Interior Point Approach Research Articles

A STUDY ON SENSITIVITY ANALYSIS FOR CONVEX QUADRATIC PROGRAMS

We extend the two similar interior-point approaches to sensitivity analysis originally developed for linear programs to those for convex quadratic programs, where the first approach is the ∊-sensitivity analysis and the other is Yildirim and Todd's. We study the relationship between the bounds on perturbation of the input parameters arising from the extension of Yildirim and Todd's approach and those from the ∊-sensitivity analysis. Furthermore, we prove that Yildirim and Todd's bounds are asymptotically the same as the symmetrized optimal tripartition bounds in the case of a special type of nondegeneracy.

A smoothing heuristic for a bilevel pricing problem

In this paper, we provide a heuristic procedure, that performs well from a global optimality point of view, for an important and difficult class of bilevel programs. The algorithm relies on an interior point approach that can be interpreted as a combination of smoothing and implicit programming techniques. Although the algorithm cannot guarantee global optimality, very good solutions can be obtained through the use of a suitable set of parameters. The algorithm has been tested on large-scale instances of a network pricing problem, an application that fits our modeling framework. Preliminary results show that on hard instances, our approach constitutes an alternative to solvers based on mixed 0–1 programming formulations.

An interior point Newton‐like method for non‐negative least‐squares problems with degenerate solution

AbstractAn interior point approach for medium and large non‐negative linear least‐squares problems is proposed. Global and locally quadratic convergence is shown even if a degenerate solution is approached. Viable approaches for implementation are discussed and numerical results are provided. Copyright © 2006 John Wiley & Sons, Ltd.

Interior-Point Algorithms, Penalty Methods and Equilibrium Problems

In this paper we consider the question of solving equilibrium problems--formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPECs)--as nonlinear programs, using an interior-point approach. These problems pose theoretical difficulties for nonlinear solvers, including interior-point methods. We examine the use of penalty methods to get around these difficulties and provide substantial numerical results. We go on to show that penalty methods can resolve some problems that interior-point algorithms encounter in general.

05/02215 Short term hydroelectric scheduling combining network flow and interior point approaches

05/02215 Short term hydroelectric scheduling combining network flow and interior point approaches

Interior point methods for large-scale nonlinear programming

In this paper we describe an algorithm for solving nonlinear nonconvex programming problems, which is based on the interior point approach. The main theoretical results concern direction determination and step-length selection. We split inequality constraints into active and inactive parts to overcome problems with instability. Inactive constraints are eliminated directly, whereas active constraints are used for defining a symmetric indefinite linear system. Inexact solution of this system is obtained iteratively using indefinitely preconditioned conjugate gradient method. Theorems confirming efficiency of the indefinite preconditioner are introduced. Furthermore, a new merit function is defined and a filter principle is used for step-length selection. The algorithm was implemented in the interactive system for universal functional optimization UFO. Results of numerical experiments are reported.

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.

Short term hydroelectric scheduling combining network flow and interior point approaches

In this paper, short term hydroelectric scheduling is formulated as a network flow optimization model and solved by interior point methods. The primal–dual and predictor–corrector versions of such interior point methods are developed and the resulting matrix structure is explored. This structure leads to very fast iterations since it avoids computation and factorization of impedance matrices. For each time interval, the linear algebra reduces to the solution of two linear systems, either to the number of buses or to the number of independent loops. Either matrix is invariant and can be factored off-line. As a consequence of such matrix manipulations, a linear system which changes at each iteration has to be solved, although its size is reduced to the number of generating units and is not a function of time intervals. These methods were applied to IEEE and Brazilian power systems, and numerical results were obtained using a MATLAB implementation. Both interior point methods proved to be robust and achieved fast convergence for all instances tested.

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.

Extremum-seeking control of state-constrained nonlinear systems

This paper treats an extremum seeking control problem for a class of state-constrained nonlinear systems with unknown parameters. Using an interior-point approach, an adaptive method is proposed for generating set-points online which converge to the feasible extremum point of an objective function containing unknown parameters. A tracking controller ensures system states remain feasible at all times. A simulation example demonstrates application of the method.

An interior-point sequential approximate optimization methodology

The use of optimization in a simulation-based design environment has become a common trend in industry today. Computer simulation tools are commonplace in many engineering disciplines, providing the designers with tools to evaluate a design’s performance without building a physical prototype. This has triggered the development of optimization techniques suitable for dealing with such simulations. One of these approaches is known as sequential approximate optimization. In sequential approximate minimization a sequence of optimizations are performed over local response surface approximations of the system. This paper details the development of an interior-point approach for trust-region-managed sequential approximate optimization. The interior-point approach will ensure that approximate feasibility is maintained throughout the optimization process. This facilitates the delivery of a usable design at each iteration when subject to reduced design cycle time constraints. In order to deal with infeasible starting points, homotopy methods are used to relax constraints and push designs toward feasibility. Results of application studies are presented, illustrating the applicability of the proposed algorithm.

Interior‐point method for non‐linear non‐convex optimization

AbstractIn this paper, we propose an algorithm for solving non‐linear non‐convex programming problems, which is based on the interior point approach. Main theoretical results concern direction determination and step‐length selection. We split inequality constraints into active and inactive to overcome problems with stability. Inactive constraints are eliminated directly while active constraints are used to define symmetric indefinite linear system. Inexact solution of this system is obtained iteratively using indefinitely preconditioned conjugate gradient method. Theorems confirming efficiency of several indefinite preconditioners are proved. Furthermore, new merit function is defined, which includes effect of possible regularization. This regularization can be used to overcome problems with near linear dependence of active constraints. The algorithm was implemented in the interactive system for universal functional optimization UFO. Results of extensive numerical experiments are reported. Copyright © 2004 John Wiley & Sons, Ltd.

Interior point methods meet simplex in L, fitting problems

AbstractInterior point methods specialized to the L∞ fitting problem are surveyed, improved, and compared with the traditional simplex approach. A primal affine‐scaling interior point method is presented, completing the affine‐scaling interior point family approach to the L∞ fitting problem. Computational complexity and data storage are reduced for interior point approaches when dealing with polynomial fitting problems. Numerical experiments indicate that interior point approaches rarely perform better than the simplex method for the tested problems. The primal affine‐scaling method presented in this paper achieved the best results among the interior point family.

Simultaneous Solution Strategies for Inclusion of Input Saturation in the Optimal Design of Dynamically Operable Plants

Recent work has considered the inclusion of input saturation in optimization-based integrated plant and control system design. Two mathematical formulations have been derived which allow saturation to be included within a simultaneous optimization framework; these are mixed-integer and bilinear formulations. The MIP formulation applied within an integrated design and control framework was found to be computationally intensive. This paper reviews these formulations and proposes an alternative solution strategy that uses an interior point approach to handle the complementarity constraints present in the bilinear formulation. A solution algorithm is presented and applied to an example problem.

Optimal active power dispatch combining network flow and interior point approaches

In this paper, the optimal active power dispatch is formulated as a network flow optimization model and solved by interior point methods. The primal-dual and predictor-corrector versions of such interior point methods are developed and the resulting matrix structure is explored. This structure leads to very fast iterations since it is possible to reduce the linear system either to the number of buses or to the number of independent loops. Either matrix is invariant and can be factored offline. As a consequence of such matrix manipulations, a linear system which changes at each iteration has to be solved; its size, however, reduces to the number of generating units. These methods were applied to IEEE and Brazilian power systems and the numerical results were obtained using a C implementation. Both interior point methods proved to be robust and achieved fast convergence in all instances tested.

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.

A Riccati-based primal interior point solver for multistage stochastic programming

We propose a new method for certain multistage stochastic programs with linear or nonlinear objective function, combining a primal interior point approach with a linear-quadratic control problem over the scenario tree. The latter problem, which is the direction finding problem for the barrier subproblem is solved through dynamic programming using Riccati equations. In this way we combine the low iteration count of interior point methods with an efficient solver for the subproblems. The computational results are promising. We have solved a financial problem with 1,000,000 scenarios, 15,777,740 variables and 16,888,850 constraints in 20 hours on a moderate computer.

A Primal-Dual Decomposition-Based Interior Point Approach to Two-Stage Stochastic Linear Programming

Decision making under uncertainty is a challenge faced by many decision makers. Stochastic programming is a major tool developed to deal with optimization with uncertainties which has found applica...

A Primal-Dual Decomposition-Based Interior Point Approach to Two-Stage Stochastic Linear Programming

Decision making under uncertainty is a challenge faced by many decision makers. Stochastic programming is a major tool developed to deal with optimization with uncertainties which has found applications in, e.g., finance, such as asset–liability and bond–portfolio management. Computationally, however, many models in stochastic programming remain unsolvable because of overwhelming dimensionality. For a model to be well solvable, its special structure must be explored. Most of the solution methods are based on decomposing the data. In this paper we propose a new decomposition approach for two-stage stochastic programming, based on a direct application of the path-following method combined with the homogeneous self-dual technique. Numerical experiments show that our decomposition algorithm is very efficient for solving stochastic programs. In particular, we apply our decomposition method to a two-period portfolio selection problem using options on a stock index. In this model the investor can invest in a money-market account, a stock index, and European options on this index with different maturities. We experiment with our model with market prices of options on the S&P500.