740 research outputs found
Bounding stationary averages of polynomial diffusions via semidefinite programming
We introduce an algorithm based on semidefinite programming that yields
increasing (resp. decreasing) sequences of lower (resp. upper) bounds on
polynomial stationary averages of diffusions with polynomial drift vector and
diffusion coefficients. The bounds are obtained by optimising an objective,
determined by the stationary average of interest, over the set of real vectors
defined by certain linear equalities and semidefinite inequalities which are
satisfied by the moments of any stationary measure of the diffusion. We
exemplify the use of the approach through several applications: a Bayesian
inference problem; the computation of Lyapunov exponents of linear ordinary
differential equations perturbed by multiplicative white noise; and a
reliability problem from structural mechanics. Additionally, we prove that the
bounds converge to the infimum and supremum of the set of stationary averages
for certain SDEs associated with the computation of the Lyapunov exponents, and
we provide numerical evidence of convergence in more general settings
Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization
We describe methods for proving upper and lower bounds on infinite-time
averages in deterministic dynamical systems and on stationary expectations in
stochastic systems. The dynamics and the quantities to be bounded are assumed
to be polynomial functions of the state variables. The methods are
computer-assisted, using sum-of-squares polynomials to formulate sufficient
conditions that can be checked by semidefinite programming. In the
deterministic case, we seek tight bounds that apply to particular local
attractors. An obstacle to proving such bounds is that they do not hold
globally; they are generally violated by trajectories starting outside the
local basin of attraction. We describe two closely related ways past this
obstacle: one that requires knowing a subset of the basin of attraction, and
another that considers the zero-noise limit of the corresponding stochastic
system. The bounding methods are illustrated using the van der Pol oscillator.
We bound deterministic averages on the attracting limit cycle above and below
to within 1%, which requires a lower bound that does not hold for the unstable
fixed point at the origin. We obtain similarly tight upper and lower bounds on
stochastic expectations for a range of noise amplitudes. Limitations of our
methods for certain types of deterministic systems are discussed, along with
prospects for improvement.Comment: 25 pages; Added new Section 7.2; Added references; Corrected typos;
Submitted to SIAD
Adjoint-based predictor-corrector sequential convex programming for parametric nonlinear optimization
This paper proposes an algorithmic framework for solving parametric
optimization problems which we call adjoint-based predictor-corrector
sequential convex programming. After presenting the algorithm, we prove a
contraction estimate that guarantees the tracking performance of the algorithm.
Two variants of this algorithm are investigated. The first one can be used to
solve nonlinear programming problems while the second variant is aimed to treat
online parametric nonlinear programming problems. The local convergence of
these variants is proved. An application to a large-scale benchmark problem
that originates from nonlinear model predictive control of a hydro power plant
is implemented to examine the performance of the algorithms.Comment: This manuscript consists of 25 pages and 7 figure
Global Optimisation for Energy System
The goal of global optimisation is to find globally optimal solutions, avoiding local optima and other stationary points. The aim of this thesis is to provide more efficient global optimisation tools for energy systems planning and operation. Due to the ongoing increasing of complexity and decentralisation of power systems, the use of advanced mathematical techniques that produce reliable solutions becomes necessary. The task of developing such methods is complicated by the fact that most energy-related problems are nonconvex due to the nonlinear Alternating Current Power Flow equations and the existence of discrete elements. In some cases, the computational challenges arising from the presence of non-convexities can be tackled by relaxing the definition of convexity and identifying classes of problems that can be solved to global optimality by polynomial time algorithms. One such property is known as invexity and is defined by every stationary point of a problem being a global optimum. This thesis investigates how the relation between the objective function and the structure of the feasible set is connected to invexity and presents necessary conditions for invexity in the general case and necessary and sufficient conditions for problems with two degrees of freedom. However, nonconvex problems often do not possess any provable convenient properties, and specialised methods are necessary for providing global optimality guarantees. A widely used technique is solving convex relaxations in order to find a bound on the optimal solution. Semidefinite Programming relaxations can provide good quality bounds, but they suffer from a lack of scalability. We tackle this issue by proposing an algorithm that combines decomposition and linearisation approaches. In addition to continuous non-convexities, many problems in Energy Systems model discrete decisions and are expressed as mixed-integer nonlinear programs (MINLPs). The formulation of a MINLP is of significant importance since it affects the quality of dual bounds. In this thesis we investigate algebraic characterisations of on/off constraints and develop a strengthened version of the Quadratic Convex relaxation of the Optimal Transmission Switching problem. All presented methods were implemented in mathematical modelling and optimisation frameworks PowerTools and Gravity
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