Approximate gradient projected condition in multiobjective optimization

In this work we present an extention of the well-known Approximated Gradient Projection (AGP) [8] property from the scalar problem with equality and inequality constraints to multiobjective problems. We prove that the condition called Multiobjective Approximate Gradient Projection (MAGP), is necessary for a point to be a local weak Pareto point and we study, under convex assumptions, sufficient conditions.Facultad de Ingenierí

ON NON-QUADRATIC PENALTY FUNCTION FOR NON-LINEAR PROGRAMMING PROBLEM WITH EQUALITY CONSTRAINTS

Purpose: The present paper focuses on the Non-Linear Programming Problem (NLPP) with equality constraints. NLPP with constraints could be solved by penalty or barrier methods. Methodology: We apply the penalty method to the NLPP with equality constraints only. The non-quadratic penalty method is considered for this purpose. We considered a transcendental i.e. exponential function for imposing the penalty due to the constraint violation. The unconstrained NLPP obtained in this way is then processed for further solution. An improved version of evolutionary and famous meta-heuristic Particle Swarm Optimization (PSO) is used for the same. The method is tested with the help of some test problems and mathematical software SCILAB. The solution is compared with the solution of the quadratic penalty method. Results: The results are also compared with some existing results in the literature

Approximate gradient projected condition in multiobjective optimization

In this work we present an extention of the well-known Approximated Gradient Projection (AGP) [8] property from the scalar problem with equality and inequality constraints to multiobjective problems. We prove that the condition called Multiobjective Approximate Gradient Projection (MAGP), is necessary for a point to be a local weak Pareto point and we study, under convex assumptions, sufficient conditions.Facultad de Ingenierí

Two new weak constraint qualifications and applications

We present two new constraint qualifications (CQs) that are weaker than the recently introduced relaxed constant positive linear dependence (RCPLD) CQ. RCPLD is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a CQ. This is done in the first new CQ, which we call the constant rank of the subspace component (CRSC) CQ. This new CQ also preserves many of the good properties of RCPLD, such as local stability and the validity of an error bound. We also introduce an even weaker CQ, called the constant positive generator (CPG), which can replace RCPLD in the analysis of the global convergence of algorithms. We close this work by extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods: sequential quadratic programming, augmented Lagrangians, interior point algorithms, and inexact restoration.Facultad de Ciencias ExactasDepartamento de Matemátic

TWO NEW WEAK CONSTRAINT QUALIFICATIONS AND APPLICATIONS

We present two new constraint qualifications (CQs) that are weaker than the recently introduced relaxed constant positive linear dependence (RCPLD) CQ. RCPLD is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a CQ. This is done in the first new CQ, which we call the constant rank of the subspace component (CRSC) CQ. This new CQ also preserves many of the good properties of RCPLD, such as local stability and the validity of an error bound. We also introduce an even weaker CQ, called the constant positive generator (CPG), which can replace RCPLD in the analysis of the global convergence of algorithms. We close this work by extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods: sequential quadratic programming, augmented Lagrangians, interior point algorithms, and inexact restoration.RONEX-Optimization (PRONEX-CNPq/FAPERJ) [E-26/171.510/2006-APQ1]Fapesp [2006/53768-0, 2009/09414-7, 2010/19720-5]CNPq [300900/2009-0, 303030/2007-0, 305740/2010-5, 474138/2008-9

On Approximate KKT Condition and its Extension to Continuous Variational Inequalities

In this work, we introduce a necessary sequential Approximate-Karush-Kuhn-Tucker (AKKT) condition for a point to be a solution of a continuous variational inequality, and we prove its relation with the Approximate Gradient Projection condition (AGP) of Garciga-Otero and Svaiter. We also prove that a slight variation of the AKKT condition is sufficient for a convex problem, either for variational inequalities or optimization. Sequential necessary conditions are more suitable to iterative methods than usual punctual conditions relying on constraint qualifications. The AKKT property holds at a solution independently of the fulfillment of a constraint qualification, but when a weak one holds, we can guarantee the validity of the KKT conditions.CNPq[503328/2009-0]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP[09/09414-7]PRONEX-CNPq/FAPERJ[E-26/171.164/2003]PRONEX-CNPq/FAPER

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