Strong duality, weak duality and penalization for a state constrained parabolic control problem

2005 ◽  
pp. 233-241 ◽  
Author(s):  
U. Mackenroth
Keyword(s):  
Control Problem ◽  
Strong Duality ◽  
Weak Duality

scholarly journals Higher-Order Generalized Invexity in Control Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
S. K. Padhan ◽  
C. Nahak
Keyword(s):  
Control Problem ◽  
Strong Duality ◽  
Higher Order ◽  
Control Problems ◽  
Weak Duality ◽  
Converse Duality ◽  
Generalized Invexity ◽  
Duality Results ◽  
Higher Order Duality

We introduce a higher-order duality (Mangasarian type and Mond-Weir type) for the control problem. Under the higher-order generalized invexity assumptions on the functions that compose the primal problems, higher-order duality results (weak duality, strong duality, and converse duality) are derived for these pair of problems. Also, we establish few examples in support of our investigation.

scholarly journals On geometric duality for state-constrained control problems

1979 ◽  
Vol 20 (2) ◽  
pp. 301-312
Author(s):  
T.R. Jefferson ◽  
C.H. Scott
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
State Constraints ◽  
Strong Duality ◽  
State Constraint ◽  
Continuous Functions ◽  
Control Problems ◽  
Absolutely Continuous Functions ◽  
Pure State Constraints ◽  
Constrained Control Problems

For convex optimal control problems without explicit pure state constraints, the structure of dual problems is now well known. However, when these constraints are present and active, the theory of duality is not highly developed. The major difficulty is that the dual variables are not absolutely continuous functions as a result of singularities when the state trajectory hits a state constraint. In this paper we recognize this difficulty by formulating the dual probram in the space of measurable functions. A strong duality theorem is derived. This pairs a primal, state constrained convex optimal control problem with a dual convex control problem that is unconstrained with respect to state constraints. In this sense, the dual problem is computationally more attractive than the primal.

scholarly journals A Kind of New Higher-Order Mond-Weir Type Duality for Set-Valued Optimization Problems

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 372
Author(s):  
Liu He ◽  
Qi-Lin Wang ◽  
Ching-Feng Wen ◽  
Xiao-Yan Zhang ◽  
Xiao-Bing Li
Keyword(s):  
Existence Theorem ◽  
Lower Order ◽  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Strong Duality ◽  
Higher Order ◽  
Weak Duality ◽  
Duality Theorems ◽  
Benson Proper Efficiency

In this paper, we introduce the notion of higher-order weak adjacent epiderivative for a set-valued map without lower-order approximating directions and obtain existence theorem and some properties of the epiderivative. Then by virtue of the epiderivative and Benson proper efficiency, we establish the higher-order Mond-Weir type dual problem for a set-valued optimization problem and obtain the corresponding weak duality, strong duality and converse duality theorems, respectively.

Strong and weak duality principles for fractal dimension in Euclidean space

1995 ◽  
Vol 118 (3) ◽  
pp. 393-410 ◽  
Author(s):  
Colleen D. Cutler
Keyword(s):  
Strong Duality ◽  
The Other ◽  
Duality Principle ◽  
Weak Duality ◽  
Pointwise Dimension ◽  
Analytic Subset ◽  
Analytic Set ◽  
Borel Measures ◽  
Dual Representations ◽  
Packing Dimensions

AbstractTricot [27] provided apparently dual representations of the Hausdorff and packing dimensions of any analytic subset of Euclidean d-space in terms of, respectively, the lower and upper pointwise dimension maps of the finite Borel measures on ℝd. In this paper we show that Tricot's two representations, while similar in appearance, are in fact not duals of each other, but rather the duals of two other ‘missing’ representations. The key to obtaining these missing representations lies in extended Frostman and antiFrostman lemmas, both of which we develop in this paper. This leads to the formulation of two distinct characterizations of dim (A) and Dim (A), one which we call the weak duality principle and the other the strong duality principle. In particular, the strong duality principle is concerned with the existence, for each analytic set A, of measures on A that are (almost) of the same exact dimension (Hausdorff or packing) as A. The connection with Rényi (or information) dimension and a variational principle of Cutler and Olsen[12] is also established.

scholarly journals Approximate dual and approximate vector variational inequality for multiobjective optimization

1989 ◽  
Vol 47 (3) ◽  
pp. 418-423 ◽  
Author(s):  
G.–Y. Chen ◽  
B. D. Craven
Keyword(s):  
Variational Inequality ◽  
Multiobjective Optimization ◽  
Duality Theorem ◽  
Optimization Problem ◽  
Strong Duality ◽  
Vector Variational Inequality ◽  
Weak Duality ◽  
Multiobjective Optimization Problem ◽  
Feasible Set

AbstractAn approximate dual is proposed for a multiobjective optimization problem. The approximate dual has a finite feasible set, and is constructed without using a perturbation. An approximate weak duality theorem and an approximate strong duality theorem are obtained, and also an approximate variational inequality condition for efficient multiobjective solutions.

On the multiobjective control problem

2018 ◽  
Vol 24 (2) ◽  
pp. 223-231
Author(s):  
Promila Kumar ◽  
Bharti Sharma
Keyword(s):  
Control Problem ◽  
Strong Duality ◽  
Efficient Solutions ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Duality Theorems ◽  
Multiobjective Control ◽  
Primal And Dual Problems ◽  
Primal And Dual

Abstract 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 number 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.

scholarly journals A Riesz decomposition theorem

1989 ◽  
Vol 114 ◽  
pp. 123-133 ◽  
Author(s):  
S. E. Graversen
Keyword(s):  
Strong Duality ◽  
Decomposition Theorem ◽  
Weak Duality ◽  
Countable Base ◽  
Standard Process ◽  
Locally Compact ◽  
Riesz Decomposition ◽  
Excessive Functions ◽  
Strong Markov ◽  
Strong Feller

The topic of this note is the Riesz decomposition of excessive functions for a “nice” strong Markov process X. I.e. an excessive function is decomposed into a sum of a potential of a measure and a “harmonic” function. Originally such decompositions were studied by G.A. Hunt [8]. In [1] a Riesz decomposition is given assuming that the state space E is locally compact with a countable base and X is a transient standard process in strong duality with another standard process having a strong Feller resolvent. Recently R.K. Getoor and J. Glover extended the theory to the case of transient Borei right processes in weak duality [6].

scholarly journals Two types of duality in multiobjective fractional programming

1996 ◽  
Vol 54 (1) ◽  
pp. 99-114 ◽  
Author(s):  
L. Coladas ◽  
Z. Li ◽  
S. Wang
Keyword(s):  
Fractional Programming ◽  
Duality Theorem ◽  
Strong Duality ◽  
Weak Duality ◽  
Multiobjective Fractional Programming ◽  
Dual Problems ◽  
Fractional Program ◽  
Mild Assumption

In this paper, we tire concerned with duality of a multiobjective fractional program. Two different dual problems are introduced with respect to the primal multiobjective fractional program. Under a mild assumption, we prove a weak duality theorem and a strong duality theorem for each type of duality. Finally, we explore some relations between these two types of duality.

scholarly journals Linear programming problems on time scales

2018 ◽  
Vol 12 (1) ◽  
pp. 192-204 ◽  
Author(s):  
Rasheed Al-Salih ◽  
Martin Bohner
Keyword(s):  
Linear Programming ◽  
Optimality Conditions ◽  
Time Scales ◽  
Duality Theorem ◽  
Strong Duality ◽  
Programming Models ◽  
Weak Duality ◽  
Continuous Linear ◽  
Arbitrary Time ◽  
Linear Programming Problems

In this work, we study linear programming problems on time scales. This approach unifies discrete and continuous linear programming models and extends them to other cases ?in between?. After a brief introduction to time scales, we formulate the primal as well as the dual time scales linear programming models. Next, we establish and prove the weak duality theorem and the optimality conditions theorem for arbitrary time scales, while the strong duality theorem is established for isolated time scales. Finally, examples are given in order to illustrate the effectiveness of the presented results.

Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem

2020 ◽  
Vol 26 ◽  
pp. 78
Author(s):  
Thirupathi Gudi ◽  
Ramesh Ch. Sau
Keyword(s):  
Finite Element ◽  
Control Problem ◽  
Dirichlet Boundary ◽  
A Priori ◽  
Diffusion Problem ◽  
Discrete Control ◽  
Optimal Order ◽  
Control Constraints ◽  
Element Analysis ◽  
Dirichlet Boundary Control

We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Laplace equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the L2 cost functional. A priori error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases. Theoretical results are illustrated by some numerical experiments.

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