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

An asymptotically superlinearly convergent semismooth Newton augmented Lagrangian method for Linear Programming

Powerful interior-point methods (IPM) based commercial solvers, such as Gurobi and Mosek, have been hugely successful in solving large-scale linear programming (LP) problems. The high efficiency of these solvers depends critically on the sparsity of the problem data and advanced matrix factorization techniques. For a large scale LP problem with data matrix $A$ that is dense (possibly structured) or whose corresponding normal matrix $AA^T$ has a dense Cholesky factor (even with re-ordering), these solvers may require excessive computational cost and/or extremely heavy memory usage in each interior-point iteration. Unfortunately, the natural remedy, i.e., the use of iterative methods based IPM solvers, although can avoid the explicit computation of the coefficient matrix and its factorization, is not practically viable due to the inherent extreme ill-conditioning of the large scale normal equation arising in each interior-point iteration. To provide a better alternative choice for solving large scale LPs with dense data or requiring expensive factorization of its normal equation, we propose a semismooth Newton based inexact proximal augmented Lagrangian ({\sc Snipal}) method. Different from classical IPMs, in each iteration of {\sc Snipal}, iterative methods can efficiently be used to solve simpler yet better conditioned semismooth Newton linear systems. Moreover, {\sc Snipal} not only enjoys a fast asymptotic superlinear convergence but is also proven to enjoy a finite termination property. Numerical comparisons with Gurobi have demonstrated encouraging potential of {\sc Snipal} for handling large-scale LP problems where the constraint matrix $A$ has a dense representation or $AA^T$ has a dense factorization even with an appropriate re-ordering.Comment: Due to the limitation "The abstract field cannot be longer than 1,920 characters", the abstract appearing here is slightly shorter than that in the PDF fil

Introducing Interior-Point Methods for Introductory Operations Research Courses and/or Linear Programming Courses

In recent years the introduction and development of Interior-Point Methods has had a profound impact on optimization theory as well as practice, influencing the field of Operations Research and related areas. Development of these methods has quickly led to the design of new and efficient optimization codes particularly for Linear Programming. Consequently, there has been an increasing need to introduce theory and methods of this new area in optimization into the appropriate undergraduate and first year graduate courses such as introductory Operations Research and/or Linear Programming courses, Industrial Engineering courses and Math Modeling courses. The objective of this paper is to discuss the ways of simplifying the introduction of Interior-Point Methods for students who have various backgrounds or who are not necessarily mathematics majors

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.Comment: 18 page

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more