Interior Point Methods For Linear Programming Research Articles

A Primal-dual algorithm for unfolding neutron energy spectrum from multiple activation foils

In this paper we propose a robust and efficient primal-dual interior-point method for a nonlinear ill-conditioned problem with associated errors which are arising in the unfolding procedure for neutron energy spectrum from multiple activation foils. Based on the maximum entropy principle and Boltzmann's entropy formula, the discrete form of the unfolding problem is equivalent to computing the analytic center of the polyhedral set \begin{document}$ P = \{x \in R^n \mid Ax = b, x \ge 0\} $\end{document} , where the matrix \begin{document}$ A \in R^{m\times n} $\end{document} is ill-conditioned, and both \begin{document}$ A $\end{document} and \begin{document}$ b $\end{document} are inaccurate. By some derivations, we find a new regularization method to reformulate the problem into a well-conditioned problem which can also reduce the impact of errors in \begin{document}$ A $\end{document} and \begin{document}$ b $\end{document} . Then based on the primal-dual interior-point methods for linear programming, we propose a hybrid algorithm for this ill-conditioned problem with errors. Numerical results on a set of ill-conditioned problems for academic purposes and two practical data sets for unfolding the neutron energy spectrum are presented to demonstrate the effectiveness and robustness of the proposed method.

Optimized choice of parameters in interior-point methods for linear programming

In this work, we propose a predictor–corrector interior point method for linear programming in a primal–dual context, where the next iterate is chosen by the minimization of a polynomial merit function of three variables: the first is the steplength, the second defines the central path and the third models the weight of a corrector direction. The merit function minimization is performed by restricting it to constraints defined by a neighborhood of the central path that allows wide steps. In this framework, we combine different directions, such as the predictor, the corrector and the centering directions, with the aim of producing a better one. The proposed method generalizes most of predictor–corrector interior point methods, depending on the choice of the variables described above. Convergence analysis of the method is carried out, considering an initial point that has a good practical performance, which results in Q-linear convergence of the iterates with polynomial complexity. Numerical experiments using the Netlib test set are made, which show that this approach is competitive when compared to well established solvers, such as PCx.

Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood

The interior-point algorithms can be classified in multiple ways. One of these takes into consideration the length of the step. In this way, we can speak about large-step and short-step methods, that work in different neighbourhoods of the central path. The large-step algorithms work in a wide neighbourhood, while the short-step ones determine the new iterates that are in a smaller neighbourhood. In spite of the fact that the large-step algorithms are more efficient in practice, the theoretical complexity of the short-step ones is generally better. Ai and Zhang introduced a large-step interior-point method for linear complementarity problems using a wide neighbourhood of the central path, which has the same complexity as the best short-step methods. We present a new wide neighbourhood of the central path. We prove that the obtained large-step primal–dual interior-point method for linear programming has the same complexity as the best short-step algorithms.

A predictor-corrector interior-point algorithm for monotone variational inequality problems

Mehrolra’s recent suggestion of a predictor-corrector variant of primal-dual interior-point method for linear programming is currently the interior-point method of choice for linear programming. In this work the authors give a predictor-corrector interior-point algorithm for monotone variational inequality problems. The algorithm was proved to be equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require the initial iteration to be feasible. Numerical results of experiments are presented.

The solution of euclidean norm trust region SQP subproblems via second-order cone programs: an overview and elementary introduction

It is well known that convex sequential quadratic programming (SQP) subproblems with an Euclidean norm trust region constraint can be reduced to second-order cone programs for which the theory of Euclidean Jordan algebras leads to efficient interior-point algorithms. Here, a brief and self-contained outline of the principles of such an implementation is given and the application to SQP subproblems as well as to cubic regularization problems is discussed. All identities relevant for the implementation are derived from scratch and are compared to interior-point methods for linear programs (LPs). Sparsity of the data of the SQP subproblem can be maintained essentially in the same way as for interior-point methods for LPs. The presentation is intended as an introduction for students and for colleagues who may have heard about Jordan algebras but did not yet find the time to get involved with them. A simple Matlab implementation is made available and the discussion of implementational aspects addresses a scaling property that is critical for SQP subproblems.

A New Primal–Dual Predictor–Corrector Interior-Point Method for Linear Programming Based on a Wide Neighbourhood

In this paper, we propose a new predictor---corrector interior-point algorithm for linear programming based on a wide neighbourhood. In each iteration, the algorithm computes the Ai-Zhang's predictor direction (SIAM J. Optim. 16(2):400---417, 2005) and a new corrector direction, in an attempt to improve its performance. We drive that the duality gap reduces in both predictor and corrector steps. Moreover, we also prove that the complexity of the algorithm coincides with the best iteration bound for small neighbourhood algorithms. Finally, some numerical experiments are provided which reveal capability and effectiveness of the proposed method.

Sparsity preserving preconditioners for linear systems in interior-point methods

Systems of normal equations arising in interior-point methods for linear programming in the case of a degenerate optimal face have highly ill-conditioned coefficient matrices. In 2004, Monteiro et al. (SIAM J Optim 15:96---100, 2004) proposed a preconditioner which guarantees uniform well-conditionedness. However, the proposed preconditioner may lead to considerable loss of sparsity. Our approach is directed towards a generalization of the proposed preconditioner which makes a balance between sparsity and well-conditionedness. Experimental results on Netlib instances show the effects of the new approach.

A robust and efficient proposal for solving linear systems arising in interior-point methods for linear programming

We introduce an efficient and robust proposal for solving linear systems arising at each iteration of primal-dual interior-point methods for linear programming. Our proposal is based on the stable system presented by Gonzalez-Lima et al. (Comput. Opt. Appl. 44:213–247, 2009). Using similar techniques as those employed in the splitting preconditioner introduced by Oliveira and Sorensen (Linear Algebra Appl. 394:1–24, 2005) we are able to express the stable system matrix in block form such that the diagonal blocks are nonsingular diagonal matrices and the off-diagonal blocks are matrices close to zero when the iterates are close to the solution set of the linear programming problem. For degenerate problems a perturbation of the diagonal is added. We use a low-cost fixed iterative method to solve this system. Numerical experiments have shown that our approach leads to very accurate solutions for the linear programming problem.

Infeasible constraint-reduced interior-point methods for linear optimization

Constraint-reduction schemes have been proposed for the solution by means of interior-point methods of linear programs with many more inequality constraints than variables in the standard dual form. Such schemes have been shown to be provably convergent and highly efficient in practice. A critical requirement of these schemes is the availability of an initial dual-feasible point. In this paper, building on a general framework (which encompasses several previously proposed approaches) for dual-feasible constraint-reduced interior-point optimization, for which we prove convergence to a single point of the sequence of dual iterates, we propose a framework for ‘infeasible’ constraint-reduced interior-point optimization. Central to this framework is an exact (ℓ1 or ) penalty function scheme endowed with a mechanism for iterative adjustment of the penalty parameter, which aims at yielding, after a finite number of iterations, a value that guarantees feasibility (for the original problem) of the minimizers. Finiteness of the sequence of penalty parameter adjustments is proved under mild assumptions for all algorithms that fit within the framework, including ‘infeasible’ extensions of a ‘dual’ algorithm proposed in the early 1990s (Dantzig and Ye, A build-up interior-point method for linear programming: Affine scaling form, Working paper, Department of Management Science, University of Iowa, 1991) and of two recently proposed ‘primal–dual’ algorithms (Tits, Absil, and Woessner, Constraint reduction for linear programs with many inequality constraints, SIAM J. Optim. 17 (2006), pp. 119–146; Winternitz, Nicholls, Tits, and O’Leary, A constraint-reduced variant of Mehrotra’s predictor–corrector algorithm, Comput. Optim. Appl. (published on-line as of January 2011), DOI: 10.1007/s10589-010-9389-4). The last one, a constraint-reduced variant of Mehrotra’s Predictor–Corrector algorithm, is then more specifically considered: further convergence results are proved, and numerical results are reported that demonstrate that the approach is of practical interest.

Enhancing the behavior of interior-point methods via identification of variables

In many linear programming models of real life problems the solution set is not bounded. The presence of unbounded variables in the solution set can severely hurt the practical performance of primal–dual interior-point methods for linear programming that generate iterates which follow closely the central path or converge to the analytic center. In this work we study the effect of the unbounded variables by analysing the numerical behavior of the LSSN algorithm proposed by González-Lima, Tapia and Potra 1. when applied to linear problems with unbounded solution sets. We discuss the numerical behavior of the algorithm and we present a numerical procedure, based on the performance of the algorithm, to identify and remove the unbounded variables and related constraints. We develop theoretical support for the procedure and experimental evidence of its performance.

Correlative Sparsity in Primal-Dual Interior-Point Methods for LP, SDP, and SOCP

Exploiting sparsity has been a key issue in solving large-scale optimization problems. The most time-consuming part of primal-dual interior-point methods for linear programs, second-order cone programs, and semidefinite programs is solving the Schur complement equation at each iteration, usually by the Cholesky factorization. The computational efficiency is greatly affected by the sparsity of the coefficient matrix of the equation which is determined by the sparsity of an optimization problem (linear program, semidefinite program or second-order cone program). We show if an optimization problem is correlatively sparse, then the coefficient matrix of the Schur complement equation inherits the sparsity, and a sparse Cholesky factorization applied to the matrix results in no fill-in.

Implementation of warm-start strategies in interior-point methods for linear programming in fixed dimension

We implement several warm-start strategies in interior-point methods for linear programming (LP). We study the situation in which both the original LP instance and the perturbed one have exactly the same dimensions. We consider different types of perturbations of data components of the original instance and different sizes of each type of perturbation. We modify the state-of-the-art interior-point solver PCx in our implementation. We evaluate the effectiveness of each warm-start strategy based on the number of iterations and the computation time in comparison with “cold start” on the NETLIB test suite. Our experiments reveal that each of the warm-start strategies leads to a reduction in the number of interior-point iterations especially for smaller perturbations and for perturbations of fewer data components in comparison with cold start. On the other hand, only one of the warm-start strategies exhibits better performance than cold start in terms of computation time. Based on the insight gained from the computational results, we discuss several potential improvements to enhance the performances of such warm-start strategies.

An exact primal–dual penalty method approach to warmstarting interior-point methods for linear programming

One perceived deficiency of interior-point methods in comparison to active set methods is their inability to efficiently re-optimize by solving closely related problems after a warmstart. In this paper, we investigate the use of a primal---dual penalty approach to overcome this problem. We prove exactness and convergence and show encouraging numerical results on a set of linear and mixed integer programming problems.

A Gravitational Interior Point Method for LP

In [3, 1] gravitational methods for linear programming (LP) have been introduced. Several versions exist, the three main versions discussed there use a ball of (a): 0 radius, (b): small positive radius, and (c): the largest possible radius with the given center that will completely fit within the polytope, with the option of changing its radius as the algorithm progresses.

A product-form Cholesky factorization method for handling dense columns in interior point methods for linear programming

Cholesky factorization has become the method of choice for solving the symmetric system of linear equations arising in interior point methods (IPMs) for linear programming (LP), its main advantages being numerical stability and efficiency for sparse systems. However in the presence of dense columns there is dramatic loss of efficiency. A typical approach to remedy this problem is to apply the Sherman-Morrison-Woodbury (SMW) update formula to the dense part. This approach while being very efficient, is not numerically stable. Here we propose using product-form Cholesky factorization to handle dense columns. The proposed approach is essentially as stable as the original Cholesky factorization and nearly as efficient as the SMW approach. We demonstrate these properties both theoretically and computationally. A key part of our theoretical analysis is the proof that the elements of the Cholesky factors of the matrices that arise in IPMs for LP are uniformly bounded as the duality gap converges to zero.

Using Indicators in Finite Termination Procedures

Early identification of the zero variables in a constrained optimization problem can be used to computational advantage. Is there a similar gain in identification of variables at their upper bounds? In this context, we study finite termination procedures in interior-point methods for linear programs with bounded variables. To prevent the computed solution from violating the bound constraints, one approach incorporates nearest bound information into a projection model through an affine scaling transformation.Using Tapia indicators, we identify variables in the active set, remove them from the subproblem, and solve a lower dimensional projection problem. Numerical evidence suggests that identifying and removing the variables at their upper bounds from the optimal face identification problem plays a more important role in finite termination procedures than the choice of affine scaling transformation.

Postponing the Choice of Penalty Parameter and Step Length

We study, in the context of interior-point methods for linear programming, some possible advantages of postponing the choice of the penalty parameter and the steplength, which happens both when we apply Newton's method to the Karush-Kuhn-Tucker system and when we apply a predictor-corrector scheme. We show that for a Newton or a strictly predictor step the next iterate can be expressed as a linear function of the penalty parameter μ, and, in the case of a predictor-corrector step, as a quadratic function of μ. We also show that this parameterization is useful to guarantee either the non-negativity of the next iterate or the proximity to the central path. Initial computational results of these strategies are shown and compared with PCx, an implementation of Mehotra's predictor-corrector method.

Interior-point methods for linear programming: a review

The paper reviews some recent advances in interior-point methods for linear programming and indicates directions in which future progress can be made. Most of the interior-point methods belong to any of three categories: affine-scaling methods, potential reduction methods and central path methods. These methods are discussed together with infeasible interior methods and homogeneous self-dual methods for linear programming. Also discussed are some theoretical issues in interior-point methods like dependence of complexity bounds on some non-traditional measures different from the input length L of the problem. Finally, the paper concludes with remarks on the comparison of interior-point methods with the simplex method based on their performance on NITLIB suite, a standard collection of test problems.

ROBUST LINEAR PROGRAMMING AND OPTIMAL CONTROL

The paper describes an efficient method for solving an optimal control problem that arises in robust model-predictive control. The problem is to design the input sequence that minimizes the peak tracking error between the ouput of a linear dynamical system and a desired target output, subject to inequality constraints on the inputs. The system is uncertain, with an impulse response that can take arbitrary values in a given polyhedral set. This problem can be formulated as a robust linear programming problem with structured uncertainty. The presented method is based on Mehrotra's interior-point method for linear programming, and takes advantage of the problem structure to achieve a complexity that grows linearly with the control horizon, and increases as a cubic polynomial as a function of the system order, the number of inputs, and the number of uncertainty parameters.

Generalization of Primal—Dual Interior-Point Methods to Convex Optimization Problems in Conic Form

We generalize primal--dual interior-point methods for linear programming (LP) problems to the convex optimization problems in conic form. Previously, the most comprehensive theory of symmetric primal--dual interior-point algorithms was given by Nesterov and Todd for feasible regions expressed as the intersection of a symmetric cone with an affine subspace. In our setting, we allow an arbitrary convex cone in place of the symmetric cone. Even though some of the impressive properties attained by Nesterov--Todd algorithms are impossible in this general setting of convex optimization problems, we show that essentially all primal--dual interior-point algorithms for LP can be extended easily to the general setting. We provide three frameworks for primal--dual algorithms, each framework corresponding to a different level of sophistication in the algorithms. As the level of sophistication increases, we demand better formulations of the feasible solution sets. Our algorithms, in return, attain provably better theoretical properties. We also make a very strong connection to quasi-Newton methods by expressing the square of the symmetric primal--dual linear transformation (the so-called scaling) as a quasi-Newton update in the case of the least sophisticated framework.