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Duality
Published in Shashi Kant Mishra, Bhagwat Ram, Introduction to Linear Programming with MATLAB®, 2017
Shashi Kant Mishra, Bhagwat Ram
From (7.16) and (7.19), we get λTb≤cTx. $$ \begin{aligned} \lambda ^{T}b \le c^{T}x. \end{aligned} $$ Note: From the weak duality theorem, we have noticed that the value of the objective function of the dual problem is always less than or equal to the value of the objective function of the primal problem. That is, the value of the objective function of the dual problem is a lower bound to the value of objective function of the primal problem. This is one important concept of the duality.
Connectivity
Published in Jonathan L. Gross, Jay Yellen, Mark Anderson, Graph Theory and Its Applications, 2018
Jonathan L. Gross, Jay Yellen, Mark Anderson
Borrowing from operations research terminology, we consider certain primal-dual pairs of optimization problems that are intimately related. Usually, one of these problems involves the maximization of some objective function, while the other is a minimization problem. A feasible solution to one of the problems provides a bound for the optimal value of the other problem (this is sometimes referred to as weak duality), and the optimal value of one problem is equal to the optimal value of the other (strong duality). Menger’s Theorems and their many variations epitomize this primal-dual relationship. The following terminology is used throughout this section.
On Fenchel c-conjugate dual problems for DC optimization: characterizing weak, strong and stable strong duality
Published in Optimization, 2023
In every approach including Fenchel conjugation to derive a dual problem for a primal one, the convexity, together with the properness and lower semicontinuity of the involved functions in the problem, make it possible to apply conjugate duality techniques to epigraphs, which ensure strong duality results. Given a primal problem, (P), and its dual, (D), it is said that the pair verifies weak duality when and strong duality when there is zero duality gap, i.e. , and the dual has an optimal solution. Connected to strong duality, stable strong duality means that there is strong duality when the objective function of the primal problem is perturbed with a linear functional. For more details on the development of regularity conditions for strong and stable strong dualities, we encourage the reader to revise [5,16,17].
Optimization of first-order Nicoletti boundary value problem with discrete and differential inclusions and duality
Published in Optimization, 2022
In Section 4, based on the infimal convolution of convex functions, a dual problem for convex problems with Nicoletti DSIs is constructed and the dual theorem is formulated. It is proved that if and are the values of primal and dual problems, respectively, then there is a weak duality, i.e. for all feasible solutions. In addition, if the standard condition for convex analysis on the existence of interior point is satisfied, then both problems have solutions and there is a strong duality .
New duality results for evenly convex optimization problems
Published in Optimization, 2021
M. D. Fajardo, S. M. Grad, J. Vidal
In a more general framework, Penot and Rubinov in [28] related the Lagrangian and the pertubational approach (or parametrization approach, as it is named in that paper) to duality for optimization problems. In order to allow a more comprehensive comparison between their work and ours, we have adapted their notation to the one used in this work. Given a set Z, a Lagrangian for is a function which must verify that , in which case, the optimal value of satisfies Note that no convexity or topological assumptions were imposed on the involved functions in this case. Defining a dual functional for , as , for every , a dual problem for is and for this primal-dual pair of optimization problems there holds weak duality. According to [28, Prop. 1 (Sect. 3.3)], if is a perturbation function for , and, for all , is -convex at 0 (see [28, Sect. 2] for a definition), then is a Lagrangian for . Here is any coupling function. As it can be observed, the function L in Definition 5.1 is not the same as (20): if we take the infimum on Y, the function in Definition 5.1 is always , except perhaps for points , because of the special structure of the coupling functions we considered.