Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.Comment: 18 page

Fast, Distributed Optimization Strategies for Resource Allocation in Networks

Many challenges in network science and engineering today arise from systems composed of many individual agents interacting over a network. Such problems range from humans interacting with each other in social networks to computers processing and exchanging information over wired or wireless networks. In any application where information is spread out spatially, solutions must address information aggregation in addition to the decision process itself. Intelligently addressing the trade off between information aggregation and decision accuracy is fundamental to finding solutions quickly and accurately. Network optimization challenges such as these have generated a lot of interest in distributed optimization methods. The field of distributed optimization deals with iterative methods which perform calculations using locally available information. Early methods such as subgradient descent suffer very slow convergence rates because the underlying optimization method is a first order method. My work addresses problems in the area of network optimization and control with an emphasis on accelerating the rate of convergence by using a faster underlying optimization method. In the case of convex network flow optimization, the problem is transformed to the dual domain, moving the equality constraints which guarantee flow conservation into the objective. The Newton direction can be computed locally by using a consensus iteration to solve a Poisson equation, but this requires a lot of communication between neighboring nodes. Accelerated Dual Descent (ADD) is an approximate Newton method, which significantly reduces the communication requirement. Defining a stochastic version of the convex network flow problem with edge capacities yields a problem equivalent to the queue stability problem studied in the backpressure literature. Accelerated Backpressure (ABP) is developed to solve the queue stabilization problem. A queue reduction method is introduced by merging ideas from integral control and momentum based optimization

Distributed online algorithms for energy management in smart grids

The 21st-century electric power grid is transitioning from a centralized structure designed for bulk-power transfer to a distributed paradigm that integrates the variable renewable energy (VRE) resources spatially distributed across the grid. This work proposes algorithmic solutions for distributed economic dispatch based on Subgradient method and Alternating Direction Method of Multipliers (ADMM), both designed to be agnostic with any initialization vector. The proposed distributed online solutions leverage a dynamic average consensus algorithm to track the time-variant linearly coupled constraint that allows an abrupt change in power demand of the network because of the high penetration of VRE resources. The problems are modeled as discrete dynamic systems to investigate the stability and convergence of the algorithm. The update procedures are designed such that the iterates converge to the optimal solution of the original optimization problem, steered by the gain parameter corresponding to the second largest eigenvalue of the system matrix

Stability and instability in saddle point dynamics Part II: The subgradient method

In part I we considered the problem of convergence to a saddle point of a concave-convex function in $C^2$ via gradient dynamics and an exact characterization was given to their asymptotic behaviour. In part II we consider a general class of subgradient dynamics that provide a restriction in a convex domain. We show that despite the nonlinear and non-smooth character of these dynamics their ω-limit set is comprised of solutions to only linear ODEs. In particular, we show that the latter are solutions to subgradient dynamics on affine subspaces which is a smooth class of dynamics the asymptotic properties of which have been exactly characterized in part I. Various convergence criteria are formulated using these results and several examples and applications are also discussed throughout the manuscript.ERC starting grant 67977

Distributed Detection and Estimation in Wireless Sensor Networks

In this article we consider the problems of distributed detection and estimation in wireless sensor networks. In the first part, we provide a general framework aimed to show how an efficient design of a sensor network requires a joint organization of in-network processing and communication. Then, we recall the basic features of consensus algorithm, which is a basic tool to reach globally optimal decisions through a distributed approach. The main part of the paper starts addressing the distributed estimation problem. We show first an entirely decentralized approach, where observations and estimations are performed without the intervention of a fusion center. Then, we consider the case where the estimation is performed at a fusion center, showing how to allocate quantization bits and transmit powers in the links between the nodes and the fusion center, in order to accommodate the requirement on the maximum estimation variance, under a constraint on the global transmit power. We extend the approach to the detection problem. Also in this case, we consider the distributed approach, where every node can achieve a globally optimal decision, and the case where the decision is taken at a central node. In the latter case, we show how to allocate coding bits and transmit power in order to maximize the detection probability, under constraints on the false alarm rate and the global transmit power. Then, we generalize consensus algorithms illustrating a distributed procedure that converges to the projection of the observation vector onto a signal subspace. We then address the issue of energy consumption in sensor networks, thus showing how to optimize the network topology in order to minimize the energy necessary to achieve a global consensus. Finally, we address the problem of matching the topology of the network to the graph describing the statistical dependencies among the observed variables.Comment: 92 pages, 24 figures. To appear in E-Reference Signal Processing, R. Chellapa and S. Theodoridis, Eds., Elsevier, 201

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Towards Adaptive and Grid-Transparent Adjoint-Based Design Optimization Frameworks

With the growing environmental consciousness, the global perspective in energy production is shifting towards renewable resources. As recently reported by the Office of Energy Efficiency & Renewable Energy at the U.S. Department of Energy, wind-generated electricity is the least expensive form of renewable power and is becoming one of the cheapest forms of electricity from any source. The aeromechanical design of wind turbines is a complex and multidisciplinary task which necessitates a high-fidelity flow solver as well as efficient design optimization tools. With the advances in computer technologies, Computational Fluid Dynamics (CFD) has established its role as a high-fidelity tool for aerodynamic design.In this dissertation, a grid-transparent unstructured two- and three-dimensional compressible Reynolds-Averaged Navier-Stokes (RANS) solver, named UNPAC, is developed. This solver is enhanced with an algebraic transition model that has proven to offer accurate flow separation and reattachment predictions for the transitional flows. For the unsteady time-periodic flows, a harmonic balance (HB) method is incorporated that couples the sub-time level solutions over a single period via a pseudo-spectral operator. Convergence to the steady-state solution is accelerated using a novel reduced-order-model (ROM) approach that can offer significant reductions in the number of iterations as well as CPU times for the explicit solver. The unstructured grid is adapted in both steady and HB cases using an r-adaptive mesh redistribution (AMR) technique that can efficiently cluster nodes around regions of large flow gradients.Additionally, a novel toolbox for sensitivity analysis based on the discrete adjoint method is developed in this work. The Fast automatic Differentiation using Operator-overloading Technique (FDOT) toolbox uses an iterative process to evaluate the sensitivities of the cost function with respect to the entire design space and requires only minimal modifications to the available solver. The FDOT toolbox is coupled with the UNPAC solver to offer fast and accurate gradient information. Ultimately, a wrapper program for the design optimization framework, UNPAC-DOF, has been developed. The nominal and adjoint flow solutions are directly incorporated into a gradient-based design optimization algorithm with the goal of improving designs in terms of minimized drag or maximized efficiency