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LAGRANGE MULTIPLIERS AND OPTIMALITY
, 1993
"... Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions ..."
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Cited by 89 (7 self)
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Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of onesided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the gametheoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows blackandwhite constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a freestanding exposition of basic nonsmooth analysis as motivated by and applied to this subject.
A simple algorithm for finding maximal network flows and an application to the Hitchcock problem
 CANADIAN JOURNAL OF MATHEMATICS
, 1957
"... ..."
Fractional Domination, Fractional Packings, and Fractional Isomorphisms of Graphs." Ph.D. dissertation
 Auburn University
"... Except where reference is made to the work of others, the work described in this dissertation is my own or was done in collaboration with my advisory committee. This dissertation does not include proprietary or classified information. Certificate of Approval: Dr. Chris Rodger ..."
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Cited by 1 (0 self)
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Except where reference is made to the work of others, the work described in this dissertation is my own or was done in collaboration with my advisory committee. This dissertation does not include proprietary or classified information. Certificate of Approval: Dr. Chris Rodger
Structure, Duality, and Randomization: Common Themes in AI and OR
"... Both the Artificial Intelligence (AI) community and the Operations Research (OR) community are interested in developing techniques for solving hard combinatorial problems. OR has relied heavily on mathematical programming formulations such as integer and linear programming, while AI has developed co ..."
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Both the Artificial Intelligence (AI) community and the Operations Research (OR) community are interested in developing techniques for solving hard combinatorial problems. OR has relied heavily on mathematical programming formulations such as integer and linear programming, while AI has developed constrainedbased search and inference methods. Recently, we have seen a convergence of ideas, drawing on the individual strengths of these paradigms. Furthermore, there is a great deal of overlap in research on local search and metaheuristics by both communities. Problem structure, duality, and randomization are overarching themes in the study of AI/OR approaches. I will compare and contrast the different views from AI and OR on these topics, highlighting potential synergistic benefits. 1
Lagrangian Duality
"... Contents 3.1. Geometric Multipliers . . . . . . . . . . . . . . . . p. 3 3.2. Duality Theory . . . . . . . . . . . . . . . . . . . p. 12 3.3. Linear and Quadratic Programming Duality . . . . . . p. 21 3.4. Strong Duality Theorems . . . . . . . . . . . . . . p. 25 3.4.1. Convex Cost  Linear Constra ..."
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Contents 3.1. Geometric Multipliers . . . . . . . . . . . . . . . . p. 3 3.2. Duality Theory . . . . . . . . . . . . . . . . . . . p. 12 3.3. Linear and Quadratic Programming Duality . . . . . . p. 21 3.4. Strong Duality Theorems . . . . . . . . . . . . . . p. 25 3.4.1. Convex Cost  Linear Constraints . . . . . . . . . p. 26 3.4.2. Convex Cost  Convex Constraints . . . . . . . . p. 30 3.5. Fritz John Conditions when there is no Optimal Solution p. 37 3.6. Notes and Sources . . . . . . . . . . . . . . . . . p. 42 1 2 Lagrangian Duality Chap. 3 In this chapter, we take a geometric approach towards Lagrange multipliers, with a view towards duality, the central notion of this chapter. Because of its geometric character, duality theory admits insightful visualization, through the use of hyperplanes, and their convex set support and separation properties. The min common point and max crossing point problems, discussed in Section 1.4.2 are principal
On the Equivalence of Linear Programming Problems
, 2010
"... In 1951, Dantzig [3] showed the equivalence of linear programming and twoperson zerosum games. However, in the description of his reduction from linear programming to zerosum games, he noted that there was one case in which his reduction does not work. This also led to incomplete proofs of the re ..."
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In 1951, Dantzig [3] showed the equivalence of linear programming and twoperson zerosum games. However, in the description of his reduction from linear programming to zerosum games, he noted that there was one case in which his reduction does not work. This also led to incomplete proofs of the relationship between the Minmax Theorem of game theory and the Strong Duality Theorem of linear programming. In this note, we fill these gaps. 1