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1
Content available On efficiency and duality for a class of nonconvex nondifferentiable multiobjective fractional variational control problems
Antczak Tadeusz, Arana-Jimenéz Manuel, Treanţă Savin
Opuscula Mathematica
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2023
|
Vol. 43, no. 3
335--391
EN
In this paper, we consider the class of nondifferentiable multiobjective fractional variational control problems involving the nondifferentiable terms in the numerators and in the denominators. Under univexity and generalized univexity hypotheses, we prove optimality conditions and various duality results for such nondifferentiable multiobjective fractional variational control problems. The results established in the paper generalize many similar results established earlier in the literature for such nondifferentiable multiobjective fractional variational control problems.
2
Content available Sufficient optimality condition and duality of nondifferentiable minimax ratio constraint problems under (p, r)-ρ-(η, θ)-invexity
Kailey Navdeep, Sethi Sonali, Saini Shivani
Control and Cybernetics
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2022
|
Vol. 51, No. 1
71--88
EN
There are several classes of decision-making problems that explicitly or implicitly prompt fractional programming problems. Portfolio selection problems, agricultural planning, information transfer, numerical analysis of stochastic processes, and resource allocation problems are just a few examples. The huge number of applications of minimax fractional programming problems inspired us to work on this topic. This paper is concerned with a nondifferentiable minimax fractional programming problem. We study a parametric dual model, corresponding to the primal problem, and derive the sufficient optimality condition for an optimal solution to the considered problem. Further, we obtain the various duality results under (p, r)-ρ-(η, θ)-invexity assumptions. Also, we identify a function lying exclusively in the class of (−1, 1)-ρ-(η, θ)- invex functions but not in the class of (1,−1)-invex functions and convex function already existing in the literature. We have given a non-trivial model of nondifferentiable minimax problem and obtained its optimal solution using optimality results derived in this paper.
3
Content available Optimality conditions in set-valued optimization using approximations as generalized derivatives
Gadhi Nazih Abderrazzak, Ichatouhane Aissam
Control and Cybernetics
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2022
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Vol. 51, No. 4
425--444
EN
The set criterion is an appropriate defining approach regarding the solutions for the set-valued optimization problems. By using approximations as generalized derivatives of set-valued mappings, we establish necessary optimality conditions for a constrained set-valued optimization problem in the sense of set optimization in terms of asymptotical pointwise compact approximations. Sufficient optimality conditions are then obtained through first-order strong approximations of data set-valued mappings.
4
Content available Necessary optimality conditions for robust nonsmooth multiobjective optimization problems
Gadhi Nazih Abderrazzak, Ohda Mohamed
Control and Cybernetics
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2022
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Vol. 51, No. 3
289--302
EN
This paper deals with a robust multiobjective optimization problem involving nonsmooth/nonconvex real-valued functions. Under an appropriate constraint qualification, we establish necessary optimality conditions for weakly robust efficient solutions of the considered problem. These optimality conditions are presented in terms of Karush-Kuhn-Tucker multipliers and convexificators of the related functions. Examples illustrating our findings are also given.
5
Content available Neumann boundary optimal control problems governed by parabolic variational equalities
Bollo Carolina M., Gariboldi Claudia M., Tarzia Domingo A.
Control and Cybernetics
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2021
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Vol. 50, No. 2
227--252
EN
We consider a heat conduction problem S with mixed boundary conditions in an n-dimensional domain with regular boundary and a family of problems Sα with also mixed boundary conditions in , where α > 0 is the heat transfer coefficient on the portion of the boundary Г1. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux q which is definite on the complementary portion Г2 of the boundary of Ω. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient α goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter λ, in relation to the parabolic problems S and Sαα
6
Second-order characterization of convex functions and its applications
Nadi Mohammad Taghi, Yao Jen Chih, Zafarani Jafar
Journal of Applied Analysis
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2019
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Vol. 25, nr 1
49--58
EN
Some developments of the second-order characterizations of convex functions are investigated by using the coderivative of the subdifferential mapping. Furthermore, some applications of the second-order subdifferentials in optimization problems are studied.
7
On the multiobjective control problem
Kumar P., Sharma B.
Journal of Applied Analysis
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2018
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Vol. 24, nr 2
223--231
EN
In this paper, sufficient optimality conditions are established for the multiobjective control problem using efficiency of higher order as a criterion for optimality. The ρ-type 1 invex functionals (taken in pair) of higher order are proposed for the continuous case. Existence of such functionals is confirmed by a numer of examples. It is shown with the help of an example that this class is more general than the existing class of functionals.Weak and strong duality theorems are also derived for a mixed dual in order to relate efficient solutions of higher order for primal and dual problems.
8
Content available Optimality conditions and duality for generalized fractional minimax programming involving locally Lipschitz (b,Ψ,Φ,ρ) -univex functions
Antczak T., Mishra S. K., Upadhyay B. B.
Control and Cybernetics
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2018
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Vol. 47, no. 1
5--32
EN
In this paper, we are concerned with optimality conditions and duality results of generalized fractional minimax programming problems. Sufficient optimality conditions are established for a class of nondifferentiable generalized fractional minimax programming problems, in which the involved functions are locally Lipschitz (b,Ψ,Φ,ρ)-univex. Subsequently, these optimality conditions are utilized as a basis for constructing various parametric and nonparametric duality models for this type of fractional programming problems and proving appropriate duality theorems.
9
Content available Necessary optimality conditions for a set valued fractional programming problem in terms of contingent epiderivatives
Gadhi A. N., Idrissi M. El.
Control and Cybernetics
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2018
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Vol. 47, no. 2
147--156
EN
In this paper, we are concerned with a multi-objective fractional extremal programming problem. Using the concept of subdifferential of cone-convex set valued mappings, introduced by Baier and Jahn (1999), together with the convex separation principle, we give necessary optimality conditions. An example illustrating the usefulness of our results is also provided.
10
Content available On the relaxation of state-constrained linear control problems via Henig dilating cones
Kogut P. I., Leugering G., Schiel R.
Control and Cybernetics
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2016
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Vol. 45, no. 2
131--162
EN
We discuss a regularization of state-constrained optimal control problems via a Henig relaxation of ordering cones. Considering a state-constrained optimal control problem, the pointwise state constraint is replaced by an inequality condition involving a so-called Henig dilating cone. It is shown that this class of cones provides a reasonable solid approximation of the typically nonsolid ordering cones which correspond to pointwise state constraints. Thereby, constraint qualifications, which are based on the existence of interior points, can be applied to given problems. Moreover, we characterize admissibility and solvability of the original problem by analyzing the associated relaxed problem. We also show that the optimality system for the original problem can be obtained through the limit passage in the corresponding optimality system for the relaxed problem. As an example of our approach, we derive the optimality conditions for a state constrained Neumann boundary optimal control problem and show that in this case the corresponding Lagrange multipliers are more regular than Borel measures.
11
Content available Existence, uniqueness and convergence of simultaneous distributed-boundary optimal control problems
Gariboldi C. M., Tarzia D. A.
Control and Cybernetics
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2015
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Vol. 44, no. 1
5--17
EN
We consider a steady-state heat conduction problem P for the Poisson equation with Mied Bondary conditions in a bounded multidimensional domain Ω. We also consider a family of problems Pα for the same Poisson equation with mixed boundary conditions, α > 0 being the heat transfer coefficient defined on a portion Γ1 of the boundary. We formulate simultaneous distributed and Neumann boundary optimal control problems on the internal energy g within Ω and the heat flux q, defined on the complementary portion Γ2 of the boundary of Ω for quadratic cost functional. Here, the control variable is the vector (g,q). We prove existence and uniqueness of the optimal control (g,q) for the system state of P, and (gα,qα) for the system state of Pα, for each α > 0, and we give the corresponding optimality conditions. We prove strong convergence, in suitable Sobolev spaces, of the vectorial optimal controls, system and adjoint states governed by the problems Pα to the corresponding vectorial optimal control, system and adjoint states governed by the problem P, when the parameter α goes to infinity. We also obtain estimations between the solutions of these vectorial optimal control problems and the solution of two scalar optimal control problems characterized by fixed g (with boundary optimal control q) and fixed q (with distributed optimal control g), respectively, for cases both of α > 0 and α = ∞.
12
Content available Sufficient optimality criteria and duality for multiobjective variational control problems with B-(p,r)-invex function
Antczak T., Jimenez M. A.
Opuscula Mathematica
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2014
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Vol. 34, no. 4
665--682
EN
In this paper, we generalize the notion of B-(p, r)-invexity introduced by Antczak in [A class of B-(p; r)-invex functions and mathematical programming, J. Math. Anal. Appl. 286 (2003), 187–206] for scalar optimization problems to the case of a multiobjective variational programming control problem. For such nonconvex vector optimization problems, we prove sufficient optimality conditions under the assumptions that the functions constituting them are B-(p, r)-invex. Further, for the considered multiobjective variational control problem, its dual multiobjective variational control problem in the sense of Mond-Weir is given and several duality results are established under B-(p, r)-invexity.
13
Content available Optimality conditions for a class of relaxed quasiconvex minimax problems
Diem H. T. H., Khanh P. Q.
Control and Cybernetics
|
2014
|
Vol. 43, no. 2
249--260
EN
A class of minimax problems is considered. We approach it with the techniques of quasiconvex optimization, which includes most important nonsmooth and relaxed convex problems and has been intensively developed. Observing that there have been many contributions to various themes of minimax problems, but surprisingly very few on optimality conditions, the most traditional and developed topic in optimization, we establish both necessary and sufficient conditions for solutions and unique solutions. A main feature of this work is that the involved functions are relaxed quasi- convex in the sense that the sublevel sets need to be convex only at the considered point. We use star subdifferentials, which are slightly bigger than other subdifferentials applied in many existing results for minimization problems, but may be empty or too small in various situations. Hence, when applied to the special case of minimization problems, our results may be more suitable. Many examples are provided to illustrate the applications of the results and also to discuss the imposed assumptions.
14
Nondifferentiable (Phi,rho)-type I and generalized (Phi,rho)-type I functions in nonsmooth vector optimization
Antczak T.
Journal of Applied Analysis
|
2013
|
Vol. 19, nr 2
247--270
EN
In this paper, new classes of nondifferentiable generalized invex functions are introduced. Further, nonsmooth vector optimization problems with functions belonging to the introduced classes of (generalized) (Phi,rho)-type I functions are considered. Sufficient optimality conditions and duality results for such classes of nonsmooth vector optimization problems are established. It turns out that the presented results are proved also for such nonconvex vector optimization problems in which not all functions constituting them possess the fundamental property of invexity.
15
Content available remote Combined Reformulation of Bilevel Programming Problems
Pilecka M.
Schedae Informaticae
|
2012
|
Vol. 21
65--79
EN
In [19] J.J. Ye and D.L. Zhu proposed a new reformulation of a bilevel programming problem which compounds the value function and KKT approaches. In [19] partial calmness condition was also adapted to this new reformulation and optimality conditions using partial calmness were introduced. In this paper we investigate above all local equivalence of the combined reformulation and the initial problem and how constraint qualifications and optimality conditions could be defined for this reformulation without using partial calmness. Since the optimal value function is in general nondifferentiable and KKT constraints have MPEC-structure, the combined reformulation is a nonsmooth MPEC. This special structure allows us to adapt some constraint qualifications and necessary optimality conditions from MPEC theory using disjunctive form of the combined reformulation. An example shows, that some of the proposed constraint qualifications can be fulfilled.
16
Content available Sterowanie optymalne systemami o parametrach rozłożonych z opóźnieniami
Kowalewski A.
Pomiary Automatyka Robotyka
|
2011
|
R. 15, nr 12
57-59
PL
W artykule dokonano przeglądu problematyki sterowania optymalnego systemami o parametrach rozłożonych z opóźnieniami w warunkach brzegowych. Zaprezentowano podstawowe zagadnienia dotyczące systemów o parametrach rozłożonych takie jak: istnienie i jednoznaczność rozwiązań, sterowanie optymalne rozłożone i brzegowe, sterowanie optymalne ze sprzężeniem zwrotnym, sterowanie czasowo-optymalne, sterowalność oraz modele matematyczne systemów o parametrach rozłożonych.
EN
In this article the review of optimal control problems for distributed parameter systems with boundary conditions involving time delays is performed. The principal problems, namely: existence and uniqueness of solutions, distributed and boundary optimal control, optimal feedback control, controllability and mathematical models of distributed parameter systems are presented.
17
Content available remote Local cone approximations in optimization
Castellani M., Pappalardo M.
Control and Cybernetics
|
2007
|
Vol. 36, no 3
583-600
EN
We show how to use intensively local cone approximations to obtain results in some fields of optimization theory such as optimality conditions, constraint qualifications, mean value theorems and error bound.
18
Content available remote Optimality properties of controls with bang-bang components in problems with semilinear state equation
Felgenhauer U.
Control and Cybernetics
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2005
|
Vol. 34, no 3
763-785
EN
In this paper we study optimal control problems with bang-bang solution behavior for a special class of semilinear dynamics. Generalizing a former result for linear systems, optimlity conditions are derived by a duality based approach. The results apply for scalar as well as for vector control functions and, in particular, for the case of the so-called multiple switches, too. Further, an iterative procedure for determining switching points is proposed, and convergence results are provided.
19
Content available remote Optimality and sensitivity for semilinear bang-bang type optimal control problems
Felgenhauer U.
International Journal of Applied Mathematics and Computer Science
|
2004
|
Vol. 14, no 4
447-454
EN
In optimal control problems with quadratic terminal cost functionals and systems dynamics linear with respect to control, the solution often has a bang-bang character. Our aim is to investigate structural solution stability when the problem data are subject to perturbations. Throughout the paper, we assume that the problem has a (possibly local) optimum such that the control is piecewise constant and almost everywhere takes extremal values. The points of discontinuity are the switching points. In particular, we will exclude the so-called singular control arcs, see Assumptions 1 and 2, Section 2. It is known from the results by Agrachev et al. (2002) stating that regularity assumptions, together with a certain strict second-order condition for the optimization problem formulated in switching points, are sufficient for strong local optimality of a state-control solution pair. This finite-dimensional problem is analyzed in Section 3 and optimality conditions are formulated (Lemma 2). Using well-known results concerning solution sensitivity for mathematical programs in Rn (Fiacco, 1983) one may further conclude that, under parameter changes in the problem data, the switching points will change Lipschitz continuously. The last section completes these qualitative statements by calculating sensitivity differentials (Theorem 2, Lemma 6). The method requires a simultaneous solution of certain linearized multipoint boundary value problems.
20
Content available Boundary optimal control problem for the phase-field transition system using fractional steps method
Moroşanu C.
Control and Cybernetics
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2003
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Vol. 32, no 1
5-32
EN
In this paper we prove the convergence of an iterative scheme of fractional steps type for boundary optimal control problem which is governed by the phase-field transition system. The existence of an optimal control and necessary optimality conditions are given for approximating problem. A gradient type algorithm and numerical implementation of this algorithm are discussed.
PL
W artykule dowodzi się zbieżności procedury iteracyjnej typu kroków ułamkowych dla zagadnienia sterowania optymalnego brzegiem w układzie pola ze zmianą fazy. Wykazano istnienie sterowania optymalnego i podano warunki konieczne optymalności dla zagadnienia aproksymującego. Rozpatrzono odpowiedni algorytm gradientowy oraz realizację numeryczną tego algorytmu.
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