Pseudonormality and a language multiplier theory for constrained optimization

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003.Includes bibliographical references (leaves 211-213).Lagrange multipliers are central to analytical and computational studies in linear and non-linear optimization and have applications in a wide variety of fields, including communication, networking, economics, and manufacturing. In the past, the main research in Lagrange multiplier theory has focused on developing general and easily verifiable conditions on the constraint set, called constraint qualifications, that guarantee the existence of Lagrange multipliers for the optimization problem of interest. In this thesis, we present a new development of Lagrange multiplier theory that significantly differs from the classical treatments. Our objective is to generalize, unify, and streamline the theory of constraint qualifications. As a starting point, we derive an enahanced set of necessary optimality conditions of the Fritz John-type, which are stronger than the classical Karush-Kuhn-Tucker conditions. They are also more general in that they apply even when there is a possibly nonconvex abstract set constraint, in addition to smooth equality and inequality constraints. These optimality conditions motivate the introduction of a new condition, called pseudonormality, which emerges as central within the taxonomy of significant characteristics of a constraint set. In particular, pseudonormality unifies and extends the major constraint qualifications. In addition, pseudonormality provides the connecting link between constraint qualifications and exact penalty functions. Our analysis also yields identification of different types of Lagrange multipliers. Under some convexity assumptions, we show that there exists a special Lagrange multiplier vector, called informative, which carries significant sensitivity information regarding the constraints that directly affect the optimal cost change.(cont.) In the second part of the thesis, we extend the theory to nonsmooth problems under convexity assumptions. We introduce another notion of multiplier, called geometric, that is not tied to a specific optimal solution and does not require differentiability of the cost and constraint functions. Using a line of development based on convex analysis, we develop Fritz John-type optimality conditions for problems that do not necessarily have optimal solutions. Through an extended notion of constraint pseudonormality, this development provides an alternative pathway to strong duality results of convex programming. We also introduce special geometric multipliers that carry sensitivity information and show their existence under very general conditions.by Asuman E. Ozdaglar.Ph.D

On constraint qualifications in mathematical programming

Статья посвящена условию R-регулярности (Error Bound Property) в задачах математического программирования. Данное условие играет важную роль в анализе сходимости численных алгоритмов оптимизации и является достаточно общим условием регулярности (constraint qualification) в задачах математического программирования. В статье получаются новые достаточные условия наличия R-регулярности в задачах математического программирования. The article is devoted to the condition of R-regularity (Error Bound Property) in problems of mathematical programming. This condition plays an important role in analyzing the convergence of numerical optimization algorithms and it is a fairly general condition of regularity (constraint qualification) in problems of mathematical programming. The article obtains new sufficient conditions for the presence of R-regularity in problems of mathematical programming

К УСЛОВИЯМ РЕГУЛЯРНОСТИ В МАТЕМАТИЧЕСКОМ ПРОГРАММИРОВАНИИ

The article is devoted to the condition of R-regularity (Error Bound Property) in problems of mathematical programming. This condition plays an important role in analyzing the convergence of numerical optimization algorithms and it is a fairly general condition of regularity (constraint qualification) in problems of mathematical programming. The article obtains new sufficient conditions for the presence of R-regularity in problems of mathematical programming.Статья посвящена условию R-регулярности (Error Bound Property) в задачах математического программирования. Данное условие играет важную роль в анализе сходимости численных алгоритмов оптимизации и является достаточно общим условием регулярности (constraint qualification) в задачах математического программирования. В статье получаются новые достаточные условия наличия R-регулярности в задачах математического программирования

Constant-Rank Condition and Second-Order Constraint Qualification

The constant-rank condition for feasible points of nonlinear programming problems was defined by Janin (Math. Program. Study 21:127-138, 1984). In that paper, the author proved that the constant-rank condition is a first-order constraint qualification. In this work, we prove that the constant-rank condition is also a second-order constraint qualification. We define other second-order constraint qualifications.Fil: Andreani, R.. Universidade Estadual de Campinas; BrasilFil: Echagüe, C. E.. Universidad Nacional de La Plata; ArgentinaFil: Schuverdt, María Laura. Universidad Nacional de La Plata; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentin

К условию R-регулярности в математическом программировании

This article is devoted to the Error Bound property (also named R-regularity) in mathematical programming problems. This property plays an important role in analyzing the convergence of numerical optimization algorithms, a topic covered by multiple publications, and at the same time it is a relatively generic constraint qualification that guarantees the satisfaction of the necessary Kuhn – Tucker optimality conditions in mathematical programming problems. In the article, new sufficient conditions for the error bound property are described, and it’s also shown that several known necessary conditions are insufficient. The sufficient conditions obtained can be used to prove the regularity of a large class of sets including sets that cannot be proven regular by other known constraints.Исследуется условие R-регулярности (Error Bound) в задачах математического программирования, которое играет важную роль в анализе сходимости численных алгоритмов оптимизации, что подтверждается многочисленными публикациями, и в то же время является достаточно общим условием регулярности (constraint qualification), обеспечивающим справедливость необходимых условий оптимальности Куна – таккера в задачах математического программирования. В статье представлены новые достаточные условия наличия R-регулярности в задачах математического программирования, а также показано, что известные необходимые условия не являются достаточными. Полученные достаточные условия позволяют доказать наличие R-регулярности у довольно широкого класса множеств, в том числе и у таких, для которых не выполняются другие известные условия

Cognitive Beamforming for Multiple Secondary Data Streams With Individual SNR Constraints

In this paper, we consider cognitive beamforming for multiple secondary data streams subject to individual signal-to-noise ratio (SNR) requirements for each secondary data stream. In such a cognitive radio system, the secondary user is permitted to use the spectrum allocated to the primary user as long as the caused interference at the primary receiver is tolerable. With both secondary SNR constraint and primary interference power constraint, we aim to minimize the secondary transmit power consumption. By exploiting the individual SNR requirements, we formulate this cognitive beamforming problem as an optimization problem on the Stiefel manifold. Both zero forcing beamforming (ZFB) and nonzero forcing beamforming (NFB) are considered. For the ZFB case, we derive a closed form beamforming solution. For the NFB case, we prove that the strong duality holds for the nonconvex primal problem and thus the optimal solution can be easily obtained by solving the dual problem. Finally, numerical results are presented to illustrate the performance of the proposed cognitive beamforming solutions.Comment: This is the longer version of a paper to appear in the IEEE Transactions on Signal Processin

Teoría y métodos para problemas de optimización multiobjetivo

En esta tesis estudiamos la posibilidad de extender el método Lagrangiano Aumentado clásico de optimización escalar, para resolver problemas con objetivos múltiples. El método Lagrangiano Aumentado es una técnica popular para resolver problemas de optimización con restricciones. Consideramos dos posibles extensiones: - mediate el uso de escalarizaciones. Basados en el trabajo consideramos el uso de funciones débilmente crecientes para analizar la convergencia global de un método Lagrangiano Aumentado para resolver el problema multiobjetivo con restricciones de igualdad y de desigualdad. - mediante el uso de una función Lagrangiana Aumentada vectorial. En este caso el subproblema en el método Lagrangiano Aumentado tiene la particularidad de ser vectorial y planetamos su resolución mediante el uso de un método del tipo gradiente proyectado no monótono. En las extensiones que presentamos en la tesis se analizan las hipótesis más débiles bajo las cuales es posible demostrar convergencia a un punto estacionario del problema multiobjetivo.Facultad de Ciencias Exacta