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Let G be a family of subsets of an n-element set. The family G is called non-trivial 3-wise intersecting if the intersection of any three subsets in G is non-empty, but the intersection of all subsets is empty. For a real number p ∈ (0 , 1) we define the measure of the family by the sum of p | G | (1 - p) n - | G | over all G ∈ G . We determine the maximum measure of non-trivial 3-wise intersecting families. We also discuss the uniqueness and stability of the corresponding optimal structure. These results are obtained by solving linear programming problems. [ABSTRACT FROM AUTHOR]

Two approaches are applied to the set-valued optimization problem. The following problems have been examined by Corley, Luc and their colleagues: Take the union of all objective values and then search for (weakly, properly, etc.) minimal points in this union with respect to the vector ordering. This approach is called the vector approach to set optimization. The concept shifted when the set relations were popularized by Kuroiwa–Tanaka–Ha at the end of the twentieth century. They introduced six types of set relations on the power set of topological vector space using a convex ordering cone C with nonempty interior. Therefore, this approach is called the set relation approach to set optimization. For a given vector optimization problem, several approaches are applied to construct a dual problem. A difficulty lies in the fact that the minimal point in vector optimization problem is not necessarily a singleton, though it becomes a subset of the image space in general. In this paper, we first present new definitions of set-valued conjugate map based on comparison of sets (the set relation approach) followed by introducing some types of weak duality theorems. We also show convexity and continuity properties of conjugate relations. Lastly, we present some types of strong duality theorems using nonlinear scalarizing technique for set that is generalizations of Gerstewitz's scalarizing function for the vector-valued case. [ABSTRACT FROM AUTHOR]

Duality theory is important in finding solutions to optimization problems. For example, in linear programming problems, the primal and dual problem pairs are closely related, i.e., if the optimal solution of one problem is known, then the optimal solution for the other problem can be obtained easily. In order for an optimization problem to be solved through the dual, the first step is to formulate its dual problem and analyze its characteristics. In this paper, we construct the dual model of an uncertain linear multi-objective optimization problem as well as its weak and strong duality criteria via conic duality. The multi-objective form of the problem is solved using the utility function method. In addition, the uncertainty is handled using robust optimization with ellipsoidal and polyhedral uncertainty sets. The robust counterpart formulation for the two uncertainty sets belongs to the conic optimization problem class; therefore, the dual problem can be built through conic duality. The results of the analysis show that the dual model obtained meets the weak duality, while the criteria for strong duality are identified based on the strict feasibility, boundedness, and solvability of the primal and dual problems. [ABSTRACT FROM AUTHOR]

An extended form of robust continuous-time linear programming problem with time-dependent matrices is formulated in this paper. This complicated problem is studied theoretically in this paper. We also design a computational procedure to solve this problem numerically. The desired data that appeared in the problem are considered to be uncertain quantities, which are treated according to the concept of robust optimization. A discretization problem of the robust counterpart is formulated and solved to obtain the ϵ -optimal solutions. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel and rigorous approach to the Fokker–Planck equation, or Kolmogorov forward equation, for the Feynman–Kac transform of non-Markov anomalous processes. The equation describes the evolution of the density of the anomalous process Y t = X E t under the influence of potentials, where X is a strong Markov process on a Lusin space X that is in weak duality with another strong Markov process X ̂ on X and { E t , t ≥ 0 } is the inverse of a driftless subordinator S that is independent of X and has infinite Lévy measure. We derive a probabilistic representation of the density of the anomalous process under the Feynman–Kac transform by the dual Feynman–Kac transform in terms of the weak dual process X ̂ t and the inverse subordinator { E t ; t ≥ 0 }. We then establish the regularity of the density function, and show that it is the unique mild solution as well as the unique weak solution of a non-local Fokker–Planck equation that involves the dual generator of X and the potential measure of the subordinator S. During the course of the study, we are naturally led to extend the notation of Riemann–Liouville integral to measures that are locally finite on [ 0 , ∞). [ABSTRACT FROM AUTHOR]

Duality theory is important in finding solutions to optimization problems. For example, in linear programming problems, the primal and dual problem pairs are closely related, i.e., if the optimal solution of one problem is known, then the optimal solution for the other problem can be obtained easily. In order for an optimization problem to be solved through the dual, the first step is to formulate its dual problem and analyze its characteristics. In this paper, we construct the dual model of an uncertain linear multi-objective optimization problem as well as its weak and strong duality criteria via conic duality. The multi-objective form of the problem is solved using the utility function method. In addition, the uncertainty is handled using robust optimization with ellipsoidal and polyhedral uncertainty sets. The robust counterpart formulation for the two uncertainty sets belongs to the conic optimization problem class; therefore, the dual problem can be built through conic duality. The results of the analysis show that the dual model obtained meets the weak duality, while the criteria for strong duality are identified based on the strict feasibility, boundedness, and solvability of the primal and dual problems.

An extended form of robust continuous-time linear programming problem with time-dependent matrices is formulated in this paper. This complicated problem is studied theoretically in this paper. We also design a computational procedure to solve this problem numerically. The desired data that appeared in the problem are considered to be uncertain quantities, which are treated according to the concept of robust optimization. A discretization problem of the robust counterpart is formulated and solved to obtain the ϵ-optimal solutions.

In this paper we associate with an infinite family of real extended functions defined on a locally convex space a sum, called robust sum, which is always well-defined. We also associate with that family of functions a dual pair of problems formed by the unconstrained minimization of its robust sum and the so-called optimistic dual. For such a dual pair, we characterize weak duality, zero duality gap, and strong duality, and their corresponding stable versions, in terms of multifunctions associated with the given family of functions and a given approximation parameter ε ≥ 0 which is related to the ε-subdifferential of the robust sum of the family. We also consider the particular case when all functions of the family are convex, assumption allowing to characterize the duality properties in terms of closedness conditions. [ABSTRACT FROM AUTHOR]

Vector is an important type of geospatial data, discretization is an important link for its fusion with raster data, and the generation of the discrete line is the basic problem. In view of the shortcomings of discrete line generation algorithm of triangular grid, this paper proposes a mathematical model for establishing the equivalent triangle grid discrete line mathematical model by means of the weak duality hexagonal grid and solving it by dimensionality reduction. Firstly, according to the weak duality relationship between the triangular grids and the hexagonal grids, an equivalent triangular grid discrete line model is established based on the hexagonal grids. Then, using the dimension reduction method, the two‐dimensional discrete line model is equivalently transformed into a one‐dimensional closed path solution. Finally, a discrete triangle conversion generation algorithm for planar triangular grids is designed and implemented. The experimental results show that the proposed algorithm is ingenious and rigorous in theory and beneficial to the programming. The operation efficiency is about 9‐10 times of the similar algorithms, with a better result. This algorithm can be applied to vector data in real‐time grid transformation, terrain modeling, spatial analysis, simulation and other fields,with broad application prospects. [ABSTRACT FROM AUTHOR]

The uncertainty for the continuous-time linear programming problem with time-dependent matrices is considered in this paper. In this case, the robust counterpart of the continuous-time linear programming problem is introduced. In order to solve the robust counterpart, it will be transformed into the conventional form of the continuous-time linear programming problem with time-dependent matrices. The discretization problem is formulated for the sake of numerically calculating the ϵ-optimal solutions, and a computational procedure is also designed to achieve this purpose.

Present research deals with more efficient solution of a multi-objective linear fractional (MOLF) optimization problem by using branch and bound method. The MOLF optimization problem is reduced into multi-objective optimization problem by a transformation. The reduced multi-objective optimization problem is converted into single objective optimization problem by giving suitable weight for each objective. The equivalency theorems are established. Weak duality concept is used to compute the bounds for each partition and some theoretical results are also established. The proposed method is motivated by the work of Shen et al. (J Comput Appl Math 223:145-158, 2009). Matlab code is designed for the proposed method to run all the simulated results and it is applied on two numerical problems. The efficiency of the method is measured by comparing with earlier established method. [ABSTRACT FROM AUTHOR]

Third order dual of a primal nonlinear programming problem is established which involves the third order derivatives of the functions constituting the primal problem. Desired duality theorems are provided for the pair of primal and the corresponding third order dual problem. Numerical examples are illustrated to justify the efficiency of the proposed method. It is also observed that some of the existing results are obtained as special cases. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish the strong duality theorem for the parametric formulation of continuous-time linear fractional programming problems. We also consider the so-called extended form of continuous-time linear fractional programming problems that assume the vector-valued functions to be measurable and bounded on the time interval $[0,T]$. The by-product of the main result is the establishment of the strong duality theorem for the extended form of continuous-time linear fractional programming problems. [ABSTRACT FROM AUTHOR]

Two different pairs of higher-order symmetric dual programs such as Wolfe type and Mond-Weir type are studied. The weak, strong and converse duality theorems are established for the higher-order symmetric dual programs under higher-order $$\rho -(\eta ,\theta )-$$ invexity and $$\rho -(\eta ,\theta )-$$ pseudo-invexity assumptions. Many examples and counterexamples are illustrated to justify our work. We also observe that several known results are obtained as special cases. [ABSTRACT FROM AUTHOR]

The concepts of both duality and fuzzy uncertainty in linear programming have been theoretically analyzed and comprehensively and practically applied in an abundance of cases. Consequently, their joint application is highly appealing for both scholars and practitioners. However, the literature contributions on duality in fuzzy linear programming (FLP) are neither complete nor consistent. For example, there are no consistent concepts of weak duality and strong duality. The contributions of this survey are (1) to provide the first comprehensive overview of literature results on duality in FLP, (2) to analyze these results in terms of research gaps in FLP duality theory, and (3) to show avenues for further research. We systematically analyze duality in fuzzy linear programming along potential fuzzifications of linear programs (fuzzy classes) and along fuzzy order operators. Our results show that research on FLP duality is fragmented along both dimensions; more specifically, duality approaches and related results vary in terms of homogeneity, completeness, consistency with crisp duality, and complexity. Fuzzy linear programming is still far away from a unifying theory as we know it from crisp linear programming. We suggest further research directions, including the suggestion of comprehensive duality theories for specific fuzzy classes while dispensing with restrictive mathematical assumptions, the development of consistent duality theories for specific fuzzy order operators, and the proposition of a unifying fuzzy duality theory. [ABSTRACT FROM AUTHOR]