Primal-dual Interior-point Approach Research Articles

Distributed optimal control of nonlinear systems using a second-order augmented Lagrangian method

In this paper, we propose a distributed second-order augmented Lagrangian method for distributed optimal control problems, which can be exploited for distributed model predictive control. We employ a primal-dual interior-point approach for the inner iteration of the augmented Lagrangian and distribute the corresponding computations using message passing over what is known as the clique tree of the problem. The algorithm converges to its centralized counterpart and it requires fewer communications between sub-systems as compared to algorithms such as the alternating direction method of multipliers. We illustrate the efficiency of the framework when applied to randomly generated interconnected sub-systems as well as to a vehicle platooning problem.

Routing Optimization with Efficient Second Order Distributed Approach Using Congestion Control Rules

Distributed joint congestion control and routing optimization has received a significant amount of attention recently. To date, however, most of the existing schemes follow a key idea called the back-pressure algorithm. Despite having many salient features, the first-order sub gradient nature of the back-pressure based schemes results in slow convergence and poor delay performance. To overcome these limitations, the present study was made as first attempt at developing a second-order joint congestion control and routing optimization framework that offers utility-optimality, queue-stability, fast convergence, and low delay. Contributions in this project are three-fold. The present study propose a new second-order joint congestion control and routing framework based on a primal-dual interior-point approach and established utility-optimality and queue-stability of the proposed second-order method. The results of present study showed that how to implement the proposed second-order method in a distributed fashion.

Joint Congestion Control and Routing Optimization: An Efficient Second-Order Distributed Approach

Distributed joint congestion control and routing optimization has received a significant amount of attention recently. To date, however, most of the existing schemes follow a key idea called the back-pressure algorithm. Despite having many salient features, the first-order subgradient nature of the back-pressure based schemes results in slow convergence and poor delay performance. To overcome these limitations, in this paper, we make a first attempt at developing a second-order joint congestion control and routing optimization framework that offers utility-optimality, queue-stability, fast convergence, and low delay. Our contributions in this paper are three-fold: i) we propose a new second-order joint congestion control and routing framework based on a primal-dual interior-point approach; ii) we establish utility-optimality and queue-stability of the proposed second-order method; and iii) we show how to implement the proposed second-order method in a distributed fashion.

Wavelet Neural Network-based Wind-power Forecasting in Economic Dispatch: A Differential Evolution, Bacterial Foraging Technology, and Primal Dual-interior Point Approach

Wind speed and wind-power generation are characterized by their inherent variability and uncertainty. To overcome this drawback, an accurate prediction of wind speed is essential. The purpose of this paper is to develop a hybrid wavelet neural network model for wind-speed forecasting and thus, in turn, for wind-power generation. The combined optimal economic scheduling of the wind generators and conventional generators has also been investigated in this paper. The solution methods, namely primal dual interior point, differential evolution, and bacterial foraging technology, are used for solving the wind-thermal economic dispatch (ED). The feasibility of the proposed algorithms is demonstrated on three-unit, 13-unit, and 40-unit systems and their performances are compared in terms of the generation cost and execution time. The results show that the proposed algorithms are indeed capable of handling ED problems.

On the interplay between interior point approximation and parametric sensitivities in optimal control

Infinite-dimensional parameter-dependent optimization problems of the form ‘ min J ( u ; p ) subject to g ( u ) ⩾ 0 ’ are studied, where u is sought in an L ∞ function space, J is a quadratic objective functional, and g represents pointwise linear constraints. This setting covers in particular control constrained optimal control problems. Sensitivities with respect to the parameter p of both, optimal solutions of the original problem, and of its approximation by the classical primal-dual interior point approach are considered. The convergence of the latter to the former is shown as the homotopy parameter μ goes to zero, and error bounds in various L q norms are derived. Several numerical examples illustrate the results.

Primal-dual interior-point approach to compute the L1 solution of the state estimation problem

A solution to the single-snapshot non-linear L1 estimation of the power transmission network is presented. The non-linear L1 estimation problem is formulated as a non-linear program and solved using a primal-dual interior-point approach. The efficiency of this approach is dependent on the numerical procedure used to solve the reduced Karush-Kuhn-Tucker (KKT) system of equations. It is shown that two mathematically equivalent formulations of the non-linear programming problem can be obtained. These formulations lend themselves to fundamentally different numerical procedures to solve the reduced KKT system. Numerical testing on IEEE systems is used to quantify the performance of the interior-point approach on both formulations. Comparisons are also carried out with a recent implementation of an iteratively reweighted least-squares method for non-linear L1 regression.

Power system Huber M-estimation with equality and inequality constraints

The solution to the power system state estimation problem using the Huber M-estimator has been previously discussed. In earlier methods, the state estimation problem has been formulated as an unconstrained nonlinear program. The power systems literature reports solution to this problem using an iteratively re-weighted least squares technique. In this paper, the Huber M-estimator is formulated as a constrained nonlinear programming problem in which all functions are twice continuously differentiable. Such a formulation entails two advantages. Firstly, it enables accurate modelling of both zero bus injections and realistic system limits via equality and inequality constraints. Secondly, the differentiability property allows the system state to be conveniently computed via a primal-dual interior-point approach. Simulations on standard power systems show that even in the presence of bad data, the equality constraints in the Huber M-estimator effectively model the zero bus injections. Moreover, the numerical results reveal that the enforcement of practical system limits via inequality constraints can be useful in the absence of complete system observability.

Nonlinear Optimization with Many Degrees of Freedom in Process Engineering

Applications of nonlinear optimization problems with many degrees of freedom have become more common in the process industries, especially in the area of process operations. However, most widely used nonlinear programming (NLP) solvers are designed for the efficient solution of problems with few degrees of freedom. Here we consider a new NLP algorithm, IPOPT, designed for many degrees of freedom and many potentially active constraint sets. The IPOPT algorithm follows a primal-dual interior point approach, and its robustness, improved convergence, and computational speed compared to those of other popular NLP algorithms will be analyzed. To demonstrate its effectiveness on process applications, we consider large gasoline blending and data reconciliation problems, both of which contain nonlinear mass balance constraints and process properties. Results on this computational comparison show significant benefits from the IPOPT algorithm.

Mathematical programs with equilibrium constraints (MPECs) in process engineering

Mathematical programs with equilibrium constraints (MPECs) form a relatively new and interesting subclass of nonlinear programming problems. In this paper we propose a novel method of solving MPECs by appropriate reformulation of the equilibrium conditions. The reformulation can be easily incorporated in a certain class of interior point algorithms for nonlinear optimization. The algorithm used in the study follows a primal-dual interior point approach and shows encouraging results on a test suite of MPECs. The algorithm is also able to perform optimization of distillation columns with phase changes and tray optimization using only continuous variables. We also consider a number of topics to improve performance of the algorithm and to identify classes of process engineering problems that can be handled as MPECs.

Local Convergence of a Primal-Dual Method for Degenerate Nonlinear Programming

In recent work, the local convergence behavior of path-following interior-point methods and sequential quadratic programming methods for nonlinear programming has been investigated for the case in which the assumption of linear independence of the active constraint gradients at the solution is replaced by the weaker Mangasarian–Fromovitz constraint qualification. In this paper, we describe a stabilization of the primal-dual interior-point approach that ensures rapid local convergence under these conditions without enforcing the usual centrality condition associated with path-following methods. The stabilization takes the form of perturbations to the coefficient matrix in the step equations that vanish as the iterates converge to the solution.

Primal-dual interior point approach for computingl 1-solutions andl ?-solutions of overdetermined linear systems

This paper concerns solving an overdetermined linear systemA T x=b in the leastl 1-norm orl ∞-norm sense, whereA∈ℝ m×n ,m<n. We show that the primal-dual interior point approach for linear programming can be applied, in an effective manner, to linear programming versions of thel 1 andl ∞-problems. The resulting algorithms are simple to implement and can attain quadratic or superlinear convergence rate. At each iteration, the algorithms must solve a linear system with anm×m positive-definite coefficient matrix of the formADA T , whereD is a positive diagonal matrix. The preliminary numerical results indicate that the proposed algorithms offer considerable promise.