Class Of Interior-point Methods Research Articles

SIGEST

The SIGEST article in this issue is “What Tropical Geometry Tells Us about the Complexity of Linear Programming,” by Xavier Allamigeon, Pascal Benchimol, Stéphane Gaubert, and Michael Joswig. Linear programming means optimizing a linear objective function with respect to linear equality constraints and linear inequality constraints. This remains a pervasive task in modern society, where decisions must be made while keeping control of multiple factors, such as staff time, raw materials, energy, financial outlay, or carbon footprint. Linear programs also arise as subtasks for more general problems, often forming computational bottlenecks. The search for linear programming algorithms with good worst-case complexity was boosted by the work of Khachiyan (1979) and Karmarker (1984), which brought interior point methods to the fore. In this SIGEST article, the authors construct a negative result: they define a linear program for which a widely used class of interior point methods has complexity that is exponential in the number of variables (Theorem A). This leads to a counterexample for the continuous analogue of the Hirsch conjecture, proposed by Deza, Terlaky, and Zinchenko in 2009. The authors' proofs use the tools of tropical geometry, a world where addition is replaced by maximization and multiplication is replaced by standard addition. Intuitively, this approach is likely to have value in scheduling-type problems because (a) for two activities that may be performed concurrently, the time required is the maximum of the individual times, and (b) for two activities that must take place consecutively, the time required is the sum of the individual times. The original version of this article appeared in the SIAM Journal on Applied Algebra and Geometry in 2018. In preparing this SIGEST version, the authors have added new material to sections 1 and 2 in order to increase accessibility, and in section 9 they have included a discussion of further work in this area and relevant open problems.

Rapid infeasibility detection in a mixed logarithmic barrier-augmented Lagrangian method for nonlinear optimization

ABSTRACTWe present a modification of a primal-dual algorithm based on a mixed augmented Lagrangian and a log-barrier penalty function. The goal of this new feature is to quickly detect infeasibility. An additional parameter is introduced to balance the minimization of the objective function and the realization of the constraints. The global convergence of the modified algorithm is analysed under mild assumptions. We also show that under a suitable choice of the parameters along the iterations, the rate of convergence of the algorithm to an infeasible stationary point is superlinear. This is the first local convergence result for the class of interior point methods in the infeasible case. We finally report some numerical experiments to show that this new algorithm is quite efficient to detect infeasibility and does not deteriorate the overall behavior in the general case.

APPLYING NEWTON ALGORITHMS WITHIN A SUPERVISED FEED FORWARD NEURAL NETWORK ARCHITECTURE TO FORECAST A MISSILE TRAJECTORY

A Neural Network is trained to forecast a moving trajectory. The neural network training is formulated as a nonlinear programming problem and a Newton method is used to find the optimal weights. The learning Algorithm is derived using a Recursive Prediction Error Method that approximates the inverse of the Hessian. Furthermore, box Constraints are added to the network weights to avoid network paralysis and a constraint nonlinear programming problem is formulated. Logarithmic Barrier methods which are a class of Interior Point Methods are presented. Interior point methods have good convergence properties because the weights move on a center path in the interior of the feasible weight. The logarithmic barrier method is combined with the Newton method to form a Newton-Barrier method. The moving missile trajectory is simulated using differential equations and the proposed algorithm is used to train the network in order to forecast the missile position at any given time.

Failure of global convergence for a class of interior point methods for nonlinear programming

Using a simple analytical example, we demonstrate that a class of interior point methods for general nonlinear programming, including some current methods, is not globally convergent. It is shown that those algorithms produce limit points that are neither feasible nor stationary points of some measure of the constraint violation, when applied to a well-posed problem.

A hybrid method for the nonlinear least squares problem with simple bounds

This paper presents a new method with trust region technique for solving the nonlinear least squares problem with lower and upper bounds on the variables. The proposed method constructs trust region constraints that are ellipses centered at the iterative points in such a way that they lie in the interior of the feasible region. Thus the method belongs to the class of interior point methods, and hence we may expect that the generated sequence approaches a solution smoothly without the combinatorial complications inherent to traditional active set methods. We establish a convergence theorem for the proposed method and show its practical efficiency by numerical experiments.