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Condições de qualificação para programação não linear / Constraint qualifications for nonlinear programmingCastillo Huamaní, Darwin, 1982 23 August 2018 (has links)
Orientador: Roberto Andreani / Dissertação (mestrado)  Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 20180823T21:59:03Z (GMT). No. of bitstreams: 1
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Previous issue date: 2013 / Resumo: Nesta dissertação, estudamse as condições de qualificação para problemas de programação não linear, essencialmente, a condição de qualificação de posto constante da componente de subespaço e a condição de qualificação de gerador positivo constante que foram recentemente introduzidas. Ademais, fazse um estudo sobre a teoria das bases positivas no espaço R??, onde vêse a demonstração do teorema da partição de uma base positiva, que é usado na demonstração de que a relaxação da condição de qualificação de dependência linear positiva constante implica a condição de qualificação de posto constante da componente de subespaço. Notase que as propriedades das bases positivas não são semelhantes às propriedades das bases lineares dos subespaços vetoriais / Abstract: In this dissertation, we study the constraint of qualifications for nonlinear programming problems, essentially, The constraint qualification of constant rank component subspace and the constraint qualification of generator positive constant, recently introduced. Furthermore, a study is done about of the theory of positive bases in the space R?? where one can see the demonstration of partition of a positive basis theorem, which is used in the proof of that the relaxation of constraint qualification of constant positive linear dependence implies the constraint qualification of constant rank component subspace. Note that the properties of the positive bases are not similar to the properties of the linear bases for vector subspaces / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

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Enhanced Optimality Conditions and New Constraint Qualifications for Nonsmooth Optimization ProblemsZhang, Jin 12 December 2014 (has links)
The main purpose of this dissertation is to investigate necessary optimality conditions for a class of very general nonsmooth optimization problems called the mathematical program with geometric constraints
(MPGC). The geometric constraint means that the image of certain mapping is included in a
nonempty and closed set.
We first study the conventional nonlinear program with equality, inequality and abstract set constraints as a special case of MPGC. We derive the enhanced Fritz John condition and from which, we obtain the enhanced KarushKuhnTucker (KKT) condition and introduce the associated pseudonormality and quasinormality condition. We prove that either pseudonormality or quasinormality with regularity implies the existence of a local error bound. We also give a tighter upper estimate for the Fr\'chet subdifferential and the limiting subdifferential of the value function in terms of quasinormal multipliers which is usually a smaller set than the set of classical normal multipliers.
We then consider a more general MPGC where the image of the mapping from a Banach space is included in a
nonempty and closed subset of a finite dimensional space. We obtain the enhanced Fritz John necessary optimality conditions in terms of the
approximate subdifferential. One of the technical
difficulties in obtaining such a result in an infinite dimensional space is
that no compactness result can be used to show the existence of local
minimizers of a perturbed problem. We employ the celebrated
Ekeland's variational principle to obtain the results instead. We then apply our results to the study of exact penalty and sensitivity analysis.
We also study a special class of MPCG named mathematical programs with equilibrium constraints (MPECs). We argue that the MPEClinear independence constraint qualification is not a constraint qualification for the strong (S) stationary condition when the objective function is nonsmooth. We derive the enhanced Fritz John Mordukhovich (M) stationary condition for MPECs. From this enhanced Fritz John Mstationary condition we introduce the associated MPEC generalized pseudonormality and quasinormality condition and build the relations between them and some other widely used MPEC constraint qualifications. We give upper estimates for the subdifferential of the value function in terms of the enhanced M and Cmultipliers respectively.
Besides, we focus on some new
constraint qualifications introduced for nonlinear extremum problems in the
recent literature. We show that, if the constraint functions are continuously
differentiable, the relaxed MangasarianFromovitz constraint qualification (or,
equivalently, the constant rank of the subspace component condition) implies
the existence of local error bounds. We further extend the new result to the MPECs. / Graduate / 0405

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Condições sequenciais de otimalidade / Sequential optimality conditionsHaeser, Gabriel 09 April 2009 (has links)
Orientador: Jose Mario Martinez / Tese (doutorado)  Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 20180814T02:27:22Z (GMT). No. of bitstreams: 1
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Previous issue date: 2009 / Resumo: Estudamos as condições de otimalidade provenientes dos algoritmos de penalidade externa, penalidade interna, penalidade internaexterna e restauração inexata, e mostramos relações com a CPLD, uma nova condição de qualificação estritamente mais fraca que a condição de MangasarianFromovitz e a condição de posto constante de Janin. Estendemos o resultado do clássico Lema de Carathéodory, onde mostramos um limitante para o tamanho dos novos multiplicadores. Apresentamos novas condições de otimalidade relacionadas à condição AGP (Approximate Gradient Projection). Quando há um conjunto extra de restrições lineares, definimos uma condição do tipo AGP e provamos relações com a CPLD e as equações KKT. Resultados similares são obtidos quando há um conjunto extra de restrições convexas. Mostramos também algumas generalizações e relações com um algoritmo de restauração inexata. / Abstract: We study optimality conditions generated by the external penalty, internal penalty, internalexternal penalty and inexact restoration algorithms, and we show relations with the CPLD, a new constraint qualification strictly weaker than the MangasarianFromovitz condition and the constant rank condition of Janin. We extend the result of the classical Carathéodory's Lemma, where we show a bound for the size of the new multipliers. We present new optimality conditions related to the Approximate Gradient Projection condition (AGP). When there is an extra set of linear constraints, we define an AGP type condition and prove relations with CPLD and KKT conditions. Similar results are obtained when there is an extra set of convex constraints. We provide some further generalizations and relations to an inexact restoration algorithm. / Doutorado / Otimização / Doutor em Matemática Aplicada

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Optimality Conditions for Cardinality Constrained Optimization ProblemsXiao, Zhuoyu 11 August 2022 (has links)
Cardinality constrained optimization problems (CCOP) are a new class of optimization
problems with many applications. In this thesis, we propose a framework
called mathematical programs with disjunctive subspaces constraints (MPDSC), a
special case of mathematical programs with disjunctive constraints (MPDC), to investigate
CCOP. Our method is different from the relaxed complementaritytype reformulation
in the literature. The first contribution of this thesis is that we study various stationarity conditions for MPDSC, and then apply them to CCOP. In particular, we recover disjunctivetype strong (S) stationarity and Mordukhovich (M) stationarity for CCOP, and then reveal the relationship between them and those from the relaxed complementaritytype reformulation. The second contribution of this thesis is that we obtain some new results for MPDSC, which do not hold for MPDC in general. We show that many constraint qualifications like the relaxed constant positive linear dependence (RCPLD) coincide with their piecewise versions for MPDSC. Based on such result, we prove that RCPLD implies error bounds for MPDSC. These two results also hold for CCOP. All of these disjunctivetype constraint qualifications for CCOP derived from MPDSC are weaker than those from the relaxed complementaritytype reformulation in some sense. / Graduate

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Condições de otimalidade, qualificação e métodos tipo Lagrangiano aumentado para problemas de equilíbrio de Nash generalizados / Optimality conditions, constraint qualifications and Augmented Lagrangian type methods for Generalized Nash Equilibrium ProblemsFrank Navarro Rojas 14 March 2018 (has links)
Esta tese é um estudo acerca do Problema de Equilíbrio de Nash Generalizado (GNEP). Na primeira parte, faremos um resumo dos principais conceitos sobre GNEPs, a relação com outros problemas já conhecidos e comentaremos brevemente os principais métodos já feitos até esta data para resolver numericamente este tipo de problema. Na segunda parte, estudamos condições de otimalidade e condições de qualificação (CQ) para GNEPs, fazendo uma analogia como em otimização. Estendemos os conceitos de cone tangente, normal, gerado pelas restrições ativas, linearizado e polar para a estrutura dos GNEPs. Cada CQ de otimização gera dois tipos de CQ para GNEPs, sendo que a denotada por CQGNEP é mais forte e útil para a análise de algoritmos para GNEPs. Mostramos que as condições de qualificação para GNEPs deste tipo em alguns casos não guardam a mesma relação que em otimização. Estendemos também o conceito de Aproximadamente KarushKuhnTucker (AKKT) de otimização para GNEPs, o AKKTGNEP. É bem conhecido que AKKT é uma genuína condição de otimalidade em otimização, mas para o caso dos GNEPs mostramos que isto não ocorre em geral. Por outro lado, AKKTGNEP é satisfeito, por exemplo, em qualquer solução de um GNEP conjuntamente convexo, desde que seja um equilíbrio bvariacional. Com isso em mente, definimos um método do tipo Lagrangiano Aumentado para o GNEP usando penalidades quadráticas e exponenciais e estudamos as propriedades de otimalidade e viabilidade dos pontos limites de sequências geradas pelo algoritmo. Finalmente alguns critérios para resolver os subproblemas e resultados numéricos são apresentados. / This thesis is a study about the generalized Nash equilibrium problem (GNEP). In the first part we will summarize the main concepts about GNEPs, the relationship with other known problems and we will briefly comment on the main methods already done in order to solve these problems numerically. In the second part we study optimality conditions and constraint qualification (CQ) for GNEPs making an analogy with the optimization case. We extend the concepts of the tangent, normal and generated by the active cones, linear and polar cone to the structure of the GNEPs. Each optimization CQ generates two types of CQs for GNEPs, with the one called CQGNEP being the strongest and most useful for analyzing the algorithms for GNEPs. We show that the qualification conditions for GNEPs of this type in some cases do not have the same relation as in optimization. We also extend the Approximate Karush KuhnTucker (AKKT) concept used in optimization for GNEPs to AKKTGNEP. It is well known that AKKT is a genuine optimality condition in optimization but for GNEPs we show that this does not occur in general. On the other hand, AKKTGNEP is satisfied, for example, in any solution of a jointly convex GNEP, provided that it is a bvariational equilibrium. With this in mind, we define Augmented Lagrangian methods for the GNEP, using the quadratic and the exponential penalties, and we study the optimality and feasibility properties of the sequence of points generated by the algorithms. Finally some criteria to solve the subproblems and numerical results are presented.

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Bilevel programmingZemkoho, Alain B. 25 June 2012 (has links) (PDF)
We have considered the bilevel programming problem in the case where the lowerlevel problem admits more than one optimal solution. It is wellknown in the literature that in such a situation, the problem is illposed from the view point of scalar objective optimization. Thus the optimistic and pessimistic approaches have been suggested earlier in the literature to deal with it in this case. In the thesis, we have developed a unified approach to derive necessary optimality conditions for both the optimistic and pessimistic bilevel programs, which is based on advanced tools from variational analysis. We have obtained various constraint qualifications and stationarity conditions depending on some constructive representations of the solution setvalued mapping of the follower’s problem. In the auxiliary developments, we have provided rules for the generalized differentiation and robust Lipschitzian properties for the lowerlevel solution setvalued map, which are of a fundamental interest for other areas of nonlinear and nonsmooth optimization.
Some of the results of the aforementioned theory have then been applied to derive stationarity conditions for some wellknown transportation problems having the bilevel structure.

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Bilevel programming: reformulations, regularity, and stationarityZemkoho, Alain B. 12 June 2012 (has links)
We have considered the bilevel programming problem in the case where the lowerlevel problem admits more than one optimal solution. It is wellknown in the literature that in such a situation, the problem is illposed from the view point of scalar objective optimization. Thus the optimistic and pessimistic approaches have been suggested earlier in the literature to deal with it in this case. In the thesis, we have developed a unified approach to derive necessary optimality conditions for both the optimistic and pessimistic bilevel programs, which is based on advanced tools from variational analysis. We have obtained various constraint qualifications and stationarity conditions depending on some constructive representations of the solution setvalued mapping of the follower’s problem. In the auxiliary developments, we have provided rules for the generalized differentiation and robust Lipschitzian properties for the lowerlevel solution setvalued map, which are of a fundamental interest for other areas of nonlinear and nonsmooth optimization.
Some of the results of the aforementioned theory have then been applied to derive stationarity conditions for some wellknown transportation problems having the bilevel structure.

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