision set X . Nevertheless, problems (1) and (2) have the same optimal value. Although there have been several techniques utilizing timeaveraged solutions [2], [3], [4], those works are limited to convex formulations. This paper is inspired by the Lyapunov optimization technique [5] which solves stochastic and time-averaged optimization problems. This paper removes the stochastic characteristic and focuses on the connection between the technique and a general convex optimization. This allows a convergence time analysis of a drift-plus-penalty algorithm that solves problem (1). Further, this paper shows that faster convergence can be achieved by starting time averages after a suitable transient period. Another area of literature focuses on convergence ti... |

32 | Delay reduction via Lagrange multiplier sin stochastic network optimization,” in
- Huang, neely
- 2009
(Show Context)
Citation Context .../) and O(1/2) respectively [8], [3], [4]. Two O(1/) first-order methods for constrained optimization are developed in [9], [10], but the results rely on special convex formulations. A second-order method for constrained optimization [11] has a fast convergence rate but rely on special a convex formulation. All of these results rely on convexity assumption that do not hold in formulation (1). This paper develops an algorithm for the formulation (1) and analyzes its convergence time. The algorithm is shown to have O(1/2) convergence time for general problems. However, inspired by results in [12], under a uniqueness assumption on Lagrange multipliers the algorithm is shown to enter two phases: a transient phase and a steady state phase. Convergence time can be significantly improved by restarting the time averages after the transient phase. Specifically, when a dual function satisfies a locally-polyhedron assumption, the modified algorithm has O(1/) convergence time (including the time spent in the transient phase), which equals the best known convergence time for constrained convex optimization via first-order methods. On the other hand, when the dual function satisfies a locally-sm... |

22 | On the O(1/k) convergence of asynchronous distributed Alternating Direction Method of Multipliers,” arXiv preprint arXiv:1307.8254,
- Wei, Ozdaglar
- 2013
(Show Context)
Citation Context ...ther, this paper shows that faster convergence can be achieved by starting time averages after a suitable transient period. Another area of literature focuses on convergence time of first-order algorithms to an -optimal solution to a convex problem, including problem (2). For unconstrained optimization, the optimal first-order method has O(1/ √ ) convergence time [6], [7], while gradient (without strong convexity of objective function) and subgradient methods take O(1/) and O(1/2) respectively [8], [3], [4]. Two O(1/) first-order methods for constrained optimization are developed in [9], [10], but the results rely on special convex formulations. A second-order method for constrained optimization [11] has a fast convergence rate but rely on special a convex formulation. All of these results rely on convexity assumption that do not hold in formulation (1). This paper develops an algorithm for the formulation (1) and analyzes its convergence time. The algorithm is shown to have O(1/2) convergence time for general problems. However, inspired by results in [12], under a uniqueness assumption on Lagrange multipliers the algorithm is shown to enter two phases: a transient phase and a st... |

9 | Distributed and secure computation of convex programs over a network of connected processors,”
- Neely
- 2005
(Show Context)
Citation Context ... CA 90089-2565, supittay@usc.edu, mjneely@usc.edu L. Huang is with Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China, 100084, longbohuang@tsinghua.edu.cn Formulation (1) has an optimal solution which can be converted (by averaging) to the following: Minimize f(x) (2) Subject to gj(x) ≤ 0 j ∈ {1, . . . , J} x ∈ X . However, an optimal solution to formulation (2) may not be in the nonconvex decision set X . Nevertheless, problems (1) and (2) have the same optimal value. Although there have been several techniques utilizing timeaveraged solutions [2], [3], [4], those works are limited to convex formulations. This paper is inspired by the Lyapunov optimization technique [5] which solves stochastic and time-averaged optimization problems. This paper removes the stochastic characteristic and focuses on the connection between the technique and a general convex optimization. This allows a convergence time analysis of a drift-plus-penalty algorithm that solves problem (1). Further, this paper shows that faster convergence can be achieved by starting time averages after a suitable transient period. Another area of literature focuses on convergence time of... |

6 | Optimal distributed gradient methods for network resource allocation problems,” to appear in
- Beck, Nedic, et al.
- 2013
(Show Context)
Citation Context .... Further, this paper shows that faster convergence can be achieved by starting time averages after a suitable transient period. Another area of literature focuses on convergence time of first-order algorithms to an -optimal solution to a convex problem, including problem (2). For unconstrained optimization, the optimal first-order method has O(1/ √ ) convergence time [6], [7], while gradient (without strong convexity of objective function) and subgradient methods take O(1/) and O(1/2) respectively [8], [3], [4]. Two O(1/) first-order methods for constrained optimization are developed in [9], [10], but the results rely on special convex formulations. A second-order method for constrained optimization [11] has a fast convergence rate but rely on special a convex formulation. All of these results rely on convexity assumption that do not hold in formulation (1). This paper develops an algorithm for the formulation (1) and analyzes its convergence time. The algorithm is shown to have O(1/2) convergence time for general problems. However, inspired by results in [12], under a uniqueness assumption on Lagrange multipliers the algorithm is shown to enter two phases: a transient phase an... |

3 |
Distributed cross-layer optimization in wireless networks: A second-order approach,” in INFOCOM,
- Liu, Xia, et al.
- 2013
(Show Context)
Citation Context ...sient period. Another area of literature focuses on convergence time of first-order algorithms to an -optimal solution to a convex problem, including problem (2). For unconstrained optimization, the optimal first-order method has O(1/ √ ) convergence time [6], [7], while gradient (without strong convexity of objective function) and subgradient methods take O(1/) and O(1/2) respectively [8], [3], [4]. Two O(1/) first-order methods for constrained optimization are developed in [9], [10], but the results rely on special convex formulations. A second-order method for constrained optimization [11] has a fast convergence rate but rely on special a convex formulation. All of these results rely on convexity assumption that do not hold in formulation (1). This paper develops an algorithm for the formulation (1) and analyzes its convergence time. The algorithm is shown to have O(1/2) convergence time for general problems. However, inspired by results in [12], under a uniqueness assumption on Lagrange multipliers the algorithm is shown to enter two phases: a transient phase and a steady state phase. Convergence time can be significantly improved by restarting the time averages after the tra... |