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

Proximity measures based on KKT points for constrained multi-objective optimization

An important aspect of optimization algorithms, for instance evolutionary algorithms, are termination criteria that measure the proximity of the found solution to the optimal solution set. A frequently used approach is the numerical verification of necessary optimality conditions such as the Karush-Kuhn-Tucker (KKT) conditions. In this paper, we present a proximity measure which characterizes the violation of the KKT conditions. It can be computed easily and is continuous in every efficient solution. Hence, it can be used as an indicator for the proximity of a certain point to the set of efficient (Edgeworth-Pareto-minimal) solutions and is well suited for algorithmic use due to its continuity properties. This is especially useful within evolutionary algorithms for candidate selection and termination, which we also illustrate numerically for some test problems

Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions

The aims of this paper are twofold. First, it is shown, for the first time, which types of nonsmooth functions are characterized by all vector critical points as being efficient or weakly efficient solutions of vector optimization problems in constrained and unconstrained scenarios on Hadamard manifolds. This implies the need to extend different concepts, such as the Karush-Kuhn-Tucker vector critical points and generalized invexity functions, to Hadamard manifolds. The relationships between these quantities are clarified through a great number of explanatory examples. Second, we present an economic application proving that Nash's critical and equilibrium points coincide in the case of invex payoff functions. This is done on Hadamard manifolds, a particular case of noncompact Riemannian symmetric spaces

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