Local Convergence of the Affine-Scaling Interior-Point Algorithm for Nonlinear Programming

This paper addresses the local convergence properties of the affine-scaling interior-point algorithm for nonlinear programming. The analysis of local convergence is developed in terms of parameters that control the interior-point scheme and the size of the residual of the linear system that provides the step direction. The analysis follows the classical theory for quasi-Newton methods and addresses q-linear, q-superlinear, and q-quadratic rates of convergence

An Interior-Point Method with Polynomial Complexity and Superlinear Convergence for Linear Complementarity Problems

For linear programming, a primal-dual interior-point algorithm was recently constructed by Zhang and Tapia that achieves both polynomial complexity and Q-superlinear convergence (Q-quadratic in the nondegenerate case). In this paper, we extend their results to quadratic programming and linear complementarity problems

Polynomial Primal-Dual Cone Affine Scaling for Semidefinite Programming

In this paper we generalize the primal--dual cone affine scaling algorithm of Sturm and Zhang to semidefinite programming. We show in this paper that the underlying ideas of the cone affine scaling algorithm can be naturely applied to semidefinite programming, resulting in a new algorithm. Compared to other primal--dual affine scaling algorithms for semidefinite programming, our algorithm enjoys the lowest computational complexity

Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research

Long step homogeneous interior point algorithm for the p* nonlinear complementarity problems

A P*-Nonlinear Complementarity Problem as a generalization of the P*-Linear Complementarity Problem is considered. We show that the long-step version of the homogeneous self-dual interior-point algorithm could be used to solve such a problem. The algorithm achieves linear global convergence and quadratic local convergence under the following assumptions: the function satisfies a modified scaled Lipschitz condition, the problem has a strictly complementary solution, and certain submatrix of the Jacobian is nonsingular on some compact set

Long-Step Homogeneous Interior-Point Method for P*-Nonlinear Complementarity Problem

A P*-Nonlinear Complementarity Problem as a generalization of the P*Linear Complementarity Problem is considered. We show that the long-step version of the homogeneous self-dual interior-point algorithm could be used to solve such a problem. The algorithm achieves linear global convergence and quadratic local convergence under the following assumptions: the function satisfies a modified scaled Lipschitz condition, the problem has a strictly complementary solution, and certain submatrix of the Jacobian is nonsingular on some compact set

Primal-dual interior-point algorithms for linear programs with many inequality constraints

Linear programs (LPs) are one of the most basic and important classes of constrained optimization problems, involving the optimization of linear objective functions over sets defined by linear equality and inequality constraints. LPs have applications to a broad range of problems in engineering and operations research, and often arise as subproblems for algorithms that solve more complex optimization problems. ``Unbalanced'' inequality-constrained LPs with many more inequality constraints than variables are an important subclass of LPs. Under a basic non-degeneracy assumption, only a small number of the constraints can be active at the solution--it is only this active set that is critical to the problem description. On the other hand, the additional constraints make the problem harder to solve. While modern ``interior-point'' algorithms have become recognized as some of the best methods for solving large-scale LPs, they may not be recommended for unbalanced problems, because their per-iteration work does not scale well with the number of constraints. In this dissertation, we investigate "constraint-reduced'' interior-point algorithms designed to efficiently solve unbalanced LPs. At each iteration, these methods construct search directions based only on a small working set of constraints, while ignoring the rest. In this way, they significantly reduce their per-iteration work and, hopefully, their overall running time. In particular, we focus on constraint-reduction methods for the highly efficient primal-dual interior-point (PDIP) algorithms. We propose and analyze a convergent constraint-reduced variant of Mehrotra's predictor-corrector PDIP algorithm, the algorithm implemented in virtually every interior-point software package for linear (and convex-conic) programming. We prove global and local quadratic convergence of this algorithm under a very general class of constraint selection rules and under minimal assumptions. We also propose and analyze two regularized constraint-reduced PDIP algorithms (with similar convergence properties) designed to deal directly with a type of degeneracy that constraint-reduced interior-point algorithms are often subject to. Prior schemes for dealing with this degeneracy could end up negating the benefit of constraint-reduction. Finally, we investigate the performance of our algorithms by applying them to several test and application problems, and show that our algorithms often outperform alternative approaches