Andreas Grothey

Local Solutions of the Optimal Power Flow Problem

The existence of locally optimal solutions to the AC optimal power flow problem (OPF) has been a question of interest for decades. This paper presents examples of local optima on a variety of test networks including modified versions of common networks. We show that local optima can occur because the feasible region is disconnected and/or because of nonlinearities in the constraints. Standard local optimization techniques are shown to converge to these local optima. The voltage bounds of all the examples in this paper are between ±5% and ±10% off-nominal. The examples with local optima are available in an online archive (http://www.maths.ed.ac.uk/optenergy/LocalOpt/) and can be used to test local or global optimization techniques for OPF. Finally we use our test examples to illustrate the behavior of a recent semi-definite programming approach that aims to find the global solution of OPF.

Reoptimization With the Primal-Dual Interior Point Method

Reoptimization techniques for an interior point method applied to solving a sequence of linear programming problems are discussed. Conditions are given for problem perturbations that can be absorbed in merely one Newton step. The analysis is performed for both short-step and long-step feasible path-following methods. A practical procedure is then derived for an infeasible path-following method. It is applied in the context of crash start for several large-scale structured linear programs. Numerical results with OOPS, a new object-oriented parallel solver, demonstrate the efficiency of the approach. For large structured linear programs, crash start leads to about 40% reduction in the number of iterations and translates into a 25% reduction of the solution time. The crash procedure parallelizes well, and speed-ups between 3.1--3.8 on four processors are achieved.

Exploiting structure in parallel implementation of interior point methods for optimization

OOPS is an object-oriented parallel solver using the primal–dual interior point methods. Its main component is an object-oriented linear algebra library designed to exploit nested block structure that is often present in truly large-scale optimization problems such as those appearing in Stochastic Programming. This is achieved by treating the building blocks of the structured matrices as objects, that can use their inherent linear algebra implementations to efficiently exploit their structure both in a serial and parallel environment. Virtually any nested block-structure can be exploited by representing the matrices defining the problem as a tree build from these objects. OOPS can be run on a wide variety of architectures and has been used to solve a financial planning problem with over 109 decision variables. We give details of supported structures and their implementations. Further we give details of how parallelisation is managed in the object-oriented framework.

A New Unblocking Technique to Warmstart Interior Point Methods Based on Sensitivity Analysis

One of the main drawbacks associated with Interior Point Methods (IPMs) is the perceived lack of an efficient warmstarting scheme which would enable the use of information from a previous solution of a similar problem. Recently there has been renewed interest in the subject. A common problem with warmstarting for IPM is that an advanced starting point which is close to the boundary of the feasible region, as is typical, might lead to blocking of the search direction. Several techniques have been proposed to address this issue. Most of these aim to lead the iterate back into the interior of the feasible region—we classify them as either “modification steps” or “unblocking steps” depending on whether the modification is taking place before solving the modified problem to prevent future problems, or during the solution if and when problems become apparent. A new “unblocking” strategy is suggested which attempts to directly address the issue of blocking by performing sensitivity analysis on the Newton step with the aim of increasing the size of the step that can be taken. This analysis is used in a new technique to warmstart interior point methods: we identify components of the starting point that are responsible for blocking and aim to improve these by using our sensitivity analysis. The relative performance of a selection of different warmstarting techniques suggested in the literature and the new proposed unblocking by sensitivity analysis is evaluated on the warmstarting test set based on a selection of NETLIB problems proposed by [Benson and Shanno, Comput. Optim. Appl., 38 (2007), pp. 371-399]. Warmstarting techniques are also applied in the context of solving nonlinear programming problems as a sequence of quadratic programs solved by interior point methods. We also apply the warmstarting technique to the problem of finding the complete efficient frontier in portfolio management problems (a problem with 192 million variables—to our knowledge the largest problem to date solved by a warmstarted IPM). We find that the resulting best combined warmstarting strategy manages to save between 50 and 60% of interior point iterations, consistently outperforming similar approaches reported in current optimization literature.

Parallel interior-point solver for structured quadratic programs: Application to financial planning problems

Many practical large-scale optimization problems are not only sparse, but also display some form of block-structure such as primal or dual block angular structure. Often these structures are nested: each block of the coarse top level structure is block-structured itself. Problems with these characteristics appear frequently in stochastic programming but also in other areas such as telecommunication network modelling.We present a linear algebra library tailored for problems with such structure that is used inside an interior point solver for convex quadratic programming problems. Due to its object-oriented design it can be used to exploit virtually any nested block structure arising in practical problems, eliminating the need for highly specialised linear algebra modules needing to be written for every type of problem separately. Through a careful implementation we achieve almost automatic parallelisation of the linear algebra.The efficiency of the approach is illustrated on several problems arising in the financial planning, namely in the asset and liability management. The problems are modelled as multistage decision processes and by nature lead to nested block-structured problems. By taking the variance of the random variables into account the problems become non-separable quadratic programs. A reformulation of the problem is proposed which reduces density of matrices involved and by these means significantly simplifies its solution by an interior point method. The object-oriented parallel solver achieves high efficiency by careful exploitation of the block sparsity of these problems. As a result a problem with over 50 million decision variables is solved in just over 2 hours on a parallel computer with 16 processors. The approach is by nature scalable and the parallel implementation achieves nearly perfect speed-ups on a range of problems.

Solving non-linear portfolio optimization problems with the primal-dual interior point method

Stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning. The deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets, time stages and scenarios per time stage. So far models treated by mathematical programming approaches have been limited to simple linear or quadratic models due to the inability of currently available solvers to solve NLP problems of typical sizes. However stochastic programming problems are highly structured. The key to the efficient solution of such problems is therefore the ability to exploit their structure. Interior point methods are well-suited to the solution of very large non-linear optimization problems. In this paper we exploit this feature and show how portfolio optimization problems with sizes measured in millions of constraints and decision variables, featuring constraints on semi-variance, skewness or non-linear utility functions in the objective, can be solved with the state-of-the-art solver.

Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds

Speed and memory requirements of branch and bound algorithms depend on the selection strategy of which candidate node to process next. The goal of this paper is to experimentally investigate this influence to the performance of sequential and parallel branch and bound algorithms. The experiments have been performed solving a number of multidimensional test problems for global optimization. Branch and bound algorithm using simplicial partitions and combination of Lipschitz bounds has been investigated. Similar results may be expected for other branch and bound algorithms.

Optimization-Based Islanding of Power Networks Using Piecewise Linear AC Power Flow

In this paper, a flexible optimization-based framework for intentional islanding is presented. The decision is made of which transmission lines to switch in order to split the network while minimizing disruption, the amount of load shed, or grouping coherent generators. The approach uses a piecewise linear model of AC power flow, which allows the voltage and reactive power to be considered directly when designing the islands. Demonstrations on standard test networks show that solution of the problem provides islands that are balanced in real and reactive power, satisfy AC power flow laws, and have a healthy voltage profile.

Direct solution of linear systems of size 10 9 arising in optimization with interior point methods

Solution methods for very large scale optimization problems are addressed in this paper. Interior point methods are demonstrated to provide unequalled efficiency in this context. They need a small (and predictable) number of iterations to solve a problem. A single iteration of interior point method requires the solution of indefinite system of equations. This system is regularized to guarantee the existence of triangular decomposition. Hence the well-understood parallel computing techniques developed for positive definite matrices can be extended to this class of indefinite matrices. A parallel implementation of an interior point method is described in this paper. It uses object-oriented programming techniques and allows for exploiting different block-structures of matrices. Our implementation outperforms the industry-standard optimizer, shows very good parallel efficiency on massively parallel architecture and solves problems of unprecedented sizes reaching 109 variables.

A structure-conveying modelling language for mathematical and stochastic programming

We present a structure-conveying algebraic modelling language for mathematical programming. The proposed language extends AMPL with object-oriented features that allows the user to construct models from sub-models, and is implemented as a combination of pre- and post-processing phases for AMPL. Unlike traditional modelling languages, the new approach does not scramble the block structure of the problem, and thus it enables the passing of this structure on to the solver. Interior point solvers that exploit block linear algebra and decomposition-based solvers can therefore directly take advantage of the problem’s structure. The language contains features to conveniently model stochastic programming problems, although it is designed with a much broader application spectrum.