Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first-order optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of one-sided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the game-theoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows black-and-white constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a freestanding exposition of basic nonsmooth analysis as motivated by and applied to this subject.

: Multi-objective optimization is known as a useful mathematical model in order to investigate some real world problems with conflicting objectives, arising from economics, engineering and human decision making. In this paper, a convex composite multi-objective optimization subject to a closed convex set constraint is studied. New first-order optimality conditions of a weakly efficient solution for the convex composite multi-objective optimization problem are established via scalarization. These conditions are then extended to derive second-order optimality conditions. Key Words: Multi-objective optimization, nonsmooth analysis, convex analysis, sufficient optimality condition. 1. Introduction This paper considers the following convex composite multi-objective optimization problem (P) V-Minimize (f 1 (F 1 (x)); \Delta \Delta \Delta ; f p (F p (x))) subject to x 2 C; 1 Research Fellow, Department of Mathemmatics, The University of Western Australia, Australia. 2 Senior Lecturer, De...

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Abstract. We introduce a notion of quasi-regularity for points with respect to the inclusion F (x) ∈ C where F is a nonlinear Frechét differentiable function from Rv to Rm. When C is the set of minimum points of a convex real-valued function h on Rm and F ′ satisfies the L-average Lipschitz condition of Wang, we use the majorizing function technique to establish the semi-local linear/quadratic convergence of sequences generated by the Gauss-Newton method (with quasi-regular initial points) for the convex composite function h ◦ F. Results are new even when the initial point is regular and F ′ is Lipschitz. Key words. The Gauss-Newton method, convex composite optimization, majorizing function, convergence. AMS subject classifications. 47J15 65H10 Secondary, 41A29 1. Introduction. The

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