Distributed approaches for solving non-convex optimizations under strong duality

This dissertation studies non-convex optimizations under the strong duality condition. In general, non-convex problems are non-deterministic polynomial-time (NP) hard and hence are difficult to solve. However, when strong duality holds, one can recover primal optimal solutions by efficiently solving their convex, associated Lagrange dual relaxations instead. For certain non-convex optimizations, such as optimal power flow (OPF), phase recovery, and etc., it has been shown that strong duality holds under most circumstances. Consequently, their associated Lagrange dual problems, usually expressed in semi-definite programming (SDP) forms, have attracted the attention of researchers. However, we notice that these SDP Lagrange duals are in general not amenable to be solved in a distributed manner. To address this issue, we propose two distributed approaches in this dissertation to study those non-convex optimizations under the condition of strong duality. Our first approach is called the continuous-time optimization dynamics approach. Rather than considering the Lagrange dual alone, this approach studies the primal and dual problems together at the same time. By viewing primal and dual variables of an optimization problem as opponents playing a min-max game, the evolution of the optimization dynamical system can be interpreted as a competition between two players. This competition will not stop until those players achieve a balance, which turns out to be an equilibrium to the optimization dynamics and is mathematically characterized as a Karush-Kuhn-Tucker (KKT) point. A convergence analysis is then developed in this dissertation, showing that under certain conditions a KKT equilibrium of the associated optimization dynamics is locally asymptotically stable. However, after the optimization dynamics converges to a KKT equilibrium, we have to further check whether or not the obtained KKT point is globally optimal, since KKT conditions are only necessary for the local optimality of non-convex problems. It motivates us to investigate under what conditions a KKT point can be guaranteed as a global optimum. Then in the dissertation we derive a global optimality condition for general quadratically constrained quadratic programmings (QCQPs). If an isolated KKT point of a general QCQP satisfies our condition, then it is locally asymptotically stable with respect to the optimization dynamics. We next apply the optimization dynamics approach to a special class of non-convex QCQPs, namely, the OPF problems. We discover that their associated optimization dynamical systems possess an intrinsic distributed structure. Simulations are also provided to show the effectiveness of our continuous-time optimization dynamics approach. Alternatively, we also propose a consensus-based, decomposed SDP relaxation approach to solve OPF problems. Here we continue to exploit the distributed structure of power networks, which finally enables us to decompose the standard SDP relaxation into a bunch of smaller size SDPs. Then each bus in the power network should locally solve its own decomposed SDP relaxation and meanwhile a global OPF solution can be achieved by running a consensus over the whole power network. Hence this approach can be implemented in a distributed manner. We show that the size of our decomposed SDP relaxation scales linearly as the power network expands, so it can greatly fasten the OPF computation speed

Optimisation of an integrated transport and distribution system

Imperial Users onl

Morse theory on spaces of braids and Lagrangian dynamics

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index and provides a gloablization of `lap number' techniques used in scalar parabolic PDEs. In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits.Comment: Revised version: numerous changes in exposition. Slight modification of two proofs and one definition; 55 pages, 20 figure

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

Inviolable energy conditions from entanglement inequalities

Via the AdS/CFT correspondence, fundamental constraints on the entanglement structure of quantum systems translate to constraints on spacetime geometries that must be satisfied in any consistent theory of quantum gravity. In this paper, we investigate such constraints arising from strong subadditivity and from the positivity and monotonicity of relative entropy in examples with highly-symmetric spacetimes. Our results may be interpreted as a set of energy conditions restricting the possible form of the stress-energy tensor in consistent theories of Einstein gravity coupled to matter.Comment: 25 pages, 3 figures, v2: refs added, expanded discussion of strong subadditivity constraints in sections 2.1 and 4.

Minimum Cost Distributed Source Coding Over a Network

This paper considers the problem of transmitting multiple compressible sources over a network at minimum cost. The aim is to find the optimal rates at which the sources should be compressed and the network flows using which they should be transmitted so that the cost of the transmission is minimal. We consider networks with capacity constraints and linear cost functions. The problem is complicated by the fact that the description of the feasible rate region of distributed source coding problems typically has a number of constraints that is exponential in the number of sources. This renders general purpose solvers inefficient. We present a framework in which these problems can be solved efficiently by exploiting the structure of the feasible rate regions coupled with dual decomposition and optimization techniques such as the subgradient method and the proximal bundle method