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Register a new userTime-Varying Optimization of LTI Systems via Projected Primal-Dual Gradient Flows
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This paper investigates the problem of regulating in real time a linear dynamical system to the solution trajectory of a time-varying constrained convex optimization problem. The proposed feedback controller is based on an adaptation of the saddle-flow dynamics, modified to take into account projections on constraint sets and output-feedback from the plant. We derive sufficient conditions on the tunable parameters of the controller (inherently related to the time-scale separation between plant and controller dynamics) to guarantee exponential and input-to-state stability of the closed-loop system. The analysis is tailored to the case of time-varying strongly convex cost functions and polytopic output constraints. The theoretical results are further validated in a ramp metering control problem in a network of traffic highways.
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We investigate a distributed optimization problem over a cooperative multi-agent time-varying network, where each agent has its own decision variables that should be set so as to minimize its individual objective subject to local constraints and global coupling constraints. Based on push-sum protocol and dual decomposition, we design a distributed regularized dual gradient algorithm to solve this problem, in which the algorithm is implemented in time-varying directed graphs only requiring the column stochasticity of communication matrices. By augmenting the corresponding Lagrangian function with a quadratic regularization term, we first obtain the bound of the Lagrangian multipliers which does not require constructing a compact set containing the dual optimal set when compared with most of primal-dual based methods. Then, we obtain that the convergence rate of the proposed method can achieve the order of $mathcal{O}(ln T/T)$ for strongly convex objective functions, where $T$ is the iterations. Moreover, the explicit bound of constraint violations is also given. Finally, numerical results on the network utility maximum problem are used to demonstrate the efficiency of the proposed algorithm.
In this work, we revisit a classical incremental implementation of the primal-descent dual-ascent gradient method used for the solution of equality constrained optimization problems. We provide a short proof that establishes the linear (exponential) convergence of the algorithm for smooth strongly-convex cost functions and study its relation to the non-incremental implementation. We also study the effect of the augmented Lagrangian penalty term on the performance of distributed optimization algorithms for the minimization of aggregate cost functions over multi-agent networks.
We consider a two-stage stochastic optimization problem, in which a long-term optimization variable is coupled with a set of short-term optimization variables in both objective and constraint functions. Despite that two-stage stochastic optimization plays a critical role in various engineering and scientific applications, there still lack efficient algorithms, especially when the long-term and short-term variables are coupled in the constraints. To overcome the challenge caused by tightly coupled stochastic constraints, we first establish a two-stage primal-dual decomposition (PDD) method to decompose the two-stage problem into a long-term problem and a family of short-term subproblems. Then we propose a PDD-based stochastic successive convex approximation (PDD-SSCA) algorithmic framework to find KKT solutions for two-stage stochastic optimization problems. At each iteration, PDD-SSCA first runs a short-term sub-algorithm to find stationary points of the short-term subproblems associated with a mini-batch of the state samples. Then it constructs a convex surrogate for the long-term problem based on the deep unrolling of the short-term sub-algorithm and the back propagation method. Finally, the optimal solution of the convex surrogate problem is solved to generate the next iterate. We establish the almost sure convergence of PDD-SSCA and customize the algorithmic framework to solve two important application problems. Simulations show that PDD-SSCA can achieve superior performance over existing solutions.
Decentralized optimization over time-varying graphs has been increasingly common in modern machine learning with massive data stored on millions of mobile devices, such as in federated learning. This paper revisits the widely used accelerated gradient tracking and extends it to time-varying graphs. We prove the $O((frac{gamma}{1-sigma_{gamma}})^2sqrt{frac{L}{epsilon}})$ and $O((frac{gamma}{1-sigma_{gamma}})^{1.5}sqrt{frac{L}{mu}}logfrac{1}{epsilon})$ complexities for the practical single loop accelerated gradient tracking over time-varying graphs when the problems are nonstrongly convex and strongly convex, respectively, where $gamma$ and $sigma_{gamma}$ are two common constants charactering the network connectivity, $epsilon$ is the desired precision, and $L$ and $mu$ are the smoothness and strong convexity constants, respectively. Our complexities improve significantly over the ones of $O(frac{1}{epsilon^{5/7}})$ and $O((frac{L}{mu})^{5/7}frac{1}{(1-sigma)^{1.5}}logfrac{1}{epsilon})$, respectively, which were proved in the original literature only for static graphs, where $frac{1}{1-sigma}$ equals $frac{gamma}{1-sigma_{gamma}}$ when the network is time-invariant. When combining with a multiple consensus subroutine, the dependence on the network connectivity constants can be further improved to $O(1)$ and $O(frac{gamma}{1-sigma_{gamma}})$ for the computation and communication complexities, respectively. When the network is static, by employing the Chebyshev acceleration, our complexities exactly match the lower bounds without hiding any poly-logarithmic factor for both nonstrongly convex and strongly convex problems.
This paper considers a time-varying optimization problem associated with a network of systems, with each of the systems shared by (and affecting) a number of individuals. The objective is to minimize cost functions associated with the individuals preferences, which are unknown, subject to time-varying constraints that capture physical or operational limits of the network. To this end, the paper develops a distributed online optimization algorithm with concurrent learning of the cost functions. The cost functions are learned on-the-fly based on the users feedback (provided at irregular intervals) by leveraging tools from shape-constrained Gaussian Processes. The online algorithm is based on a primal-dual method, and acts effectively in a closed-loop fashion where: i) users feedback is utilized to estimate the cost, and ii) measurements from the network are utilized in the algorithmic steps to bypass the need for sensing of (unknown) exogenous inputs of the network. The performance of the algorithm is analyzed in terms of dynamic network regret and constraint violation. Numerical examples are presented in the context of real-time optimization of distributed energy resources.
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