Path-following Interior-point Method Research Articles

How good are interior point methods? Klee–Minty cubes tighten iteration-complexity bounds

By refining a variant of the Klee–Minty example that forces the central path to visit all the vertices of the Klee–Minty n-cube, we exhibit a nearly worst-case example for path-following interior point methods. Namely, while the theoretical iteration-complexity upper bound is $$O(2^{n}n^{\frac{5}{2}})$$, we prove that solving this n-dimensional linear optimization problem requires at least 2 n −1 iterations.

A Primal-Dual Interior-Point Method to Solve the Optimal Power Flow Dispatching Problem

This paper presents a primal-dual path-following interior-point method for the solution of the optimal power flow dispatching (OPFD) problem. The underlying idea of most path-following algorithms is relatively similar: starting from the Fiacco-McCormick barrier function, define the central path and loosely follow it to the optimum solution. Several primal-dual methods for OPF have been suggested, all of which are essentially direct extensions of primal-dual methods for linear programming. Nevertheless, there are substantial variations in some crucial details which include the formulation of the non-linear problem, the associated linear system, the linear algebraic procedure to solve this system, the line search, strategies for adjusting the centring parameter, estimating higher order correction terms for the homotopy path, and the treatment of indefiniteness. This paper discusses some of the approaches that were undertaken in implementing a specific primal-dual method for OPFD. A comparison is carried out with previous research on interior-point methods for OPF. Numerical tests on standard IEEE systems and on a realistic network are very encouraging and show that the new algorithm converges where other algorithms fail.

Differential-Algebraic Approach to Linear Programming

This paper presents a differential-algebraic approach for solving linear programming problems. The paper shows that the differential-algebraic approach is guaranteed to generate optimal solutions to linear programming problems with a superexponential convergence rate. The paper also shows that the path-following interior-point methods for solving linear programming problems can be viewed as a special case of the differential-algebraic approach. The results in this paper demonstrate that the proposed approach provides a promising alternative for solving linear programming problems.

Self-regular functions and new search directions for linear and semidefinite optimization

In this paper, we first introduce the notion of self-regular functions. Various appealing properties of self-regular functions are explored and we also discuss the relation between selfregular functions and the well-known self-concordant functions. Then we use such functions to define self-regular proximity measure for path-following interior point methods for solving linear optimization (LO) problems. Any self-regular proximity measure naturally defines a primal-dual search direction. In this way a new class of primal-dual search directions for solving LO problems is obtained. Using the appealing properties of self-regular functions, we prove that these new large-update path-following methods for LO enjoy a polynomial, O n q+1 2q log n " iteration bound, where q 1 is the so-called barrier degree of the self-regular proximity measure underlying the algorithm. When q increases, this bound approaches the best known complexity bound for interior point methods, namely O p nlog n . Our unified analysis provides also the O p nlog n best known iteration bound of small-update IPMs. At each iteration, we need only to solve one linear system. As a byproduct of our results, we remove some limitations of the algorithms presented in [24] and improve their complexity as well. An extension of these results to semidefinite optimization (SDO) is also discussed.

Parallel Birge and Qi Factorization of Three-stage Stochastic Linear Programs

Many practical problems, such as portfolio management, problems in manufacturing, transportation or power generation, can be modeled as stochastic programs. Stochastic programs provide an effective framework for sequential decision problems with uncertain data, when uncertainty can be modeled by a discrete set of scenarios. In this paper we present an algorithm for solving a three-stage stochastic linear program based on the Birge and Qi factorization of a constraint matrix product in the frame of the primal-dual path-following interior point method. Moreover, we discuss the parallelization of this method for distributed-memory machines using Fortran/MPI and the linear algebra package LAPACK. Performance experiments on a 64-processor Beowulf cluster show the effectiveness of the proposed parallelization strategy, which exploits two levels of parallelism.

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.

Interior-point methodology for 3-D PET reconstruction.

Interior-point methods have been successfully applied to a wide variety of linear and nonlinear programming applications. This paper presents a class of algorithms, based on path-following interior-point methodology, for performing regularized maximum-likelihood (ML) reconstructions on three-dimensional (3-D) emission tomography data. The algorithms solve a sequence of subproblems that converge to the regularized maximum likelihood solution from the interior of the feasible region (the nonnegative orthant). We propose two methods, a primal method which updates only the primal image variables and a primal-dual method which simultaneously updates the primal variables and the Lagrange multipliers. A parallel implementation permits the interior-point methods to scale to very large reconstruction problems. Termination is based on well-defined convergence measures, namely, the Karush-Kuhn-Tucker first-order necessary conditions for optimality. We demonstrate the rapid convergence of the path-following interior-point methods using both data from a small animal scanner and Monte Carlo simulated data. The proposed methods can readily be applied to solve the regularized, weighted least squares reconstruction problem.

Combined Entropic Regularization and Path-Following Method for Solving Finite Convex Min-max Problems Subject to Infinitely Many Linear Constraints

In this paper, we study the minimization of the max function of q smooth convex functions on a domain specified by infinitely many linear constraints. The difficulty of such problems arises from the kinks of the max function and it is often suggested that, by imposing certain regularization functions, nondifferentiability will be overcome. We find that the entropic regularization introduced by Li and Fang is closely related to recently developed path-following interior-point methods. Based on their results, we create an interior trajectory in the feasible domain and propose a path-following algorithm with a convergence proof. Our intention here is to show a nice combination of minmax problems, semi-infinite programming, and interior-point methods. Hopefully, this will lead to new applications.

On self-concordant convex–concave functions

In this paper, we introduce the notion of a self-concordant convex-concave function, establish basic properties of these functions and develop a path-following interior point method for approximating saddle points of “sufficiently well-behaved” convex-concave functions—those which admit natural self-concordant convex-concave regularizations. The approach is illustrated by its applications to developing an exterior penalty polynomial time method for Semidefinite Programming and to the problem of inscribing the largest volume ellipsoid into a given polytope.

Fuzzy Heuristics for Sequential Linear Programming

Numerical implementations of optimization algorithms often use parameters whose values are not strictly determined by the derivation of the algorithm, but must fall in some appropriate range of values. This work describes how fuzzy logic can be used to “control” such parameters to improve algorithm performance. This concept is shown with the use of sequential linear programming (SLP) due to its simplicity in implementation. The algorithm presented in this paper implements heuristics to improve the behavior of SLP based on current iterate values of design constraints and changes in search direction. Fuzzy logic is used to implement the heuristics in a form similar to what a human observer would do. An efficient algorithm, known as the infeasible primal-dual path-following interior-point method, is used for solving the sequence of LP problems. Four numerical examples are presented to show that the proposed SLP algorithm consistently performs better than the standard SLP algorithm.

A predictor-corrector method for extended linear-quadratic programming

The saddle point form of extended linear-quadratic programs can be solved by an interior point path-following method in polynomial time. The algorithm may take advantage of the block structures of certain problems arising from optimal control and stochastic programming. In addition, it needs no line searches and treats fully or not fully quadratic problems equally. Preliminary computational results apparently show that the algorithm is effective in solving a class of two-stage stochastic programming problems.

Properties of an Interior-Point Mapping for Mixed Complementarity Problems

Using a unified theory of local homeomorphic maps, we establish some basic properties of a fundamental mapping associated with the family of path-following interior-point methods for solving a mixed nonlinear complementarity problem.

A primal-dual interior point method whose running time depends only on the constraint matrix

We propose a primal-dual "layered-step" interior point (LIP) algorithm for linear programming with data given by real numbers. This algorithm follows the central path, either with short steps or with a new type of step called a "layered least squares" (LLS) step. The algorithm returns an exact optimum after a finite number of steps--in particular, after O(n3.5c(A)) iterations, wherec(A) is a function of the coefficient matrix. The LLS steps can be thought of as accelerating a classical path-following interior point method. One consequence of the new method is a new characterization of the central path: we show that it composed of at mostn2 alternating straight and curved segments. If the LIP algorithm is applied to integer data, we get as another corollary a new proof of a well-known theorem by Tardos that linear programming can be solved in strongly polynomial time provided thatA contains small-integer entries.

The complexity of an infeasible interior-point path-following method for the solution of linear programs

Interior point methods that follow the primal-dual central path of a dual pair of linear programs (P o),(D o) require that these problems are strictly feasible. To get around this difficulty, one technique is to embed (P o),(D o) into a family of suitably perturbed strictly feasible linear programs (P r),(D r),r>0 and to follow the path (x(r),s(r)),r>0, of all strictly feasible solutions of (P r),(D r) with xi (r)s i(r)=r,i= 1…,n. Path following methods compute approximations to this path at parameters R=r o>r 1>…, and their complexity is measured by the number N=N(r, R) of steps needed to reduce a given R>0 to a desired r>0. It is shown for a standard method that N(r, R) is estimated by a bound where C o is a universal constant but K=K([bbar],[cbar])is a constant depending [bbar],[cbar] on with K(0,0)=0, and on the size of the path.

Acceleration and Parallelization of the Path-Following Interior Point Method for a Linearly Constrained Convex Quadratic Problem

In this paper, the strategies for acceleration of the path-following polynomial time interior point method for linear and linearly constrained quadratic programming problems are studied. These strategies are based on (i) exploiting the results of computations done at the previous iterations (Karmarkar’s acceleration scheme and a scheme based on the preconditioned conjugate gradient method); (ii) implementation of “fast” linear algebra routines; (iii) parallel computations.