Second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems

2005 ◽  
pp. 13-21 ◽  
Author(s):  
H. Maurer ◽  
J. Zowe
Keyword(s):  
Optimality Conditions ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Infinite Dimensional ◽  
Necessary And Sufficient

Second order optimality conditions for optimal control problems on Riemannian manifolds

2019 ◽  
Vol 25 ◽  
pp. 35 ◽  
Author(s):  
Qing Cui ◽  
Li Deng ◽  
Xu Zhang
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Optimal Control Problems ◽  
Second Order ◽  
Control Problems ◽  
Sufficient Optimality Conditions ◽  
Control Set ◽  
Necessary And Sufficient

This work is concerned with optimal control problems on Riemannian manifolds, for which two typical cases are considered. The first case is when the endpoint is free. For this case, the control set is assumed to be a separable metric space. By introducing suitable dual equations, which depend on the curvature tensor of the manifold, we establish the second order necessary and sufficient optimality conditions of integral form. Particularly, when the control set is a Polish space, the second order necessary condition is reduced to a pointwise form. As a key preliminary result and also an interesting byproduct, we derive a geometric lemma, which may have some independent interest. The second case is when the endpoint is fixed. For this more difficult case, the control set is assumed to be open in a Euclidian space, and we obtain the second order necessary and sufficient optimality conditions, in which the curvature tensor also appears explicitly. Our optimality conditions can be used to recover the following famous geometry result: the shortest geodesic connecting two fixed points on a Riemannian manifold satisfies the second variation of energy; while the existing optimality conditions in control literatures fail to give the same result.

scholarly journals Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1152
Author(s):  
Gabriel Ruiz-Garzón ◽  
Jaime Ruiz-Zapatero ◽  
Rafaela Osuna-Gómez ◽  
Antonio Rufián-Lizana
Keyword(s):  
Optimality Conditions ◽  
Optimization Problems ◽  
Second Order ◽  
Stationary Points ◽  
Directional Derivatives ◽  
Sufficient Optimality Conditions ◽  
General Notion ◽  
Hadamard Manifolds ◽  
Scalar Optimization ◽  
Necessary And Sufficient

This work is intended to lead a study of necessary and sufficient optimality conditions for scalar optimization problems on Hadamard manifolds. In the context of this geometry, we obtain and present new function types characterized by the property of having all their second-order stationary points be global minimums. In order to do so, we extend the concept convexity in Euclidean space to a more general notion of invexity on Hadamard manifolds. This is done employing notions of second-order directional derivatives, second-order pseudoinvexity functions, and the second-order Karush–Kuhn–Tucker-pseudoinvexity problem. Thus, we prove that every second-order stationary point is a global minimum if and only if the problem is either second-order pseudoinvex or second-order KKT-pseudoinvex depending on whether the problem regards unconstrained or constrained scalar optimization, respectively. This result has not been presented in the literature before. Finally, examples of these new characterizations are provided in the context of “Higgs Boson like” potentials, among others.

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

scholarly journals A note on the paper "Necessary and sufficient optimality conditions using convexifactors for mathematical programs with equilibrium constraints"

2021 ◽  
Author(s):  
Nazih Abderrazzak Gadhi
Keyword(s):  
Optimality Conditions ◽  
Sufficient Optimality Conditions ◽  
Equilibrium Constraints ◽  
Mathematical Programs ◽  
Necessary And Sufficient

In this work, some counterexamples are given to refute some results in the paper by Kohli (RAIRO-Oper. Res. 53, 1617-1632, 2019). We correct the faulty in some of his results.

scholarly journals Strict lower subdifferentiability and applications

1999 ◽  
Vol 40 (3) ◽  
pp. 379-391 ◽  
Author(s):  
H. Xu ◽  
A. M. Rubinov ◽  
B. M. Glover
Keyword(s):  
Global Convergence ◽  
Optimality Conditions ◽  
Convex Subset ◽  
Sufficient Optimality Conditions ◽  
Descent Direction ◽  
Minimization Problems ◽  
Convergence Results ◽  
Lower Subdifferential ◽  
Necessary And Sufficient ◽  
Convex Subdifferential

AbstractWe investigate the strict lower subdifferentiability of a real-valued function on a closed convex subset of Rn. Relations between the strict lower subdifferential, lower subdifferential, and the usual convex subdifferential are established. Furthermore, we present necessary and sufficient optimality conditions for a class of quasiconvex minimization problems in terms of lower and strict lower subdifferentials. Finally, a descent direction method is proposed and global convergence results of the consequent algorithm are obtained.

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