An Alternating Trust Region Algorithm for Distributed Linearly Constrained Nonlinear Programs, Application to the AC Optimal Power Flow

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

Variational Analysis In Second-Order Cone Programming And Applications

This dissertation conducts a second-order variational analysis for an important class on nonpolyhedral conic programs generated by the so-called second-order/Lorentz/ice-cream cone. These second-order cone programs (SOCPs) are mathematically challenging due to the nonpolyhedrality of the underlying second-order cone while being important for various applications. The two main devices in our study are second epi-derivative and graphical derivative of the normal cone mapping which are proved to accumulate vital second-order information of functions/constraint systems under investigation. Our main contribution is threefold: - proving the twice epi-differentiability of the indicator function of the second-order cone and of the augmented Lagrangian associated with SOCPs, and deriving explicit formulae for the calculation of the second epi-derivatives of both functions; - establishing a precise formula-entirely via the initial data-for calculating the graphical derivative of the normal cone mapping generated by the constraint set of SOCPs without imposing any nondegeneracy condition; - conducting a complete convergence analysis of the Augmented Lagrangian Method (ALM) for SOCPs with solvability, stability and local convergence analysis of both exact and inexact versions of the ALM under fairly mild assumptions. These results have strong potentials for applications to SOCPs and related problems. Among those presented in this dissertation we mention characterizations of the uniqueness of Lagrange multipliers together with an error bound estimate for second-order cone constraints; of the isolated calmness property for solutions maps of perturbed variational systems associated with SOCPs; and also of (uniform) second-order growth condition for the augmented Lagrangian associated with SOCPs

Sequential quadratic programming with indefinite Hessian approximations for nonlinear optimum experimental design for parameter estimation in differential–algebraic equations

In this thesis we develop algorithms for the numerical solution of problems from nonlinear optimum experimental design (OED) for parameter estimation in differential–algebraic equations. These OED problems can be formulated as special types of path- and control- constrained optimal control (OC) problems. The objective is to minimize a functional on the covariance matrix of the model parameters that is given by first-order sensitivities of the model equations. Additionally, the objective is nonlinearly coupled in time, which make OED problems a challenging class of OC problems. For their numerical solution, we propose a direct multiple shooting parameterization to obtain a structured nonlinear programming problem (NLP). An augmented system of nominal and variational states for the model sensitivities is parameterized on multiple shooting intervals and the objective is decoupled by means of additional variables and constraints. In the resulting NLP, we identify several structures that allow to evaluate derivatives at greatly reduced costs compared to a standard OC formulation. For the solution of the block-structured NLPs, we develop a new sequential quadratic programming (SQP) method. Therein, partitioned quasi-Newton updates are used to approximate the block-diagonal Hessian of the Lagrangian. We analyze a model problem with indefinite, block-diagonal Hessian and prove that positive definite approximations of the individual blocks prevent superlinear convergence. For an OED model problem, we show that more and more negative eigenvalues appear in the Hessian as the multiple shooting grid is refined and confirm the detrimental impact of positive definite Hessian approximations. Hence, we propose indefinite SR1 updates to guarantee fast local convergence. We develop a filter line search globalization strategy that accepts indefinite Hessians based on a new criterion derived from the proof of global convergence. BFGS updates with a scaling strategy to prevent large eigenvalues are used as fallback if the SR1 update does not promote convergence. For the solution of the arising sparse and nonconvex quadratic subproblems, a parametric active set method with inertia control within a Schur complement approach is developed. It employs a symmetric, indefinite LBL T -factorization for the large, sparse KKT matrix and maintains and updates QR-factors of a small and dense Schur complement. The new methods are complemented by two C++ implementations: muse transforms an OED or OC problem instance to a structured NLP by means of direct multiple shooting. A special feature is that fully independent grids for controls, states, path constraints, and measurements are maintained. This provides higher flexibility to adapt the NLP formulation to the characteristics of the problem at hand and facilitates comparison of different formulations in the light of the lifted Newton method. The software package blockSQP is an implementation of the new SQP method that uses a newly developed variant of the quadratic programming solver qpOASES. Numerical results are presented for a benchmark collection of OED and OC problems that show how SR1 approximations improve local convergence over BFGS. The new method is then applied to two challenging OED applications from chemical engineering. Its performance compares favorably to an available existing implementation

On choosing mixture components via non-local priors

Choosing the number of mixture components remains an elusive challenge. Model selection criteria can be either overly liberal or conservative and return poorly-separated components of limited practical use. We formalize non-local priors (NLPs) for mixtures and show how they lead to well-separated components with non-negligible weight, interpretable as distinct subpopulations. We also propose an estimator for posterior model probabilities under local and non-local priors, showing that Bayes factors are ratios of posterior to prior empty-cluster probabilities. The estimator is widely applicable and helps set thresholds to drop unoccupied components in overfitted mixtures. We suggest default prior parameters based on multi-modality for Normal/T mixtures and minimal informativeness for categorical outcomes. We characterise theoretically the NLP-induced sparsity, derive tractable expressions and algorithms. We fully develop Normal, Binomial and product Binomial mixtures but the theory, computation and principles hold more generally. We observed a serious lack of sensitivity of the Bayesian information criterion (BIC), insufficient parsimony of the AIC and a local prior, and a mixed behavior of the singular BIC. We also considered overfitted mixtures, their performance was competitive but depended on tuning parameters. Under our default prior elicitation NLPs offered a good compromise between sparsity and power to detect meaningfully-separated components

Fast numerical methods for mixed--integer nonlinear model--predictive control

This thesis aims at the investigation and development of fast numerical methods for nonlinear mixed--integer optimal control and model- predictive control problems. A new algorithm is developed based on the direct multiple shooting method for optimal control and on the idea of real--time iterations, and using a convex reformulation and relaxation of dynamics and constraints of the original predictive control problem. This algorithm relies on theoretical results and is based on a nonconvex SQP method and a new active set method for nonconvex parametric quadratic programming. It achieves real--time capable control feedback though block structured linear algebra for which we develop new matrix updates techniques. The applicability of the developed methods is demonstrated on several applications. This thesis presents novel results and advances over previously established techniques in a number of areas as follows: We develop a new algorithm for mixed--integer nonlinear model- predictive control by combining Bock's direct multiple shooting method, a reformulation based on outer convexification and relaxation of the integer controls, on rounding schemes, and on a real--time iteration scheme. For this new algorithm we establish an interpretation in the framework of inexact Newton-type methods and give a proof of local contractivity assuming an upper bound on the sampling time, implying nominal stability of this new algorithm. We propose a convexification of path constraints directly depending on integer controls that guarantees feasibility after rounding, and investigate the properties of the obtained nonlinear programs. We show that these programs can be treated favorably as MPVCs, a young and challenging class of nonconvex problems. We describe a SQP method and develop a new parametric active set method for the arising nonconvex quadratic subproblems. This method is based on strong stationarity conditions for MPVCs under certain regularity assumptions. We further present a heuristic for improving stationary points of the nonconvex quadratic subproblems to global optimality. The mixed--integer control feedback delay is determined by the computational demand of our active set method. We describe a block structured factorization that is tailored to Bock's direct multiple shooting method. It has favorable run time complexity for problems with long horizons or many controls unknowns, as is the case for mixed- integer optimal control problems after outer convexification. We develop new matrix update techniques for this factorization that reduce the run time complexity of all but the first active set iteration by one order. All developed algorithms are implemented in a software package that allows for the generic, efficient solution of nonlinear mixed-integer optimal control and model-predictive control problems using the developed methods