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A novel trust region method for solving linearly constrained nonlinear programs is presented. The proposed technique is amenable to a distributed implementation, as its salient ingredient is an alternating projected gradient sweep in place of the Cauchy point computation. It is proven that the algorithm yields a sequence that globally converges to a critical point. As a result of some changes to the standard trust region method, namely a proximal regularisation of the trust region subproblem, it is shown that the local convergence rate is linear with an arbitrarily small ratio. Thus, convergence is locally almost superlinear, under standard regularity assumptions. The proposed method is successfully applied to compute local solutions to alternating current optimal power flow problems in transmission and distribution networks. Moreover, the new mechanism for computing a Cauchy point compares favourably against the standard projected search as for its activity detection properties.

Mean-Field is an efficient way to approximate a posterior distribution in complex graphical models and constitutes the most popular class of Bayesian variational approximation methods. In most applications, the mean field distribution parameters are computed using an alternate coordinate minimization. However, the convergence properties of this algorithm remain unclear. In this paper, we show how, by adding an appropriate penalization term, we can guarantee convergence to a critical point, while keeping a closed form update at each step. A convergence rate estimate can also be derived based on recent results in non-convex optimization., Comment: Submitted to Colt 2015

A novel decomposition scheme to solve parametric non-convex programs as they arise in Nonlinear Model Predictive Control (NMPC) is presented. It consists of a fixed number of alternating proximal gradient steps and a dual update per time step. Hence, the proposed approach is attractive in a real-time distributed context. Assuming that the Nonlinear Program (NLP) is semi-algebraic and that its critical points are strongly regular, contraction of the sequence of primal-dual iterates is proven, implying stability of the sub-optimality error, under some mild assumptions. Moreover, it is shown that the performance of the optimality-tracking scheme can be enhanced via a continuation technique. The efficacy of the proposed decomposition method is demonstrated by solving a centralised NMPC problem to control a DC motor and a distributed NMPC program for collaborative tracking of unicycles, both within a real-time framework. Furthermore, an analysis of the sub-optimality error as a function of the sampling period is proposed given a fixed computational power., 16 pages, 9 figures

A novel Nonlinear Model Predictive Control (NMPC) scheme is proposed in order to shape the harmonic response of constrained nonlinear systems. The salient ingredient is the short-time Fourier transform (STFT) of the system's output signal, which is constrained in an NMPC problem, leading to the novel formulation of so-called spectrum constraints. Recursive feasibility and asymptotic stability of the proposed NMPC scheme with such spectrum constraints are guaranteed by means of an appropriate ellipsoidal terminal invariant set. The efficacy of the proposed approach is demonstrated on a nonlinear vibration damping problem.

This thesis deals with the development of numerical methods for solving nonconvex optimisation problems by means of decomposition and continuation techniques. We first introduce a novel decomposition algorithm based on alternating gradient projections and augmented Lagrangian relaxations. A proof of local convergence is given under standard assumptions. The effect of different stopping criteria on the convergence of the augmented Lagrangian loop is investigated. As a second step, a trust region algorithm for distributed nonlinear programs, named TRAP, is introduced. Its salient ingredient is an alternating gradient projection for computing a set of active constraints in a distributed manner, which is a novelty for trust region techniques. Global convergence as well as local almost superlinear convergence are proven. The numerical performance of the algorithm is assessed on nonconvex optimal power flow problems. The core of this thesis is the development and analysis of an augmented Lagrangian algorithm for tracking parameter-dependent optima. Despite their interesting features for large-scale and distributed optimisation, augmented Lagrangian methods have not been designed and fully analysed in a parametric setting. Therefore, we propose a novel optimality-tracking scheme that consists of fixed number of descent steps computed on an augmented Lagrangian subproblem and one dual update per parameter change. It is shown that the descent steps can be performed by means of first-order as well as trust region methods. Using the Kurdyka-Lojasiewicz property, an analysis of the local convergence rate of a class of trust region Newton methods is provided without relying on the finite detection of an optimal active set. This allows us to establish a contraction inequality for the parametric augmented Lagrangian algorithm. Hence, stability of the continuation scheme can be proven under mild assumptions. The effect of the number of primal iterations and the penalty is analysed by means of numerical examples. Finally, the efficacy of the augmented Lagrangian continuation scheme is successfully demonstrated on three examples in the field of optimal control. The first two examples consists of a real-time NMPC algorithm based on a multiple-shooting discretisation. In particular, it is shown that our C++ software package is competitive with the state-of-the-art codes on NMPC problems with long prediction horizons, and can address a more general class of real-time NMPC problems. The third case study is the distributed computation of solutions to multi-stage nonconvex optimal power flow problems in a real-time setting.

A novel trust region method for solving linearly constrained nonlinear programs is presented. The proposed technique is amenable to a distributed implementation, as its salient ingredient is an alternating projected gradient sweep in place of the Cauchy point computation. It is proven that the algorithm yields a sequence that globally converges to a critical point. As a result of some changes to the standard trust region method, namely a proximal regularisation of the trust region subproblem, it is shown that the local convergence rate is linear with an arbitrarily small ratio. Thus, convergence is locally almost superlinear, under standard regularity assumptions. The proposed method is successfully applied to compute local solutions to alternating current optimal power flow problems in transmission and distribution networks. Moreover, the new mechanism for computing a Cauchy point compares favourably against the standard projected search, as for its activity detection properties.

A novel model predictive control (MPC) scheme that allows one to enforce hard constraints on the spectrum of a constrained system’s output signal is presented. The approach is based on a time-local analysis of the spectrum of output signals by means of the short-time Fourier transform (STFT), and its squared magnitude, called the spectrogram. It is shown that an MPC problem with spectrogram constraints can be formulated as a quadratically constrained quadratic program (QCQP). We prove recursive feasibility and stability of the proposed spectrogram MPC scheme via an ellipsoidal invariant set, including spectrogram constraints. Moreover, it is pointed out how the proposed spectrogram MPC approach can be extended to MPC for tracking while ensuring recursive feasibility. Finally, we present simulation results of spectrogram MPC applied to a resonant system. Our simulations show that, by employing the proposed spectrogram MPC approach, oscillations can be attenuated in the system output, tracking a reference signal, by explicitly enforcing hard constraints on its spectrum.

In this paper, a novel optimality-tracking algorithm for solving Economic Nonlinear Model Predictive Control (ENMPC) problems in real-time is presented. Developing online schemes for ENMPC is challenging, since it is unclear how convexity of the Quadratic Programming (QP) problem, which is obtained by linearisation of the NMPC program around the current iterate, can be enforced efficiently. Therefore, we propose addressing the problem by means of an augmented Lagrangian formulation. Our tracking scheme consists of a fixed number of inexact Newton steps computed on an augmented Lagrangian subproblem followed by a dual update per time step. Under mild assumptions on the number of iterations and the penalty parameter, it can be proven that the sub-optimality error provided by the parametric algorithm remains bounded over time. This result extends the authors' previous works from a theoretical and a computational perspective. Efficacy of the approach is demonstrated on an ENMPC example consisting of a bioreactor.

A novel decomposition scheme to solve parametric non-convex programs as they arise in Nonlinear Model Predictive Control (NMPC) is presented. It consists of a fixed number of alternating proximal gradient steps and a dual update per time step. Hence, the proposed approach is attractive in a real-time distributed context. Assuming that the Nonlinear Program (NLP) is semi-algebraic and that its critical points are strongly regular, contraction of the sequence of primal-dual iterates is proven, implying stability of the sub-optimality error, under some mild assumptions. Moreover, it is shown that the performance of the optimality-tracking scheme can be enhanced via a continuation technique. The efficacy of the proposed decomposition method is demonstrated by solving a centralised NMPC problem to control a DC motor and a distributed NMPC program for collaborative tracking of unicycles, both within a real-time framework. Furthermore, an analysis of the sub-optimality error as a function of the sampling period is proposed given a fixed computational power.

A novel splitting scheme to solve parametric multi-convex programs is presented. It consists of a fixed number of proximal alternating minimisations and a dual update per time step, which makes it attractive in a real-time Nonlinear Model Predictive Control (NMPC) framework and for distributed computing environments. Assuming that the parametric program is semi-algebraic and that its critical points are strongly regular, a contraction estimate is derived and it is proven that the sub-optimality error remains stable under some mild assumptions. Efficacy of the method is demonstrated by solving a bilinear NMPC problem to control a DC motor. In particular, the effect of the sampling period on the optimality tracking error is analysed for a fixed computational power.

A novel trust region method for solving linearly constrained nonlinear programs is presented. The proposed technique is amenable to a distributed implementation, as its salient ingredient is an alternating projected gradient sweep in place of the Cauchy point computation. It is proven that the algorithm yields a sequence that globally converges to a critical point. As a result of some changes to the standard trust region method, namely a proximal regularisation of the trust region subproblem, it is shown that the local convergence rate is linear with an arbitrarily small ratio. Thus, convergence is locally almost superlinear, under standard regularity assumptions. The proposed method is successfully applied to compute local solutions to alternating current optimal power flow problems in transmission and distribution networks. Moreover, the new mechanism for computing a Cauchy point compares favourably against the standard projected search as for its activity detection properties.

The optimal power flow (OPF) problem, a fundamental problem in power systems, is generally nonconvex and computationally challenging for networks with an increasing number of smart devices and real-time control requirements. In this paper, we first investigate a fully distributed approach by means of the augmented Lagrangian and proximal alternating minimization method to solve the nonconvex OPF problem with a convergence guarantee. Given time-critical requirements, we then extend the algorithm to a distributed parametric tracking scheme with practical warm-starting and termination strategies, which aims to provide a closed-loop sub-optimal control policy while taking into account the grid information updated at the time of decision making. The effectiveness of the proposed algorithm for real-time nonconvex OPF problems is demonstrated in numerical simulations.

A novel augmented Lagrangian method for solving non-convex programs with nonlinear cost and constraint couplings in a distributed framework is presented. The proposed decomposition algorithm is made of two layers: The outer level is a standard multiplier method with penalty on the nonlinear equality constraints, while the inner level consists of a block-coordinate descent (BCD) scheme. Based on standard results on multiplier methods and recent results on proximal regularised BCD techniques, it is proven that the method converges to a KKT point of the non-convex nonlinear program under a semi-algebraicity assumption. Efficacy of the algorithm is demonstrated on a numerical example.