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

An elastic primal active-set method for structured QPs

[no abstract

Practical Enhancements in Sequential Quadratic Optimization: Infeasibility Detection, Subproblem Solvers, and Penalty Parameter Updates

The primary focus of this dissertation is the design, analysis, and implementation of numerical methods to enhance Sequential Quadratic Optimization (SQO) methods for solving nonlinear constrained optimization problems. These enhancements address issues that challenge the practical limitations of SQO methods. The first part of this dissertation presents a penalty SQO algorithm for nonlinear constrained optimization. The method attains all of the strong global and fast local convergence guarantees of classical SQO methods, but has the important additional feature that fast local convergence is guaranteed when the algorithm is employed to solve infeasible instances. A two-phase strategy, carefully constructed parameter updates, and a line search are employed to promote such convergence. The first-phase subproblem determines the reduction that can be obtained in a local model of constraint violation. The second-phase subproblem seeks to minimize a local model of a penalty function. The solutions of both subproblems are then combined to form the search direction, in such a way that it yields a reduction in the local model of constraint violation that is proportional to the reduction attained in the first phase. The subproblem formulations and parameter updates ensure that near an optimal solution, the algorithm reduces to a classical SQO method for constrained optimization, and near an infeasible stationary point, the algorithm reduces to a (perturbed) SQO method for minimizing constraint violation. Global and local convergence guarantees for the algorithm are proved under reasonable assumptions and numerical results are presented for a large set of test problems.In the second part of this dissertation, two matrix-free methods are presented for approximately solving exact penalty subproblems of large scale. The first approach is a novel iterative re-weighting algorithm (IRWA), which iteratively minimizes quadratic models of relaxed subproblems while simultaneously updating a relaxation vector. The second approach recasts the subproblem into a linearly constrained nonsmooth optimization problem and then applies alternating direction augmented Lagrangian (ADAL) technology to solve it. The main computational costs of each algorithm are the repeated minimizations of convex quadratic functions, which can be performed matrix-free. Both algorithms are proved to be globally convergent under loose assumptions, and each requires at most $O(1/\varepsilon^2)$ iterations to reach $\varepsilon$-optimality of the objective function. Numerical experiments exhibit the ability of both algorithms to efficiently find inexact solutions. Moreover, in certain cases, IRWA is shown to be more reliable than ADAL. In the final part of this dissertation, we focus on the design of the penalty parameter updating strategy in penalty SQO methods for solving large-scale nonlinear optimization problems. As the most computationally demanding aspect of such an approach is the computation of the search direction during each iteration, we consider the use of matrix-free methods for solving the direction-finding subproblems within SQP methods. This allows for the acceptance of inexact subproblem solutions, which can significantly reduce overall computational costs. In addition, such a method can be plagued by poor behavior of the global convergence mechanism, for which we consider the use of an exact penalty function. To confront this issue, we propose a dynamic penalty parameter updating strategy to be employed within the subproblem solver in such a way that the resulting search direction predicts progress toward both feasibility and optimality. We present our penalty parameter updating strategy and prove that does not decrease the penalty parameter unnecessarily in the neighborhood of points satisfying certain common assumptions. We also discuss two matrix-free subproblem solvers in which our updating strategy can be readily incorporated

Optimization and Applications

Proceedings of a workshop devoted to optimization problems, their theory and resolution, and above all applications of them. The topics covered existence and stability of solutions; design, analysis, development and implementation of algorithms; applications in mechanics, telecommunications, medicine, operations research

Interior-point methods for PDE-constrained optimization

In applied sciences PDEs model an extensive variety of phenomena. Typically the final goal of simulations is a system which is optimal in a certain sense. For instance optimal control problems identify a control to steer a system towards a desired state. Inverse problems seek PDE parameters which are most consistent with measurements. In these optimization problems PDEs appear as equality constraints. PDE-constrained optimization problems are large-scale and often nonconvex. Their numerical solution leads to large ill-conditioned linear systems. In many practical problems inequality constraints implement technical limitations or prior knowledge. In this thesis interior-point (IP) methods are considered to solve nonconvex large-scale PDE-constrained optimization problems with inequality constraints. To cope with enormous fill-in of direct linear solvers, inexact search directions are allowed in an inexact interior-point (IIP) method. This thesis builds upon the IIP method proposed in [Curtis, Schenk, Wächter, SIAM Journal on Scientific Computing, 2010]. SMART tests cope with the lack of inertia information to control Hessian modification and also specify termination tests for the iterative linear solver. The original IIP method needs to solve two sparse large-scale linear systems in each optimization step. This is improved to only a single linear system solution in most optimization steps. Within this improved IIP framework, two iterative linear solvers are evaluated: A general purpose algebraic multilevel incomplete L D L^T preconditioned SQMR method is applied to PDE-constrained optimization problems for optimal server room cooling in three space dimensions and to compute an ambient temperature for optimal cooling. The results show robustness and efficiency of the IIP method when compared with the exact IP method. These advantages are even more evident for a reduced-space preconditioned (RSP) GMRES solver which takes advantage of the linear system's structure. This RSP-IIP method is studied on the basis of distributed and boundary control problems originating from superconductivity and from two-dimensional and three-dimensional parameter estimation problems in groundwater modeling. The numerical results exhibit the improved efficiency especially for multiple PDE constraints. An inverse medium problem for the Helmholtz equation with pointwise box constraints is solved by IP methods. The ill-posedness of the problem is explored numerically and different regularization strategies are compared. The impact of box constraints and the importance of Hessian modification on the optimization algorithm is demonstrated. A real world seismic imaging problem is solved successfully by the RSP-IIP method