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LAGRANGE MULTIPLIERS AND OPTIMALITY
, 1993
"... Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions ..."
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Cited by 89 (7 self)
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Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder 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 onesided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the gametheoretic 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 blackandwhite 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.
First And SecondOrder Optimality Conditions For Convex Composite MultiObjective Optimization
 J. Optim. Theo. Appli. 95
, 1997
"... : Multiobjective 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 multiobjective optimization subject to a closed conve ..."
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Cited by 6 (0 self)
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: Multiobjective 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 multiobjective optimization subject to a closed convex set constraint is studied. New firstorder optimality conditions of a weakly efficient solution for the convex composite multiobjective optimization problem are established via scalarization. These conditions are then extended to derive secondorder optimality conditions. Key Words: Multiobjective optimization, nonsmooth analysis, convex analysis, sufficient optimality condition. 1. Introduction This paper considers the following convex composite multiobjective optimization problem (P) VMinimize (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...
Amenable functions in optimization
 IN NONSMOOTH OPTIMIZATION METHODS AND APPLICATIONS
, 1992
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Majorizing functions and convergence of the GaussNewton method for convex composite optimization
 SIAM J. Optim
"... Abstract. We introduce a notion of quasiregularity 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 realvalued function h on Rm and F ′ satisfies the Laverage Lipschitz conditi ..."
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Cited by 4 (2 self)
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Abstract. We introduce a notion of quasiregularity 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 realvalued function h on Rm and F ′ satisfies the Laverage Lipschitz condition of Wang, we use the majorizing function technique to establish the semilocal linear/quadratic convergence of sequences generated by the GaussNewton method (with quasiregular 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 GaussNewton method, convex composite optimization, majorizing function, convergence. AMS subject classifications. 47J15 65H10 Secondary, 41A29 1. Introduction. The
J. Math. Anal. Appl. •• • (••••) •••–••• Contents lists available at SciVerse ScienceDirect Journal of Mathematical Analysis and
"... www.elsevier.com/locate/jmaa Convergence analysis of the Gauss–Newton method for convex inclusion ..."
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www.elsevier.com/locate/jmaa Convergence analysis of the Gauss–Newton method for convex inclusion