Interior-point methods for linear programming: a review

Entropy optimization: Interior point methods; Interior point methods for semidefinite programming; Linear programming: Interior point methods; Linear programming: Karmarkar projective algorithm; Potential reduction methods for linear programming; Sequential quadratic programming: Interior point methods for distributed optimal control problems; Successive quadratic programming: Solution by active sets and interior point methods HOMOGENEOUS SELFDUAL METHODS FOR LINEAR PROGRAMMING

New Advances in Convex Optimization and Control Applications

This paper presents some computational experience with a primal-dual interior-point method for smooth convex programming. The algorithm is a direct extension of a primal-dual interior-point method for linear programming. In particular, it does not involve any line search, a definite advantage when function evaluations are costly. The test problems are geometric programming problems and randomly generated convex quadratically constrained problems. We study the influence of the size of the problem (number of inequality constraints and numbers of free variables) on the total number of iterations.

Primal-dual potential reduction methods for semidefinite programming using affine-scaling directions

We study, in the context of interior-point methods for linear programming, some possible advantages of postponing the choice of the per. "'lty parameter and the step length, which happens both when we apply Newton's method to the Karush-Kuhn-Thcker 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 J-L, and, in the case of a predictor-corrector step, as a quadratic function of 1-L 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. Computational results of these strategies are shown and compareci with PCx, an implementation of Mehrotra's predictor-corrector method.

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.

Splitting dense columns in sparse linear systems

In this paper we consider a linear programming problem with the underlying matrix unimodular, and the other data integer. Given arbitrary near optimum feasible solutions to the primal and the dual problems, we obtain conditions under which statements can be made about the value of certain variables in optimal vertices. Such results have applications to the problem of determining the stopping criterion in interior point methods like the primal-dual affine scaling method and the path following methods for linear programming.

A barrier method for dynamic Leontief-type linear programs

This paper studies the semidefinite programming SDP problem, i.e., the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First the classical cone duality is reviewed as it is specialized to SDP is reviewed. Next an interior point algorithm is presented that converges to the optimal solution in polynomial time. The approach is a direct extension of Ye’s projective method for linear programming. It is also argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion. Finally, the significance of these results is studied in a variety of combinatorial optimization problems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k-partite subgraph problem in graph...

First-order primal-dual methods are appealing for their low memory overhead, fast iterations, and effective parallelization. However, they are often slow at finding high accuracy solutions, which creates a barrier to their use in traditional linear programming (LP) applications. This paper exploits the sharpness of primal-dual formulations of LP instances to achieve linear convergence using restarts in a general setting that applies to alternating direction method of multipliers (ADMM), primal-dual hybrid gradient method (PDHG) and extragradient method (EGM). In the special case of PDHG, without restarts we show an iteration count lower bound of \(\Omega (\kappa ^2 \log (1/\epsilon ))\), while with restarts we show an iteration count upper bound of \(O(\kappa \log (1/\epsilon ))\), where \(\kappa \) is a condition number and \(\epsilon \) is the desired accuracy. Moreover, the upper bound is optimal for a wide class of primal-dual methods, and applies to the strictly more general class of sharp primal-dual problems. We develop an adaptive restart scheme and verify that restarts significantly improve the ability of PDHG, EGM, and ADMM to find high accuracy solutions to LP problems.

The algorithm proposed in this paper is the affine polynomial potential reduction method, with new procedures for updating the lower bounds for an optimal solution of the linear programming problem. A method is developed for updating the lower bounds by large steps, with strict control over the duality gaps associated with each iterate. Two algorithms are obtained by this approach: the first one has complexity $O(nL)$ iterations, and a very simple updating procedure; the second one updates the lower bounds at points very near the central trajectory and achieves a complexity of $O(\sqrt{n} L)$ iterations.

The success of the implementation of the normal equations approach of interior point methods (IPM) for linear programming depends on the quality of its analysis phase, i.e. reordering for sparsity. The goal of this analysis is to find a permutation matrix P such that the Cholesky factor of PAD 2 A T P T is the sparsest possible. In practice, heuristics are used to solve this problem because finding an optimal permutation is an NP-complete problem. Two such heuristics, namely the minimum degree and the minimum local fill—in orderings are particularly useful in the context of IPM implementations. In this paper a parametric set of symbolic orderings is presented, which connects these two major approaches. It will be shown that in the “neighborhood” of the minimum degree ordering a practically efficient method exist. Implementation details will be discussed as well, and on a demonstrative set of linear programming test problems the performance of the new method will be compared with Sparspak’s GENQMD subroutine which was for a long time public accesible from NETLIB, and with the minimum local fill—in ordering implementation of CPLEX version 4.0.KeywordsMinimum DegreeInterior Point MethodCholesky FactorSparsity PatternTrue DegreeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper we study the complexity of primal-dual path-following methods which are a particular kind of interior point method for solving linear programs. In particular we establish a relationship between the complexity and the integral of a weighted curvature along the central trajectory (path) defined in the interior of the feasible solution space of a linear program. An important property of the trajectory, viz., that its higher order derivatives are bounded by a geometric series, is presented. Applying this property we show that the complexity of such methods is bounded from below and from above by the curvature integral. This theorem reduces the complexity analysis to a relatively simple problem of estimating the curvature integral.

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