The necessary optimality condition for controlled systems with delay in problems with constraints

1970 ◽  
Vol 13 (11) ◽  
pp. 1252-1258
Author(s):  
M. I. Gordion ◽  
V. I. Plotnikov ◽  
A. M. Sterlin
Keyword(s):  
Optimality Condition ◽  
Necessary Optimality Condition ◽  
Controlled Systems

Optimality of Approximate Quasi-Weakly Efficient Solutions for Vector Equilibrium Problems via Convexificators

2021 ◽  
Author(s):  
Guolin Yu ◽  
Siqi Li ◽  
Xiao Pan ◽  
Wenyan Han
Keyword(s):  
Optimality Condition ◽  
Generalized Convexity ◽  
Equilibrium Problems ◽  
Efficient Solutions ◽  
Sufficient Optimality Condition ◽  
Necessary Optimality Condition ◽  
Weakly Efficient Solutions ◽  
Convexity Assumption ◽  
Pseudoconvex Function ◽  
Vector Equilibrium

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.

scholarly journals A note on the paper "Optimality conditions for nonsmooth interval-valued and multiobjective semi-infinite programming"

2020 ◽  
Author(s):  
Nazih Abderrazzak Gadhi ◽  
Aissam Ichatouhane
Keyword(s):  
Optimality Conditions ◽  
Programming Problem ◽  
Optimality Condition ◽  
Short Proof ◽  
Necessary Optimality Condition ◽  
Main Result Theorem ◽  
Correct Formulation ◽  
Result Theorem ◽  
Infinite Programming ◽  
Interval Valued

A nonsmooth semi-infinite interval-valued vector programming problem is solved in the paper by Jennane et all. (RAIRO-Oper. Res. doi: 10.1051/ro/2020066, 2020). The necessary optimality condition obtained by the authors, as well as its proof, is false. Some counterexamples are given to refute some results on which the main result (Theorem 4.5) is based. For the convinience of the reader, we correct the faulty in those results, propose a correct formulation of Theorem 4.5 and give also a short proof.

scholarly journals Necessary optimality condition for trilevel optimization problem

2020 ◽  
Vol 16 (1) ◽  
pp. 55-70
Author(s):  
Gaoxi Li ◽  
◽  
Zhongping Wan ◽  
Jia-wei Chen ◽  
Xiaoke Zhao ◽  
...  
Keyword(s):  
Optimality Condition ◽  
Optimization Problem ◽  
Necessary Optimality Condition

Noether’s theorem for fractional variational problems of variable order

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Tatiana Odzijewicz ◽  
Agnieszka Malinowska ◽  
Delfim Torres
Keyword(s):  
Optimality Condition ◽  
Variational Problems ◽  
Time Variable ◽  
Variable Order ◽  
Necessary Optimality Condition ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Fractional Variational Problems ◽  
Derivatives Of

AbstractWe prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether’s theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.

scholarly journals A multiplier rule in set-valued optimisation

2003 ◽  
Vol 68 (1) ◽  
pp. 93-100 ◽  
Author(s):  
Akhtar A. Khan ◽  
Fabio Raciti
Keyword(s):  
Optimality Condition ◽  
Necessary Optimality Condition ◽  
Dini Derivative ◽  
Multiplier Rule

A multiplier rule is given as a necessary optimality condition for proper minimality in set-valued optimisation. We use derivatives in the sense of the lower Dini derivative for the objective set-valued map and the set-valued maps defining the constraints.

scholarly journals Optimal design for eigen-frequencies of a longitudinal bar using Pontryagin's maximum principle considering the influence of concentrated mass

2017 ◽  
Vol 39 (1) ◽  
pp. 1-12
Author(s):  
Bui Hai Le ◽  
Tran Minh Thuy
Keyword(s):  
Maximum Principle ◽  
Optimal Design ◽  
Optimality Condition ◽  
Pontryagin’S Maximum Principle ◽  
State Variables ◽  
Necessary Optimality Condition ◽  
Concentrated Mass ◽  
Pontryagin's Maximum Principle ◽  
Eigen Frequencies ◽  
Objective Functional

In this paper, the problem of optimal design for eigen-frequencies of a longitudinal bar using Pontryagin's maximum principle (PMP) considering the influence of concentrated mass is presented. The necessary optimality condition when simultaneously maximizing system's eigen frequencies and minimizing system's weight considering the influence of concentrated mass is established by using Maier objective functional in order to control the final state of the objective functional. By considering eigen frequencies as state variables, the analogy coefficient k in the necessary optimality condition is explicitly determined. Numerical results obtained in this paper include: (1) the bar's optimal configurations as well as frequency responses in different cases of objective functions; (2) the Pareto front for the system's first eigen frequency and weight; (3) the influence of concentrated mass on the bar's optimal configuration.

scholarly journals A non-standard class of variational problems of Herglotz type

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Natália Martins
Keyword(s):  
Variational Problem ◽  
Optimality Condition ◽  
Unknown Function ◽  
Variational Problems ◽  
Standard Theory ◽  
Necessary Optimality Condition ◽  
Natural Boundary ◽  
Special Cases ◽  
Independent Variable ◽  
Natural Boundary Conditions

<p style='text-indent:20px;'>In this paper, we extend the variational problem of Herglotz considering the case where the Lagrangian depends not only on the independent variable, an unknown function <inline-formula><tex-math id="M1">\begin{document}$ x $\end{document}</tex-math></inline-formula> and its derivative and an unknown functional <inline-formula><tex-math id="M2">\begin{document}$ z $\end{document}</tex-math></inline-formula>, but also on the end points conditions and a real parameter. Herglotz's problems of calculus of variations of this type cannot be solved using the standard theory. Main results of this paper are necessary optimality condition of Euler-Lagrange type, natural boundary conditions and the Dubois-Reymond condition for our non-standard variational problem of Herglotz type. We also prove a necessary optimality condition that arises as a consequence of the Lagrangian dependence of the parameter. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases. In addition, two examples are given in order to illustrate our results.</p>

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