Book Chapter10.1007/BFB0120982
A unified theory of first and second order conditions for extremum problems in topological vector spaces
Aharon Ben-Tal,Jochem Zowe +1 more
- 01 Jan 1982
- pp 39-76
200
TL;DR: In this paper, an abstract optimization problem (P) in infinite dimensional spaces is studied and necessary conditions of first and second order sufficient conditions with or without differentiability assumptions are derived for special cases of the general problem.
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Abstract: The paper studies an abstract optimization problem (P) in infinite dimensional spaces. From a general extremality condition, a variety of necessary conditions of first and second order, with or without differentiability assumptions, are derived for special cases of the general problem (P). Classical results are refined and new ones are added. Second order sufficient condition, under differentiability assumptions, are derived as well.
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Citations
Metric regularity, tangent sets, and second-order optimality conditions
TL;DR: In this paper, a strong regularity theorem is proved, which shows that the usual constraint qualification conditions ensuring the regularity of the set-valued maps expressing feasibility in optimization problems, are in fact minimal assumptions.
272
First- and second-order epi-differentiability in nonlinear programming
TL;DR: In this paper, the essential objective function, which is the sum of the given objective and the indicator of the constraints, is shown to be twice epi-differentiable at any point where the active constraints satisfy the Mangasarian-Fromovitz qualification.
A generalized second-order derivative in nonsmooth optimization
Roberto Cominetti,R. Correa +1 more
TL;DR: In this paper, a new notion of generalized second-order directional derivatives and generalized Hessian for nonsmooth real-valued functions is studied, together with some calculus rules that may facilitate their practical computation.
144
References
•Book
Nonlinear Programming: Sequential Unconstrained Minimization Techniques
Anthony V. Fiacco,Garth P. McCormick +1 more
- 01 Jan 1968
TL;DR: This report gives the most comprehensive and detailed treatment to date of some of the most powerful mathematical programming techniques currently known--sequential unconstrained methods for constrained minimization problems in Euclidean n-space--giving many new results not published elsewhere.
•Book
Linear operators leaving invariant a cone in a Banach space
M. G. Kreĭn,M. A. Rutman +1 more
- 01 Jan 1950
1.2K