266 research outputs found
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
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