Implementation of a continuation method for nonlinear complementarity problems via normal maps

Ankara : Department of Industrial Engineering and Institute of Engineering and Sciences, Bilkent Univ., 1997.Thesis (Master's) -- Bilkent University, 1997.Includes bibliographical references.In this thesis, a continuation method for nonlinear complementarity problems via normal maps that is developed by Chen, Harker and Pinar [8] is implemented. This continuation method uses the smooth function to approximate the normal map reformulation of nonlinear complementarity problems. The algorithm is implemented and tested with two different plussmoothing functions namely interior point plus-smooth function and piecewise quadratic plus-smoothing function. These two functions are compared. Testing of the algorithm is made with several known problems.Erkan, AliM.S

Optimization and Applications

[no abstract available

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

Second order strategies for complementarity problems

Orientadores: Sandra Augusta Santos, Roberto AndreaniTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientificaResumo: Neste trabalho reformulamos o problema de complementaridade não linear generalizado (GNCP) em cones poliedrais como um sistema não linear com restrição de não negatividade em algumas variáveis, e trabalhamos na resolução de tal reformulação por meio de estratégias de pontos interiores. Em particular, definimos dois algoritmos e provamos a convergência local de tais algoritmos sob hipóteses usuais. O primeiro algoritmo é baseado no método de Newton, e o segundo, no método tensorial de Chebyshev. O algoritmo baseado no método de Chebyshev pode ser visto como um método do tipo preditor-corretor. Tal algoritmo, quando aplicado a problemas em que as funções envolvidas são afins, e com escolhas adequadas dos parâmetros, torna-se o bem conhecido algoritmo preditor-corretor de Mehrotra. Também apresentamos resultados numéricos que ilustram a competitividade de ambas as propostas.Abstract: In this work we reformulate the generalized nonlinear complementarity problem (GNCP) in polyhedral cones as a nonlinear system with nonnegativity in some variables and propose the resolution of such reformulation through interior-point methods. In particular we define two algorithms and prove the local convergence of these algorithms under standard assumptions. The first algorithm is based on Newton's method and the second, on the Chebyshev's tensorial method. The algorithm based on Chebyshev's method may be considered a predictor-corrector one. Such algorithm, when applied to problems for which the functions are affine, and the parameters are properly chosen, turns into the well-known Mehrotra's predictor corrector algorithm. We also present numerical results that illustrate the competitiveness of both proposals.DoutoradoOtimizaçãoDoutor em Matemática Aplicad

Active-set prediction for interior point methods

This research studies how to efficiently predict optimal active constraints of an inequality constrained optimization problem, in the context of Interior Point Methods (IPMs). We propose a framework based on shifting/perturbing the inequality constraints of the problem. Despite being a class of powerful tools for solving Linear Programming (LP) problems, IPMs are well-known to encounter difficulties with active-set prediction due essentially to their construction. When applied to an inequality constrained optimization problem, IPMs generate iterates that belong to the interior of the set determined by the constraints, thus avoiding/ignoring the combinatorial aspect of the solution. This comes at the cost of difficulty in predicting the optimal active constraints that would enable termination, as well as increasing ill-conditioning of the solution process. We show that, existing techniques for active-set prediction, however, suffer from difficulties in making an accurate prediction at the early stage of the iterative process of IPMs; when these techniques are ready to yield an accurate prediction towards the end of a run, as the iterates approach the solution set, the IPMs have to solve increasingly ill-conditioned and hence difficult, subproblems. To address this challenging question, we propose the use of controlled perturbations. Namely, in the context of LP problems, we consider perturbing the inequality constraints (by a small amount) so as to enlarge the feasible set. We show that if the perturbations are chosen judiciously, the solution of the original problem lies on or close to the central path of the perturbed problem. We solve the resulting perturbed problem(s) using a path-following IPM while predicting on the way the active set of the original LP problem; we find that our approach is able to accurately predict the optimal active set of the original problem before the duality gap for the perturbed problem gets too small. Furthermore, depending on problem conditioning, this prediction can happen sooner than predicting the active set for the perturbed problem or for the original one if no perturbations are used. Proof-of-concept algorithms are presented and encouraging preliminary numerical experience is also reported when comparing activity prediction for the perturbed and unperturbed problem formulations. We also extend the idea of using controlled perturbations to enhance the capabilities of optimal active-set prediction for IPMs for convex Quadratic Programming (QP) problems. QP problems share many properties of LP, and based on these properties, some results require more care; furthermore, encouraging preliminary numerical experience is also presented for the QP case

Structure Exploitation in Mixed-Integer Optimization with Applications to Energy Systems

Das Ziel dieser Arbeit ist neue numerische Methoden für gemischt-ganzzahlige Optimierungsprobleme zu entwickeln um eine verbesserte Geschwindigkeit und Skalierbarkeit zu erreichen. Dies erfolgt durch Ausnutzung gängiger Problemstrukturen wie separierbarkeit oder Turnpike-eigenschaften. Methoden, die diese Strukturen ausnutzen können, wurden bereits im Bereich der verteilten Optimierung und optimalen Steuerung entwickelt, sie sind jedoch nicht direkt auf gemischt-ganztägige Probleme anwendbar. Um verteilte Rechenressourcen zur Lösung von gemischt-ganzzahligen Problemen nutzen zu können, sind neue Methoden erforderlich. Zu diesem Zweck werden verschiedene Erweiterungen bestehender Methoden sowie neuartige Techniken zur gemischt-ganzzahligen Optimierung vorgestellt. Benchmark-Probleme aus Strom- und Energiesystemen werden verwendet, um zu demonstrieren, dass die vorgestellten Methoden zu schnelleren Laufzeiten führen und die Lösung großer Probleme ermöglichen, die sonst nicht zentral gelöst werden können. Die vorliegende Arbeit enthält die folgenden Beiträge: - Eine Erweiterung des Augmented Lagrangian Alternating Direction Inexact Newton-Algorithmus zur verteilten Optimierung für gemischt-ganzzahlige Probleme. - Ein neuer, teilweise-verteilter Optimierungsalgorithmus für die gemischt-ganzzahlige Optimierung basierend auf äußeren Approximationsverfahren. - Ein neuer Optimierungsalgorithmus für die verteilte gemischt-ganzzahlige Optimierung, der auf branch-and-bound Verfahren basiert. - Eine erste Untersuchung von Turnpike-Eigenschaften bei Optimalsteuerungsproblemen mit gemischten-Ganzzahligen Entscheidungsgrößen und ein spezieller Algorithmus zur Lösung dieser Probleme. - Eine neue Branch-and-Bound Heuristik, die a priori Probleminformationen effizienter nutzt als aktuelle Warmstarttechniken. Schließlich wird gezeigt, dass die Ergebnisse der vorgestellten Optimierungsalgorithmen für verteilte gemischt-ganzzahlige Optimierung stark Partitionierungsabhängig sind. Zu diesem Zweck wird auch eine Untersuchung von Partitionierungsmethoden für die verteilte Optimierung vorgestellt

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum