A Morse-type index for critical points of vector functions

In this work we study the critical points of vector functions form Rn to Rm with n m, following the definition introduced by S. Smale in the context of vector optimization. The local monotonicity properties of a vector function around a critical point which are invariant with respect to local coordinate changes are considered. We propose a classification of critical points through the introduction of an index for a critical point consisting of a triple of nonnegative integers. The proposed index is based on the ”sign” of an appropriate vector-valued second-order differential, that is proved to be invariant with respect to local coordinate changes. In order to avoid anomalous behaviours of the Jacobian matrix, the analysis is partially restricted to the proper critical points, a subset of critical points which enjoy stability properties with respect to perturbations of the order structure. Under nondegeneracy conditions, the index is proved to be locally constant. Moreover, the stability properties of the index with respect to perturbations both of the ordering cone and of the function are considered. Finally, the consistency of the proposed classification with the one given by Whitney for stable maps from the plane into the plane is proved. Keywords: Copula; Fréchet class; positive dependence stochastic ordering; right-tail decreasing (RTI); left-tail decreasing (LTD)

Set-convergence of convex sets and stability in vector optimization

This work establishes the lower convergence (in the sense of Kuratowski - Painlevé) of the set of minimal points of An to the set of minimal points of A, whenever An is a sequence of convex subsets of an euclidean space satisfying the dominance property and converging to A. Using this result and introducing a property for a function f that guarantees the convergence of the image f(An) to f(A) when An converges to A, we obtain some stability results in the decision space for a class of suitable perturbations of the feasible region of a vector optimization problem.

Well-posedness and scalarization in vector optimization

In this paper we study several existing notions of well-posedness for vector optimization problems. We distinguish them into two classes and we establish the hierarchical structure of their relationships. Moreover, we relate vector well-posedness and well-posedness of an appropriate scalarization. This approach allows us to show that, under some compactness assumption, quasiconvex problems are well-posed.well-posedness, vector optimization problems, nonlinear scalarization, generalized convexity.

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Box-constrained vector optimization: a steepest descent method without “a priori” scalarization

In this paper a notion of descent direction for a vector function defined on a box is introduced. This concept is based on an appropriate convex combination of the “projected” gradients of the components of the objective functions. The proposed approach does not involve an “apriori” scalarization since the coefficients of the convex combination of the projected gradients are the solutions of a suitable minimization problem depending on the feasible point considered. Subsequently, the descent directions are considered in the formulation of a first order optimality condition for Pareto optimality in a box-constrained multiobjective optimization problem. Moreover, a computational method is proposed to solve box-constrained multiobjective optimization problems. This method determines the critical points of the box constrained multiobjective optimization problem following the trajectories defined through the descent directions mentioned above. The convergence of the method to the critical points is proved. The numerical experience shows that the computational method efficiently determines the whole local Pareto front.Multi-objective optimization problems, path following methods, dynamical systems, minimal selection.

Set-Convergence and Linear Operators: Some Results with Applications

In this work we study the convergence of the images of a converging sequence of convex sets {An} through a converging sequence of bounded linear operators Ln. The assumptions used here are especially fit for applications in three distinct fields. First, we study the convergence of the kernels of a class of bounded linear operators and of their adjoints. Then, we establish a stability results for the so-called abstract spilne problem, a relevant interpolation tool used, for instance, in econometric and financial applications. Finally, we deal with the stability of the set of efficient solutions for a linear multiobjective optimization problem.

Well-posedness and convexity in vector optimization

We study a notion of well-posedness in vector optimization through the behaviour fo minimizing sequences of sets, defined in terms of Hausdorff set-convergence. We prove the involvement of strict efficiency, a refinement of the notion of efficiency, in the definition of well-posedness. Using the previous results we identify a class of well-posed vector optimization problems: the convex problems with compact efficient frontiers.vector-optimization, well-posedness, stability, Hausdorff set-convergence