Optimality conditions for nonconvex problems over nearly convex feasible sets

Necessary And Sufficient ◽

Slater’S Condition

AbstractIn this paper, we study the optimization problem (P) of minimizing a convex function over a constraint set with nonconvex constraint functions. We do this by given new characterizations of Robinson’s constraint qualification, which reduces to the combination of generalized Slater’s condition and generalized sharpened nondegeneracy condition for nonconvex programming problems with nearly convex feasible sets at a reference point. Next, using a version of the strong CHIP, we present a constraint qualification which is necessary for optimality of the problem (P). Finally, using new characterizations of Robinson’s constraint qualification, we give necessary and sufficient conditions for optimality of the problem (P).

Necessary and Sufficient Conditions for Optimality

Periodic Optimization ◽

10.1007/978-3-7091-2652-3_9 ◽

1972 ◽

pp. 183-217 ◽

Cited By ~ 2

Author(s):

Kun S. Chang

Keyword(s):

Sufficient Conditions ◽

Invexity of supremum and infimum functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700020335 ◽

2002 ◽

Vol 65 (2) ◽

pp. 289-306 ◽

Cited By ~ 2

Author(s):

Nguyen Xuan Ha ◽

Do Van Luu

Keyword(s):

Sufficient Conditions ◽

Infinite Family ◽

Lipschitz Functions ◽

Directional Derivatives ◽

Mathematical Programs ◽

Invex Function ◽

Under suitable assumptions we establish the formulas for calculating generalised gradients and generalised directional derivatives in the Clarke sense of the supremum and the infimum of an infinite family of Lipschitz functions. From these results we derive the results ensuring such a supremum or infimum are an invex function when all functions of the invex. Applying these results to a class of mathematical programs, we obtain necessary and sufficient conditions for optimality.

Necessary and Sufficient Conditions for Optimality of Singular Controls in a Certain Class of Multi-Input Systems

Transactions of the Society of Instrument and Control Engineers ◽

10.9746/sicetr1965.14.633 ◽

1978 ◽

Vol 14 (6) ◽

pp. 633-639

Author(s):

Hiroo YAMAURA ◽

Kazuo OCHIAI ◽

Kazuhide NAKAJI

Keyword(s):

Sufficient Conditions ◽

Singular Controls ◽

Generalized semi-Markov decision processes

Journal of Applied Probability ◽

10.1017/s0021900200107740 ◽

1979 ◽

Vol 16 (03) ◽

pp. 618-630

Author(s):

Bharat T. Doshi

Keyword(s):

Markov Decision Processes ◽

Sufficient Conditions ◽

Decision Processes ◽

The State ◽

Service Rate ◽

Storage Model ◽

Sample Paths ◽

Markov Decision ◽

Various authors have derived the necessary and sufficient conditions for optimality in semi-Markov decision processes in which the state remains constant between jumps. In this paper similar results are presented for a generalized semi-Markov decision process in which the state varies between jumps according to a Markov process with continuous sample paths. These results are specialized to a general storage model and an application to the service rate control in a GI/G/1 queue is indicated.

Necessary and sufficient conditions for optimality for singular control problems: A limit approach

Journal of Mathematical Analysis and Applications ◽

10.1016/0022-247x(71)90111-9 ◽

1971 ◽

Vol 34 (2) ◽

pp. 239-266 ◽

Cited By ~ 64

Author(s):

David H Jacobson ◽

Jason L Speyer

Keyword(s):

Sufficient Conditions ◽

Singular Control ◽

Control Problems ◽

NECESSARY AND SUFFICIENT CONDITIONS FOR DYNAMIC OPTIMIZATION

Macroeconomic Dynamics ◽

10.1017/s1365100514000546 ◽

2014 ◽

Vol 20 (3) ◽

pp. 667-684 ◽

Cited By ~ 2

Author(s):

A. Kerem Coşar ◽

Edward J. Green

Keyword(s):

Optimization Problems ◽

Infinite Horizon ◽

Sufficient Conditions ◽

Transversality Condition ◽

Infinite Horizon Optimization ◽

The Government ◽

Duality Approach ◽

We characterize the necessary and sufficient conditions for optimality in discrete-time, infinite-horizon optimization problems with a state space of finite or infinite dimension. It is well known that the challenging task in this problem is to prove the necessity of the transversality condition. To do this, we follow a duality approach in an abstract linear space. Our proof resembles that of Kamihigashi (2003), but does not explicitly use results from real analysis. As an application, we formalize Sims's argument that the no-Ponzi constraint on the government budget follows from the necessity of the tranversality condition for optimal consumption.

Minimization of nonsmooth integral functionals

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171292000899 ◽

1992 ◽

Vol 15 (4) ◽

pp. 673-679

Author(s):

Nikolaos S. Papageorgiou ◽

Apostolos S. Papageorgiou

Keyword(s):

Sobolev Spaces ◽

Convex Analysis ◽

Optimization Problems ◽

Sufficient Conditions ◽

Integral Functionals ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Dimensional State Space ◽

In this paper we examine optimization problems involving multidimensional nonsmooth integral functionals defined on Sobolev spaces. We obtain necessary and sufficient conditions for optimality in convex, finite dimensional problems using techniques from convex analysis and in nonconvex, finite dimensional problems, using the subdifferential of Clarke. We also consider problems with infinite dimensional state space and we finally present two examples.