Hande Y. Benson

Interior-Point Methods for Nonconvex Nonlinear Programming: Filter Methods and Merit Functions

Recently, Fletcher and Leyffer proposed using filter methods instead of a merit function to control steplengths in a sequential quadratic programming algorithm. In this paper, we analyze possible ways to implement a filter-based approach in an interior-point algorithm. Extensive numerical testing shows that such an approach is more efficient than using a merit function alone.

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.

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

Abstract 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.

Solving Problems with Semidefinite and Related Constraints Using Interior-Point Methods for Nonlinear Programming

Abstract. In this paper, we describe how to reformulate a problem that has second-order cone and/or semidefiniteness constraints in order to solve it using a general-purpose interior-point algorithm for nonlinear programming. The resulting problems are smooth and convex, and numerical results from the DIMACS Implementation Challenge problems and SDPLib are provided.

Interior-point methods for nonconvex nonlinear programming: regularization and warmstarts

Abstract In this paper, we investigate the use of an exact primal-dual penalty approach within the framework of an interior-point method for nonconvex nonlinear programming. This approach provides regularization and relaxation, which can aid in solving ill-behaved problems and in warmstarting the algorithm. We present details of our implementation within the loqo algorithm and provide extensive numerical results on the CUTEr test set and on warmstarting in the context of quadratic, nonlinear, mixed integer nonlinear, and goal programming.

A COMPARATIVE STUDY OF LARGE-SCALE NONLINEAR OPTIMIZATION ALGORITHMS

In recent years, much work has been done on imple- menting a variety of algorithms in nonlinear programming soft- ware. In this paper, we analyze the performance of several state- of-the-art optimization codes on large-scale nonlinear optimization problems. Extensive numerical results are presented on dierent classes of problems, and features of each code that make it ecient or inecient for each class are examined.

Interior-point methods for nonconvex nonlinear programming: Jamming and numerical testing

Abstract.The paper considers an example of Wächter and Biegler which is shown to converge to a nonstationary point for the standard primal–dual interior-point method for nonlinear programming. The reason for this failure is analyzed and a heuristic resolution is discussed. The paper then characterizes the performance of LOQO, a line-search interior-point code, on a large test set of nonlinear programming problems. Specific types of problems which can cause LOQO to fail are identified.

Mathematical programming for Multi-Vehicle Motion Planning problems

Real world Multi-Vehicle Motion Planning (MVMP) problems require the optimization of suitable performance measures under an array of complex and challenging constraints involving kinematics, dynamics, communication connectivity, target tracking, and collision avoidance. The general MVMP problem can thus be formulated as a mathematical program (MP). In this paper we present a mathematical programming (MP) framework that captures the salient features of the general MVMP problem. To demonstrate the use of this framework for the formulation and solution of MVMP problems, we examine in detail four representative works and summarize several other related works. As MP solution algorithms and associated numerical solvers continue to develop, we anticipate that MP solution techniques will be applied to an increasing number of MVMP problems and that the framework and formulations presented in this paper may serve as a guide for future MVMP research.

Multi-vehicle path coordination under communication constraints

We generate time-optimal velocity profiles for a group of path-constrained vehicles with fixed and known initial and goal locations. Each vehicle robot must follow a fixed path, arrive at its goal as quickly as possible (or at least not increase the time for the last robot to arrive at its goal) and stay in communication with other robots in the arena throughout its journey. We seek to solve this multi-objective optimization problem by generating optimal velocities along the paths. The problem is formulated as a nonlinear programming problem (NLP) with constraints on the kinematics, dynamics, collision avoidance and communication. Solutions demonstrate the trade off between the arrival time, the required transmission power and the communication connectivity requirements. Typically the optimization improved connectivity at no appreciable cost in journey time (as measured by the time of arrival of the last-arriving robot).

Using Interior-Point Methods within an Outer Approximation Framework for Mixed Integer Nonlinear Programming

Interior-point methods for nonlinear programming have been demonstrated to be quite efficient, especially for large scale problems, and, as such, they are ideal candidates for solving the nonlinear subproblems that arise in the solution of mixed-integer nonlinear programming problems via outer approximation. However, traditionally, infeasible primal-dual interior-point methods have had two main perceived deficiencies: (1) lack of infeasibility detection capabilities, and (2) poor performance after a warmstart. In this paper, we propose the exact primal-dual penalty approach as a means to overcome these deficiencies. The generality of this approach to handle any change to the problem makes it suitable for the outer approximation framework, where each nonlinear subproblem can differ from the others in the sequence in a variety of ways. Additionally, we examine cases where the nonlinear subproblems take on special forms, namely those of second-order cone programming problems and semidefinite programming problems. Encouraging numerical results are provided.