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A Brief Introduction to Linear and Dynamic Programming
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
An LP has a specific algebraic form. There is a corresponding algebraic form, called the dual LP. The original problem is called the primal problem, and the primal together with the dual is called a primal–dual pair; the primal is the first LP in the pair and the dual LP is the second LP in the pair. When an LP is a model of a “real-world” situation, very often there is a different (dual) perspective of the situation which is modeled by the dual LP. Knowing the existence and form of the dual LP provides a vantage point from which to look for a dual interpretation of the situation; examples are provided below by exhibiting dual linear programs for Examples 1 and 3.
Linear Programming
Published in Michael W. Carter, Camille C. Price, Ghaith Rabadi, Operations Research, 2018
Michael W. Carter, Camille C. Price, Ghaith Rabadi
It follows from this property that, if feasible objective function values are found for a primal and dual pair of problems, and if these values are equal to each other, then both of the solutions are optimal solutions.
A proximal iterative algorithm for system of generalized nonlinear variational-like inequalities and fixed point problems
Published in Applicable Analysis, 2023
Javad Balooee, Shih-sen Chang, Jen-Chih Yao
For each , let be a real Banach space with the dual space , and be the dual pair between and its dual space . Assume that for , , and are the mappings. Suppose further that for , are extended real-valued bifunctions such that for each , is a proper, lower semicontinuous and -subdifferential functional on with . We consider the problem of finding such that and which is called a system of generalized nonlinear variational-like inequalities ().
Normalized solutions to the fractional Kirchhoff equations with a perturbation
Published in Applicable Analysis, 2023
Lintao Liu, Haibo Chen, Jie Yang
C denotes a universal positive constant.For and , we denote ..Set denoting the homogeneous fractional sobolev space with the norm Denote by the symmetric decreasing rearrangement of a function.The symbol denotes weak convergence and the symbol denotes strong convergence.The symbol denotes the dual pair for any Banach space and its dual space.
Concentrating ground state solutions for quasilinear Schrödinger equations with steep potential well
Published in Applicable Analysis, 2021
Throughout this paper, we use standard notations. For simplicity, for any . C and will denote positive constants unless specified. We use ‘’ and ‘’ to denote the strong and weak convergence in the related function spaces respectively. is the usual Lebesgue space with the standard norm . We write to mean the Lebesgue integral of over a Lebesgue measurable set . For a Lebesgue measurable set A, we denote the Lebesgue measure of A by . is the dual space of E. denote the dual pair for any Banach space and its dual space.