An Improved Predictor-Corrector Interior-Point Algorithm for Linear Complementarity Problems with -Iteration Complexity

This paper proposes an improved predictor-corrector interior-point algorithm for the linear complementarity problem (LCP) based on the Mizuno-Todd-Ye algorithm. The modified corrector steps in our algorithm cannot only draw the iteration point back to a narrower neighborhood of the center path but also reduce the duality gap. It implies that the improved algorithm can converge faster than the MTY algorithm. The iteration complexity of the improved algorithm is proved to obtain √() which is similar to the classical Mizuno-Todd-Ye algorithm. Finally, the numerical experiments show that our algorithm improved the performance of the classical MTY algorithm

An infeasible interior-point arc-search method with Nesterov's restarting strategy for linear programming problems

An arc-search interior-point method is a type of interior-point methods that approximates the central path by an ellipsoidal arc, and it can often reduce the number of iterations. In this work, to further reduce the number of iterations and computation time for solving linear programming problems, we propose two arc-search interior-point methods using Nesterov's restarting strategy that is well-known method to accelerate the gradient method with a momentum term. The first one generates a sequence of iterations in the neighborhood, and we prove that the convergence of the generated sequence to an optimal solution and the computation complexity is polynomial time. The second one incorporates the concept of the Mehrotra-type interior-point method to improve numerical performance. The numerical experiments demonstrate that the second one reduced the number of iterations and computational time. In particular, the average number of iterations was reduced compared to existing interior-point methods due to the momentum term.Comment: 33 pages, 6 figures, 2 table

Unified Analysis of Kernel-Based Interior-Point Methods for \u3cem\u3eP\u3c/em\u3e *(κ)-LCP

We present an interior-point method for the P∗(κ)-linear complementarity problem (LCP) that is based on barrier functions which are defined by a large class of univariate functions called eligible kernel functions. This class is fairly general and includes the classical logarithmic function and the self-regular functions, as well as many non-self-regular functions as special cases. We provide a unified analysis of the method and give a general scheme on how to calculate the iteration bounds for the entire class. We also calculate the iteration bounds of both long-step and short-step versions of the method for several specific eligible kernel functions. For some of them we match the best known iteration bounds for the long-step method, while for the short-step method the iteration bounds are of the same order of magnitude. As far as we know, this is the first paper that provides a unified approach and comprehensive treatment of interior-point methods for P∗(κ)-LCPs based on the entire class of eligible kernel functions

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

Interior-Point Algorithms Based on Primal-Dual Entropy

We propose a family of search directions based on primal-dual entropy in the context of interior point methods for linear programming. This new family contains previously proposed search directions in the context of primal-dual entropy. We analyze the new family of search directions by studying their primal-dual affine-scaling and constant-gap centering components. We then design primal-dual interior-point algorithms by utilizing our search directions in a homogeneous and self-dual framework. We present iteration complexity analysis of our algorithms and provide the results of computational experiments on NETLIB problems

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

Computational analysis of real-time convex optimization for control systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2000.Includes bibliographical references (p. 177-189).Computational analysis is fundamental for certification of all real-time control software. Nevertheless, analysis of on-line optimization for control has received little attention to date. On-line software must pass rigorous standards in reliability, requiring that any embedded optimization algorithm possess predictable behavior and bounded run-time guarantees. This thesis examines the problem of certifying control systems which utilize real-time optimization. A general convex programming framework is used, to which primal-dual path-following algorithms are applied. The set of all optimization problem instances which may arise in an on-line procedure is characterized as a compact parametric set of convex programming problems. A method is given for checking the feasibility and well-posedness of this compact set of problems, providing certification that every problem instance has a solution and can be solved in finite time. The thesis then proposes several algorithm initialization methods, considering the fixed and time-varying constraint cases separately. Computational bounds are provided for both cases. In the event that the computational requirements cannot be met, several alternatives to on-line optimization are suggested. Of course, these alternatives must provide feasible solutions with minimal real-time computational overhead. Beyond this requirement, these methods approximate the optimal solution as well as possible. The methods explored include robust table look-up, functional approximation of the solution set, and ellipsoidal approximation of the constraint set. The final part of this thesis examines the coupled behavior of a receding horizon control scheme for constrained linear systems and real-time optimization. The driving requirement is to maintain closed-loop stability, feasibility and well-posedness of the optimal control problem, and bounded iterations for the optimization algorithm. A detailed analysis provides sufficient conditions for meeting these requirements. A realistic example of a small autonomous air vehicle is furnished, showing how a receding horizon control law using real-time optimization can be certified.by Lawrence Kent McGovern.Ph.D

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