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We show in this thesis that several existing solution algorithms, for specially structured quadratic programming problems, apparently unrelated to each other or general quadratic programming algorithms, may all be viewed as specialized... more
We show in this thesis that several existing solution algorithms, for specially structured quadratic programming problems, apparently unrelated to each other or general quadratic programming algorithms, may all be viewed as specialized active set algorithms which exploit the structure of the problem they solve. We also present several new efficient algorithms for solving specially structured quadratic programming problems. In particular, we show that the active set approach provides a unified description of several solution algorithms, for various versions of the isotonic regression problem, relates them to quadratic programming theory and significantly simplifies their analysis. We also present several new efficient active set algorithms for various versions of this problem. We also present a new, quadratic time, active set solution algorithm for a certain parametric quadratic programming problem and discuss an application to economics. Finally, we present an efficient reduced vari...
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AbstractMorton et al. [1] describe a greedy algorithm for solving a class of convex programming problems. The algorithm is validated by the use of polymatroid theory. In the present paper we establish the validity of the algorithm by a... more
AbstractMorton et al. [1] describe a greedy algorithm for solving a class of convex programming problems. The algorithm is validated by the use of polymatroid theory. In the present paper we establish the validity of the algorithm by a more elementary argument using the well known Karush-Kuhn-Tucker optimality conditions. In [1] it is shown that a variant of the above mentioned greedy algorithm solves a related quadratic programming problem. Here we show that the variant in fact solves a slightly more general problem.
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Research Interests: Mathematics, Applied Mathematics, Computer Science, Computational Complexity, Combinatorial Optimization, and 10 moreComplexity Theory, Sensitivity Analysis, Optimization Problem, Business and Management, Mathematical Optimization, Radius, Numerical Analysis and Computational Mathematics, Objective function, Polynomial Time, and Approximate solution
ABSTRACT Minimizing a separable convex objective subject to an ordering restriction on its variables is a generalization of a class of problems in statistical estimation and inventory control. It is shown that a pool adjacent violators... more
ABSTRACT Minimizing a separable convex objective subject to an ordering restriction on its variables is a generalization of a class of problems in statistical estimation and inventory control. It is shown that a pool adjacent violators (PAV) algorithm can be used to compute an optimal solution of this problem as well as the minimal and maximal extended solutions, which provide lower and upper bounds on all optimal solutions and solve certain subproblems. These results unify and extend several previously known results. In addition, it is shown that a PAV algorithm can be applied to solving the problem with integer constraints on the variables.
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The isotonic median regression problem arises in statistics. It is known that the isotonic median regression problem, with respect to a complete order, may be solved by a “Pool Adjacent Violators” algorithm. In this paper we show that... more
The isotonic median regression problem arises in statistics. It is known that the isotonic median regression problem, with respect to a complete order, may be solved by a “Pool Adjacent Violators” algorithm. In this paper we show that this algorithm is a dual method for solving a linear programming formulation of the problem. The linear programming approach provides additional insight into the algorithm as well as a simple proof of its validity. We also analyze the computational complexity of the algorithm and discuss its significance from the standpoint of linear programming theory.
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Page 1. Mathematical Programming 47 (1990) 425-439 425 North-Holland ACTIVE SET ALGORITHMS FOR ISOTONIC REGRESSION; A UNIFYING FRAMEWORK Michael J. BEST Department of Combinatorics and Optimization ...
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ABSTRACT Linear constraints are used to model diverse phenomena. The number of constraints is large in many practical applications, often running into several thousands. A system of linear constraints is quite often inconsistent... more
ABSTRACT Linear constraints are used to model diverse phenomena. The number of constraints is large in many practical applications, often running into several thousands. A system of linear constraints is quite often inconsistent especially at the stage of model formulation. A number of different approaches are possible when a system of linear constraints turns out to be inconsistent. We prove some results on some plausible approaches and their limitations.
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,w ELSEVIER European Journal of Operational Research 89 (1996) 609617 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Theory and Methodology LP relaxation of the two dimensional knapsack problem with box and GUB constraints Ansuman Bagchi a,.,... more
,w ELSEVIER European Journal of Operational Research 89 (1996) 609617 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Theory and Methodology LP relaxation of the two dimensional knapsack problem with box and GUB constraints Ansuman Bagchi a,., Nalinaksha ...
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Chv atal (1980) describes a class of zero-one knapsack problems provably di cult for branch and bound and dynamic programming algorithms. Chung et al.(1988) identi es a class of integer knapsack problems hard for branch and bound... more
Chv atal (1980) describes a class of zero-one knapsack problems provably di cult for branch and bound and dynamic programming algorithms. Chung et al.(1988) identi es a class of integer knapsack problems hard for branch and bound algorithms. We show that for both classes of problems local search provides optimal solutions quickly. Keywords: knapsack problem* local search* computational complexity
Suppose that we are given an instance of a combinatorial optimization problem with min-max objective along with an optimal solution for it. Let the cost of a single element be varied. We refer to the range of values of the element's... more
Suppose that we are given an instance of a combinatorial optimization problem with min-max objective along with an optimal solution for it. Let the cost of a single element be varied. We refer to the range of values of the element's cost for which the given optimal solution remains optimal as its exact tolerance. In this paper we examine the problem of determining the exact tolerance of each element in combinatorial optimization problems with min-max objectives. We show that under very weak assumptions, the exact tolerance of ...